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From Computer to Brain
William W. Lytton
From Computer to Brain
Foundations of Computational Neuroscience
Springer
William W. Lytton, M.D.
Associate Professor, State University of New York, Downstato, Brooklyn, NY
Visiting Associate Professor, University of Wisconsin, Madison
Visiting Associate Professor, Polytechnic University, Brooklyn, NY
Staff Neurologist., Kings County Hospital, Brooklyn, NY
In From Computer to Brain: Foundations of Computational Neuroscience,
William Lytton provides a gentle but rigorous introduction to the art
of modeling neurons and neural systems. It is an accessible entry to the
methods and approaches used to model the brain at many different levels,
ranging from synapses and dendrites to neurons and neural circuits.
Different
types of questions are asked at each level that require different types
of models.
Why learn this?
One of the reasons why someone might want to learn about computational
neuroscience is to better predict the outcomes of experiments. The process
of designing an experiment to test a hypothesis involves making predictions
about what the possible outcomes of the experiment might be and to
work out the implications of each possible result. This is a difficult task
in
most biological systems, especially ones like the brain that involve many
interacting parts, some of which are not even known. A model may reveal
assumptions about the system that were not fully appreciated.
One of the earliest and most successful models is the Hodgkin-Huxley
model of the action potential (Chap. 12). For their classic papers on the
giant squid axon, they integrated the differential equations on a
handpowered
mechanical calculator. Computers today are millions of times
faster than those in the 1950s and it is now possible to simulate cortical
neurons having thousands of dendritic compartments and dozens of different
types of ions channels, and networks with thousands of interacting
neurons. The complex dynamics of these networks is exceptionally difficult
to predict without computational tools from computer science, and
mathematical tools from dynamical systems theory.
But there is another reason to delve into this book. Computational
neuroscience
also provides a framework for thinking about how brain mechanisms
give rise to behavior. The links between facts and consequences are often
subtle. We are told, for example, that lateral inhibition enhances contrast
discontinuities, but without a quantitative model, such as that for the
Limulus compound eye (Chap. 8), it is not at all obvious how this occurs,
especially when the signals are varying in time as well as space. The
jump from mechanism to behavior becomes even more difficult to understand
for the cellular basis of learning and memory, where the memory of
a single item can be distributed over millions of synapses in distant parts
of the brain.
Why read this book?
Computational neuroscience is such a young field that ten years ago there
were no good books for someone who was getting started. That has now
changed and there are now several excellent textbooks available, but most
of them focus on one type of model, such as Hodgkin-Huxley style models
or abstract neural networks, or presume a high level of mathematical
sophistication.
This book gives a balanced view of the wide range of modeling
techniques that are available, in a way that is accessible to a wide
audience.
Another reason for reading this book is to enjoy the playfulness that the
author brings to a subject that can be dry and technical. His imaginative
use of examples brings mathematical ideas to life. This is a book that will
bring a smile to your face as well as inspire your imagination.
Terrence J. Sejnowski
Howard Hughes Medical Institute
Salk Institute for Biological Studies
University of California at San Diego
Preface
As a college student, I got interested in the big questions of brain science:
the origin of language and thought, the nature of memory, the integration of
sensation and action, the source of consciousness, and the mind-body problem.
Although I worked on the mind-body problem all through sophomore
year, I didn’t arrive at a solution. After studying some more psychology,
some biology and some physics, I took a wonderful computer science course.
I decided that the way to understand the mind’s brain was to analyze it in
detail as a computational device.
When I graduated from college and looked at graduate programs. I explained
to interviewers that I wanted to apply computers to brain science.
No one I spoke to in biomedicine or in computer science knew of anyone
doing such things. In retrospect, such people were around, mostly located
in departments of mathematics and physics. Because I couldn’t find what
I was looking for, I put off my research training for almost a decade. When
I came back, a new field had emerged and was gaining recognition. Shortly
afterwards, this field, whose various branches and tributaries had gone by
a variety of names, was dubbed Computational Neuroscience.
As I got involved in research, I began to appreciate that massive
theoretical
and informational frameworks had to be, and were being, built in order
to approach these grandly challenging problems that interested me. So,
like many of my colleagues, I became a bricklayer, laboriously adding brick
upon brick to the wall, without always looking up or down. Later, given
the opportunity to teach an undergraduate course, I was happy to have a
chance to reflect more expansively on the problems, techniques, aspirations
and goals of my field. I found that the students asked basic questions that
required me to peak out from under the residue of brick-dust and think
about the architecture.
When I started to teach my course, I reviewed the various textbooks in
computational neuroscience, most of them brand new and many excellent.
I realized that many of these new texts were committed to one particular
angle or theoretical bias, and did not reveal the broad scope and wide
interplay of ideas that I found so exciting. Additionally, they generally
required too much math for most of the students in the class.
In fact, most students came into the course either without strong math
background, or without strong biology background, or without either. At
first, I was concerned that I would be unable to stuff in enough “remedial”
material to bring everyone up to a point where they could understand
what I was talking about. However, I found that I could cover a wide
swath of topics by only teaching the little pieces that I needed out of each
one. Additionally, for students who were not particularly comfortable with
math, I made an effort to explain things a lot: first in metaphor, then
in pictures, then directly in words, then in equations, and finally with
computer simulations that brought it to life interactively. I have included
the former set of approaches directly in this book, and have made available
programs to allow a student to directly interact with, and alter, all of the
figures, at least those that aren’t just pictures of things.
As I wrote this book, I was sort of hoping that by the end of it I might
finally figure out the answer to the mind-body problem. Alas, no such luck.
But I’ve tried to convey some of the information that will be required for
a solution. The rest is left as an exercise for the reader.
Bill Lytton
East Flatbush, NY
June, 2002
Acknowledgments
I first want to thank the folks at University of Wisconsin who helped me
organize, launch, and teach the course on which this book is based: Zoology
400, Introduction to Computational Neuroscience. In particular a number of
students helped with the development of the course, the lecture notes, and
the software: Kevin Hellman, Stephen Cowan, William Annis (Fig. 10.4),
Ben Zhao, Winston Chang, Kim Wallace (Fig. 14.3), Adam Briska, Luke
Greeley, Sam Sober, and Sam Williams.
Version 1.0 of the software for this book, written in Perl CGI, was
supported by the University of Wisconsin Office of Medical Education,
Research, and Development. I particularly want to thank Mark Albanese
and John Harting for their encouragement, and thank Jeff Hamblin, who
wrote the software the first time around.
A number of longtime collaborators engaged me in many helpful discussions
that found their way into this book: Dan Uhlrich, Peter Lipton, Josh
Chover, Jay Rosenbek, Lew Haberly, Sonny Yamasaki, and Peter Van Kan.
Many thanks to the enablers of Springer-Verlag: Paula Callaghan, Jenny
Wolkowicki, and Frank Ganz. They were unfailingly helpful and friendly
and gave me plenty of leeway.
The book was written in Latex using Emacs running under Redhat Linux.
As a fan of software, I particularly wish to acknowledge Neuron, a wonderful
program that was used to produce all of the figures in the book. The
software available for this book is written in the Neuron simulator. I thank
Mike Hines and Ted Carnevale for developing and supporting Neuron, and
responding to my requests for language augmentations.
I need to acknowledge my biological and intellectual patrimony. The
hybridization
of Dad, a psychoanalyst, and Mom, an applied mathematician,
produced a computational neuroscientist. Intellectually, I am most strongly
indebted to Terry Sejnowski, who trained me.
I want to thank my wife, Jeeyune, who put up with a lot — my working
on this book during our most intimate moments, and my son, Barry, who
put up with a fair amount – my working on it while I was supposed to be
watching “SpongeBob SquarePants.” Oh, and Barry drew Fig. 9.6.
I especially must thank two colleagues who helped out enormously by
reading the manuscript in detail and making numerous suggestions that
improved the book greatly: Mark Stewart and Ning Qian.
I’d like to place the blame for any remaining errors on someone else, but
I can’t find a fall guy. So for whatever errors remain, I’m the guy.
Contents
Foreword vii
Preface ix
Acknowledgments xi
1 Introduction 1
1.1 For whom is this book? . . . . . . . . . . . . . . . . . . . 1
1.2 What is in the book? . . . . . . . . . . . . . . . . . . . . 2
1.3 Do I need a computer for this book? . . . . . . . . . . . 3
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Why learn this now? . . . . . . . . . . . . . . . . . . . . 4
1.5 What is the subtext? . . . . . . . . . . . . . . . . . . . . 5
1.6 How is the book organized? . . . . . . . . . . . . . . . . 6
I Perspectives 9
2 Computational Neuroscience and You 11
2.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Brainmetaphors . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Compare and contrast computer and brain . . . . . . . . 12
2.4 Origins of computer science and neuroscience . . . . . . 14
2.5 Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Levels of organization . . . . . . . . . . . . . . . . . . . . 16
Levels of investigation . . . . . . . . . . . . . . . . . . . . 18
2.6 New engineering vs. old engineering . . . . . . . . . . . . 18
2.7 The neural code . . . . . . . . . . . . . . . . . . . . . . . 20
2.8 The goals and methods of computational neuroscience . 22
2.9 Summary and thoughts . . . . . . . . . . . . . . . . . . . 23
3 Basic Neuroscience 25
3.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Microscopic view of the nervous system . . . . . . . . . . 26
3.3 Macroscopic view of the nervous system . . . . . . . . . 29
Slicing the brain . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Parts of the brain . . . . . . . . . . . . . . . . . . . . . . 31
3.5 How do we learn about the brain? . . . . . . . . . . . . . 34
Anatomical methods . . . . . . . . . . . . . . . . . . . . 35
3.6 Neurophysiology . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Molecular biology and neuropharmacology . . . . . . . . 37
3.8 Psychophysics . . . . . . . . . . . . . . . . . . . . . . . . 38
3.9 Clinical neurology and neuropsychology . . . . . . . . . . 39
Ablative diseases . . . . . . . . . . . . . . . . . . . . . . 39
Intrinsic diseases . . . . . . . . . . . . . . . . . . . . . . . 40
3.10 Summary and thoughts . . . . . . . . . . . . . . . . . . . 41
II Computers 43
4 Computer Representations 45
4.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Calculator or typewriter . . . . . . . . . . . . . . . . . . 47
4.3 Punch cards and Boolean algebra . . . . . . . . . . . . . 48
4.4 Analog vs. digital representations . . . . . . . . . . . . . 50
4.5 Types of computer representations . . . . . . . . . . . . 51
4.6 Representation of numbers . . . . . . . . . . . . . . . . . 52
Representation of letters and words . . . . . . . . . . . . 55
4.7 Representation of pictures . . . . . . . . . . . . . . . . . 56
4.8 Neurospeculation . . . . . . . . . . . . . . . . . . . . . . 59
4.9 Summary and thoughts . . . . . . . . . . . . . . . . . . . 62
5 The Soul of an Old Machine 63
5.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 The art of the hack . . . . . . . . . . . . . . . . . . . . . 64
5.3 Software and hardware . . . . . . . . . . . . . . . . . . . 65
5.4 Basic computer design . . . . . . . . . . . . . . . . . . . 66
Pointers come fromcomputermemory design . . . . . . 67
Sequential algorithms come from computer control flow . 67
CPU:machine commands . . . . . . . . . . . . . . . . . 69
5.5 Programs and hacks . . . . . . . . . . . . . . . . . . . . . 69
Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6 Pointermanipulation . . . . . . . . . . . . . . . . . . . . 74
A kludge . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A computer virus . . . . . . . . . . . . . . . . . . . . . . 76
5.7 Neurospeculation . . . . . . . . . . . . . . . . . . . . . . 77
5.8 Summary and thoughts . . . . . . . . . . . . . . . . . . . 82
III Cybernetics 83
6 Concept Neurons 87
6.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 History and description of McCulloch-Pitts neurons . . . 88
6.3 Describing networks by weights and states . . . . . . . . 90
Calculating total-summed-input by dot product . . . . . 92
Calculating state . . . . . . . . . . . . . . . . . . . . . . 94
6.4 From single unit to network of units . . . . . . . . . . . . 95
6.5 Network architecture . . . . . . . . . . . . . . . . . . . . 102
6.6 Summary and thoughts . . . . . . . . . . . . . . . . . . . 104
7 Neural Coding 105
7.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Coding in space: ensemble codes . . . . . . . . . . . . . . 106
Local vs. distributed ensemble coding . . . . . . . . . . . 108
7.3 Coding with volts and chemicals: neural state code . . . 110
7.4 Coding in time: temporal and rate codes . . . . . . . . . 111
Temporal integration . . . . . . . . . . . . . . . . . . . . 112
Clocking . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.5 Frequency coding . . . . . . . . . . . . . . . . . . . . . . 114
7.6 Summary and thoughts . . . . . . . . . . . . . . . . . . . 118
8 Our Friend the Limulus 121
8.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 The biology . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.3 What we can ignore . . . . . . . . . . . . . . . . . . . . . 122
8.4 Why the eye lies: the problem . . . . . . . . . . . . . . . 123
8.5 Design issues . . . . . . . . . . . . . . . . . . . . . . . . . 126
Making themodel small—scaling . . . . . . . . . . . . 126
Making the model small — dimensional reduction . . . . 127
Eliminating edge effects—wraparound . . . . . . . . . . 128
Presenting the input — parameterization . . . . . . . . . 130
Parameterizing the activation function . . . . . . . . . . 132
Parameterizing the weight matrix . . . . . . . . . . . . . 132
8.6 The limulus equation . . . . . . . . . . . . . . . . . . . . 134
8.7 State calculation . . . . . . . . . . . . . . . . . . . . . . . 135
8.8 Life as a limulus . . . . . . . . . . . . . . . . . . . . . . . 137
8.9 Summary and thoughts . . . . . . . . . . . . . . . . . . . 139
9 Supervised Learning: Delta Rule and Back-Propagation 141
9.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 141
9.2 Supervised learning . . . . . . . . . . . . . . . . . . . . . 142
9.3 The delta rule . . . . . . . . . . . . . . . . . . . . . . . . 145
The energy analogy . . . . . . . . . . . . . . . . . . . . . 146
The delta rule solves AND . . . . . . . . . . . . . . . . . 147
9.4 Backward propagation . . . . . . . . . . . . . . . . . . . 149
9.5 Distributed representations . . . . . . . . . . . . . . . . . 151
9.6 Distributed representation in eye movement control . . . 152
Design of themodel . . . . . . . . . . . . . . . . . . . . . 154
Results fromthemodel: generalization . . . . . . . . . . 157
Exploration of themodel: hidden unit analysis . . . . . . 159
Computer modeling vs. traditional mathematical modeling 161
9.7 Summary and thoughts . . . . . . . . . . . . . . . . . . . 162
10 Associative Memory Networks 163
10.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 163
10.2 Memories in an outer product . . . . . . . . . . . . . . . 164
Association across a single synapse . . . . . . . . . . . . 164
The outer product of two vectors . . . . . . . . . . . . . 165
Making hetero- and autoassociativememories . . . . . . 167
Limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . 171
Instantaneous vs. gradual learning and recall . . . . . . . 174
10.3 Critique of the Hopfield network . . . . . . . . . . . . . . 176
10.4 Summary and thoughts . . . . . . . . . . . . . . . . . . . 177
IV Brains 179
11 From Soap to Volts 189
11.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 189
11.2 Basic cell design . . . . . . . . . . . . . . . . . . . . . . . 190
11.3 Morphing soap and salt to batteries and resistors . . . . 192
11.4 Converting the RC circuit into an equation . . . . . . . . 194
Capacitance and current . . . . . . . . . . . . . . . . . . 195
Adding up the currents . . . . . . . . . . . . . . . . . . . 196
11.5 Parameter dependence . . . . . . . . . . . . . . . . . . . 198
Advantages and disadvantages of numerical integration . 200
11.6 Time constant and temporal summations . . . . . . . . . 201
11.7 Slow potential theory . . . . . . . . . . . . . . . . . . . . 205
Averaging by adding PSPs . . . . . . . . . . . . . . . . . 208
11.8 Summary and thoughts . . . . . . . . . . . . . . . . . . . 211
12 Hodgkin-Huxley Model 213
12.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 213
12.2 From passive to active . . . . . . . . . . . . . . . . . . . 213
The resting membrane potential is about −70 mV . . . . 214
The membrane is insulator, capacitor, and battery . . . . 214
Synaptic inputs aren’t current injections . . . . . . . . . 215
12.3 History of the action potential . . . . . . . . . . . . . . . 216
Hodgkin and Huxley . . . . . . . . . . . . . . . . . . . . 216
12.4 The parallel-conductance model . . . . . . . . . . . . . . 217
The circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 220
Where do the batteries come from? . . . . . . . . . . . . 221
12.5 Behavior of the active channels . . . . . . . . . . . . . . 223
Feedback systems . . . . . . . . . . . . . . . . . . . . . . 224
Particle duality . . . . . . . . . . . . . . . . . . . . . . . 226
Particle dynamics . . . . . . . . . . . . . . . . . . . . . . 227
12.6 The particle equations . . . . . . . . . . . . . . . . . . . 228
State variables define a state . . . . . . . . . . . . . . . . 229
12.7 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.8 Implications for signaling . . . . . . . . . . . . . . . . . . 233
The threshold and channelmemory . . . . . . . . . . . . 233
Rate coding redux . . . . . . . . . . . . . . . . . . . . . . 234
12.9 Summary and thoughts . . . . . . . . . . . . . . . . . . . 237
13 Compartment Modeling 239
13.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 239
13.2 Dividing into compartments . . . . . . . . . . . . . . . . 241
Building the model . . . . . . . . . . . . . . . . . . . . . 242
13.3 Chemical synapsemodeling . . . . . . . . . . . . . . . . 245
Shunting inhibition . . . . . . . . . . . . . . . . . . . . . 246
GABA and glutamate . . . . . . . . . . . . . . . . . . . . 248
13.4 Passive neuronmodel . . . . . . . . . . . . . . . . . . . . 250
Synaptic responses . . . . . . . . . . . . . . . . . . . . . 253
13.5 Back-propagating spikes and the Hebb synapse . . . . . 254
13.6 Summary and thoughts . . . . . . . . . . . . . . . . . . . 257
14 From Artificial Neural Network to Realistic Neural
Network 259
14.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 259
14.2 Hopfield revisited . . . . . . . . . . . . . . . . . . . . . . 259
14.3 Suppression model for reducing interference . . . . . . . 261
14.4 A digression into philosophy . . . . . . . . . . . . . . . . 262
14.5 Acetylcholine has multiple effects . . . . . . . . . . . . . 264
The dual-matrix hypothesis . . . . . . . . . . . . . . . . 265
True confessions . . . . . . . . . . . . . . . . . . . . . . . 266
14.6 Summary and thoughts . . . . . . . . . . . . . . . . . . . 267
15 Neural Circuits 269
15.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 269
15.2 The basic layout . . . . . . . . . . . . . . . . . . . . . . . 269
15.3 Hippocampus . . . . . . . . . . . . . . . . . . . . . . . . 270
15.4 Thalamus . . . . . . . . . . . . . . . . . . . . . . . . . . 272
15.5 Cerebellum. . . . . . . . . . . . . . . . . . . . . . . . . . 273
15.6 Basal ganglia . . . . . . . . . . . . . . . . . . . . . . . . 275
15.7 Neocortex . . . . . . . . . . . . . . . . . . . . . . . . . . 277
15.8 Summary and thoughts . . . . . . . . . . . . . . . . . . . 278
16 The Basics 281
16.1 Why learn this? . . . . . . . . . . . . . . . . . . . . . . . 281
16.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Scientific notation . . . . . . . . . . . . . . . . . . . . . . 282
Numerical prefixes . . . . . . . . . . . . . . . . . . . . . . 282
Units and abbreviations . . . . . . . . . . . . . . . . . . 283
Unit conversions . . . . . . . . . . . . . . . . . . . . . . . 285
Dimensional analysis . . . . . . . . . . . . . . . . . . . . 286
16.3 Binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Translating back and forth . . . . . . . . . . . . . . . . . 287
Addition and subtraction . . . . . . . . . . . . . . . . . . 289
Octal and hex . . . . . . . . . . . . . . . . . . . . . . . . 289
Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . 290
16.4 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . 290
What is algebra?Why linear? . . . . . . . . . . . . . . . 291
Addition and subtraction . . . . . . . . . . . . . . . . . . 292
Dot product . . . . . . . . . . . . . . . . . . . . . . . . . 292
Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 293
Outer product . . . . . . . . . . . . . . . . . . . . . . . . 293
Matrixmultiplication . . . . . . . . . . . . . . . . . . . . 294
16.5 Numerical calculus . . . . . . . . . . . . . . . . . . . . . 295
Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . 296
Numerical solutions . . . . . . . . . . . . . . . . . . . . . 297
Mathematical symbols . . . . . . . . . . . . . . . . . . . 299
Analytic solution to the charging curve . . . . . . . . . . 299
16.6 Electrical engineering . . . . . . . . . . . . . . . . . . . . 300
The three big laws: Ohm, Kirchhoff, and the other one . 301
Ohm’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . 304
References 307
Glossary 313
Index 357
1
Introduction
1.1 For whom is this book?
This book is for undergraduates and beginning graduate students. Of
course, other graduate students, postdoctoral students, and anyone concerned
with the fate of Western Civilization should read this book as well.
Why focus on undergraduates? Because the malleable mind of the
undifferentiated
stem-person of college age is likely to be capable of making
conceptual leaps that more ossified brains cannot. A theme that will recur
throughout the book is the difficulty of unifying the compute and the
neuro.
There are two major barriers to grand unification. First, computer science
(like math, physics, and engineering) is made up of grand unifiers and
their all-encompassing schemes, while neuroscience (like the rest of biology)
is boatloads of facts. The differentiated mind of the engineer cannot
swallow so many facts without a unifying framework. The differentiated
mind of the biologist knows too much and is distracted by the many facts
that contradict any preliminary framework someone tries to build. The fabled
undergraduate mind, however, is notoriously unburdened by facts and
yet willing to accommodate them. It (that mind, whether he or she) also
typically seeks big pictures and is willing to take the leaps of faith
required
to acquire them.
The second barrier: these are early days. The field is a newly emerging
hybrid and is itself still undifferentiated (an undifferentiated field needs
an undifferentiated mind). It is a field still driven more by passion and
2 1. Introduction
fashion than by cool reason, emerging like a star from a gaseous cloud (OK
that’s a bit fanciful). Anyway, some of my colleagues will read this book
and say that I’ve missed the whole point of the field. I’ve said the same
of their books. Specifically, many researchers in the field come from the
aforementioned engineering, physics, and math tradition. Their efforts are
directed at developing a theoretical framework and their work tends to be
colored by the framework they have chosen. I, on the other hand, come from
the bio side. As a consequence, this book is a largely atheoretical approach
to a theoretical science. I present stuff that is either fun or interesting
or
important and maybe sometimes all three.
In this computational neuroscience funhouse, I have included biological
facts, computer science facts, equations, and theories both true and false.
In some cases, I have included false theories because I don’t know that
they’re false, though I may suspect it. In other cases, I introduce
unlikely
hypotheses just to roll them around and play with them. This interplay of
facts and ideas makes up much of the work (or play) of modeling. Through
experiencing it, the student will get a better idea of how and why modeling
is done. In the last chapter, in particular, I bring the reader to my own
particular circle of purgatory — the hall of perpetual mystification. I’ve
tried to illustrate the complexities and contradictions of the field without
unduly confusing the reader.
1.2 What is in the book?
Although my target undergraduate students are pluripotential, they are not
yet omniscient. Specifically, they are majoring in philosophy, physics, math,
engineering, biology, zoology, psychology, physical education, or business
administration. As a result they know a lot about some things and little or
nothing at all about others. For this reason, I have tried to cover all the
basic
bases. I have relegated much of this to a final chapter (Chap. 16), which
can
be read piecemeal as needed. In addition, many of the fundamentals have
seeped into the main text as well. I have included a lot of basic computer
science since an understanding of computational neuroscience requires a
fairly sophisticated working knowledge of computers. This broad, blanket
coverage means that certain chapters may seem trivial and others overly
demanding for the particular student.
For the nonmathematical reader, the biggest challenge will be the
advanced math topics, notably matrix algebra and numerical calculus.
Though these topics are hard and can easily fill up a year of classroom
instruction,
I have tried to extract just the parts needed for present purposes
and to make this accessible to anyone with mastery of high-school algebra.
I have written out most equations in words so as to make them more
accessible
to anyone who is allergic to math. Additionally, the computer is a
1.3. Do I need a computer for this book? 3
great leveler in this regard. Many once abstruse concepts in mathematics
can now be quickly illustrated graphically. Computer in hand, this book can
be enjoyed as a lighthearted romp through calculus, electrical engineering,
matrix algebra, and other sometimes-intimidating topics.
For the nonbiological reader, the biggest challenge will be the profusion of
facts and jargon words. This onslaught of information can be intimidating
and discouraging. There is so much to know that it can be hard to know
where to start. Often it is impossible to connect one set of facts to
another
set of facts. That is the goal of computational neuroscience. When you first
encounter these facts, it will be without the benefit of such a model.
Although I have tried to write clearly and comprehensibly, I have also
tried to use a lot of jargon. This can be annoying. I try to use jargon
kindly and responsibly, to help the reader learn the words needed to read
and converse knowledgeably in the many subfields that make up computational
neuroscience. I have tried to always define jargon words immediately
upon use in the text. As further assistance, I’ve provided a glossary. In
addition
to introducing jargon words, I also introduce some jargon concepts
— touchstone ideas that are frequently referenced by people in a particular
field. Having so much to introduce, concepts and phrases are sometimes
mentioned, but not followed up on. They are presented to provide the
reader with vocabulary and mental reference points for further reading or
just plain thinking.
1.3 Do I need a computer for this book?
This book is meant to be read independently of any computer work. I have
not put explicit exercises in the book but have made them available online
(see below). One of the neat things about computational neuroscience is
that it is so readily accessible. It is hard to get hold of the particle
accelerators,
centrifuges, and chimpanzees needed for most scientific study. But
computers are everywhere, making computer-based research accessible to
undergraduates and even to nonacademic folks. This is more true of
computational
neuroscience than it is of other computer modeling fields. To do
weather prediction you need a supercomputer. A simple desktop PC will
do for most of the material in this book. If the first stage of learning a
field
is to talk the talk by learning vocabulary, running the computer exercises
will enable you to walk the walk as well.
Software
All of the figures in this book were put together using Neuron, a computer
simulation program written by Mike Hines at Yale University. This program
is freely available at http://www.neuron.yale.edu. Although Neuron is pri4
1. Introduction
marily designed for running the type of realistic simulations highlighted in
the latter part of the book, it is flexible enough that I was able to use it
for all the other simulations as well.
Software to produce all of the simulations and to run the emulator of
Chap. 5 is available at these sites:
http://www.springer-ny.com/computer2brain
http://www.cnl.salk.edu/fctb
http://www.neuron.yale.edu/fctb
I will be pleased to consider additions or augmentations to this software,
particularly if the contributor has already coded them.
Examples in the software are primarily presented through a graphical
user interface. The reader or teacher who is interested in pursuing or
presenting
the subject in depth will want to become familiar with the Neuron
program and with HOC, the Neuron programming language. This will allow
the programs to be manipulated more flexibly in order to look at different
aspects of a particular modeling problem.
Many of the examples presented in this book could also be readily programmed
up in Matlab or Mathematica, or in other simulation programs
such as Genesis or PDP++.
1.4 Why learn this now?
The genetic code was cracked in the mid-20th century. The neural code will
be cracked in the mid-21st. Genetic science has given way to its
applications
in biotechnology and bioengineering. Neuroscience is still up-and-coming,
the next big thing. Furthermore, as genetic manipulations and basic
neuroscience
add more raw facts to the broth, the need for meaning, structure,
and theory becomes greater and greater. Enough information is coming
together
that the next generation of computational neuroscientists will make
the leap into understanding. That means grant money, prizes, and fancy
dinners! (If the movies are a guide, it also means evil robots, mind control,
and dystopia, but let’s not ruin the moment.)
There’s such a variety of things to learn about in computational
neuroscience
that the student is in the position of the proverbial kid in the
confectionery: so many problems to work on; so many amazing facts and
theories from so many interrelated fields. Of course, this profusion of
riches
can also be frustrating. One doesn’t know which gaudy bauble to pick up
first, and, having picked one and discovered that it is not quite gold,
strong
is the temptation to drop it and pick up one that seems gaudier still.
1.5. What is the subtext? 5
1.5 What is the subtext?
In an age of ubiquitous computers, any topic can be discussed in their
context, as attested to by the recent publication of From Computer to Stain:
Dry Cleaning in a Digital Age, the inspiration for the title of this book.
However, computational dry-cleaning is still just dry-cleaning done with
computers. In fact, most of computational biology is just biology done with
computers. Computational neuroscience is a little different. The computer
itself represents the state of our knowledge about how complex information
processing devices like the brain might work. For this reason, I have
covered
more computer science than would usually show up in a neuroscience book.
Present computer science curricula generally emphasize sophisticated
abstractions
that pull one away from the machine. Similarly, a branch of
computational neuroscience has concerned itself with finding general
principles
of neural computation and has shied away from the messy meat of
the brain. My contention is that the meat is the message for the brain
and for the computer. Lovely abstract theories must grow out of an
understanding
of the machine. The most useful theories will be different for
different machines. If there is a grand unified theory, it will stand
abstract
and austere away from the daily marketplace of synapses or transistors.
For this reason I have gone into some detail about the design and operation
of an ancient computer, the PDP-8, a machine with the power of a
modern pocket calculator. Such a simple machine can be readily described
in a chapter. It also is small enough that one quickly runs into its limits
and
has to overcome them with programming tricks, commonly called “hacks.”
Hacking is now frowned upon in computer science, since it is mostly used to
break into other people’s machines. Biological evolution is one long
history
of hacking — using a piece of machinery for a new purpose and gradually
working it into shape so that it seems to have been engineered from scratch
for that purpose.
Understanding the process and products of evolution means understanding
the problems of engineering with limited resources (and unlimited time).
Programming a PDP-8 or rebuilding a diesel engine with pieces of scrap in
a Third-World country requires ingenuity, ability to compromise, and
willingness
to make mistakes and start again. This process may leave us with
a program or an engine with vestigial organs, tangled distribution routes,
and inefficient procedures. Just as building the machine was a study in
frustration, so examination of the machine will be a frustrating study that
will also lead to dead-ends and back-tracking.
In this book, I repeatedly contrast my emphasis on the brain with the
tendency of others in the field to focus on theory rather than detail.
Perhaps
I occasionally disparage these poor theoretical guys as cyborgs and
hedgehogs. This is all in fun. Integration of theory and fact is a necessary
goal in computational neuroscience. I try to give both their due in this
6 1. Introduction
book but I have not been successful in integrating them. The section titles
— Computers, Cybernetics, and Brains — demonstrate this.
Our brains are full of contradictions but we learn to live with them. If we
want to study the brain, we must be prepared for the kinds of ambiguities
and occasional false leads that characterize life with our own brains.
1.6 How is the book organized?
Computational neuroscience is a new field whose essential paradigms are
still the subject of debate. For this reason, it is not possible to present
the basic material with the conceptual coherence of an introduction to
well-established fields like chemistry or physics. The field remains a
hodgepodge
of exciting ideas and remarkable facts, some of which cannot be
neatly conjoined. Instead of progress in a neat sequence from one idea to
another, this book will at times seem to jump back and forth from one
thing to another. This is an inherent difficulty of trying to teach both the
computational and biological approaches in a single text. In general, I try
to fill in the gaps where they can be filled in and point them out where
they remain unbridged.
The organization of the book is as follows. I start with a brief
introduction
to neuroscience, touching on many but not all of the subfields that may need
to be considered. I then go top-down with a description of how computers
work. I start by noting how computers represent information. I then go
into still more detail about the bits-and-bytes level of computer function.
From there, I switch from transistors to neural network units and explore
the concepts of artificial neural networks. This will entail a comparison
between transistors and neurons and an explanation of how the artificial
neural network units represent a compromise position between the two.
From there, I show how these units can be connected together into artificial
neural networks. I further expand on the artificial neural network paradigm,
showing the use of these networks to explain the retina of a simple sea
creature, the horseshoe crab. I look at another simple brain system, but
this time one found in humans, the brainstem reflex that stabilizes the eyes
in the head when the head is moved. Then, in a more speculative vein, I
go still higher in the brain, looking at how artificial neural networks can
be used to emulate aspects of human memory. This will involve an explicit
compare-and-contrast with computer memory design.
Following this, I turn bottom-up, more seriously exploring the biological
concepts of nervous system function. I start with a detailed description
of the neuron with some ideas of how the different parts of the neuron
can be modeled. I then explore in greater detail the two major techniques
of realistic neuronal modeling: compartment modeling and the Hodgkin-
Huxley equations. I then look at an example of how artificial neural network
1.6. How is the book organized? 7
models of learning can inform our understanding of the brain and how
study of the brain leads us to reconsider the details of these artificial
neural
networks.
The final chapter covers some details of the mathematical and scientific
approaches and techniques used in this book. It includes unit analysis,
binary arithmetic, linear algebra, calculus, and electronics. Comfort with
handling units and scientific notation is needed for finding your way around
science. Knowledge of binary is important for finding your way around a
computer. Linear algebra is useful for finding your way around a network.
Calculus is important for assessing movement and change. Electronics is
needed for understanding electrical signaling in neurons. In each case, the
material has been presented graphically and algebraically to make the
subject
accessible to those who do not feel comfortable with mathematical
notation. It is expected that many readers will be unfamiliar with some
or all of these areas and will want to read these sections as the technique
comes up in the main text.
Since there are many topics touched on that do not always relate cleanly
to one another, I tried to provide additional guidance. Each chapter begins
with a brief introduction entitled “Why learn this?” Similarly, each
chapter
ends with “Summary and thoughts,” meant to synthesize concepts and
remind the reader of what was learned. This section is “... and thoughts,”
rather than “... and conclusions” because in many cases the conclusions
await.
Part I
Perspectives
2
Computational Neuroscience and You
2.1 Why learn this?
A major goal of computational neuroscience is to provide theories as to how
the brain works. Such mind–body theorizing has been a subject of
philosophical,
theological, and scientific debate for centuries. The new theories
and taxonomies for organizing information about the brain will be built
upon this historical foundation. It is valuable to see where we are starting.
2.2 Brain metaphors
Mechanical models or metaphors for the brain date back to the time when
the brain first beat out the heart as leading candidate for siting the soul.
Plato likened memory to the technique of imprinting a solid image onto a
block of wax. Over the centuries, the nervous system has been compared
to a hydraulic system, with pressurized signals coursing in and out; a post
office, with information packets being exchanged; or a telephone switchboard,
with multiple connecting wires to be variously assorted. Today, the
digital computer, or sometimes the Internet, is cited as a model for brain
function. Do these modern mechanisms hold greater promise than prior
metaphors for helping us understand our most intimate organ?
In many ways, the brain is not much like the standard digital computer.
Yet, both as a direct model of certain aspects of brain functioning and as
a tool for exploring brain function, the computer enjoys many advantages
12 2. Computational Neuroscience and You
over previous models. Take, for example, the post office. The difficulties
of
actually utilizing the postal service to test out the feasibility of a
particular
brain model must give one pause. (However, below I discuss a similar
human-based system that was proposed more than a century ago as a
calculating
technique for weather prediction.) The telephone switchboard, on the
other hand, is a considerably more manipulable organizational and
technological
artifact. In fact, the early analog computers of the 1930s and 1940s
were, in appearance and in some functional aspects, aggrandized telephone
switchboards. Although simple neural models were run on such machines,
the technical difficulties of programming them made them far less useful
than digital computers as a tool. However, the basic concepts of analog
computing may be useful for understanding brain function.
2.3 Compare and contrast computer and brain
When we liken the brain to a computer, we mean several things. First,
we mean that several definable computer actions are analogues of things
that the brain appears to do. Such computer actions include memory, input/
output, and representation. Second, we mean that computers have
been used to do a variety of tasks that were previously believed to be
exclusively
the province of human intelligence: playing chess, reading books
aloud, recognizing simple objects, performing logical and mathematical
symbol manipulations. Finally, although no machine has yet passed the
Turing test (a machine passes if it fools a conversation partner into
thinking
that it is a person), those who work intensively with computers develop
a distinct sense of communicating or even communing with the machine.
Modeling is the work and play of computational neuroscience, as it is
for much of physics, engineering, business, and applied mathematics. It’s a
tricky thing. To learn something about the thing being modeled, we need
to reduce the model to the essentials. If we reduce too far, however, we may
miss a critical component that is responsible for interesting properties.
For
the Wright brothers and other early aviators, the process of building a
heavier-than-air flying machine was a task of bird emulation. To those who
said that heavier-than-air flight was impossible, they could point to birds
as
a counterexample. As they evaluated the basic bird, it would have seemed
clear that many aspects of bird design were not required for flight. For
example, the beak seems quite clearly designed more for eating than for
flying. However, the beak’s aerodynamic design might still tell us
something
valuable about fuselage design. The importance of other bird features for
flight would not have been as apparent. For example, it might a priori
seem likely that wing beating was critical for flight. It is critical for
smallcreature
flight but not for the flight of large birds or airplanes. Many early,
misguided attempts were made to design a full-sized ornithopter (e.g., that
2.3. Compare and contrast computer and brain 13
aircraft with flapping wings that beats itself to death). On the other hand,
the Wright brothers had a key insight when they noticed that birds steered
by tilting their body to the side (rolling) rather than by using a rudder
like
a boat.
When we model birds we know what we want to model. The function of
interest is flying.We can focus on flying and ignore feeding, foraging,
fleeing,
etc. The brain, however, is doing many things simultaneously and using
hidden processes to do them. Therefore, we can model a brain function, such
as chess playing, and yet gain little or no insight into how the brain plays
chess. The brain is utilizing unconscious properties that we are not aware
of when we play chess. In this example, I would guess that an important
underlying ability used in chess is the capacity of the brain to complete
partial patterns. This ability is seen in the normal unawareness of the
blind
spot. It is also seen in the abnormal confabulatory tendency of demented
or psychotic individuals to forge links between false perceptions so as to
build an internally consistent, although irrational, story.
In this book, as we look in detail at how a computer works, and compare
and contrast its functioning with that of the brain, a variety of
differences will become apparent. We consider various brain features and
wonder whether or not these are critical features for information process,
for memory, or for thought. Certainly, many aspects of brain design
are not critical for brain information processing but are there for other
purposes: metabolism, growth and differentiation, cell repair, and general
maintenance.
If we wanted to use a modern jet aircraft as a model to help us better
understand birds and the phenomenon of flight, we would want to take
note of similarities and differences that might clarify essential concepts.
Both have wings; it seems reasonable to expect that wings are essential for
heavier-than-air flight (note that helicopters are considered rotating wing
aircraft). However, the wings are made of very different materials so there
is
apparently nothing critical in the design of feathers. Closer analysis would
reveal that airplanes and large birds like albatrosses have similarly shaped
wings (smaller birds and insects use different-style wings suited to their
small size).
Using the computer as a model to understand the brain raises questions
about similarities both in detail and in function. Airplanes fly like
albatrosses
but computers don’t think like brains. Both brains and computers
process information, but information processing may not be central to the
process of thinking. Therefore, we will wish to explore not only differences
from the bottom, differences in materials and design principles, but also
differences from the top, differences in capacity and capability.
Starting with the manufacturing side, there are already a variety of
differences
that can be explored. Computers are made of sand and metal,
while brains are made of water, salt, protein, and fat. The computer chip is
built onto a two-dimensional matrix, while the brain fills three dimensions
14 2. Computational Neuroscience and You
with its wiring. The time and size scales involved are also believed to be
vastly different. Of course, this depends on exactly what is being compared
to what. As we will see, typically a transistor in the computer is compared
to a neuron in the brain. With this comparison, the time scales are about
1 ms for the neuron vs. 1 ns for the transistor (see Chap. 16, Section 16.2
for discussion of units). The spatial scale is about 1 mm for the largest
neuron vs. less than 1 μm for a modern CMOS transistor. Thus the neuron
is much bigger and much slower. However, if it eventually turns out that
the proper analogue for the transistor is the synapse, or a particular type
of ion channel or a microtubule, then we would have to reevaluate this
comparison.
Additional differences arise when one considers functional issues. Brains
take hints; computers are remarkably stupid if given a slightly misspelled
command or incomplete information. The digital computer has a generalpurpose
architecture that is designed to run many different programs. The
brain, on the other hand, has dedicated, special-purpose circuits that
provide
great efficiency at solving particular problems quickly. Calculations on
a digital computer are done serially, calculating step by step in a cookbook
fashion from the beginning to the end of the calculation. The brain, on
the other hand, performs many calculations simultaneously, using parallel
processing. Digital computers use binary; transistors can take on only two
values: 0 or 1. In this book, we utilize binary extensively, and consider
its
applicability to the brain. This may not be a fair approximation since the
brain uses a variety of elements that take on a continuum of analog values.
2.4 Origins of computer science and neuroscience
Neuroscience and computer science came into being at about the same time
and influenced each other heavily in their formative stages. Over time, the
fields have diverged widely and have developed very different notions of
seemingly shared concepts such as memory, cognition, and intelligence.
D.O. Hebb proposed over 40 years ago that a particular type of usedependent
modification of the connection strength of synapses might
underlie learning in the nervous system. The Hebb rule predicts that
synaptic
strength increases when both the presynaptic and postsynaptic neurons
are active simultaneously. Recent explorations of the physiological
properties
of neuronal connections have revealed the existence of long-term
potentiation, a sustained state of increased synaptic efficacy consequent to
intense synaptic activity. The conditions that Hebb predicted would lead
to changes in synaptic strength have now been found to cause long-term
potentiation in some neurons of the hippocampus and other brain areas.
As we will see, similar conditions for changing synaptic strength are used
2.5. Levels 15
in many neural models of learning and memory. These models indicate the
great computational potential of this type of learning rule.
One difference between the neuroscience and computer science viewpoints
has to do with the necessary adoption of a big-picture approach
by computer scientists and a reductionist approach by many neuroscientists.
These two approaches are typically called top-down and bottom-up,
respectively. The top-down approach arises from an engineering perspective:
design a machine to perform a particular task. If you’re interested in
intelligence, then design an artificial intelligence machine. The bottom-up
perspective is the province of the phenomenologist or the taxonomist:
collect
data and organize it. Even granting that most U.S. science today is
federally mandated to be hypothesis-driven, an essential element of biology
is the discovery of facts. Hypotheses are then designed to fit these facts
together. As outlined here, these positions are caricatures. Most biologists
want to consider how the brain thinks, and many computer and cognitive
scientists are interested in what goes on inside the skull.
In this book, we concern ourselves with many ideas that have been
promulgated
for understanding higher levels of nervous system function such
as memory. However, it is important to note that much of the data on real
nervous systems has been gathered from either the peripheral nervous systems
of higher animals or from the nervous systems of invertebrates such as
worms, leeches, and horseshoe crabs. The genesis of the action potential or
neuron spike, one of the most important ideas to come out of computational
study of the brain (Chap. 12), was the result of studying the peripheral
nervous
system of the squid. The low-level source of much of our knowledge
of the nervous system contrasts sharply with the ambition to understand
the highest levels of mental functioning, and helps explain why some of the
topics to be discussed may seem quite remote from human neural function,
while other subjects will be very relevant but highly speculative.
2.5 Levels
The notions of bottom-up and top-down approaches to the problem of
nervous system function, and the corresponding contrast between acknowledged
facts at the lower level and uncertain hypotheses at the higher, lead
naturally to hierarchical divisions. Two such divisions that are commonly
used are called the levels of organization and levels of investigation. Each
of
these divisions into levels creates a hierarchy for brain research that
leads
between the reductionist bottom and the speculative top.
The levels-of-investigation analysis was historically a product of topdown
thinking. This approach, pioneered by computationalists, starts at the
top with the big-picture problem of brain function and drips down to the
implementation in neurons or silicon. The levels-of-organization analysis
16 2. Computational Neuroscience and You
was in part a reaction to this. By putting all of its levels on an equal
footing, the levels-of-organization approach invited the investigator to
start
anywhere and either build up or hypothesize down.
Levels of organization
Levels of organization is fundamentally a bottom-up perspective. The basic
observation that leads to this division of the knowledge comes from
the “grand synthesis” that connected the physical with the vital world.
Modern biology explains genetics and physiology in terms of the interactions
of molecules. This allows connections to be made all the way from
physics to physiology. Physics is the more fundamental science. The basic
concepts of biology can be understood from the concepts of physics, while
the converse is not the case. However, understanding biology directly from
physics would be a hopeless task for two reasons. First, there is no way one
can predict what would occur in a biological system using knowledge of
atoms and electron orbits. Second, the conceptual leap from physics to
biology
is simply too great to be made without interposed models from other
fields. Specifically, much of biology can be understood from cell biology,
which can be understood from molecular biology, which can be understood
from biochemistry, which can be understood from organic chemistry, which
can be understood from physical chemistry, which can be understood from
physics. In comparison with this known hierarchy of knowledge, the levels
of organization of the nervous system remain tentative. Any hierarchy will
likely embody a fundamental, trivial law: big things are built out of
smaller
things.
Following the scheme of others, we can build a hierarchy of levels of
organization and levels of study. From smallest to largest:
study method object of study
physics ions
chemistry transmitters and receptors
cell biology neurons
computer science networks
neurology systems
psychology behavior or thought
Although the general order of dependencies in the nervous system can be
assumed to be based on size and the simple inclusion of one structure
within another, the exact structures that are of functional importance are
not clear. The ambiguity starts when one considers the appropriate items
for anchoring the two ends. At the top, one can choose to regard either
behavior or internal mental representation as the highest level suitable for
scientific investigation. There is a long history of debate in the
psychology
literature between proponents of these two positions. Behavioralists believe
that since physical movement is the only measurable evidence of nervous
2.5. Levels 17
system function, this is the only appropriate area of high-level functional
study. Other psychologists believe that putative internal representations of
the external world are also suitable subjects of investigation, even though
these cannot be measured directly. Computational approaches generally
make the latter assumption, not only postulating internal representations
but often making them the central question for further study.
At the small end of the organizational scale, most investigators would
consider the concentrations of ions and neurotransmitters and their channels
and receptors to be the smallest pieces of nervous system that are worth
paying any attention to. A dissenter from this view is the physicist Roger
Penrose, who believes that the underlying basis of neural function will lie
in
quantum mechanics and that it’s necessary to study the subatomic realm.
In between quantum mechanics and behavior, there is still more room
for debate both as to which levels are relevant, and as to which levels can
be adequately built on a previous level without further investigation at an
intermediate level. To go back to the physics-to-biology spectrum described
above, the conceptual jump from the concepts of physics to the concepts of
organic chemistry would not be possible without the intermediate concepts
developed by physical chemistry. This is because the representations of
electron orbitals and chemical bonds used in physical chemistry provide
conceptual links between the detailed equations describing electron orbitals
used in physics, and the schematic stick diagrams used for bonds in organic
chemistry. Similarly, the neuroscience levels of organization suggests that
neurons can be adequately described by taking account of properties at the
level of transmitters and receptors. That’s probably not going to turn out
to be true. It’s likely that intermediate-sized ultrastructural components
of the neuron such as spines, dendrites, and synapses may have their own
critical properties that cannot be understood without independent study
of these structures in themselves.
As we move up the scale, higher levels of neural organization are less well
understood and can be farmed out somewhat arbitrarily to various interested
specialty areas. Much study of networks has come out of computer
science, but the organization of networks is also studied in mathematics by
geometry and topology. The level of cortical columns is not shown in this
diagram. It is unclear whether this level would go below or above the level
of the network. I gave systems to neurology, a clinical field that
subdivides
brain function into motor, sensory, and various cognitive systems based
on changes seen with brain damage. Engineers mean something different
when they study systems in the field called “signals and systems.” Systems
neuroscience has yet another connotation, referring to neurophysiological
techniques related to investigating the origins of perception and behavior.
18 2. Computational Neuroscience and You
Levels of investigation
The levels-of-investigation approach comes from David Marr, a
computationalist
who produced some very influential early models of different brain
areas. This viewpoint is from the top down. The top level is the level of
problem
definition (this was called the computational-theoretic level by Marr).
Marr suggested that understanding any particular brain function requires
that we first understand what problem the brain is solving. Problem in
hand, we can deduce what additional information the brain would need to
solve it. The next level is that of algorithm definition. An algorithm is
like
a cookbook recipe, defining a step-by-step approach to accomplish some
task. The third and final level is the level of implementation, where the
algorithm is finally translated into machinery, whether neural or silicon,
that can actually perform the task.
Marr’s three levels of problem, algorithm, and implementation are the
current approach a software engineer would take in designing a big program
(e.g., a word processor or a Net browser) using a modern computer
language. If writing a Web browser like Netscape or Explorer, for example,
we would first define the problem — delivering information from remote
sites to a user in a user-palatable form. We would then write algorithms
that would simply assume that we have or can develop the underlying tools
needed for the subsidiary processes. For example, a basic algorithm for
processing
a Web page would be 1) request the page, 2) wait and accept the
data, 3) confirm that a full data set was received, 4) parse the data to
determine
content type, and 5) parse fully to present in a graphical form on
the screen. Individual steps would then be implemented. It is important to
avoid considering details of implementation in working out the algorithm
since we are interested in readily porting our browser between machines
that use different low-level implementations.
This Marr trinity of problem, algorithm, and implementation can be collapsed
into the familiar concepts of software and hardware. A problem is
provided. Algorithms are written into the software. The software is compiled
so as to run on a computer — the physical implementation level. A
software engineer using modern computing machinery doesn’t routinely run
into the limits of what the machine can do. The Marr top-down approach
is ideal in this engineering environment. However, as will be discussed in
the next chapter, when the limitation of the machine becomes part of the
problem, another engineering approach is needed.
2.6 New engineering vs. old engineering
Over time, science and technology have advanced from being based on everyday
commonplace observations to being based on sophisticated theories.
Similarly, engineering has moved away from the tinkerer or hacker mental2.6.
New engineering vs. old engineering 19
ity toward reasoned conceptual approaches to technical problems. Working
from theory, rather than empirically, the modern engineering approach is
close to David Marr’s notions of levels of investigation: from problem to
method to implementation.
Modern building design is predicated on principles of tension and stress.
By contrast, the great cathedrals of Europe were largely built using rules
of thumb and intuition born of experience. Sometimes they fell down.
Similarly,
computer science has given up ad hoc hacking and developed tools
and theories to allow software design problems to be addressed from basic
principles.
From one perspective, Marr’s insistence on first defining the problem is
unavoidable. Until we know that the brain can do a certain thing, we cannot
study it. A blind man who has never had sight, and had not spoken with
someone who has, would have an impossible task trying to study vision
based simply on being told that it represented an alternative to hearing.
On the other hand, insistence on an initial problem definition can lead to
what has been called premature definition. Fondly held hypotheses can be
blinders that preclude appreciation of new facts that could shed light on
the
problem. This risk is particularly great in the general area of brain/mind
studies, where the appeal to intuition is hard to resist.
The complexity of the unconscious workings of the many subdivisions of
brain function makes them resistant to an introspective, intuitive
understanding
of what is going on behind the scene. This can make it impossible
to frame the problem correctly. For example, Marr believed that vision
primarily performed the task of taking the two-dimensional retinal
representation
of the world and re-creating a three-dimensional internal model
of the world, just as one might look at a family photograph and determine
that your cousin was standing to the side and in front of your aunt. This
definition of the problem of vision seems intuitively reasonable to a
sighted
person. This clearly describes something that the brain can do and that
needs to be done under some circumstances. However, this turns out to
be a bad start for studying vision in the brain. As it turns out, any single
problem definition for vision will turn out to be a bad choice, since the
brain does not simply have one type of vision but instead utilizes many
different types of vision that are processed simultaneously.
Marr’s is a sophisticated engineering approach to vision. Given the
complexity
of brain and our limited intuition, a naive engineering approach
is more reasonable. A congenitally blind man starts with no clue as to
where to begin studying this mysterious phenomenon called “vision.” He
might therefore start asking questions about different things that vision
can do. “Can it detect objects behind other objects? Can it detect motion?
Can it determine shapes?” This set of questions would put the blind-
manexplaining-
vision in the proverbial blind-men-with-elephant situation (each
feels a different body part and each has a different idea of what an
elephant
is). The blind man might conclude that vision was not one thing but was
20 2. Computational Neuroscience and You
made up of separate detecting systems that handled various detection tasks.
This sort of piecemeal understanding would bring him closer to the
underlying
mechanism of visual perception in the brain than was Marr with his
sighted person’s intuition of a unitary process. Although the blind man
would have no appreciation of the personal experience (the qualia) of seeing,
he would have some notion of how the brain actually performs the
task. Lacking access to the myriad unconscious processes of our brain, we
are nonetheless blessed with the illusion of introspection. Sight makes us
blind to vision.
Modern, sophisticated engineering takes place in big laboratories. Discovery
proceeds in reasoned steps according to quarterly plans. The old
engineering was, to paraphrase Edison, all inspiration and perspiration.
This is the engineering of tinkerers and hackers, who take discarded bits
and pieces of machinery and cobble them together so as to make them do
things they were not originally meant to do. When it doesn’t work, the
tinkerer bends and hammers and makes it work. The workman’s ideal is
always to have the right tool for the job. The tinkerer is cursed with the
wrong tools, the wrong materials, the wrong job.
This discrepancy between available material and the exigencies of the
task is also the plight of the evolutionary process. For example, a gill is
not
well suited to life on land. Since water must be moved across the gill in
order to replenish oxygen, the gill needs to be exposed externally. However,
gills must also be kept wet at all times. Keeping wet becomes a problem
when we move the gill onto dry land. Although the modern lung looks
great, early versions would have been crude hacks that managed to barely
satisfy these contradictory needs: invaginating the gill to prevent drying,
while exposing it enough to prevent asphyxiation.
2.7 The neural code
The brain denies the philosopher’s, the mathematical modeler’s, and the
guy-on-the-street’s desire for clarity and simplicity. Since there is no
single
overarching task for the brain to do, different facets of brain function
must
be studied separately. This does not necessarily mean that there are no
unifying principles. There may well be basic neural codes that are used
throughout the animal kingdom. However, the discovery of neural codes
will no more free us from the need for further research into different brain
areas than the discovery of the genetic code revealed all the functions of
all enzymes and structural proteins.
The analogy between the search for the genetic code and the search for a
neural code has been highlighted by Francis Crick, discoverer of the former
and pursuer of the latter. To doubters, he points out that the quest for
a simple genetic code seemed quixotic to anyone who considered that the
2.7. The neural code 21
complexity of the natural design encompasses enzymes and organs, growth
and development. Of course, the discovery of a simple genetic code did not
in any way provide an understanding of all the things that are coded for.
It did, however, provide a powerful new tool for exploring these things.
Similarly, the discovery of neural code or codes will not tell us how any
part of the brain works, but will enable us to start to understand what we
see when we amplify electrical signals from different parts of the brain.
Several neural signals are well established. However, some of these signals
probably carry no information at all, while other signals carry information
that is not used by the brain or body. For example, the electroencephalogram
(EEG) is a very well studied signal that is emitted by the brain. There
is information in the EEG that permits an outside observer to determine
whether a brain is awake or asleep or even, after some signal processing,
whether the brain is hearing clicks or seeing a flashing checkerboard. These
field signals are generally an epiphenomenon, a side effect that has no
functional
relevance. These signals are not used within the brain under normal
circumstances, and are too weak to be used for telepathy, no matter how
close you put the two heads together. There are some cases where the field
is used or misused. Some neurons in the goldfish communicate internally
with such field effects. Field effects are used to communicate between
individuals
in the weakly-electric fish (the strongly electric fish use their fields
to stun prey). Field effects are also responsible for pathological signaling
in cases of epilepsy and multiple sclerosis. However, in general the EEG
can be considered an information-carrying neural signal that is not used
internally as a neural code.
Various signals are used directly by the brain and therefore can be
considered
to be codes. For example, the rate of spiking of neurons carries
information that determines how powerfully a muscle will contract. This is
a code that has been cracked: the nerve tells the muscle “squeeze ...
squeeze
harder.” It appears likely that similar rate coding is also used in the
central
nervous system. Rate coding has also been suggested to be the primary
code in parts of sensory cortex. Neurons in visual cortex spike fastest when
presented with oriented bars of a certain configuration, and auditory cortex
neurons will spike faster in response to particular sound frequencies.
There are an enormous number of electrical and chemical signals that
influence neuron firing. Many of these can be considered to have a coding
function as well. Most neurons use chemical synapses to communicate.
The presence of neurotransmitter is a coding signal at these synapses.
Synapses are typically viewed as passive information conduits connecting
complicated information-processing neurons. An alternative view is that
a synaptic complex may itself be a sophisticated information processor.
Neurotransmitter concentration may vary and be a relevant signal in some
cases. Within the postsynaptic cell, ions and molecules function as second
and third messengers in cascades of chemical reactions. These chemical re22
2. Computational Neuroscience and You
actions can be very rapid. It may be that sequences of chemical reactions
are as important as electrical activity for neural information processing.
2.8 The goals and methods of computational
neuroscience
Conferences in computational neuroscience often feature energetic debates
about what constitutes the correct approach to the problem of understanding
brain function. Generally, it’s biologists against computationalists,
bottom-uppers versus top-downers. To caricature, the most rabid biologists
believe that a model that deviates from the details of known physiology
is inadmissibly inaccurate. Meanwhile, the computer scientists, physicists,
and mathematicians feel that models that fail to simplify aggressively do
not allow any useful generalizations to be made. The view presented here
is one of compromise. Both perspectives are in part correct. Leaving out
biological details will lead to models that can no longer make the
connections
with physiological experiment. However, failure to simplify at all can
produce models that may not generalize at all. For example, it is possible
to model a specific experiment with such fidelity to detail that one just
has
another copy of the experiment. Such a model would not be able to generalize
so as to explain other related experiments in the same brain area.
Also duplicating the system will not by itself give you any insight into how
the system works.
In addition to the inherent intellectual tension between dry computers
and wet biology, there are also historical tensions between traditional
applied
mathematics and the newer computational approaches. Traditionally,
applied mathematics and theoretical physics were done with paper and
pencil. The resulting formulations embedded complex physical phenomena
in simple attractive formulae that could be disseminated by T-shirt and
coffee cup. The Maxwell equations and E = mc2 are examples that have
been translated into both of these media. Although these equations are
mysterious to most people, their elegance and aesthetic appeal is evident.
They look like a key to the mysteries of the universe. Computer modeling,
on the other hand, has little of the elegance and none of the generality of
the
traditional great equations. Although it is possible that neuroscience may
someday yield clear-cut defining equations of this sort, it seems to me more
likely that it will not. Just as with wet biology experiments, the results
of
computer simulations are rarely definitive and perhaps never canonical in
the way of the great physics equations.
Computer modeling or simulation can be considered to be experimental
mathematics. Simulations are themselves so complex that they must
be studied by using virtual experiments to try to understand them. The
simulated complex system, like the original, shows emergent behaviors
2.9. Summary and thoughts 23
whose origins and implications are not immediately obvious and must be
explored experimentally. Traditional mathematics provides clean translations
of reality. Simulation provides an alternative reality with advantages
of manipulability and accessibility.
Simulation is used to assess large sets of complex mathematical formulae
that cannot be solved by traditional analytic (paper-and-pencil) means.
Since the single simulation never unequivocally represents the biology, it
is often necessary to cross-check results among several simulations that
represent the same system with different levels of detail or scale or simply
with different choices for undefined parameters.
On the bright side, simulation also produces a variety of very nice benefits.
Simply transforming a notion about how something works into an
explicit computer model requires a complete accounting for all system
parameters.
Compiling this list often reveals basic, critical aspects of the
system that are not known. Sometimes this is simply because no one ever
bothered to look. Additionally, running computer simulations permits one
to test specific questions about causality that can only be guessed at in
paper-and-pencil modeling. Finally, working with computer simulations
provides a way of getting a very intimate view of a complex system. The
next time you take a commercial airliner flight, consider that this may be
your pilot’s first flight in this aircraft type, since many airlines now do
all step-up training on a simulator. Just as flight simulators provide an
intuitive feel for flight, neural simulators can provide intuition and
understanding
of the dynamics of neural systems. If I swim with the neurons
long enough, maybe I’ll learn to think like a neuron.
2.9 Summary and thoughts
I have presented this brief history of brain thoughts partly to present my
own view and place it in perspective. The view in this book is particularly
contrasted with Marr’s ideal separation of ends from means. The present
evolutionary view of implementation entangled with task is comparable
to the mainstream programming practices of Marr’s era. In Chap. 5, I
present basic computer science through exploration of computer practices
from that era, when hacking was necessary to perform complex computations
despite hardware limitations. These practices have been lost with the
growing power of computers and increasing sophistication of programming
tools.
This book focuses on the interface between task and machine, where
tricks and shortcuts, hacks in software parlance, are used to optimize a
function on a particular architecture. The assumption is that the brain
uses a thousand tiny hacks, each cleverly evolved to do some little task
very well.
3
Basic Neuroscience
3.1 Why learn this?
I come from a biology background. For this reason, I am sometimes surprised
that there are people in the field who do not care where the thalamus
is. On the other hand, those who come from a physics background would
be shocked to hear that I don’t use Hamiltonians.
So there is a debate. If you want to understand the brain, do you need to
know what the thalamus is, or the amygdala, or even the cortex? The most
theoretical theorists would say no. Their belief is that fundamental
theories
will emerge independent of the details of evolutionary history that gave us
a cortex and gave birds a wulst (a cortex-like structure). They will note
that understanding the emergence of parabolic trajectories from Newton’s
second law is only obscured by close study of the principles of cannonry,
with its attention to the details of combustion and barrel hardening. If the
basic principles of neural computation are already known, as is assumed by
some theorists, then close attention to brain details may not be necessary.
If this is the case, then our current understanding of brain function is
an adequate framework to build upon for future research. My guess is
that the current intellectual framework (the paradigm) is basically sound.
However, I wager that it will be so heavily built upon that it will be
barely
recognizable 100 years from now. Even if we don’t suffer a paradigm shift,
we may still get a major paradigm face-lift.
The viewpoint of this book is that new concepts of computational
neuroscience
will largely emerge from study of the brain. If readers of this book
26 3. Basic Neuroscience
intend to study the brain directly or to study the research of those who
study it, they must know standard terminology, coordinate systems, and
concepts. This chapter provides an initial introduction to these. It is
hoped
this will be sufficient so that the interested reader can segue painlessly
to
neuroscience textbooks.
3.2 Microscopic view of the nervous system
Living tissue is made up of cells. A cell has a fatty membrane and is filled
with liquid and proteins known as cytoplasm as well as smaller functional
parts called organelles. In the bodies of humans and other multicellular
organisms, these cells are typically specialized and organized. Livers are
made of liver cells and brains are made of brain cells. There are two major
types of brain cells: neurons and glia. Neurons are usually identified by
their
ability to produce action potentials, but not all neurons can produce action
potentials. It is believed that neurons are the principal elements involved
in information processing in the brain.
In the classical model, a neuron has dendrites, a cell body, and an axon.
According to Cajal’s century-old “neuron doctrine,” information comes
into
the dendrites of the neuron. Signals then travel to the cell body, which in
turn activates the axon, which can send a signal out of the neuron. Note
that this is a classical view that may be true for many but certainly not
all
neurons.
Fig. 3.1 shows a typical pyramidal cell of the mammalian cortex. These
cells are considered principal cells because of their large size and long
projections
to other areas. The cell body or soma (small oval), which would
be the bulk of the cell in other cell types, is dwarfed by the dendrites,
which extend out in all directions. In this cell type, there are lots of
small
dendrites and then one major long dendrite with only a few branches. This
major, apical dendrite can be a millimeter in length. The axon, which is
thinner and much longer than the dendrites, is shown here coming out
from the lower right. It would continue off the page. The axon is typically
much longer than the dendrites — several of them go from your lower back
to your big toe, a distance of about a meter. Axons also branch to connect
with multiple targets. An axon branch from another cell is shown at
the upper left forming a synapse on the cell (rectangle). The synapse is
shown enlarged below the label, illustrating the presynaptic bouton at the
end of the axon and a spine on the dendrite. The terminology of presynaptic
and postsynaptic defines the direction of signal flow. Transmitter is
released presynaptically, floats across the synaptic cleft, and activates
receptors
postsynaptically. The two neurons are not directly connected but
communicate via this cleft.
3.2. Microscopic view of the nervous system 27
axon
dendrites
soma
synapse
axon
Presynaptic
Postsynaptic
Spine
synaptic
cleft
bouton spine
Fig. 3.1: A classical cortical neuron showing synaptic connectivity.
The neuron is explored in more detail in Chap. 11.
The information in the neuron is in the form of electrical potentials
(voltages) across the membrane. Information is conveyed via the synapse
through arrival of neurotransmitter on receptors. This triggers postsynaptic
potentials (PSPs). These can be excitatory postsynaptic potentials
(EPSPs) or inhibitory postsynaptic potentials (IPSPs). These are graded
potentials, meaning that they vary in size. Fig. 3.2 shows a tracing of
potentials
in a model neuron. The trace starts at a resting membrane potential
(RMP) of about −70 mV. An IPSP pushes the membrane potential down
to more negative values and away from its firing potential. A pair of EPSPs
of increasing size move the potential in a positive direction, toward firing
threshold. Finally a larger EPSP reaches the firing threshold and the cell
fires a spike (action potential, AP). This spike is all-or-none. It is of
stereo28
3. Basic Neuroscience
5 mV
5 ms
IPSP
EPSP EPSP
AP
−70 mV
(RMP)
Fig. 3.2: Postsynaptic and action potentials. Arrow shows where
3rd EPSP triggers the AP.
typed shape and amplitude and is not graded. The action potential is the
signal that can be sent down the axon to create a PSP in another neuron.
Most synapses are chemical: neurotransmitters are released from one
neuron and picked up by receptors on another neuron, generating an
electrical
signal in the latter neuron. The classical synapse is axodendritic; it
connects an axon to a dendrite. There also exist dendrodendritic synapses,
which connect dendrites to dendrites. In addition to chemical synapses,
there are electrical synapses with specialized channels (gap junctions) that
allow current to flow directly from one neuron to another. Complementing
these synaptic mechanisms are various nonsynaptic interneuronal signals.
These signals includes volume transmission, where transmitters are broadcast
through extracellular space to a lot of neurons. Glia, as well as neurons,
may be involved in volume transmission. There is also ephaptic transmission,
where an electric field is generated by one cell and influences
another. As our understanding of the nervous system continues to grow,
more nonsynaptic transmission will need to be included in models.
Glia are presumed to be supporting cells but they may also play a role
in information processing. Long regarded as passive onlookers, it is now
appreciated that glia also have electrical potentials and chemical receptors.
(Actually, pretty much all cell types, including liver cells and kidney
cells,
have electrical potentials and chemical receptors, so this doesn’t
necessarily
mean much.) Glia selectively take up and release neurotransmitters and
extracellular ions. In doing this they cannot help but influence neuronal
3.3. Macroscopic view of the nervous system 29
activity. It is not clear whether this influence could play a part in
information
processing. Astrocytes are the major glial type that take up and release
chemicals. There are also other types of glia. Oligodendroglia are notable
since they provide the myelin sheaths that insulate axons and speed up
action potential transmission. Microglia protect the brain against bacteria
and other invaders.
3.3 Macroscopic view of the nervous system
The nervous system can be structurally subdivided in several ways. There
is the peripheral nervous system, which reaches out to hands and feet, and
the central nervous system (CNS), which lies within the bony protection
of spine and skull. The CNS is divided up into forebrain, brainstem, and
spinal cord. Within the CNS one can distinguish gray matter, made up of
primarily of cell bodies and dendrites, and white matter, made up of axons.
The white matter is white due to the fatty white myelin. Gray matter at
the surface of the brain is cortex. Gray matter masses deep in the brain
are called nuclei.
In addition to dividing the brain by appearance and location, we can also
divide it into different areas that have shared or common connectivity and
presumably shared function (e.g., thalamus, basal ganglia). We can also
discuss different directions in which the brain can be sliced.
Slicing the brain
To describe a structure, it is helpful to have a coordinate system.
Unfortunately,
the brain, and the body in general, has two major coordinate
systems that are often used together in describing a single slice. The first
anatomical coordinate system is based on the external world, using the
common terms left, right, lateral, medial, anterior, and posterior. “Up”
and “down” are the other cardinal directions in this system, but they are
not used. The other coordinate system is based on the body axis and uses
the terms rostral, caudal, ventral, dorsal, lateral, and medial. In the
context
of the nervous system, we use the neuraxis. The neuraxis is the axis of the
CNS as it curves around from tail (caudal) to head (rostral). The front
part of this curve is the ventral (belly) part and the back part is the
dorsal
(back) part. Because of the curvature, ventral and anterior are synonyms at
spinal cord levels. However, ventral is downward at the brain. Thus, the two
coordinate systems have different correspondences at different locations.
Left and right would seem to be easy. However, in the context of
neuroscience,
left and right mean anatomical left and anatomical right. These
coordinates always refer to the subject’s viewpoint and not to the viewer’
s
viewpoint. It is assumed that the subject is facing the viewer so that the
30 3. Basic Neuroscience
Horizontal
Sagittal
Coronal
Fig. 3.3: The three planes of body sectioning.
right side is on the viewers left. When asked to look for something on the
left side of a sliced brain (or a CAT scan or MRI), look to your right.
There is a standard anatomical position that differs depending on the
animal. It’s generally a standing position. In the case of our favorite
animal,
the human, it is standing up, face forward, with hands hanging at the side,
palms forward. From there, we can conceptually (or actually) slice the
body in three orthogonal planes (Fig. 3.3): horizontally (a plane horizontal
to the ground), sagittally (a vertical plane passing from belly to back),
and
coronally (a vertical plane running from ear to ear).
Each plane has its own pertinent coordinates from the three-dimensional
body structure. I’ll refer to each of these primarily in the context of the
brain though they also apply to the spinal cord, which has different
coordinate
axes due to CNS curvature. In the horizontal plane, directions are
anterior vs. posterior (toward or away from the nose), left vs. right, and
medial vs. lateral (toward the middle or toward the edge). For the coronal
plane, left vs. right and medial vs. lateral also pertain. The other
direction
would be up and down, but the neuraxis coordinate system is usually used.
For the brain coronal, up is dorsal and down is ventral. In the sagittal
plane, we have anterior to posterior and dorsal to ventral. Sagittal planes
can be mid-sagittal or parasagittal (off to the side). Coronal planes can be
more anterior or more posterior. Horizontal planes are more dorsal or more
ventral in the brain but are more caudal or more rostral in the spinal cord.
3.4. Parts of the brain 31
THALAMUS
SPINAL CORD
CEREBELLUM
BRAINSTEM
HIPPOCAMPUS
BASAL
GANGLIA
CORTEX
Fig. 3.4: Labeled schematic of the brain in sagittal section.
3.4 Parts of the brain
The brain is itself subdivided into an outer rind called the cortex and
multiple internal nuclei (Fig. 3.4). The cerebellum is an additional “
minibrain”
stuck on behind; it has its own nuclei and cortex. Major deep nuclear
complexes are the thalamus and the basal ganglia. Along with the wiring to
and from the cortex, these fill up most of the brain. The nuclei and cortex
are gray matter, the subcortical connections are white matter.
32 3. Basic Neuroscience
Computational neuroscience seeks to answer organizational and functional
questions about different areas of the brain. This goal is often
frustrated by the fact that, for many sections of the brain, basic functions
are still a mystery. As an analogy, imagine that, in the far future,
technical know-how has been lost after one of the usual apocalypse-movie
scenarios. Someone finds a radio, an artifact of ancient civilization. The
effort to understand what it does will be much frustrated by the fact that
when turned on it just hisses, having no radio transmissions to pick up.
Similarly, many brain areas are in the midst of a great system of other
brain areas of equally unknown function. We take one of these areas out to
study, shock one end, and look at the mysterious hissing coming out the
other end.
For this reason, many of the best-studied parts of the brain are those
that interact directly with the environment. The retina is such a part. We
know that it developed to perceive light, and we can control the amount
of light that hits it. By contrast, although we know that the hippocampus
has something to do with memory, we don’t really know what it does.
It is connected to structures on either side whose function is even more
mysterious. Therefore, as we study the basic anatomy and physiology of
the hippocampus, we need to hypothesize global as well as local functions.
The cortex is one of the most popular brain bits for humans since it is a
major part that makes us different from other creatures. The major parts of
the cortex, called neocortex (new cortex), form the outer layer of the brain.
The cortex is heavily folded, giving it greater surface area. Human cortices
are considerably more folded than those of most monkeys. This is important;
babies who are born with smooth, unfolded brains (lissencephaly)
suffer serious retardation.
Moving across the cortex, there is evidence of functional organization into
columns laid out in a two-dimensional array. Moving down into neocortex,
six layers of interconnected axons, dendrites, and cell bodies can be
identified.
Cortex can also be subdivided into different areas that are responsible
for different aspects of sensory processing, motor control, and cognition.
Occipital cortex, in the back, does vision. Frontal cortex, in the front,
does
socializing and judgment. Left temporal cortex (lateral and ventral) does
speech recognition. Right parietal cortex (more dorsal) does spatial
orientation.
Grossly, the left brain is literate and numerate and the right brain
is good at pictures. This might well remind you of phrenology, the old
science
of feeling the skull to figure out personality. In fact, areas of the brain
may increase in size depending on their usage so that highly musical people
will have a larger auditory area. Cortical areas are generally represented
as
forming a hierarchy from the simple perceptions of primary sensory areas,
through perceptual integration, up to the multimodality representations of
association areas. However, cortical areas typically feature strong
recurrent
loops between connected areas, and there is much evidence to suggest that
even lower areas participate in high-level perception.
3.4. Parts of the brain 33
In addition to neocortex, there are other cortical areas that are regarded
as more primitive because they are found in lesser creatures and because
they appear to be more simply organized. Two of these areas are favorites
for modeling: the piriform cortex and the hippocampus. Piriform or olfactory
cortex is responsible for smell. Olfactory cortex is the only sensory
cortex that receives its input directly from the end organ rather than
receiving
sensory information via the thalamus (described below). In addition
to its simplicity, the closeness of olfaction to the periphery makes
piriform
cortex particularly popular for studying structure–function relations.
The hippocampus is one of the most heavily studied and modeled areas
of brain. It is the area responsible for episodic memory. Episodic memory
means memory for specific incidents and can be contrasted with procedural
memory (memory of how to do things like ride bikes and play simple games)
and semantic memory (how many states in the U.S.). The hippocampus is
characterized by largely feedforward excitatory connections and the presence
of long-term potentiation, a form of synaptic weight change suggestive
of the types of changes used in artificial neural networks. Additionally,
the
hippocampus is involved in certain types of epilepsy, making it a favorite
modeling locus for clinically oriented studies as well.
Another favorite area for modeling is the thalamus. The thalamus is
thought of as the gateway to the cortex. All sensory information except
smell comes into the brain through individual nuclei of the thalamus, which
are called relay nuclei. One of the best studied of the relay nuclei is the
one for vision, the lateral geniculate nucleus (LGN). In addition to sensory
nuclei, there are other divisions of the thalamus that are involved in
memory
and with movement. In general, the thalamus and neocortex are heavily
interconnected with excitatory feedback loops. There are also a number
of inhibitory feedback loops within the thalamus. Overall, the thalamus’s
relay function remains obscure. One hypothesis is that the thalamus may
be involved in directing attention to a particular stimulus. In addition to
its role in sensation, the thalamus is also heavily involved in sleep. Again,
its role here remains obscure, partially because the role of sleep remains
obscure.
The basal ganglia is a central motor region in the brain. It appears to
be involved in initiation and planning of movement. Diseases of the basal
ganglia include Parkinson disease, which causes people to move too little,
and Huntington and Tourette diseases, which cause people to move too
much. Unusual features of basal ganglia organization include the facts that
it is largely feedforward, that projections are largely inhibitory, and that
there is massive convergence from cortex. The basal ganglia is part of a big
loop involving motor cortex → basal ganglia → thalamus → cortex.
The cerebellum is also generally regarded as a motor area, though it
may better be thought of as being a site of sensorimotor organization.
It gets a lot of limb and body position information and helps coordinate
movement and posture. The cerebellum, like the basal ganglia, is notable
34 3. Basic Neuroscience
for the dominant role of inhibition. Purkinje cells are massive inhibitory
cells in the cerebellum. They are arrayed in long rows and look like power
stanchions with 10,000 wires running by them making synapses. This very
regular organization has made the area a favorite for modeling as well.
Other, more remote provinces of the nervous system can also be modeled.
The autonomic nervous system, with both central and peripheral branches,
is largely responsible for nonvoluntary activities such as heart function,
blood flow regulation, and pupillary response to light. These systems
generally
maintain the level of some physiological variable such as blood pressure
or light hitting the retina, just as a thermostat maintains temperature in
a house. Such negative feedback systems are well explored in engineering,
and these models have been applied to the autonomic nervous system.
The enteric nervous system runs the gut. Although the enteric nervous
system has connections that allow it to communicate with the brain, it
can function independently. An isolated gastrointestinal tube, disconnected
from the body, will still eat and defecate quite happily. Fancifully, I
think
of the enteric nervous systems as a separate creature within (i.e., an alien
),
with the approximate sentient capabilities of a large worm. The vague
abdominal
notions of fear, anger, or satiety are perhaps the closest we can
come to understanding what a far simpler organism feels and knows about
his surroundings: the worldview of a worm.
The brain is a complicated place. Here are a few of the many other
brain areas that I haven’t mentioned: cingulate cortex, claustrum, amygdala,
Forel’s field H3, red nucleus, substantia nigra (black stuff), locus
ceruleus (blue place), nucleus solitarius, nucleus ambiguus. Some of these
have an approximately known function, or at least known affiliations with
areas of approximately known function. Others remain enmeshed in utter
puzzlement and enigmatic mystery.
3.5 How do we learn about the brain?
In the above section, I emphasized how little we know about the brain.
In this section, I want to talk about how much we do know and how we
know it. As discussed in the first chapter, the levels to be investigated
range
from studying the flow of ions through membranes to studying cognition. In
computational neuroscience, major areas of interest are information,
representation,
and calculation. It’s likely that many of the levels of investigation
will turn out to be relevant to understanding these areas. Other levels may
not be relevant to these functional issues but will still be of interest as
we try to understand neural dynamics. For example, our understanding of
epilepsy has benefited greatly from computer models of brain activity.
Although any and all levels are of potential interest, a few levels get most
of the attention. These are generally the lower levels, from the gene to the
3.5. How do we learn about the brain? 35
single cell. Like the parable of the drunk looking for his keys (under the
light rather than where he dropped them), researchers look where they can
see. Researchers are even more strongly motivated than drunks: they have
to get grants.
The good news is that our light on the brain expands as new techniques
are developed. The development of physiological probes such as positron
emission tomography (PET) and functional magnetic resonance imaging
(fMRI) have permitted the viewing of activity in the living, thinking human
brain. However, there still remains a large gap in the scale at which
activity
is accessible. It is possible to record from a single cell in an animal
brain
but not usually in a human brain. It is generally not possible to record
from a neuronal ensemble (hundreds to hundreds of thousands of neurons).
An important distinction to be aware of in discussing research methods
is the distinction between anatomy and physiology. Anatomical methods
provide static measures of structure: classical slice-and-dice man-in-the-
pan
methods. Anatomy does not show activity. Physiological methods do show
activity. In general, when we are interested in information transfer, we are
more interested in physiological measures. However, anatomical measures
can help us with the wiring and basic layout of the brain.
Anatomical methods
Major anatomical methods involve using some sort of radiation, mostly
visible light, infrared, radio, x-ray, and electrons. Anatomy is the science
of figuring out how to look and then figuring out what has been seen.
The major anatomical tool is the microscope. Another major type of tool
is the imaging device. This includes computer tomography (CT or CAT
scan), which uses x-rays, and magnetic resonance imaging (MRI), which
uses the nuclear magnetic resonance (NMR) that results when radio waves
are applied to atoms in a strong magnetic field.
Light microscopy gives a nice view — one can zoom in from the naked
eye to about 1000-fold magnification. Light microscopy is very convenient
since we have an organ that can directly detect and image radiation in
this range. The limitation to light microscopy is due to the wavelength of
light, about half a micron. It is impossible to clearly resolve anything
much
smaller than a micron. Cell somas are about 5 to 30 microns across and can
be easily seen under a microscope. Synapses, however, are tiny structures
of about 1 micron and cannot be seen in any detail with light.
Another problem with most types of microscopy is that the raw brain
doesn’t provide much contrast — it’s mostly white on white with some gray
on gray. To get contrasting images, various stains are used. The Golgi stain
is historically important because it was used by Cajal in his turn-of-the-
20th-century studies that identified the neuron and its role. Ironically,
this
stain permitted Cajal to see the separation between neurons and correctly
counter Golgi’s contention that neurons formed a continuous syncytium
36 3. Basic Neuroscience
of tissue. The Golgi stain is usually picked up by only a few cells in a
slice of tissue and turns these cells black. Some modern techniques involve
direct injection of dyes though electrodes placed in specific cells. Since
these
electrodes can also be used to measure electrical signals, this offers a
nice
opportunity to correlate electrical activity with cell shape.
Electron microscopy (EM) works at a shorter wavelength and permits
photography of objects down to about 10 nanometers. A disadvantage is
the inability to zoom in and out so as to identify the connectivity of the
structure seen. This means that it is not always obvious whether one is
looking at a piece of a neuron or a piece of some other cell. For example,
it
took researchers a long time to pick synapses out of the many little circles
and lines seen on the black and white images provided by EM. Additionally,
EM requires difficult staining, embedding, and cutting procedures.
Imaging methods, CT and MRI, are used to look at macroscopic structures.
Presently these methods cannot be brought down to microscopic
levels except in small pieces of tissue. These imaging views can be
extremely
clear, sometimes better than looking at sliced brain after death.
Of course, they have the further advantage of not requiring either death or
slicing. These are in vivo techniques, used in the living organism, to be
contrasted
with in vitro techniques, used in tissue that has been removed and
placed in a dish. These techniques are particularly useful clinically since
pathology such as brain tumors and stroke can be seen. However, since
current
imaging technology does not permit visualization down to microscopic
levels, there remains a large gap between what can be seen in life and what
can be seen after death
In addition to the neuroanatomical imaging techniques, there are also
physiological techniques that give some measure of activity in the brain
rather than just recording structure. The most prominent of these techniques
are PET and fMRI. These can be used to measure ions or
metabolites in a living animal or person.
3.6 Neurophysiology
Neuroanatomy is the study of form. Neurophysiology would like to be the
study of function. Unfortunately, it only occasionally rises to this ideal.
We
are often so clueless as to what’s going on that neurophysiology becomes
more a description of the dynamics than an explanation of its meaning
or intent. Physiology measures things that change. In neurophysiology, a
major thing that changes is electrical potential; hence, much of
neurophysiology
is electrophysiology. Other things that are measured include the flux
of specific ions such as calcium, the uptake of nutrients such as glucose,
and the binding of neurotransmitters such as glutamate.
3.7. Molecular biology and neuropharmacology 37
The techniques of electrophysiology generally involve the use of
microelectrodes
that measure either voltage or current (see Chap. 16, Section 16.6
for description of electrical concepts, and Chap. 11 for details of
electrophysiology).
The electricity measured in the nervous system is generated
by the equivalent of a battery that produces voltage across the cell
membrane.
This battery keeps the inside of the membrane at a negative potential
(potential and voltage are synonyms), called the resting membrane potential
(RMP). Deviations from the RMP are electrical signals in the neuron
(Fig. 3.2). Signals that make the membrane more negative inside are called
hyperpolarizing. Hyperpolarization means further polarizing the membrane
beyond its already negative polarization. These signals are considered
inhibitory,
although they do not always reduce cell firing (see Fig. 12.9).
Signals that make the membrane more positive inside are called depolarizing
(they relieve some of the negative polarization) and are usually
excitatory. The action potential or spike is a brief (1 millisecond)
duration
depolarizing signal that briefly reverses membrane polarity, making
the membrane positive inside. The action potential can travel along the
axon and produce release of chemical (neurotransmitter) at a synapse.
The most direct way to measure potential is to insert a glass microelectrode
through the membrane and compare the potential inside to the
potential of a wire placed outside. This is intracellular recording.
Electrodes
used for intracellular recording are tiny hollow straws of glass called
micropipettes.
The glass itself doesn’t conduct electricity but the hollow tube
is filled with a salt solution that does. Alternatively, metal electrodes
are
placed near the cell for extracellular recording. Although these electrodes
do not directly measure a neuron’s intrinsic voltage, they do detect
electrical
fields that closely reflect this voltage. If an extracellular electrode is
close enough to a cell, it can record activity from that cell alone. This is
called single-unit recording. If you pull the electrode further away, it
will
pick up signals from multiple cells — multiunit recording. The signals from
the different cells will have slightly different shapes that make them
distinguishable
from one another. If you pull the electrode still further away,
all of the signals blur into an average field potential. Field potentials
can
be strong enough to be detected from outside the head. This is the potential
measured by electroencephalography (EEG), which detects very small
potentials by using electrodes glued to the head.
3.7 Molecular biology and neuropharmacology
Molecular biology has in recent years been the most rapidly advancing field
of biomedicine. The capability exists not only to knock out specific genes
but in some cases to knock out specific genes in specific brain areas at
specific
times in development. The well-known “central dogma” of molecular
38 3. Basic Neuroscience
biology is that DNA makes RNA makes protein. By creating a “knockout,”
a particular gene is inactivated, permitting us to evaluate changes in
an animal’s behavior that reflects the function of that gene. The problem
with this approach from the modeling perspective is that these behavior
changes are unlikely to be a direct and obvious consequence of function.
By way of a hoary example, if you stick a screwdriver into your radio and
it starts making a high-pitched whine, this does not necessarily imply that
you have cleverly ablated the “whine-suppressor” for this radio.
Neuropharmacology is the study of the effects of drugs on the nervous
system. Neuropharmacology is mostly the study of receptors — figuring
out where in the brain different compounds bind and what their binding
activates. In general there are many different subtypes of receptor for each
endogenous ligand and there are usually many different non-endogenous
compounds that also bind to specific receptor subtypes. Some of these
non-endogenous compounds are synthetic, having been developed by drug
companies to try to treat disease. Many more, however, are natural products
that one organism makes in order to kill or disable another organism.
General examples of this are apple seeds, which contain cyanide, and fungi
that make penicillin to kill bacteria. Neuroactive compounds are quite
common
poisons since the nervous system is a particularly vulnerable spot in
animals. The puffer fish makes tetrodotoxin, a sodium ion channel blocker.
Bees make apamin, a potassium channel blocker. Scorpions produce
charybdotoxin,
another potassium channel blocker. The deadly nightshade plant
makes belladonna, deadly at high dose but formerly used at low doses to
make a woman beautiful by increasing the size of her pupils.
3.8 Psychophysics
Psychophysics uses precisely defined stimuli and then asks how they are
perceived by a person, often the investigator himself. Early triumphs in
this area involved the use of specific colors to probe the way that the eye
perceives light. Even before detailed molecular and cellular studies could
be done, this led to the understanding of the three different types of color
receptors in the retina, each tuned to a different frequency. The relation
between these frequencies explains the effects of color combinations: why
blue and yellow appear green. More recent psychophysical research has
used computer-generated images to probe higher levels of visual function.
For example, it has been shown that rapid projection of images can lead to
breakdown in binding of stimulus attributes such that the color and shape
of an object are not correctly correlated. For example, rapid presentation
of a red square and a blue circle can lead to perceptual error such that a
blue square and a red circle are seen (illusory conjunction).
3.9. Clinical neurology and neuropsychology 39
3.9 Clinical neurology and neuropsychology
Medical study of people with brain disorders has played a central role in
our
understanding of normal brain function. Damage to the brain produces in
the victim a very personal experience of the dependence of mind on brain.
Brain damage can produce peculiar states of depersonalization and
derealization.
Patients with obvious brain pathology are treated by neurology,
while those with grossly normal brains are classified as having mental
disorders
and treated by psychiatry. A disease like stroke is easily seen to be
a brain disease; if you take the brain out, you find a hole in it. In a
disease
like schizophrenia, abnormalities in brain structure are much more subtle
and are only now being discovered. Schizophrenia was traditionally viewed
as a mind disease, due to having a bad mother rather than a bad brain. In
addition to the increasing awareness of brain abnormalities in psychiatric
disease, there is also increasing appreciation of the changes in personality
and thinking that occur in association with brain diseases such as multiple
sclerosis and Parkinson disease.
The object of clinical research is to alleviate illness rather than to
understand
human function. Nonetheless, studies of patients have contributed
substantially to our understanding of the brain. Disease can be regarded as
an experiment of nature that may reveal important insights about neural
organization. From this perspective, ablative and intrinsic brain diseases
represent different kinds of experiments. Just as the design of an
experiment
determines what can be and what is discovered, the kinds of brain
diseases that occur have had a large influence on our view of the brain.
Ablative diseases
An ablation knocks out a piece of brain. Stroke, tumor, and trauma are
major ablative disorders. These disorders come from outside of the
functioning
brain itself. They are imposed on the brain. War injuries were the
first brain diseases studied in the early days of neurology, during the
Civil
War. More recently, brain dysfunction after stroke has been heavily studied.
A stroke occurs when a blood vessel is blocked off, starving a part of
the brain and killing it. A stroke of the left middle cerebral artery
typically
causes an aphasia, a disorder of language. Such aphasias have been widely
studied to learn how the brain processes language. However, the specific
patterns of aphasia may have as much to do with the particular patterns
of middle cerebral artery organization as with the brain’s organization for
language production.
Since ablative diseases are imposed on the brain from outside, these
insults can produce idiosyncratic effects that do not reflect attributes of
brain organization. In general, the modular nature of ablative disease has
suggested a modular view of brain function. After a stroke, an area of
brain is lost and the patient has a particular dysfunction. It is then
natural
40 3. Basic Neuroscience
to conclude that the piece of brain that was lost was the “center” for
the function that was lost. From there, one builds up a view of the brain
as a series of these centers that pass information to one another. Before
the development of brain imaging, such modular brain theories were very
useful for the localization of strokes. Modular theories continue to be
highly
influential in brain science as well as in neurology. Certainly, the brain
is
not equipotential: some brain areas are dedicated to vision, while others
are used for hearing. However, while there is merit to modular theories,
they tend to be oversimplified and can tell us only a limited amount about
how the brain functions.
A remarkable set of stroke-related mental disorders are those seen with
damage to the right hemisphere. Neglect is a neurological syndrome that
occurs frequently, albeit often transiently, following large strokes in that
area. The neglect is of the left side of the world and the left side of the
patient’s body. Patients ignore people and things situated to their left.
Most remarkably, they may be unable to identify their own left arm. If
pressed, they confabulate absurdly, concocting ever more elaborate stories
to explain why a strange arm would be found in such close contact with
their body. One patient, for example, was asked so frequently about his
arm that he developed the peculiar habit of throwing one cigarette across
his body whenever he smoked. When asked, he explained that his brother
was sharing the bed with him and he wanted to share the cigarette with
him.
The bright side of ablative diseases is that they are not progressive.
They damage the brain and stop. The brain can then start to recover.
It is remarkable how well the brain can recover. A few months after a
stroke, a person may be left with no noticeable deficit, despite having a
hole in his brain that can be seen on MRI. The process that leads to
recovery of function is similar to the process of functional organization
during development and to the normal process of cortical reorganization
that occurs with learning. In all of these cases, modeling has suggested
that
an initial phase of altered dynamics is followed by synaptic plasticity. The
brain is a dynamic structure, with functional circuits that are repeatedly
organized and reorganized.
Intrinsic diseases
Diseases caused by alterations in cellular organization, metabolism,
neurotransmitters,
or electrical conduction are intrinsic diseases. Correlation
of mental dysfunction in these diseases with underlying cellular or chemical
abnormalities will likely provide important insights into how the brain
works. For example, schizophrenia produces peculiar patterns of thought
that are also seen between seizures in some forms of epilepsy. These
syndromes are likely to involve widespread abnormalities in intercerebral
3.10. Summary and thoughts 41
communication. Intrinsic diseases are typically progressive, making them
degenerative diseases
Intrinsic disease are generally harder to study than ablative disease. As
experiments of nature, they correspond to more recent microscopic
experiments
that classify various cellular and molecular components of the
nervous system. Study of these diseases tends to suggest a global
interconnected
view of brain function. Intrinsic diseases of the brain are caused
by disorders of the elements of brain organization: synapses, ion channels,
or particular neuronal types. The prototypical intrinsic diseases are
those involving intoxication with neurotransmitter agonists or antagonists.
Disorders of mentation associated with various drugs are examples of this.
Parkinson disease is another example of an intrinsic disease. Loss of a
particular type of cell results in a decline in release of dopamine, a
neurotransmitter.
This has profound effects on movement and thinking, causing
a marked slowing as well as other problems. Giving a dopamine precursor
as a drug can restore these functions. This finding suggests a diffuse
network of interacting units somehow glued together by use of a common
neurotransmitter.
3.10 Summary and thoughts
Even without considering computational neuroscience, one finds many
different
ways of thinking about the brain. These come from the different
types of experimental techniques available, the different kinds of training
of the people doing the experiments, and even from the varying types of
diseases that people get.
Much of the research that has been done has gone into elucidating the
chemical and electrical properties of individual neurons. Far less has been
done at higher levels. In particular, the technology does not exist to
determine
detailed neuronal connectivity or to assess the activity in large
neuronal ensembles in behaving animals or people. Historically, notions
about the highest levels of perception and behavior have mostly been
deduced from strokes and personal intuition. The recent development of
functional imaging should give more solid information about brain activity
during mentation.
Now that techniques for studying most of the levels of organization are
becoming available, it is tempting to look forward to the day when we can
just look at all levels of neural activity and see how the brain works. With
the experimental means to identify everything everywhere all the time, we
could build the perfect brain model. It would do everything that the brain
does in just the way the brain does it. Even with this, understanding would
still elude us. We still wouldn’t understand any more about how the brain
does what it does. Complex systems like the brain show emergent proper42
3. Basic Neuroscience
ties, properties that cannot be explained by just knowing the properties of
the constituent parts. The classic example is the emergence of the laws of
thermodynamics from the bulk behavior of very many gas particles, each
of which follows Newtonian rules. Newton’s laws (or quantum mechanical
laws) do not directly suggest the thermodynamic laws.
Similarly, a small ganglion in the lobster runs the lobster stomach with
only about 30 neurons. These neurons have all been studied thoroughly;
their responses mapped. Given the simplicity of the network, it would seem
that knowledge of the elements would yield understanding of the network
dynamics. This has not been the case. Computer modeling is being done
to understand the emergent properties of this simple network. There are
several Web sites devoted to this thing — look for stomatogastric ganglion.
Neuroscience subfields like neurophysiology, neuroanatomy, and
neurochemistry
each cover a particular area of neuroscience understanding.
Explicit computer models build bridges across the gaps between these fields.
Every researcher is a modeler. Without a model, whether held in the head
or written out in words, pictures, or equations, data do not confer
understanding.
The brain is a place of myriad wonders and many different models
will be needed to explain it.
Part II
Computers
4
Computer Representations
4.1 Why learn this?
In the 1950s, biologists were homing in on the chemical source of heredity.
Skeptics pointed out the enormous variety of things that had to be coded
for: enzymes and structural proteins, the shapes of faces and bodies, the
number and the form of limbs, the structure of the brain, and the structure
of the kidney. It seemed inconceivable that all these representations could
be reduced to any single process or principle, much less to a single code or
a small set of symbolic chemicals. The skeptics were wrong. Turns out that
there are four chemicals that are arranged in triplets to code for 20 amino
acids plus stop and start codes. Admittedly, this hasn’t solved the problem
of understanding where an arm comes from. However, it’s remarkable that
at some level everything can be expressed with an alphabet of four letters,
word lengths fixed at three, and some awfully long sentences, all of which
can now be read.
Many people are skeptical that any simple code will be extracted from
the brain that will allow us to congeal our understanding of vision,
audition,
volition, and respiration, to name just a few. Actually, I’m skeptical.
However, some of the pioneers who unraveled the genetic code subsequently
switched over to the search for the neural code. Perhaps their experience in
genetics taught them something of the possibilities in impossible problems.
46 4. Computer Representations
The classical behavioralists held that, as far as the brain was concerned,
“There is no there there”: 1 everything could be explained by simple loops
connecting a stimulus with a response. If something happened out in the
world, the brain would respond appropriately; there was no need to postulate
complex internal representations or internal processing. I haven’t read
much behavioralist literature, but even behavioralists must have admitted
that people and animals can remember things. Anyway, nowadays it’s
generally
accepted that there are representations in the brain. At some level,
the brain has to represent or model the world out there. A lot of research
in computational neuroscience, cognitive neuroscience, and neurology has
to do with hypothesizing what these representations might be and then
coming up with tests to see if these representations can be validated.
In this chapter, I discuss basic computer representations. Representations
in the brain most likely differ considerably from computer representations.
Nonetheless, learning about computer representations is valuable for a
couple
of reasons. First, these representations are very influential in neural
network research. For example, most neural network models assume that
a neuron can be approximated by a unit with a scalar state (a scalar, a
single number, is contrasted with a vector or array of numbers). Some go
further and suggest that this scalar state will only take on two values (i.e
.,
a binary device — like a transistor).
Another reason to study computer representations is that these
representations
are the best-quantifiable examples of complex information storage
strategies that we have access to. Note the qualifier “quantifiable” here.
One
would expect language, poetry, music, and painting – complex products of
the free-range brain — to be closer to brain representations. Linguists
argue
that the structure of language is a good indication of the structure
of the brain and that commonalities among different languages reflect this
structure. At some level this is likely to be so. However, language is a
rare, unusual brain capacity seen only in one species. The human brain has
probably developed far in a unique direction to perform this clever trick. I
suspect that language will turn out to involve a peculiar mapping, remote
from any essences of neural representation.
Representations in a computer are called data structures. They are
designed to be efficient for some purpose. Such design always involves
tradeoffs.
In this chapter, we look at typical computer representations for the
alphabet, for numbers, and for pictures. In the next chapter, I discuss
computer design and the design of programs, demonstrating how a program
is just another type of representation. The design of appropriate data
structures is a key aspect of the art of computer programming. Similarly,
competing needs and pressures have formed representations in the brain as
well.
1Gertrude Stein’s comment about Oakland, her hometown.
4.2. Calculator or typewriter 47
4.2 Calculator or typewriter
Many people think of computers as overgrown calculators. However, the
major thing most people do with a computer is word processing, which
is not calculation at all. The computer, a mysterious object for many,
thereby seems all the more mysterious: a glorified calculator or a glorified
typewriter? This has also been a problem for computer experts. A lot
of debate about computer architecture had to do with dual design goals:
data processing for business markets, and number crunching for scientific
and engineering markets. When the founder of International Business Machines
(IBM) made the famous prediction of a world market for only five
or six computers (1958), it reflected the difficulty of seeing any advantage
to using an expensive, and at that time unreliable, machine as a substitute
for file cabinets and punch cards.
The central concept that reconciles the computer’s calculating and word
processing functions is the notion of representation of information. Ideas
about representation come as much or more from business as from mathematics.
Computers emerged as tools for business and government under the
auspices of IBM. Historically, the early “information revolutions” came
out
of business. (And perhaps prehistorically, counting and writing probably
developed for trade.) In the early Renaissance, double-entry bookkeeping
developed as a data management technique to organize debits and
credits in an accessible data structure. Other early revolutions included
both well-known high technologies such as printing, and more obscure low
technologies such as pigeonholes and alphabetical filing.
The epitome of turn-of-the-20th-century information processing technology
is the often caricatured large room filled with clerks, each of whom has
an in-box and an out-box. This technology was widely used for “statistics,”
a field that originally meant keeping track of information about the state
or
nation. At the time, this labor-intensive technology was proposed as a way
to do weather forecasting, using the same algorithms as those now used by
supercomputers for this purpose. The idea was that each clerk would be
provided with information about air temperature, pressure, and winds at
a particular part of the globe for which he was responsible. By comparing
his information with that of neighboring clerks responsible for neighboring
areas of the globe, he could determine whether air would be flowing into or
out of his zone, and thereby determine weather. This was an early discovery
of the finite element algorithm that is used today. The algorithm was never
implemented using office workers.
48 4. Computer Representations
4.3 Punch cards and Boolean algebra
At the end of the 19th century, standardized data forms and punch cards
came into use, introduced in order to tabulate counts for the United States
census. Early data processing machines were developed to count holes in
punch cards. These were simple mechanical computers but differed from
modern electronic computers in that they were not user programmable.
Their programming was built in, and was generally fairly straightforward
— e.g., count the number of times that a hole in row 15, column 5 is
present
along with a hole in row 23, column 73.
A simple version of this basic computing device can be built with index
cards. Prepare each card by punching 10 holes uniformly along the top.
Binary information can then be entered by cutting out the top of a hole to
indicate “true” and leaving it closed over to indicate “false.” For
example,
let’s say you have a file of recipe cards. Some recipes require sugar, some
require flour, and some require broccoli. You assign each of the punched
holes to a particular ingredient. If that ingredient is present in the recipe,
you cut open the hole to the edge of the card. If the ingredient is not
used in that dish, you leave the hole alone. The intact hole (the default)
then means false (represented as F or 0 in Boolean algebra); the open hole
represents true (represented as T or 1).
If a dish requires sugar and flour, but no broccoli, the sugar and flour
holes should be opened up to indicate that it is true that the recipe requires
sugar and it is true that the recipe requires flour. Now comes the clever bit.
You have a stack of recipe cards and you have discovered that you have
broccoli on hand, but no sugar or flour. You take a knitting needle and pass
it through the “sugar” hole and the “flour hole.” You pick up the stack by
the two needles and shake slightly. Those cards that require sugar or flour
will fall out. You put aside the cards that fell out. You now do the same
thing with the broccoli hole, but in this case you keep the cards that fall out.
This is the set of recipes that require broccoli but do not require sugar or
flour. In Boolean notation, you have performed the logical operation “(not
(sugar or flour)) and broccoli.” which can also be written “(not sugar and
not flour) and broccoli.” In Boolean notation, using the first letter of each
word as the symbol, this is (∼ (S∨F))∧B or, equivalently, (∼ S∧ ∼ F)∧B.
The needle always pulls out cards that don’t use that ingredient (False) and
leave behind cards that do use it (True). Complex Boolean operations can
be performed using the proper combination of needles.
This recipe card scheme was marketed as a practical database management
technique in the 1960s. Although it has been made obsolete with
the increased availability of databases on electronic computers, the algorithm
remains a viable one for some database implementations. Multiple
true/false fields can be stored in a single word with each true/false choice
taking up only a single bit. These can then be combined using Boolean
operators such as “and,” “or,” and “not.”
4.3. Punch cards and Boolean algebra 49
Any information storage strategy has advantages and limitations. Different
techniques will differ in the type of information most readily stored, the
storage capacity, the ease of entering new information, the speed of accessing
old information, and other factors. Trade-offs depend on what aspect of
data manipulation needs to be optimized. For example, a database that has
to have frequent changes will be optimized far differently than one that gets
set up once and then accessed many times. One well-studied trade-off is
that between space and time: big memory-consuming data structures typically
offer faster access to information. Although the specific data storage
issues are likely to be far different in the brain, the general trade-offs may
well be similar. In particular, we might expect that brain areas that require
rapid access to information might be bulkier than those where speed
is not important. This might be a contrast between the memory functions
of cerebellum and hippocampus.
A common data format that illustrates the time/space trade-off is the
scantron test form that’s used for the Scholastic Aptitude Test (SAT) and
other standardized tests. These forms have been optimized for ease of input
by the untrained user. Although it always feels like they take a lot of time to
fill out, they are actually rather quick when one considers the training that
would be required for use of a more complex, more compact data structure.
At the beginning of these tests, you fill in your name and student ID with
the famous number 2 pencil. Each circle on the scantron form represents a
single bit. It can either be filled in or not. If it is filled in, it represents a 1
and can be considered a set bit. If it is empty, it represents a 0 and can be
considered a clear bit.
The bits on the scantron form represent not only binary numbers but
also represent your name, social security number, and answers to the test
questions. This is where the choice of representation comes in. One needs to
represent the many digits and many possible values of each name field using
bits. Let’s take the binary representation of a last name as an example. Each
letter of the name is represented by a set of 26 bits (in a column), out of
which one bit is set and the rest are clear. The name is represented across a
fixed number of columns, usually 20. This is a representation of your name
in binary. It uses 20 ・ 26 = 520 bits of storage.
The scantron representation of your name is inefficient but is an easy one
for untrained people to understand and quickly master. This representation
is therefore a good one for the person–machine interface. Since we know
that the first letter of the name is a capital and the rest are small letters,
we only need the 26 letters and can simply translate the first letter into its
capital form and the others in the small form. Including all 52 large and
small letters would be a redundant representation for this purpose. This is
an example of how the information content of a message can’t be viewed
in isolation but is also dependent on the knowledge base of the transmitter
and receiver of that message.
50 4. Computer Representations
A more efficient name representation would be a direct bit code for each
letter. Given that there are only 26 letters, only five bits would be required
to store a letter in binary since 25 = 32. As we see below, the standard
computer encoding for a letter actually uses seven to eight bits. Given that
your last name is probably less than 20 characters in length, bits are being
wasted there as well. If we instead wanted to use a five bit code for each
letter of the alphabet, the user would have to look up each code in a table.
This is poor design for a human interface; there would likely be many errors.
An additional inefficiency comes from having a fixed-length representation
of 20 characters for the last name. This inefficiency represents a limitation
of the medium: paper can’t grow and shrink as needed. Although hard disks
and memory also don’t physically grow and shrink, many data structures
do so. If you want a data structure to be arbitrarily sized, you typically
use a marker to indicate where the field ends or a variable counter to give
the field size for each instance. A compact code for my six-letter last name
might use 36 bits: six bits to give the number of letters (this allows counts
up to 63) and 5 ・ 6 = 30 bits for the name itself. This is much smaller than
the 520 bits of the scantron form.
4.4 Analog vs. digital representations
The word digital comes from the word digit, meaning finger. Though this
would suggest that it refers to the base 10 number system, the word is
used to refer to any equipment that uses discrete rather than continuous
states. Thus, the modern computer is a digital computer that uses binary
representations. This is likely to be a major difference from brain function.
Although Von Neumann and other early computationalists took the all-ornone
nature of the axonal spike as evidence that the brain was primarily a
binary digital device, this viewpoint is not currently popular.
Binary means 2-valued: each element (a binary digit or bit) takes on
either a value of 0 or a value of 1 (see Chap. 16, Section 16.3). One can
produce a discrete system using any number of symbols. For example, a
computer could use decimal instead of binary by dividing up the 5 volts
into 10 ranges instead of 2, each of which would then represent a digit
from 0 to 9. Discrete representations can be readily expressed in symbols.
Each symbol of a set of symbols corresponds to a particular value. For
example, the two physical values of the binary digital representation used
in a computer are nominally 5 volts and 0 volts. These values are symbolized
as HIGH vs. LOW, True vs. False (using positive true), T vs. F, or 1 vs. 0.
Using binary symbols is a good idea for discussing computer systems since
the underlying signal, voltage, is actually an analog measure. Because of
this, a computer has to accept a range of values for each of the symbolic
4.5. Types of computer representations 51
values: typically anywhere from 3.5 to 5 volts for 1 and anywhere from 0
to 1.5 volts for 0.
A digital representation uses discrete values. An analog representation
uses continuous values. A property of a continuous value is that there is
always another value between any two values that you pick. For example,
there are temperatures between 53o and 54o. In fact, there are an
infinite number of temperatures between these values. Like temperature,
other measurable macroscopic values in the world are analog values, although
quantum values are, by definition, discrete. Voltage is the main
quantity used to encode information in the computer. Digital information
is extracted from this analog signal by thresholding. Discrete representations
have the advantage of being precise. The field of information theory
is based on the ability to “count” the amount of information by translating
it into binary representations.
Analog representations are more complicated, since they can theoretically
contain infinite amounts of information, limited by the ability to read
out the information, determined by measurement resolution and noise. This
infinite precision doesn’t imply good accuracy, however. Early analog computers
were used to model the trajectories of artillery shells. Since the
equations that described the behavior of the electrical components in the
computer are identical to those that describe the behavior of cannon shells,
these machines could theoretically produce perfect simulations. However,
the electronic components were temperamental. Their parameters would
change with change in temperature and this would change the predictions.
The behavior of the artillery shells would also change slightly with temperature
change but this change was not be as great and perhaps not even in
the same direction.
4.5 Types of computer representations
I now describe some typical representations used in the computer. The
point is to illustrate that the nature of the computer’s design dictates the
forms of its data structures. The most substantial single requirement is that
everything must be reduced to binary. In the computer, binary provides
a single common representation from which all other representations are
built. Because of this, all representations are interconvertible. A picture
on the computer screen can be mistakenly printed out as gibberish text.
It can also be executed as a computer program, which will most likely
crash the machine. This interconvertibility of representations is a significant
difference between computer and brain. In the nervous system, information
is stored in a variety of chemicals and electrical forms. It is unclear how
these different representation systems can be related or if they may in part
52 4. Computer Representations
operate largely independently. The complexities of neurons presented later
in the book make computer data structures appear simple and elegant.
As we saw in the case of the scantron, computer representations may
be designed to allow the computer to interact with people and the outside
world or to communicate efficiently with each other, with other data
structures, and with programs. In addition to the environmental interactions
required by input and output to people, computers may also have
to directly sense or manipulate the physical world. Development of such
programs, the realm of robotics, is of great interest for our understanding
of brain function. Getting a machine to do something practical in the
outside world often reveals the limitations of representations that otherwise
seemed adequate. For example, a digitized photograph seems a pretty
good representation for planning navigation around a room. However, extracting
the needed information about the three-dimensional world from
this two-dimensional representation is a difficult task that has perplexed
researchers for years. One trick that brains appear to use is to actively seek
out sensory information on an as-needed basis rather than depending on
the passive absorption of information that happens to hit receptors. In the
context of seeing, this is called “active vision.” Your eyes are always moving
around, building and rebuilding your image of the world rather than just
waiting for something interesting to drift into view.
Representations are also specialized depending on extraneous factors
having to do with communications media or conditions. For example,
long-distance communication is expensive and makes a virtue of brevity
in the representation. This has led to development of data-compression
algorithms. Under circumstances where security is a concern, such as
communicating credit card numbers over the Internet, brevity is be sacrificed
in order to assure obscurity and prevent eavesdropping. This is the
realm of cryptography. Under conditions where transmission lines may be
unreliable, redundancy is used to ensure accuracy of transmission using
error-correcting codes and check sums.
4.6 Representation of numbers
It may seem funny that one has to worry about the representation of
numbers on a computer, a machine that was designed to handle numbers.
Even here, however, compromises must be made to balance the demands
of input/output (here communication with the human user) and internal
functionality (arithmetic operations). Arithmetic and logic operations are
performed in the computer with strings of zeros and ones. Modern computers
are designed to do this as efficiently as possible. To achieve this
goal various number representations have been developed to cleanly handle
negative numbers, fractions, decimals, complex numbers, logarithms, and
4.6. Representation of numbers 53
exponents. Operations range from addition of integers to transcendental
functions on complex numbers.
Early on, some computers were designed to work directly in base 10.
This meant easy communication with users but inefficient representations
with many wasted bits. It was also difficult to program these machines to
do arithmetic. Since the underlying hardware uses binary values (0 volts,
5 volts) on the transistors, it became clear that everyone would be much
happier if all internal operations were done in binary. All modern digital
electronic equipment uses binary. When you use a calculator to multiply
two numbers, the calculator translates the base 10 numbers that you enter
into bits.
Addition and subtraction are the most basic arithmetic operation.
Subtraction is, of course, just the addition of a negative number. The development
of various representations of negative numbers in order to permit
efficient subtraction illustrates the use of a machine-dependent hack to optimize
performance. In particular, the now-standard representation, two’s
complement, uses to advantage an architectural quirk that could otherwise
be regarded as a limitation on the computer’s ability to do arithmetic. As
far as I know, every computer nowadays uses two’s complement.
Two’s complement uses the fact that binary numbers are stored in physical
memory locations of finite length. Computer memory is divided up into
words and bytes. Depending on the specific computer architecture, words
can be of different sizes. However, a byte is now pretty much standardized
at 8 bits in length. A word will typically be 4 or 8 bytes, hence 32 or 64
bits. Using straight binary storage an 8-bit byte can store up to 111111112
(FF16) which is 1000000002 − 1 or 1 ・ 29 − 1 = 255. If we needed to store
bigger numbers, we could just take bigger and bigger chunks of memory.
Of course, we would then need some kind of additional tag to indicate how
many bytes were being used to store the number. Perhaps the number of
bytes used could be stored as the first byte of the data structure. This data
structure would be fairly efficient in terms of memory use but would make
it difficult to do additions and subtractions since the numbers would have
to be aligned correctly first. This representation also could not readily handle
negative numbers. Since this sort of elastic representation is not used,
numbers must fit within the limited space allotted. This means that numbers
that are too large cannot be represented. In particular, if you add two
big numbers, you will get a number that won’t fit. This process will lead
to an overflow condition, and bits will be lost. As we will see, this is the
machine limitation that is used in the two’s complement representation.
First let us look at a couple of alternative negative number representations
that have been used in computers. One way to represent negative
numbers is to take one bit away from the numerical representation and
make it a sign bit. This is called the sign magnitude representation. With
this representation, a byte only has 7 bits to represent the number, up to
12710. If the first bit of the byte is 0, that would indicate a positive number
54 4. Computer Representations
and if 1, a negative number. This method has the advantage of relatively
easy translation for entry or print-out since the presence or absence of a
negative sign translates directly into one bit. Unfortunately, addition and
subtraction are not particularly easy using sign magnitude.
Another representation is called one’s complement. This representation
takes advantage of the fact that it is easy to design hardware that will
invert all the bits of a number, changing all the 1s to 0s and all the 0s to
1s. This inversion of all bits is called the bit-wise complement of a number.
The bit-wise complement can be used as the negative of a number. Since
the bit-wise complement creates a 1 wherever there was a 0 and vice versa,
any number plus its complement will yield a byte with all 1s: 11111111.
Since a number plus its negative is zero, this means that 11111111 will be
a representation for zero in one’s complement. Since the complement of
11111111 is 00000000, this is another zero. The former is considered −0
and the latter +0. Having two zeros is a bit of an inconvenience for input
and output. An additional complexity arises in the need to add back the
overflow when doing subtraction with one’s complement. As an example,
let’s subtract 710 − 110 using 4-bit one’s complement. The bit-wise complement
of 0001 (110) is 1110. 710 is 0111. We add 710 + −110 in one’s
complement by doing 0111 + 1110 = 10101. We are using a 4-bit representation
and the extra bit on the left represents an overflow bit. Using one’s
complement, the overflow is added back to the 4-bit result to get the final
answer: 0101 + 1 = 0110 or 610.
An improvement on the one’s complement scheme is two’s complement.
Under two’s complement, there is only one zero value. There is also no need
to add back overflow bits. Instead, overflow can be discarded. The two’s
complement is formed by adding one to the one’s complement. Remember
that the sum of a number and its one’s complement is 11111111. Since the
two’s complement is the one more that the one’s complement, the sum of a
number and its two’s complement is 11111111 + 1. Since this produces an
overflow condition, 11111111 + 1 gives 00000000, which is the unique zero.
This overflow situation is familiar from experience with car odometers.
These can only represent a certain number of digits and will eventually
overflow to go back to all zeros. On an odometer, we can consider the two’s
complement of any positive number to be an equal negative excursion on
the odometer (i.e., driving backward). If we start at that negative point
and then drive forward the same distance, the odometer will register zero
again.
Let’s look at the previous problem, 710−110, using two’s complement. As
before, the bit-wise complement of 0001 is 1110, so the two’s complement is
1111. Then 710+−110 is 0111+1111 = 10110. We throw away the left-most
bit and get the desired answer: 0110 = 610.
To give another example, the calculations for 6 − 5 compared:
4.6. Representation of numbers 55
One’s complement Two’s complement
5 0101 0101
negation complement complement + 1
−5 1010 1011
6 0110 0110
sum: low bits 0000 0001
sum: overflow 1 1
overflow handling add back discard
final result 0001 0001
Representation of letters and words
Many of us have had our primary exposure to computers through the use of
a word processor. In this case, the primary representations used are those
for letters and symbols. Ancillary codes are used for things like formatting
and choice of font. The standard code for letters and symbols is called
the Ascii code. This assigns a unique 7-bit number for each letter, digit or
symbol (e.g., , . ; + =, etc.) as well as codes for non-printing symbols like
“newline” (the end of the line where you start up again at the next line).
Computer representations are generally designed for efficiency but the
standard for efficiency may differ widely depending on the purpose of the
representation. The computer I am writing on has a 64-bit architecture.
Since there are 8 bits to the byte, my computer has 8 bytes to a word.
The first sentence of this paragraph (italics) happens to be 160 characters
long, including the period. It will therefore fit into 20 words of memory. We
can therefore lay the sentence out as it would appear in the memory of my
computer (Fig. 4.1, top). This is a bitmap: each bit in a region of memory
is shown with a black square for a 1 and a white square for a 0. Each of
these squares is considered a pixel or picture element. As in the paintings
of the pointillists, if you make the pixels small enough they blend together
and produce a picture. In this figure, each word of memory is read from left
to right and consecutive words are read from bottom to top. The sentence
begins at a word boundary.
With a little effort, it is possible to read the text off of the bitmap. At
the lower left corner 01000011 is the capital C in the word computer —
the string of four 0s gives the horizontal black bar four squares long. It
is easy to see the layout of letters because all of the symbols of printable
Ascii begin with a 0 — Ascii is really a 7-bit code that is commonly stored
in the 8 bits of a standard byte. The vertical black bars in the bitmap,
representing 0s that line up in every word, show where characters begin.
Spaces are easily found because they begin with 00, as does punctuation,
for example the period in the upper right hand corner.
Since printable Ascii is basically a 7-bit code, we could store it in less
space. At the bottom of Fig. 4.1, I stored the same sentence 7 bits at a time.
This saves about two and a half words. There is 0-padding at the top right
56 4. Computer Representations
01000011
Ascii: ‘C’
Fig. 4.1: Two computer memory representations of the italicized
sentence in the text.
and lower left. The C starts at the first white pixel from the left on the
lowest row. The final period appears in mid-row at the top. Most printable
characters have a 1 in the first bit. Nonprinting Ascii representations are
used for monitor control, clearing or flashing the screen, or resetting the
cursor, and are available from the keyboard. For example, the lowest Ascii
values are used for control characters (entered from the keyboard by holding
down the control key and then pressing one of the alphabetic keys): CTL-A
is 1 (0000001) and CTL-C is 3 (0000011). The 7-bit codes for the letters A
to Z and a to z all begin with 1. Just using the lower 6 bits is an adequate
code for letters, numbers, and some punctuation.
If you want to start a new religious cult, take some text from a major
religious work and turn it into a bitmap that happens to look like something,
preferably a face. You can use Ascii or any other arbitrary encoding.
I tried doing this with the Book of Genesis (the one from the Bible) but
wasn’t able to come up with anything remarkable.
4.7 Representation of pictures
Above, we have used a bitmap as a way of displaying the contents of memory.
More commonly, bitmaps are used to paint pictures on the screen. For
example, the pictures one sees on the World Wide Web are color bitmaps,
4.7. Representation of pictures 57
@@@@@@N@@@@@x@Cp@C‘G‘@@C}@@@C@p@@A‘F@@@P@‘@
@D@H@@A@B@@XP@p@Ad@Ix@G@Cp@@XA‘@@C@p@@@_x@@@
C|@@@@ @@@@pL@@@XA‘@@D^H@@A@B@@@P@‘@@D@L@@AD
R@@@P@‘@@F@P@@@pL@@@G~@@@@@@@@@@@@@@@@@@
Fig. 4.2: A bitmap picture of a baby represented in a 32 × 32
bitmap. Two incorrect bitmap decodings are shown: a 33 × 33
bitmap and a 34 × 34 bitmap. Bottom: the same image decoded
as Ascii.
with a value for each pixel, which gives its color. This is more easily appreciated
in a black and white picture, the way pictures are reproduced in
newspapers. To turn a real-world scene into a pixilated picture, one divides
the picture into a grid and then thresholds each grid location for luminosity.
For black and white, a simple bit code suffices, e.g., 0 for white and 1
for black. Copying the values of each pixel row by row turns a picture into
a bitmap or bitstring.
The integrity of a bitmap depends on knowing its proper dimensions. For
example, the bitmap picture of a baby (Fig. 4.2) is 32 bits by 32 bits for a
total of 1024 bits. Displayed incorrectly as a 33 by 33 bitmap, the picture
is obscured (middle bitmap) and it is unrecognizable in a 34 by 34 format
(right bitmap). If read out as Ascii (below in 6-bit groups left-padded with
01) instead of as a bitmap, all identity is of course lost.
A vector plot is an alternative to bitmaps for picture representation. In
a vector representation, the picture is laid out using a set of commands
that describe where lines are to be drawn. Strictly speaking, a vector plot
draws everything as stick figures made up of line segments. The individual
commands tell where each line segment is to be drawn. Most vector
languages include more complex commands in addition to line segments.
For example, the vector language used to produce Fig. 4.2 uses circle, rectangle,
and filled-rectangle commands in addition to a line command. The
following commands were used to produce Fig. 4.2.
58 4. Computer Representations
// 32 32 −1
circ 15 23 6
circ 15 10 6
frect 12 15 17 17
line 4 1 11 4
line 19 4 25 2
setbit 13 24
setbit 17 24
line 3 10 9 13
line 20 13 27 11
line 13 20 16 20
The first line gives the size of the bitmap to draw: 32×32. Subsequent lines
tell what to draw using a coordinate system starting at pixel x = 0, y = 0
in the lower left-hand corner and going up to 31,31 at the upper right-hand
corner. For example “circ 15 10 6” says to draw a circle centered at 15, 10,
about halfway across and one-third of the way up in the bitmap. The circle
has radius 6. This circle is the lower circle in Fig. 4.2. Note that the outline
of the circle isn’t very circular. At this resolution, the bitmap is too coarse
to draw a very smooth curve. The other commands are also fairly obvious:
“line” draws a line between two points, “frect” fills in a rectangle between
two points. “Setbit” just changes a single bit from 0 to 1, black to white in
the picture.
The above vector description is a simple graphical program that serves
as an alternative to the bitmap representation of the picture. Bitmaps and
vector maps are complementary and are both used extensively in computer
graphics. Vector maps have the advantage of usually providing a much
smaller representation. A bitmap has to represent all of the 0s as well as
the 1s. Line drawings like this one are usually sparse. Vector maps give
data to represent only what is to be drawn, not what is in the background.
A vector map scales well and is well suited to animation. Unlike the bitmap
representation of Fig. 4.2, the commands will still function correctly if the
size of the bitmap is misconstrued. Also the whole picture can be expanded
by simply multiplying the values given as arguments to the commands by
a number. Vector languages are commonly used for printer languages such
as Postscript. It’s critical to use a vector language when working with a
vector device such as a plotter (a device that moves a pen across the paper
to draw graphics). Although it is easy to generate a bitmap from a vector
language, it is very difficult to go the other way and figure out what vector
commands were used to make a particular bitmap.
Although bitmaps are generally bigger than vector maps, they are easier
to print out to screen or paper. To present a bitmap on the screen, the bits
can be drawn out in the sequential order visited by the scanning electron
beam on a television or cathode-ray computer monitor. A vector map,
however, requires interpretation and does not provide all of the bits in
4.8. Neurospeculation 59
order. For example, the command “circ 15 23 6” requires that the program
calculate radial locations using the equation for a circle (r2 = x2 +y2) and
then round off where each of these positions falls on a square grid.
Description length is one concept that is used to consider the relative advantages
of various representations. Description length takes into account
not only the size of the data structure for the information but also the size
both in space and in time for the algorithm needed to decode and recode
the information. The complexity of this algorithm will depend not only on
the complexity of the representation, but also on the design of the input
and output organs and on the design of the computer itself. As we’ve noted
above, bitmaps draw directly onto a screen, while vector maps translate directly
into plotter commands. For the vector language given above, the
program for interpreting the language requires about 100 lines of programming
code. By contrast, a simple program to interpret the corresponding
bitmap takes only four lines of code. The vector language interpreter requires
both greater space for storage of the program and greater time for
its execution.
As you can see, the choice of bitmap or vector plot as a data structure
depends on the requirements of the program that will use the data structure.
In modern computers, memory is cheap. Graphics software commonly
maintains both representations so that they are immediately available for
interactive use. For example, an animation program will use vector representations
for its primary store to make it easy to move objects around
independently. However, it will need to calculate a bitmap for each image
in order to present it on the screen.
4.8 Neurospeculation
Just as complex computer representations are all built on top of the lowlevel
binary of the transistor, so brain codes are compounded on top of lowlevel
codes suited to the chemicals and voltages available. Then there are
mid-level codes such as Ascii and two’s complement. In the nervous system,
these codes involve spikes and synaptic function: rate coding, burst coding,
or synaptic co-transmitter codes. Vector-plot graphic representations are
examples of high-level data structures in the computer. In the brain, this
would involve circuitry-level codes such as cell assemblies. All codes will
be adapted for ease of manipulation, for rapid translation from stimulus to
response, or for compact storage. Pressures of multiple use may also dictate
hybrid or redundant data structures.
It is interesting to speculate how biological data structures might be
fitted to the needs of the organism. In industrial robots, the machine sees
with a scanner or CCD (charge-couple device) camera. In either case, the
natural representation is a bitmap. However, to move an arm toward a
60 4. Computer Representations
visually identified target, the bitmap has to be translated into a vector
representation that maps direction in space. Then this vector has to be
translated to a different coordinate system that provides angles for the
joints of the arm. There may also be a sensory feedback system that reports
back whether the arm actually arrives in position. In the biological context,
this is an issue of sensorimotor integration.
The concept of description length may provide insight into possible
strategies for brain storage. If there is a lot of information to be stored,
then it makes sense to develop compact data structures even at the expense
of utilizing a lot of brain to construct and deconstruct it. However, in some
cases the need for survival will require that speed take precedence over size
in the space–time trade-off. This means choosing representations that are
close to those of primary receptors or primary effectors so as to translate
in or out rapidly.
Language is an example where massive storage requirements compete
with the demand for speed. Neuropsychological evidence suggests that language
is stored in a variety of ways for different modalities of input and
output. There are phonemic representations for rapid auditory comprehension.
There are facial motor representations for manipulation of tongue,
lips, and palate. There are visual representations for reading and additional
motor representations for writing. None of these representations even
touches on real language use: semantics, grammar, meaning. With regard
to word meaning, there is evidence for different semantic storage schemes
depending on the type of word being stored. Words for objects that are
primarily defined by their appearance (e.g., animals) are stored separately
from words that are primarily defined by their use (e.g., tools).
Given the profusion of representations, it is necessary to have a lot of
translation programs. There are two approaches to producing translators.
One is to develop a central master data structure. The format of this data
structure doesn’t have to be useful for anything in particular, but should
contain information or pointers that make it easy to translate into all of
the other data structures. The alternative approach is to provide many little
translators to go between pairs of representations. Neuropsychological
evidence suggests that the brain takes the latter route for language. There
are a variety of acquired dyslexias (reading disorders) that are seen after
stroke. These provide evidence for separate grapheme (written letters and
words) to phoneme (pronunciation) and grapheme to meaning pathways.
On the other hand, it is probably the case that direct translation is not
necessary for some representations, allowing translation to take place via
a shared central format. For example, it is fairly easy to read aloud. This
suggests that grapheme to phoneme translation is immediately available
online. It is hard, and rarely necessary, to go straight from reading to writing.
Therefore, grapheme to orthography might go via a central phonetic
representation.
4.8. Neurospeculation 61
A similar case, in terms of the requirement for massive online data storage
with rapid access, is the motor system. Movements must be produced
fast and unerringly: riding a bike, throwing rocks at tigers, etc. Also, once
learned, they are not forgotten. Motor programs are often used inside of
other motor programs, a situation that in computer science would dictate
the use of subroutines and pointers. It is not clear whether or how the brain
could make use of such strategems.
Episodic memory, memory of things that happened to you over the years,
also has massive storage requirements, more and more massive as the years
go by. However, this database does not have to be online — you do not
have to access a full description of last year’s July 4th picnic in one or two
hundred milliseconds. This would suggest the use of compression formats,
which would save space but take longer to store and access (space–time
trade-off). There is evidence to suggest that the hippocampus is involved
in preparing these representations and perhaps coordinating their transfer
to other storage sites in neocortex.
By contrast, some sensory systems may have relatively little need for
storage, but need to optimize rapid processing strategems. Most of us, as
nonmusicians, probably have little raw auditory signal storage requirement
or storage ability. However, we must still process auditory signals quickly.
Music comes in at high rates. Primary auditory decodings must be passed
quickly to language centers.
An enormous amount of brain is devoted to vision. It is known that
the brain breaks up vision into many different processing pathways and
different representations used for different purposes. One major pathway
seems to be responsible for detecting and mapping motion in the visual
field. It would seem natural to use some sort of vector representation for
this. Other systems process shape, color, and other attributes of the visual
scene.
In both vision and audition, the design of internal structures is likely
adapted to the needs of input circuitry. However, there is enormous variation
in the ability of individuals to store and reproduce visual or auditory
data. Perhaps certain people are gifted with internal data structures that
are well suited to store particular sensory system patterns. This would presumably
reflect some mixture of nature and nurture. Given the enormous
plasticity of brain, it seems plausible that individuals develop their data
structures in slightly different ways during development. Visual cortex is
built somewhat differently from auditory cortex. It is not optimized to
process sound information. Yet functional imaging shows that congenitally
blind people stuff extra auditory processing into their unused visual cortex.
Somehow, their brains have built auditory data structures and auditory
data processing structures in this alien cortical area.
62 4. Computer Representations
4.9 Summary and thoughts
In this chapter I have presented common computer representations: binary
number encodings, Ascii text encoding, bitmap, and vector-plot picture
encodings. Each is a tool suited to the needs of use. However, all of these
representations are at heart determined by the structure of the computer
— we ultimately get down to bits. All computers now use binary number
encodings for numerical calculations, allowing them to run directly on the
machine. At the other end, the highest-level encoding that we looked at
was the vector-plot graphical language. Like other computer programs,
this is a compound encoding built upon the basic underlying binary coding
schemes. The vector language used in this chapter is stored as Ascii, but
just decoding Ascii would not tell you anything about the picture being
represented.
This layering of representations provides an indication as to why it may
be difficult to figure out the representations in the brain. We can probably
assume that they have been pretty well optimized, but we can’t be sure
exactly what they’ve been optimized to do. A single common code in the
brain would have the advantage of permitting easy communication between
different brain areas. On the other hand, it seems more likely that different
brain areas that must perform very different tasks have each developed
their own complex codes suitable for these tasks. The many different circuit
designs used in the brain (thalamus, basal ganglia, cortex) also suggest that
different representations will be used as information is passed around from
area to area. Different circuits will also have different roles with regard to
speed of processing. Circuits such as thalamic nuclei get only a single brief
shot at the data as they relay masses of information on to other areas.
Other circuits can take their time, chewing over information in protracted
loops of recurrent activity. Perhaps the phenomenon of coming up with
new ideas during sleep is due to such slow brain rumination.
5
The Soul of an Old Machine
5.1 Why learn this?
Computer science pioneers had an exciting idea: human intelligence was a
result of sophisticated information processing and could be emulated by any
machine that could handle enough information speedily enough. Artificial
intelligence was the effort to produce software that would validate this
hypothesis. Forty years later, they’re still trying. I don’t believe they’ll ever
find this particular grail, because I believe that there is something special
about brain design that allows it to display intelligence. Theologians are
typically dualists, believing that this special thing is immaterial. I think
that it’s a set of clever design features.
If brain function, intelligence, memory, thought, etc. are the results of
design features, then we wish to understand both how the brain is designed
and how design constraints are likely to influence function. To put this in
a computer science context, we want to understand the hardware and then
see how hardware constrains software. This approach differs substantially
from Marr’s top-down approach, which started with function and worked
down to implementation. However, it will not be possible to simply start
with implementation and infer function. Only by probing from the top and
the bottom simultaneously are we likely to catch something interesting
between our pincers.
In this chapter, I want to make two points: 1) hardware molds and constrains
software, and 2) a hacker can exploit the idiosyncrasies of a machine
in order to push the machine beyond its design limitations. Extending
64 5. The Soul of an Old Machine
on the example of two’s complement in the previous chapter, I want to
show how hardware has molded software in the modern digital computer.
Programmers exploit computer design to obtain surprising results in programs.
My contention is that nature has similarly made the most of design
constraints, utilizing tissue in new ways to obtain novel results.
5.2 The art of the hack
In computer lingo, a hack is a trick. A trick performed with mirrors takes
advantages of optical anomalies to produce unexpected images. A hack
uses computer design anomalies to make programs behave in unusual ways.
Nowadays, computer hackers are generally bad guys. The primary reason
for using hacks is subterfuge, to break into or simply break a computer
system by making it do something other than what the designers intended.
In the early days of computers, hacking was a standard programming skill,
as the limitations of the computer hardware made it impossible to solve
many problems in a straightforward way. The primary limitation was that
of space; programs had to be short. Old machines were also limited due
to their small instruction sets. These computers could not directly perform
basic tasks like multiplication. A lot of programming had to be done in
order to perform simple but necessary tasks.
With limited commands and limited space for both data and programs,
programmers looked for shortcuts and tricks, many of which took advantage
of quirks of a particular machine. This made for very obscure, hard-to-read
programs that could not be readily moved from one machine to another. It
also made for programs that were not very robust. For example, the Y2K
problem arose as the year 2000 approached because many old programs had
only alloted two bytes to store the year, assuming that the two digits would
always be preceded by 19. Although this saved bytes, it led to confusion and
the need for massive reprogramming as 2000 neared. Now, programmers
have switched them all over to four digits. There will be even more hassles
when the year 10000 rolls around.
Below, I introduce a simple machine that is modeled on the historic
PDP-8. I explain how it works. Then I show how hacking was used to
extend the machine. The first example is the two’s complement hack given
in the previous chapter. Then I show how a program can alter itself while
it is running. Finally I give a simple example of a computer virus. Such
minimalist early examples of virtual reproduction inspired the study of
artificial life.
5.3. Software and hardware 65
5.3 Software and hardware
Before the invention of programming languages such as FORTRAN and
BASIC, programs were written in machine language or assembler language.
Machine language is the pattern of bits that the machine works with. It
reflects the way that transistors are put together to perform specific simple
binary operations such as addition. Machine language sits in memory. It is
loaded into an instruction register and from there to the central processing
unit (CPU), where the individual bits are interpreted. For the purpose of
this book, I generally represent machine language in octal, base 8, rather
than binary simply for ease of representation (see Chap. 16, Section 16.3).
I also use assembler language. Assembler language is the same as machine
language but uses symbolic names as stand-ins for commands. For example,
in the language we will explore, the name “ADD” is used in assembler as an
obvious label that translates directly to the number 08 (= 010, the little “8”
tells that the number is given in base 8). This is 0002 in machine language.
The assembler language for a particular computer has exactly the same
commands as the machine language for that computer. Different computing
machines will have different machine languages, so that something written
in machine language on one will not run on the other.
By contrast with the one-to-one translation of assembler to machine language,
single lines in a higher-level language, such as FORTRAN, C, or
Pascal, are translated, or compiled, into many steps of machine language.
As we will show, the statement x = x + 5 (augmenting the variable x by
5) may take many machine language steps, and therefore many machine
cycles, to execute. An advantage of a high-level language is portability.
FORTRAN compilers have been written for almost all computers and
a FORTRAN program can therefore be easily ported (moved) from one
computer to another.
Translation of a high-level language into machine language is called compilation.
Another way that a higher-level language can run on a computer
is through interpretation. With compilation, the higher language is translated
once and machine code is produced. This machine code can then be
run again and again without making reference to the original program. If
the original program is changed, then it has to be compiled again. With
interpretation, the program does not get translated into machine code. The
program itself does not run directly on the machine. Instead an interpreter,
which has been either written in or compiled into machine code, runs on
the machine. The interpreter reads commands from the high-level language
one by one and simply does what each command tells it to do. Interpreters
do not translate programs but only interpret what the program says. A
previously compiled program no longer requires the compiler in order to
run, but an interpreter must always be present for an interpreted program
to run. Common interpreted languages include BASIC, JAVA, and Matlab.
66 5. The Soul of an Old Machine
5.4 Basic computer design
Representations in the computer all boil down to groupings of 1s and 0s in
patterns that the CPU can make some sense of. Whether the 0s and 1s are a
compiled program or a document written on a word processor, the computer
just sees bits laid down in consecutive main memory. When a program runs,
memory is further manipulated as a result of the machine instructions. If a
word processor is used to edit a manuscript, binary Ascii code is replaced
and rearranged. On the machine there is no intrinsic distinction between the
binary of the machine language program and the binary of the paper’s Ascii.
If pointers get mixed up, as sometimes happens, the machine will execute
the Ascii as if it were a program or will display the machine language of
the program on the monitor as if it were Ascii. Either way, the results will
be highly unsatisfactory.
At the lowest level, computers are organizations of switches. Once these
were vacuum tubes; now they are transistors. These switches maintain an
electrical lead at either a high or a low voltage. The low voltage is interpreted
as a 0, the high as a 1. Binary representation is thus only slightly
abstracted from the reality of high and low voltages. Having boiled down
higher-level computer representations to groupings of 0s and 1s, we can
then probe more deeply into computer design and ask about the machine
language, the way that specific patterns of 0s and 1s will be interpreted by
the computer.
To explore this low-level “mind–body” interface of a computer, we will
look at a simple computer loosely modeled on the PDP-8, a computer
manufactured by Digital Equipment Corporation (DEC) starting in the
mid-1960s. The PDP-8 is often used as an example machine in computer
science because of its simplicity. I have simplified it still further by paring
away some design details to make a few core points. I refer to this
imaginary machine as the PDP8– to distinguish it from the real PDP-8. A
language emulator for the PDP8– is available on the Web site. There is a
lot of information on the real PDP-8 available on the Web, including full
instruction sets and copies of old programming manuals from DEC.
A new breed of small machine, the PDP-8 was dwarfed at the time by the
giant calculating behemoths sold by IBM. Calling the machine a parallel
data processor (PDP) instead of a computer was a marketing notion. Computers
at that time were immense machines that required a large staff to
run and maintain. DEC wanted to distinguish the PDP as a machine that
didn’t require dedicated rooms or dedicated teams of technicians. Its price
made it affordable to a small company or university. PDPs were the first
step towards personal computing, and the PDP-8 was the smallest, simplest
machine in the PDP series. By modern standards, it was not very powerful,
with a complexity comparable to that of modern programmable calculator.
Nonetheless, some of these machines remained in use for decades. I spotted
one in 1992 in the betting parlor of a jai alai fronton in Tijuana.
5.4. Basic computer design 67
Pointers come from computer memory design
Although the CPU is the brain of the computer, the design of computer
memory was the big breakthrough that determined the course of modern
computing. John von Neumann, the pioneering Hungarian mathematician,
formalized the design of the stored program computer by noting that internal
memory could be used to indistinguishably store both commands
and data to be accessed by the CPU. This basic design feature, present
in almost all modern computers, is now known as the “von Neumann architecture,”
although his collaborators deserve equal credit. This design is
also called the “stored-program concept.”
The von Neumann architecture determines much of computer programming
strategy. Computer memory is referred to as main memory, core, core
memory, or, as here, random-access memory (RAM). RAM is where programs
and data reside. Random access means that any location in memory
can be retrieved at any time. It is contrasted with the sequential access
of some storage media, such as magnetic tape. Although RAM is laid out
in sequential words, any of these words can be addressed using a pointer.
Each word has a fixed number of bits. In the case of the PDP-8, each word
has 12 bits. Words are numbered sequentially from 0 up to the total size
of memory. A modern computer will have millions of words of memory
(several megabytes where a word will be 2 to 4 bytes). A typical PDP-8
had about 4000 words of main memory. A word of RAM is accessed by its
address, which is simply its number in the sequence from 0 to the number
of words of memory. In the PDP8–, I have only allowed 24 (308) words
of memory, numbered from 008 to 278 (remember how to count in octal:
25,26,27,30,31, . . .).
Pointers are a prominent feature, explicitly or implicitly, in all programming
languages because of the need to address locations in RAM. For
example, x = 5, an assignment, creates a pointer named x to a location
in memory that stores the number 5. Pointers are fundamental to the way
programs are written because they reflect the way computers are built. At
the machine level, a pointer is the address of a location in memory. Of
course, most memory aids that we use in everyday life employ some sort
of pointer to access information. For example, the alphabetical ordering of
files in a file cabinet is a pointer system.
Sequential algorithms come from computer control flow
The PDP-8 has only a few major components (Table 5.1): a CPU doing the
computing, RAM storing the program, a bus connecting the two, and a set
of specialized registers used by the CPU as local memory. The CPU does the
actual calculating; it is the computer within the computer. Any programs
and all data for programs are stored in RAM. Registers are separate words
of memory that are not random access, but are instead hard-wired to be
68 5. The Soul of an Old Machine
Abbrev Name Description
CPU Central processing unit The computer in the computer —
performs the command passed from
the IR.
RAM Random access memory Also called “core memory” or “main
memory.” Contains program and
data accessed by the CPU.
BUS Bus The backbone with the wires that
connect CPU to RAM and external
computer components.
Registers
ACC Accumulator Used to process (accumulate) numbers.
PC Program counter Points to the next instruction to be
executed.
IR Instruction register Holds word from RAM indicated
by PC and passes it to CPU for
execution.
Table 5.1. Computer parts
used for particular purposes. In the PDP-8, three registers are important:
the program counter (PC), the accumulator (ACC), and the instruction
register (IR).
The program counter is a memory pointer that indicates the location in
memory that will be loaded into the instruction register. When the program
counter points to a word of random-access memory, it will be fetched and
used as a command. However, when this same word of RAM is pointed to
by another word in RAM, it will typically be considered to be data. There
is no intrinsic difference between data and command in the structure of the
computer. This, as we will see, is an essential feature utilized by hackers.
The accumulator is a register that is used as a temporary storage point for
doing arithmetic. The instruction register is a register used as a temporary
storage point for commands before they are accessed by the CPU.
In the normal cycle of operation, the PC gives an address whose contents
are moved from RAM into the IR. In the PDP8–, we start running most
programs at memory location 0. This is similar to the way the bootstrap
sequence works on most computers: a memory block from location 0 of the
5.5. Programs and hacks 69
disk is loaded into location 0 in RAM and execution is then started from
there. After the word is fetched into the IR, a command component is split
off from an address component. The first 3 bits, a number from 0 to 7, is
the command. The other 9 bits are generally an address that the command
will act on. The CPU will then interpret and execute the command. The
execution of the command will change the state of the machine, usually
by changing a word in RAM or changing a register. After this, the PC is
automatically incremented by 1 and the cycle repeats.
Computer programs can be easily laid out in algorithms — step-by-step
instructions similar to the recipe in a cookbook. Algorithms are a major focus
in computer science. Algorithms may or may not be fundamental to the
brain, but it is apparent why they are fundamental to the von Neumann
computer architecture: the machine is designed to proceed in sequential
steps. Thus the machine cycle not only determines control flow in the computer
but also defines the way that programs are written and the way that
computation is regarded.
CPU: machine commands
As a 12-bit word is moved out of the IR into the CPU, it is broken into two
parts. The first 3 bits are the command. In the real PDP-8, some commands
used the lower 9 bits of memory to microprogram other commands, but here
I only discuss the PDP8–, in which the lower 9 bits are either an address
or else are irrelevant and are ignored. Since 3 bits are used to define the
command, there can be only eight commands: 000, 001, 010, 011, 100, 101,
110, 111 (0–7 octal). The PDP8– instruction set that corresponds to these
commands will be given the following mnemonic names: 0 — ADD, 1 —
DEC, 2 — INC, 3 — SKP, 4 — JMP, 5 — CLA, 6 — LDA, 7 — HLT.
These names are meant to be easy to remember: add, decrement, increment,
skip, jump, clear accumulator, load accumulator, halt. When using these
names instead of the numbers, one is using assembler language. Thus the
only difference between assembler language and machine language is that
the former is human-readable and the latter machine-readable. Otherwise,
they are the same language. We are now about three steps removed from
the machine representation: the machine uses voltages, which we symbolize
with binary, and then translate into octal and now further into assembler.
The language of the PDP8– is summarized in Table 5.2.
5.5 Programs and hacks
Of the eight commands, three commands alter RAM (LDA, DEC, INC),
two alter the ACC (ADD, CLA), and two alter the PC (SKP, JMP). The re70
5. The Soul of an Old Machine
Inst. Code Description Usage
ADD 0 Add address contents to ACC ADD [address]
DEC 1 Decrement address contents by 1 DEC [address]
INC 2 Increment address contents by 1 INC [address]
SKP 3 Skip next instruction if address
contains 0
SKP [address]
JMP 4 Jump to specified address JMP [address]
CLA 5 Clear the accumulator CLA – – –
LDA 6 Load accumulator to address LDA [address]
HLT 7 Halt the program HLT – – –
Table 5.2. Assembler and machine code definition. ‘– – –’ means that the bits are
ignored
maining command, HLT, stops processing but does not change any memory.
Program 1 is an addition program using this instruction set.
ADDR ASSM OCTAL BINARY COMMENT
00 CLA 00 5000 101000000000 clear ACC
01 [ADD 05] 0005 000000000101 add contents of
05 (13) to ACC
02 ADD 06 0006 000000000110 add contents of
06 (5) to ACC
03 LDA 05 6005 110000000101 put ACC
contents in 05
04 HLT 00 7000 111000000000 stop
05 ADD 13 0013 000000001011 data: x
06 ADD 05 0005 [000000000101] data: 5
07 ADD 00 0000 000000000000 not used
Program 1: Addition
The command at each address (ADDR) is shown in assembler (ASSM)
as well as in machine language, showing the number in both octal (e.g.,
5000) and binary (e.g., 101000000000). In the machine’s main memory, the
transistors will hold high or low voltages according to the values shown for
each bit in this word.
The accumulator is the location where arithmetic is done. Before doing
any arithmetic operation, it is good to clear the accumulator in case a
previous program left a number there. This is done with the CLA command.
The address argument 00 is in this case meaningless. The next two steps
are the heart of the program. ADD 5 superficially looks as if it would add
the number 5 directly to the accumulator. This is not correct since all of the
commands use pointers to other locations in memory. Therefore, the ADD
5.5. Programs and hacks 71
5 command instructs the CPU to access memory location 5. The contents
of address 05 is ADD 11, which appears to be another command rather
than a number. This apparent paradox is resolved when we remember that
data and commands are both represented by binary numbers in memory
— there is no fundamental difference between a datum and a command.
The content of memory location 05 is 138 = 1110. Notice that the contents
of locations 01 and 06 are identical. In location 01, I have bracketed the
assembler command name to indicate that this is being used as a command.
In location 06, I have bracketed the binary to indicate that this is being
used as data. In step 02, ADD 6 adds the contents of address 6 to the
accumulator, yielding 13 + 5 = 208. Step 03 copies the accumulator back
into location 05, overwriting the contents and replacing it with the number
208. Finally, in step 04, HLT 00 halts the processing. As with the CLA
command, the 00 argument is unused.
If the halt command weren’t there, the data in locations 05 to 07 would
be taken as commands, adding the pointed-to contents to the accumulator
in each case. Notice that location 07 is empty; it is not being used as
either data or program. Nonetheless, it can be run and will be handled as a
command that will add the contents of location 00 to the accumulator. In
the absence of a HLT statement, the machine will continue executing until
it encounters an illegal command (e.g., a command that points outside of
memory, a so-called segmentation error), or runs out of memory. If there is
a jump command that loops control back, the machine will enter an infinite
loop, repeatedly executing these inconsequential commands.
This program, without the halt, would be typical of compiler output for
a higher-level language expression such as x = x + 6 (or x = x + y where
y was previously set to 6). The compiler would perform a first pass where
it would find a word of memory to assign for x and another word where
the 6 was stored. It would create a symbol table to keep track of where
these variables and fixed numbers were located. The compiler would then
be able to translate the algebraic expression x + 6 into steps 0 through
2 in Program 1 and translate the assignment x = into step 3. The name
of the first compiled language FORTRAN, means FORmula TRANslator
because it was able to translate algebraic formulae into machine code in
this fashion.
Conditionals
Memory addressing dictates the use of pointers in computer languages. The
sequentially incrementing program counter dictates serial processing. Another
key feature of computer languages emerges from the SKP command,
or, more fundamentally, from the association between binary representation
and Boolean algebra. Boolean algebra is the arithmetic of truth and
falsehood, represented in the computer as 1 and 0, respectively. In the context
of a computer language, truth and falsehood dictate the fundamental
72 5. The Soul of an Old Machine
usage of branching of control flow in computer programs. The true path
will be the road taken and the false path will be the road not taken.
The high-level computer language manifestation of this aspect of machine
representation is branching at a conditional statement. In high-level
programming languages, the conditional is usually written as an if-then
statement. For example, one frequently sets the value of one variable based
on the value of another variable: if x equals 5, then set y to 3 else set it to
1. In the C computer language, this could be written if (x==5) y=3; else
y=1; or y=(x==5?3:1);. Note the use of a double equal sign. This represent
the normal meaning of equal, querying the truth or falsehood of an equality:
5==5 is true. The single equal sign in C represents the assignment
of a number to a variable: x = 5 makes x a pointer to 5 in memory. Every
computer language uses different symbols for these two very different
operations. Let’s compile these conditional assignments into our machine
language (Program 2). We’ll say that x is located at address 10 and y
is located at address 11. We’ll also need to provide storage space for the
constant numbers 5, 3, and 1: let’s store these in locations 12 to 14.
The assignments of y to 3 or 1 are easy. Each of these just involves copying
memory from one location to the other. In each case, this requires three
steps at the machine level: 1) clear the accumulator, 2) copy from memory
to the accumulator, and 3) copy from the accumulator to memory. The
conditional itself is more complicated. The only conditional command we
have in this machine language is the SKP command (the actual PDP-8 had
a few more). It checks whether a given memory address contains a 0. There
are two ways that we can turn a 5 into a 0 with our instruction set: either
add −5 or decrement by 1, five times. We’ll look at both techniques.
Program 2 shows a conditional program using decrementing.
ADDR ASSM OCTAL COMMENT
(x == 5)? section
00 CLA 00 5000 clear to start
01 DEC 10 1010 decrement unknown value
02 DEC 12 1012 decrement the counter
03 SKP 12 3012 done when counter is 0
04 JMP 01 4001 otherwise do again
05 JMP 20 4020 finished: skip over data section
Data section
10 ADD 05 0005 x — the “unknown”
11 ADD 06 0006 y to be set
12 ADD 05 0005 5 for decrementing
13 ADD 03 0003 number 3 (for y = 3)
14 ADD 01 0001 number 1 (for y = 1)
...
5.5. Programs and hacks 73
y assignment section
20 SKP 10 3010 x==0? (i.e., did x start out as 5)
21 JMP 24 4024 if x _= 0 (if x not equal to 0)
22 ADD 14 0014 if x==0 ACC = 1
23 JMP 25 4025 set y and finish
24 ADD 13 0013 if x _= 0 ACC = 3
25 LDA 11 6011 y = ACC (set y)
26 HLT 00 7000 stop
Program 2: Conditional program: if (x==5) y=3; else y=1;
The main point of this example is to show how the simplest line of a
program becomes horribly complicated and convoluted when it actually
gets performed by the computer. Similarly, we may be disappointed as we
look for the nervous system to do things in simple, straightforward ways.
In Program 2, we set up x in location 10, y in 11, and the test number 5
in 12. Program 2 does not show binary but instead includes comments in
pseudo-code (similar to but not identical to the C programming language).
I have also not listed all of memory here, only those words that are used.
Steps 1 and 2 decrement the unknown x and the known 5 in tandem. When
the decrementing reaches 0 in location 12 (the known value), control flow
jumps over the data section to location 20, having set location 10 (the
unknown value x) to x − 5. At address 20, location 10 is evaluated to see
if it contains 0 (which would mean that x was 5 originally). If it contains
0, control jumps to address 22, and the contents of address 14 are put
in the accumulator. Otherwise, the contents of address 13 are put in the
accumulator. The accumulator is then copied to address 11.
This is an inefficient way to write the program. The loop to do a comparison
by decrementing could take a while if the number was very large
instead of being 5. However, since the machine does not have a subtraction
command, decrementing would be the obvious way to do subtraction. As
discussed in Chap. 4, an efficient, direct way to do subtraction is to use two’s
complement as the representation of a negative number, using machine architecture
to do something that cannot be easily done in a straightforward
way.
In Program 3, we use the two’s complement overflow hack to represent
−5 as 77738. This allows us to write our conditional program in a slightly
more compact fashion that would run much faster if a large number was
given. Effectively, instead of performing decrements in steps 1 and 2, we
use an appropriately pre-decremented value in address 12 to complement
the unknown x in address 10. Then we can add the numbers and proceed
with the conditional as before.
74 5. The Soul of an Old Machine
ADDR ASSM OCTAL COMMENT
00 CLA 00 5000 clear to start
01 ADD 10 0010 “unknown” x value
02 ADD 12 0012 add in the −5 from address 12
03 LDA 10 6010 x = x − 5 store result in x location
04 CLA 00 5000 clear accumulator again
05 JMP 20 4020 skip over data section
Data section
10 ADD 05 0005 x — the “unknown”
11 ADD 06 0006 y to be set
12 HLT 773 7773 −5 in two’s complement
13 ADD 03 0003 number 3 (for y = 3)
14 ADD 01 0001 number 1 (for y = 1)
Test and set — same as Program 2
20 SKP 10 3010 x == 0? (i.e., did x start out as 5)
21 JMP 24 4024 if x _= 0 (if x not equal to 0)
22 ADD 14 0014 if x == 0 ACC = 1
23 JMP 25 4025 set y and finish
24 ADD 13 0013 if x _= 0 ACC = 3
25 LDA 11 6011 y = ACC (set y)
26 HLT 00 7000 stop
Program 3: Conditional program with two’s complement
5.6 Pointer manipulation
In modern computer programming technique, the distinction between program
and data should always be maintained. A program should not operate
on itself as if it were data. Modern virus scanning programs look for exactly
this kind of hack. Early programmers did not have room in memory
to maintain this clean separation. It was common for a program to alter
its own body. This is shown in Program 4, which adds a list of numbers.
The numbers to be added are in memory locations 10 to 12. The counter
for the number of words to be added is in location 07. The program alters
its own text by using the code at location 03 (INC 01) to increment
the address of the ADD command at location 01. This moves the pointer
for this ADD command so that it points to the next location in memory.
This pointer will be incremented as long as the counter in 07 is being
decremented. Each of the sequential locations will be added into the accumulator.
The incrementing of a pointer is a standard technique that is still
used today. The archaic aspect of this program is that the pointer and the
command are all part of the same word so that the program itself has to
be modified in order to access the sequential data.
5.6. Pointer manipulation 75
ADDR ASSM OCTAL COMMENT
00 CLA 00 5000 clear
01 ADD 10 0010 this command (pointer) will be altered
02 DEC 07 1007 decrement the counter
03 INC 01 2001 increment address in 01 (the hack)
04 SKP 07 3007 if counter==0 finished
05 JMP 01 4001 else go back to 01 for next number
06 HLT 00 7000 stop
Data section
07 ADD 03 0003 the counter (three numbers)
10 ADD 33 0033 value #1
11 ADD 27 0027 value #2
12 ADD 42 0042 value #3
Program 4: Add list of numbers in locations 10 to 12
A kludge
A really egregious hack is called a kludge (with the long vowel of “clue”).
An example of this is the notorious and archaic programming technique
whereby a program modifies itself while running and then loops back and
runs the modified code. When another programmer tries to change such a
program later, the altered software would behave in bizarre ways since the
modifications would also alter this unseen program that is created on the
fly. This type of hack is often highly machine-dependent since it takes advantage
of the specific numbers used for the instruction set. An example of
this that we will now show is to use overflow or underflow on an instruction
to create a different instruction.
ADDR ASSM OCTAL COMMENT
00 ADD 13 0013 value #1
01 ADD 22 0022 value #2
02 ADD 17 0017 value #3
03 CLA 00 5000 start here
04 ADD 02 0002 add/halt statement
05 DEC 04 1004 manipulate the program
06 JMP 04 4004 loop back
Program 5: Kludge to add list of numbers
In Program 5, I managed to condense a program to add a list of numbers
into a three-step loop, as compared to the five-step loop of Program 4. In
the two’s complement example we saw how numeric overflow could be used
to manipulate numbers. It is also possible to use numeric overflow or underflow
to manipulate commands. In this example, the hack uses underflow.
76 5. The Soul of an Old Machine
Decrementing 0000 gives an underflow condition that changes the word to
7777. If this word is being used as a command, the underflow changes the
command from an ADD to a HLT statement. The decrementing of the address
to change the pointer would be considered reasonable programming
practice in the PDP-8 era; continuing the decrement to an underflow would
not.
The functioning of Program 5 depends on the data being stored at the
beginning of memory. Although the program code is relocatable, the data
section is not. After the accumulator is cleared (step 03), the program adds
in the contents of location 02 (step 04). This is the pointer, which is then
decremented in step 05. The jump takes us to the next iteration at 04, which
adds in the contents of location 01. On the next iteration, the contents of
00 are added in. The next decrement step changes the ADD 00 (0000)
command to the HLT 777 command (7777). The lower, address, bits in
this case have no meaning. The jump to location 04 halts the program,
leaving the summed contents of locations 00 to 02 in the accumulator.
This program is meant to be started at 03. This is a small adding routine
that would be used by other code. Numbers to be summed are placed in
the data section, followed by a JMP to 03. If someone mistakenly JMPed to
00 and began execution in the data section, the program would still work
fine for small positive numbers. The commands at 00 to 03 just point to
numbers elsewhere in memory that get added to the accumulator; the CLA
at 04 clears the accumulator anyway. On the other hand, if large or negative
numbers were entered to be summed, one of the new numbers could be a
less innocuous command that would crash the computer or intermittently
produce weird results.
This hack uses a machine-dependent trick: a halt and an add command
happen to lie only one arithmetic bit apart. Similar hacks can be designed
to use different commands since adding a multiple of 10008 will switch from
one command to another.
A computer virus
Program 6 is a simple but defective prototype of a common malicious hack
— a virus. A virus is a piece of code that copies itself into a different
location in memory. On the Internet, such viruses not only copy themselves
on one machine but then propagate themselves to other machines as well.
After clearing the accumulator, the next two steps (ADD and LDA) shift
consecutive memory from RAM to ACC back to RAM at a higher address.
Each of these two commands is decremented to access the entire program
from high to low addresses. The program will stop after an underflow at
address 01 after six iterations that have copied the program from addresses
00 – 05 to addresses 12 – 17.
This particular virus is defective. For the virus to run properly in its
new location, it would have to be updated so that the ADD, DEC, and
5.7. Neurospeculation 77
JMP commands all correctly address the new location of the code. Also, a
proper virus would transfer control to its new location instead of halting.
Trying to hack programs that will do this sort of thing is an entertaining
sort of puzzle, which is one of the reasons that this activity is so popular
among misguided youths.
ADDR ASSM OCTAL COMMENT
00 CLA 00 5000 clear
01 ADD 05 0005 move code into ACC — becomes HLT
02 LDA 17 6017 move code from ACC to higher memory
03 DEC 01 1001 alter the address to copy from
04 DEC 02 1002 alter the address to copy to
05 JMP 00 4000 loop back to get another address
After running:
ADDR ASSM OCTAL COMMENT
00 CLA 00 5000 unchanged
01 HLT 777 7777 ADD turned into a HLT
02 LDA 06 6006 now points to below relocated code
03 DEC 01 1001 unchanged
04 DEC 02 1002 unchanged
05 JMP 00 4000 unchanged
06 ADD 00 0000
07 CLA 00 5000
10 ADD 01 0001
11 LDA 11 6011 program relocated
12 DEC 01 1001 commands here
13 DEC 02 1002
14 JMP 00 4000
Program 6: Relocating code — before ... and after.
5.7 Neurospeculation
The brain is sometimes referred to as a “black box.” A black box, in system
engineering jargon, is a component that is bought from an outside vendor
and just plugged in. The system engineer doesn’t have to know how it
works. He is content to know that it takes available inputs and provides
needed outputs. For example, if you’re building a Boeing 747, you might
pick up your inertial guidance system from Lockheed. They send you a box.
It might be black. You go to the cockpit and plug it into the navigation
system. Then off you go.
The brain has long been mysterious in this way. It was purchased from
an outside vendor. It fits neatly into a slot at the top of the case. It picks up
78 5. The Soul of an Old Machine
signals from the rest of the system and provides output signals that are used
to navigate and maneuver the entire device. In a classic “Star Trek” episode,
a nasty alien stole Spock’s brain and plugged it into a central control station
so that it would run the air conditioning or something. Complex plot twists
led to the immortal line: “Brain, brain ... what is brain?” This question
resonates at two levels. We generally know neither what the circuits are
doing nor how they are doing it. There is no instruction manual. A good
black-box instruction manual tells not only what the whole box is for, but
also what kind of signals go into each input plug and what kind of signals
come out of each output plug.
In electrical engineering jargon, the next step is to “open the black box.”
As we poke around inside the brain, we have to start with the assumption
that systems are performing calculations similar to the ones we are familiar
with. This is not always a good assumption. A brain system is likely to be
designed very differently from comparable artificial technologies. The visual
system does not work like a camera or video camera, or like an electric eye.
Despite this concern, below I’ll look at some brain analogies for various
computer concepts and components: the software/hardware dichotomy, the
distinction between CPU and memory, and the use of a central bus.
The notion of a hardware/software distinction is often used by linguists
and cognitive psychologists who do not want to open up the box. This
is a version of the Marr approach that I have been complaining about.
My contention is that the hardware develops so that it readily performs
certain algorithms in particular ways. However, it seems possible that we
could consider natural language to be the brain’s software.
Above, we discussed the distinction between compiled and interpreted
software. A compiler translates programs into machine language so that
they can run on the machine, while an interpreter leaves the program as
it is and performs the steps of the program one at a time. Code compiled
onto a PDP-8 would look like the machine code programs presented in this
chapter. Compilation is done when you need speed but don’t care about
being able to change the program. Once it is compiled, you can even throw
away the original higher-level code, making it nearly impossible to alter
the program in the future.
In the brain, motor learning would seem to be a kind of compilation.
Motor tasks are compiled so that they can run with optimal speed, in real
time. Once the task has been compiled down into the brain, it is extremely
difficult to alter. This is most clearly seen in the case of speech. After learning,
as a child, the pronunciation patterns of one’s own language, it is very
hard to learn new speech patterns in order to correctly pronounce a foreign
language learned as an adult. Of course, brain compilation goes beyond
computer compilation. In computer compilation, a program compiles down
to the machine level but cannot alter the machine itself. In brain development
and learning, the machine is changed, and underlying circuitry is
5.7. Neurospeculation 79
reorganized. The brain program is compiled into machine circuits rather
than into a predetermined machine language.
The brain is also an interpreter. Natural language (e.g., English) is
the programming language that the interpreter processes. Thought is the
process of interpretation. In computer science, the interpreter is itself a
compiled program that runs in machine language. (It is possible to write
an interpreter inside of another interpreter. With time to waste, I did this
for an old Hewlett-Packard computer and then waited 47 seconds to add 5
to 3.) Because the interpreter itself runs in machine language, I would contend
that it is still necessary to understand the brain to truly understand
natural language.
There are some languages, like LISP, that are both interpreted and compiled.
In LISP, when you write a subroutine, you initially use the code in
its interpreted form while debugging it. Then, once you’ve got it right, the
subroutine is compiled. This way, you don’t have to do repeated compilations
while making many changes to the program. Once it’s finished, you
can put it in a compiled form and it runs faster. This process is analogous
to the shift from interpreted to compiled that is seen when learning a motor
task. Take the example of learning a tennis serve. You start out with a natural
language description, an algorithm, for stepping through the process.
As you say the algorithm to yourself and attempt to execute it, you are
interpreting the code. You are also very clumsy. Over time, you eventually
compile the code directly onto the machine. After this is done, you can
throw away the old program. You no longer have to think about doing the
task, it is fast, automatic, and hard to change. The discrete steps that were
used to learn it are forgotten. Subsequently, you may have to reconstruct
the algorithm by monitoring yourself while you serve in order to teach it
to someone else.
In years back, CPUs were designed to either be RISC (reduced instruction
set chip) or CISC (complex instruction set chip). In a RISC
architecture, the CPU can perform only simple instructions but performs
them fast. On a RISC, basic routines have to be written in software.
Although the PDP-8 far predates the RISC/CISC distinction (it has a
comically meager instruction set by modern standards), the RISC concept
is illustrated by the need to write a multiplication instruction rather than
have one built into the chip. By contrast, a CISC performs a lot of complicated
processing on-chip. Using this analogy, I would suggest that brain
CPUs are likely to be CISCs. They develop massive built-in instruction
sets. In the motor system particularly, learning builds long hardware routines
that can be triggered and then allowed to play out without conscious
intervention.
In a computer, the bus is a set of shared wires that carries information
between different components: RAM, CPU, disks, etc. This is an efficient
design since the same wires can be used for different signals. It is not necessary
to have dedicated wires that just run from RAM to CPU and from
80 5. The Soul of an Old Machine
RAM to hard disk. The downside of this design is that signals on the bus
must include markers or labels indicating their destination address. In general,
everything that is connected to the bus must look at every signal,
reading the address to see if that signal is theirs. This extra reading represents
a time inefficiency. The need for labels on the signals expands the data
structure and represents a coding inefficiency. These minor inefficiencies are
worth it since the savings on physical wire is enormous.
The brain appears to use primarily labeled lines rather than labeled signals.
Labeled lines are wires that are dedicated to carrying only certain
information between specified sites. Brain wires, axons, run primarily one
way (although they can be back-fired), and are not easily shared (although
there are gap junctions between axons). Pyramidal cells are the large cortical
cells whose projection axons connect remote parts of the central nervous
system. If the brain has a bus-equivalent, these wires are it. It is possible
that the brain could use both labeled lines and labeled signals. In that case,
a projection axon (or set of projection axons) would be conceived as connecting
cortex A to cortex B rather than simply connecting cell a to cell
b. This would require that the projecting pyramidal cell show resonance at
the cellular level. The receiving region could show resonance at either the
circuit or cell level. In this scenario, a particular pattern of activity in A
would activate the projecting pyramidal cell in a distinct manner, causing
it to fire a distinct spike sequence. This spike sequence would be the labeled
message. At the other end the spike sequence would be read out through
its tendency to activate a particular neuron or particular subcircuit. It is
even possible that these labeled messages could be multiplexed, allowing
the triggering of more than one target circuit. Although this model is highly
speculative (i.e., there is not an ounce of data to support it), the pattern
of broad convergence and divergence in cortex would make such a model
possible.
Modern computers define a hierarchy of memory storage based on accessibility
and speed. At the very top are the CPU registers, such as the
program counter and accumulator. A modern CPU will have many more
of these registers, allowing it to temporarily store and repeatedly run brief
machine-level subroutines on-chip. At the next level of accessibility will be
a cache. This is particularly useful for graphics chips that need to maintain
screen information in order to update the screen quickly. The third level of
memory is RAM. This is considered main memory. As in our PDP-8 model,
this is where programs and data reside before use. At the fourth level is
the disk. This also offers random-access but is much slower than RAM;
data retrieval depends on the physical movement of an arm that samples
the magnetically charged surface of the spinning disk. At the fifth level is
some kind of slow offline media used for backups — paper punch tape in
the case of older PDP-8s.
So what are the layers of memory management for the brain? In the previous
chapter I alluded to the widely shared concept of the hippocampus as
5.7. Neurospeculation 81
an encoder of episodic memory for subsequent storage in other cortical areas.
However, for most of the brain we encounter a problem that we’ve been
skirting all along — the brain does not generally distinguish between data
representation and data processing. The two are ubiquitously intertwined.
This has analogues in various types of software data structures that use
programs or mix executable code and data. The vector plot data structure
presented in the last chapter was an interpreted program. As another example,
an industrial robot might utilize a data structure that lists common
target locations in both a vector format and a camera-centered bit format
along with the actual code that is needed to move the arm to that location.
Minimal description length describes the trade-off between processing
description length and data description length. In general, the existence of
some kind of processing technique is implicit for any data structure.
So rather than have levels of memory, the brain can be viewed as having
levels of memory/processing. The brain typically divides a sensory signal
shortly after its arrival in the central nervous system (CNS). One track
leads to nearby cells optimized for speed. These cells mediate reflexes that
immediately alter behavior to improve survival. Slower tracks take the input
to more advanced processing centers, arrayed both in parallel and in
series. When you stub your toe, lower centers register the mishap and convey
a sense of unease to conscious processing before higher centers identify
the misery location. Higher processing pathways are presumably also divided
up according to speed needs. For example the “magno” pathway that
mediates visual motion perception is faster than the “parvo” pathway that
mediates object recognition. This may reflect a need to know “where?”
before you know “what?”
An additional problem faced by the brain is that it receives much too
much data, most of which is redundant and unneeded. So in addition to
dividing the input into streams according to different processing needs, the
CNS will also want to split off input that can simply be discarded.
Above, I’ve mostly thought about the situation on the sensory side. Comparably,
the motor side must often initiate movement before it’s entirely
worked out the details of how the movement will end. In a reach-to-grasp
task you start the arm movement before the fingers are configured for the
grab.
There are several other aspects of computer design that can be considered
in the context of the brain. These include the existence of a central clock,
use of pointers and other addressing schemes, and the use of stacks, queues,
heaps, and other standard data structures. One could go on and on, whether
fruitfully or fruitlessly.
82 5. The Soul of an Old Machine
5.8 Summary and thoughts
In this chapter I’ve presented the procedures and process of hacking in
the context of its original value before its modern-day descent into evil. I
may have indulged myself a bit or two; the programs were fun to write. I
encourage the reader to get hold of my software and try it out.
Although different computer programming languages have different features,
the fundamental design constraints of the underlying hardware
appear again and again. The design of the data processing machine determines
how we use and how we can use that machine. These computer
design features include 1) pointers, 2) conditionals, 3) control flow, and 4)
equivalence of program and data.
A message of this chapter is that hardware molds software. The other
message is that cleverness can overcome apparent hardware limitations.
Development of various types of channels, neurons, and patterns of connectivity
has served to impose a higher algorithmic order on the basic
underlying architecture. Consider a jellyfish or a sponge. These animals
have simple nervous systems that don’t do a heck of a lot more than the
nervous system of the venus flytrap, a plant. Imagine that you are trapped
on a desert island, surrounded by these creatures. Now, starting with jellyfish
and sponges, using a penknife and a magnifying glass, build a robot
that can see, hear, walk, talk, and think.
Part III
Cybernetics
Cybernetics 85
Cybernetics is an old, perhaps archaic term for developing machines that
can behave like nervous systems. It dates from a time when hardware,
rather than software, was preeminent. The focus of cybernetics was on
system analysis and system control, closer to the current study of robotics
than to neural networks. Nowadays, the term cybernetics is perhaps more
common in science fiction than in science: the borg of “Star Trek” or the
cyborg of The Terminator.
Cybernetics is a relatively broad term that denotes studies of the man–
machine or animal–machine interface. I use this term in preference to neural
networks because it is broader, and because it emphasizes the contrast between
the machine and system perspective, and the biological perspective.
Cybernetics is a top-down approach, in that one looks for ways to design
machines to do a particular function. The word cybernetics, like the word
artificial in artificial neural network, emphasizes the aggressive rationality
of the approach, as distinct from the detailed, deductive thinking of biological
research. The fictional construct of the nasty unfeeling borg or cyborg
embodies this contrast between cybernetics and biology.
Using the term cybernetics also brings into focus one of its central tenets.
This is cybernetic’s goal of developing a general, domain-independent theory
that can be used to understand any large system, from robots to
spacecraft to ecosystems to brains to social networks. This perspective,
or academic field, is sometimes called general systems theory. It is very
big-picture and quite seductive, since it would be nice to be able to explain
pretty much everything with a few well-chosen concepts. My feeling,
though, is that this universalist impulse to dissolve academic boundaries
trivializes every field that it touches. Abstractions, theories, and concepts
have their own aesthetic, which can lead to theory dominating reality. For
example, back-propagation, discussed in Chap. 9, was a major advance in
neural networks. However, it was taken too seriously by many of its early
proponents. They suggested that it was such a good algorithm that it had
to be used by the brain and insisted that neurobiological research should
focus on demonstrating its existence.
In Part IV, I switch to a primary biological focus and approach again
this contrast between the mindset of cybernetics and biology, mindsets that
must be reconciled in individual minds as the field moves forward.
6
Concept Neurons
6.1 Why learn this?
We now embark on the study of one of the two major strands of computational
neuroscience: artificial neural networks (ANNs). This is the more
popular strand, since it touches more closely on the big questions of learning
and memory. The ANN movement is heir to cybernetics and to AI (artificial
intelligence), the attempt to use standard programming techniques to
develop machines that could compete with human intellectual activities.
In some areas, like chess, this effort paid off. In 1997, an IBM computer
defeated Garry Kasparov, the reigning chess champion. In other areas, such
as doctoring, lawyering, and translation of natural language, AI programs
have had only limited success. Still less successful have been efforts to replicate
the more elementary aspects of human, and animal intelligence, such
as understanding visual scenes (i.e., seeing). Indeed, although humans are
justifiably proud of higher cortical functions such as language and thought,
visual areas make up about two-thirds of our cortex.
We begin our study of both artificial neural networks and realistic neural
networks at the level of the neuron and its behavior. In the 1950s,
two neuroscientists, Warren McCulloch and Walter Pitts, proposed a basic
neuron model. Although McCulloch and Pitts were not the first to consider
neurons as calculation elements, they were pioneers in their attempt
to formally define neurons as computational elements and to explore the
consequences of neural properties. Neurobiology has advanced enormously
since the 1950s. The original McCulloch-Pitts neuron was a binary element.
88 6. Concept Neurons
The modern artificial neural network unit is no longer binary but is otherwise
little changed. The artificial neuron remains an insubstantial shadow
of a real neuron. These simplified neural models can be considered concept
neurons, explicitly designed based on a concept of how neural processing
takes place. The words unit and neuron are often used interchangeably in
discussing neural networks, although some biologists object that these units
are so abstract that it’s unfair to call them neurons.
In this chapter, we emphasize the differences of implementation and interpretation
between various ways of modeling neurons. We show how concept
neurons differ from real neurons. Although a recitation of these differences
makes it look like these are lousy models, they are not. The concept neurons
are attempts to get to the essence of neural processing by ignoring irrelevant
detail and focusing only on what is needed to do a computational
task. The complexities of the neuron must be aggressively pared in order to
cut through the biological subtleties and really understand what is going
on. As more information comes to light, some of these models will turn out
to be right, in that they capture some critical aspect of neural information
processing, while others will turn out to be wrong, meaning that they are
so far off base that they obscure more than they illuminate. Even the wrong
models can be valuable, however. They illustrate different approaches to
simplifying the neuron and can be used as meta-models, models of how to
build, and how not to build, models.
6.2 History and description of McCulloch-Pitts
neurons
In their 1943 paper, McCulloch and Pitts considered the computational
power of simple binary units. Their paper is more mathematical than it is
biological and was therefore more influential in computer engineering than
in neuroscience. John von Neumann, one of the great mathematicians of
the 20th century, became intrigued by the brain and by McCulloch and
Pitts’s view of it. This neural model had a substantial influence on the
thoughts and studies that led to modern digital computer design.
McCulloch and Pitts knew that neurons had spikes (action potentials)
that involved sudden, transient shifts of membrane voltages from negative
to positive. These spikes somehow carry information through the brain.
McCulloch and Pitts assumed a simple coding scheme for this information
carrying: each spike would represent a binary 1 (or Boolean true), and
the lack of a spike would represent a binary 0 (or Boolean false). They
showed how these spikes could be combined to do logical and arithmetical
operations. From the perspective of modern computer design, there is one
glaring problem with the design of these circuit elements — the brevity
of the spike. A duration of 1 millisecond (ms), compared with inter-spike
6.2. History and description of McCulloch-Pitts neurons 89
axon
(output)
1
0
− 0 +
Summed Input
State
transistor
base
(input)
collector
bias V
5
0
emitter
V
emitter
output
MP neuron
dendrite
(input)
Fig. 6.1: Comparison of transistor and McCulloch-Pitts neuron.
Standard symbols are used: a perpendicular line segment
is excitatory (+) and a filled circle is inhibitory (−).
intervals of 50 ms or longer, means a duty cycle (signal duration as percent
of the period) of only 2% (1 ms/50 ms). This means that it is easy to detect
the signal for 0, which sticks around for 49 ms, but hard to pick up the
signal for 1, which is there for only this small percent of the time. There is
a high risk of missing or failing to process such a short signal.
A McCulloch-Pitts neuron functions much like a transistor (Fig. 6.1).
At the time of McCulloch and Pitts, computers used vacuum tubes or
electromechanical switches. I will briefly discuss the transistor, the modern
analogue of the vacuum tube. A transistor has a “base” lead that controls
the flow of current between the other two leads. Generally, a high voltage
is considered a logical one or true (positive true in computer engineering)
and a low voltage a logical zero or false. If the base is activated so as
to permit current flow between emitter and collector, the voltage on the
emitter will go to low and the transmitter output will be logical false (0). If
90 6. Concept Neurons
the base does not permit this flow, the voltage on the emitter will remain
at high. The base can thereby set or clear (turn on or off) this single
bit device, acting as a switch with which to change the output from 1 to
0. A certain voltage, called the bias, is required to trigger the base. The
bias is a threshold that changes the output state of the transistor. Since the
transistor is controlled through voltage applied to the base, voltages can be
routed from the collector of one transistor to the base of another transistor
in order to build up complicated logical or arithmetical operations based on
a domino effect: highs and lows from one transistor produce highs and lows
in the following transistors. In the example shown, the transistor performs a
logical NOT operation: input 1 produces output 0; input 0 produces output
1.
Like the transistor, the original McCulloch-Pitts neuron had a threshold
that needed to be reached in order to activate the unit. Like the transistor,
the output was completely stereotyped, being a binary 1 if threshold was
reached and remaining at a resting value of binary 0 if threshold was not
reached. In an artificial neural network, these inputs come from the outputs
of other McCulloch-Pitts neurons, just as the inputs to a transistor come
from other transistors.
Transistors are typically organized as sets of switches. The output from
one transistor will turn a follower transistor on or off. The McCulloch-Pitts
neuron, like the transistor, handles incoming inhibitory input as a switch:
any inhibitory input is sufficient to shut down the McCulloch-Pitts neuron.
This is called veto inhibition, a highly nonlinear process. By contrast,
excitatory inputs to a McCulloch-Pitts neuron just add up linearly.
6.3 Describing networks by weights and states
In neural networks, a critical but sometimes confusing distinction is that
between weights and states. A weight is the strength of the connection between
two neurons. Biologically, this can have several interpretations, but
is thought to most closely correspond to the size of a synaptic conductance
change. We use the biological term synapse to refer to the location of a connection
between units. The term presynaptic refers to the unit projecting
into that connection. Postsynaptic refers to the unit receiving that connection.
Since most biological synapses are unidirectional, it makes sense to
speak of pre- and postsynaptic units.
A state is the degree of activation of a single neuron. The state is an
artificial neural network abstraction. Biologically, a neuron cannot be adequately
described by a single scalar state. However, to make comparisons
between real neurons and artificial neural network units, it will be useful
to figure out how best to reduce the complexities of real neuron activity to
6.3. Describing networks by weights and states 91
Weighted
Outputs
0
0 1
1 1
Inputs
−3
0 2.1 1.7 −0.5
0 x (−3) + 0 x 1 + 1 x 2.1 + 0 x 1.7+ 1 x (−.5) =
summed input =
1.6
Fig. 6.2: Calculation of input into a single unit. Line segments
represent connections between presynaptic units (small open circles)
and this unit (large open circle). Filled circles = inhibitory
connections (negative weights); perpendicular lines = excitatory
connections (positive weights). Weight values are analog and will
be determined by a learning algorithm.
a single value. As we see in Chap. 11, average membrane potential or spike
rate would be two typical choices for a scalar state in a real neuron.
In an artificial neural network, the state of a neuron is just a number that
serves as the output of that neuron. An update rule determines how the input
to a unit is translated into the state of that unit. Generally the update
rule will requires two steps. First, determine the unit’s total-summed-input
based on presynaptic states. The total-summed-input to a unit is the summation
of products of presynaptic states with corresponding presynaptic
weights. Second, determine the state based on the total-summed-input. The
first step of the update rule, the summation of products, is fairly standard
across a wide variety of network models. The second step, determination
of states, differs between models.
92 6. Concept Neurons
In Fig. 6.2, we depict binary units that take on values of either 0 or 1.
The numbers in the circles at the top are state values of units that are
presynaptic to the unit represented by the large circles in the center. Each
of these state values serves as an output from that unit. Each output is
then multiplied by the corresponding weight value, which is represented in
the figure by a number to the side of the line connecting them. Negative
numbers represent negative weights that are inhibitory — activity in a
presynaptic neuron operating through a negative weight will tend to reduce
activity in the postsynaptic neuron. The weight values are analog. Those
shown were chosen arbitrarily. In an artificial neural network, weights will
initially be randomized and will then be adjusted by a learning algorithm.
The main unit in Fig. 6.2 projects to other units via the weighted outputs
shown at the bottom of the figure. Specific weight values are not given
in the figure, but the symbols indicate that both negative and positive
projections are coming from the same unit. This is another nonbiological
feature; real neurons are strictly excitatory or strictly inhibitory (Dale’s
principle).
Looking at the leftmost input, the weight is irrelevant, since the presynaptic
cell is inactive (i.e., its output is zero). Conversely, the cell second
from left is active but it also has no effect on the neuron at the center since
the weight connecting to this cell is zero. The only cells that do have an
effect in this case are the center cell and the rightmost cell. In this example,
activity in an active cell is represented by a state value of one, so that the
weights are all multiplied by either 1 or 0. The calculation is shown at the
center of the figure with the multiplications by zero explicitly shown.
Calculating total-summed-input by dot product
Arithmetically, each input coming into a unit is a multiplication of state and
weight. A shortcut for representing this state times weight multiplication
comes from the dot product of linear algebra (see Chap. 16, Section 16.4).
Linear algebra uses scalars, vectors, and matrices (matrices is the plural
of matrix). A scalar is just a single number. A vector is a one-dimensional
array of numbers. A matrix is a two-dimensional array of numbers that is
described by the number (M) of rows and number (N) of columns.
An arrow over an italic letter (e.g., _x) is used to indicate that the variable
being represented is a vector, rather than a scalar. In the standard vector,
numbers are arranged from top to bottom in a column. This is called a
column vector. In the other vector orientation, numbers are listed from left
to right. This is called a row vector. (A vector can also be represented by
a magnitude and angle without worrying about whether it’s a column or a
row, but I won’t use that representation.) The transpose is the operation
that switches a column vector to a row vector or vice versa. Transpose is
symbolized by a superscript T. Therefore, _xT is a row vector. In a spread6.3.
Describing networks by weights and states 93
sheet, a single row can be thought of as a row vector, a single column as a
column vector, and the entire spreadsheet as a matrix.
The dot product (also called inner product) is the scalar result of multiplying
a row vector times a column vector. This scalar result is created by
taking the sum of the pairwise products of the elements of the two vectors.
The dot product can be represented by a dot (・) or by specifying the orientation
of the vectors by using the transpose symbol and leaving out the dot.
For example, for _xT = _x1 x2_and _y = _y1
y2_, _x・_y = _xT _y = x1 ・y1+x2 ・y2.
Note that the subscripted variables, x1 and x2, are scalars that are arrayed
to make the vector _x. Using numbers, with x1 = 2, x2 = 5, y1 = 3 and
y2 = 7, then _x = _2 5_, _y = _37
_, and _x ・ _y = 2・3+5・ 7 = 6+35 = 41, a
scalar.
In our application, the two vectors will be the weight vector and the
state vector. We create the weight vector by simply producing an ordered
row of numbers that are the weights coming into the neuron in question.
Similarly we list all of the corresponding states of presynaptic units in a
column to make the state vector. It is critical that the ordering be the
same so that the proper weight lines up with the corresponding state. The
dot product is defined as the sum of the pairwise products of the elements
of the two vectors. Therefore, the dot product of the state vector and the
weight vector is the total-summed-input to the unit.
For example, in Fig. 6.2, the weight vector _wT is _ −3 0 2.1 1.7 −0.5and the state vector _s is _ _0 1 1 0 1_T . Note that I’ve written out
both of the vectors in row form to save space. In the case of the state vector,
the transpose symbol is used next to the row of numbers to indicate
that this is actually a column vector. Now we can do the dot product _w ・ _s
_−3 0 2.1 1.7 −0.5_
⎛⎜⎜⎜⎜⎝
01101
⎞⎟⎟⎟⎟⎠
= 1.6
The dot product is commutative: _w ・ _s = _s ・ _w or _wT_s = _s T _w. However,
multiplying a column vector times a row vector is NOT the same as multiplying
a row vector times a column vector. Therefore, to reduce confusion,
we will do the multiplications in one way: weight times state ( _w ・_s = totalsummed-
input). This may seem counterintuitive since when diagrammed,
state comes before weight. The reason for this order will become clear when
we introduce the weight matrix.
94 6. Concept Neurons
Calculating state
The first step of the update rule was the calculation of total-summedinput.
Depending on how many inputs there are and how large the weight
numbers, total-summed-input could be a very large number. The state of a
single unit is restricted to lie within a certain range. The activation function
is used to convert the total-summed-input to unit state. The activation
function is sometimes called a squashing function because it takes a large
range of values and squashes them down to fit within the required state
range.
Different artificial neural network models use different squashing functions
to produce either an analog or a binary state. An analog state can
be any value between two endpoints. The range of a particular squashing
function will determine a range of possible state values. A binary state is
restricted to being one of only two values. The extrema of the squashing
function will be the two possible state values. For example, allowed states
for an analog unit might be real numbers between 0 and 1, while a comparable
binary unit would take on one of two states: inactive (0) or active
(1). It is also common to used ranges of −1 to 1 or binary states or −1 and
1. Generally, the squashing function will be monotonically increasing (i.e.,
only go up). In Chap. 9, I use a squashing function that provides analog
states between −2 and 1. The state of a unit serves as the output from that
unit. This output gets multiplied by weights to provide inputs to follower
neurons.
To produce a binary unit, the squashing function uses a sharp threshold
that determines whether an input value is translated to a state value of 0 or
1 (Fig. 6.3). The discontinuous function shown here can be readily defined
on the computer: if (total summed input <0) then state=0 else state=1. This
particular function, named after Oliver Heaviside, is called the Heaviside
function or the step function. Given an input of 1.6, the unit state will be
1. Alternatively, to produce an analog state, a continuous function is used,
as shown to the right of Fig. 6.3. This sigmoid curve produces a continuous
value between 0 and 1: for the total-summed-input of 1.6, the unit state
will be 0.7.
Now we can summarize the update rule for a single unit: si = σ ( _w ・ _s),
where si is the state of unit i and σ(x) is the squashing function applied
to a scalar x. This is the standard generalization of the cyborg neuron. It
differs from the McCulloch-Pitts neuron since inhibition is added in rather
than having veto power. It is a generalization since the squashing function
can be chosen to yield either binary or continuous values. As a shorthand,
I call this basic unit the sum-and-squash unit.
6.4. From single unit to network of units 95
− 0 +
Total summed input
State
1
0
0.7
1
0
− 0 +
State
Total summed input
Fig. 6.3: Calculation of state of a single unit. The input calculation
is the same as in Fig. 6.2. The state calculation differs
depending on the squashing function chosen. A sharp threshold
gives a binary state of 0 for negative inputs and 1 for positive
inputs. In this case the state is 1. A sigmoid function gives an
analog state between 0 and 1. In this case the state is 0.7.
6.4 From single unit to network of units
In the previous section we considered a single unit and its inputs. For
one unit, introducing the dot product to do such a simple calculation may
seem unnecessary. However, when we put together a lot of units and start
connecting them up, the notions of linear algebra become valuable. This is
particularly true when it comes time to enter these neural networks into
a computer, which can readily store a spreadsheet of numbers but not a
diagram of circles and line segments.
This simple update rule for the single unit (si = σ ( _w ・ _s)) allows us to
generate an update rule for the entire network. Instead of using the weight
vector _w, we use a weight matrix W, which represents all of the weights
in the network. The update rule for the state vector is _s t+1 = σ (W ・ _s t).
Superscripted t and t+1 indicate passage of time by one time step. Applying
a function (σ) to a vector is simple: the rule is to simply apply the function
to each element of the vector. For example, if we have a function f(x) =
5 ・ x + 2 and a vector _a = _3 5 7 1_T, then f(_a) = _17 27 37 7_T .
We simply multiply each element by 5 and add 2.
96 6. Concept Neurons
Each element of the W matrix is the value of a weight between two units.
W ・ _s (multiplication of the weight matrix times the state vector) gives a
vector of inputs into each element. This works because the weight matrix
is a pile of weight vectors, and matrix multiplication produces a pile of dot
products:
W =
⎛⎜⎜⎜⎜⎜⎝
_wT
1
_wT
2
_wT
3
_wT
4
...
⎞⎟⎟⎟⎟⎟⎠
hence ⇒ W ・ _s =
⎛⎜⎜⎜⎜⎜⎝
_w1 ・ _s
_w2 ・ _s
_w3 ・ _s
_w4 ・ _s
...
⎞⎟⎟⎟⎟⎟⎠
If there are nine units, the weight matrix will have nine rows and there
will be nine dot products (between each row and the state vector) giving
a vector of length 9. Each element of this vector is the summed input into
a unit. This vector is then squashed to get the state of each unit after one
update step.
To see an example, let’s look at a circle-and-stick drawing of a simple
network (Fig. 6.4). This is a fairly small network by artificial neural network
standards and is very sparse, meaning that of the possible 92 = 81
connections shown in the table below Fig. 6.4, only a few are present. While
it is easy enough to examine the circle-and-stick diagram in this small network,
the scribble can quickly become overwhelming, making a table a more
easily read representation.
With the table (and matrix) laid out in this way, each row represents
convergence onto a particular unit. For example, unit c (third row), has 3
non-empty entries, indicating a convergence of 3 inputs onto unit c, which
receives projections from presynaptic units a, b, and f. Similarly, each column
represents divergence from a single unit. For example, unit f (column
6), has a divergence of 3 with projections to postsynaptic units c, e, and
i. I have placed black squares along the main diagonal. These are the locations
of self-connects. Units are sometimes connected to themselves in a
neural network, as they may be in biological networks. In this case there
are no self-connects so the diagonal is all zeros. Given the simplicity of
Fig. 6.4, all of the connections can be readily confirmed by looking at the
circle-and-stick diagram.
We are using the formula σ(W ・_s) as our update rule. _s will be a column
vector. Each row of W will represent the ordered sequence of weights converging
onto the unit with that row number. Equivalently, each column of
W will represent the ordered sequence of weights diverging from the unit
with that column number. Any row, column number pair i, j will give the
strength of the connection from unit j to unit i. Therefore, the connectivity
table above can be directly copied into a weight matrix W (also called the
connectivity matrix). Each number is located in the same place in W as it
6.4. From single unit to network of units 97
−.3
.7
−.1
.4
.3
.2
−.5
.2
.1
.9
−.5
.5
.2
.2
.1
.2
.7
b
a c
f
d
e
i
h
g
Fig. 6.4: A nine-unit sparse recurrent network. Each neuron
is labeled to make it easy to keep track of their identities. I
have used arrows for connections instead of circles and lines
to indicate that the sign of these weights is changeable with
network learning. This diagram can be translated into the table
below:
FROM ⇒ a b c d e f g h i
TO ⇓ a -.5
b -.5
c .4 -.1 .2
d .5 .2
e .2 .7
f .3 .2
g .1 .9
h .7 .2
i -.3 .1
This table can then be copied directly into a weight matrix W:
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0 0 0 0 0 0−.5 0 0
0 0 0 −.5 0 0 0 0 0
.4 −.1 0 0 0 .2 0 0 0
.5 0 0 0 0 0 0 .2 0
0 0 0 0 0 .2 0 0 .7
0 0 .3 0 0 0 .2 0 0
0 0 0 0 .1 0 0 .9 0
0 0 0 0 .7 0 0 0 .2
0 0 −.3 0 0 .1 0 0 0
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
98 6. Concept Neurons
is in the table. Zeros are used in the matrix where there is no connection
between corresponding units.
Matrix times vector multiplication is done by taking the dot products of
each row (here the convergent weights onto a unit) times the column vector
(the convergent states of presynaptic units). (This is the right-hand side
matrix multiplication rule—the vector is on the right. There is also the lefthand
side vector times matrix multiplication using a row vector on the left
and resulting in a row vector. I will only use right-hand side multiplications
in this book.) Because each row is simply the list of projecting weights, the
dot product gives the input for each unit as illustrated in Fig. 6.2 and the
corresponding dot-product equation. By doing each of these dot products
in turn, we generate an input vector where each element corresponds to
the input for one unit. If we then take this input vector and squash it, we
have the state for each unit after one time step.
To simulate the network, we have to set not only the weights but also
a starting point for the network state. This set of values is referred to as
initial conditions. Let’s start with unit c active (set to 1) and all other units
inactive. We use sharp thresholding at 0: any input greater than 0 becomes
a state of 1, and any input less than or equal to 0 becomes a state of 0.
The first multiplication step is:
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0 0 0 0 0 0−.5 0 0
0 0 0 −.5 0 0 0 0 0
.4 −.1 0 0 0 .2 0 0 0
.5 0 0 0 0 0 0 .2 0
0 0 0 0 0 .2 0 0 .7
0 0 .3 0 0 0 .2 0 0
0 0 0 0 .1 0 0 .9 0
0 0 0 0 .7 0 0 0 .2
0 0 −.3 0 0 .1 0 0 0
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
001000000
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
=
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 00000
0.3
00
−0.3
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
This multiplication is very simple since a single active unit will simply
project forward to its postsynaptic cell. Algebraically, this is equivalent
to saying that the lone 1 in the state vector will pick out the divergence
values for that unit. Indeed, it’s easy to compare and see that the resultant
vector (the right-hand side of the equation) is identical to the 3rd column
of the matrix. This result serves as the input to each of the units. Repeated
application of the update rule (matrix multiplication to produce inputs and
thresholding to produce states) produces a sequence of vectors:
6.4. From single unit to network of units 99
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
001000000
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
0
W・_s
→
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
00000
0.3
00
−0.3
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_i
1
σ(_i)
⇒
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
000001000
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
1
W・_s
→
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
00
0.2
0
0.2
000
0.1
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_i
2
σ(_i)
⇒
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
001010001
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
2
W・_s
→
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0000
0.7
0.3
0.1
0.9
−0.3
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_i
3
σ(_i)
⇒
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
000011110
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
3
W・_s
→
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
−0.5
0
0.2
0.2
0.2
0.2
1
0.7
0.1
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_i
4
σ(_i)
⇒
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
001111111
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
4
W・_s
→
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
−0.5
−0.5
0.2
0.2
0.9
0.5
1
0.9
−0.2
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_i
5
σ(_i)
⇒
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
001111110
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
5
W・_s
→
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
−0.5
−0.5
0.2
0.2
0.2
0.5
1
0.7
−0.2
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_i
6
σ(_i)
⇒
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 001111110
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_s
6
Since the thresholding is at 0, all of the inputs that are less than or equal
to 0 become 0 and any that are greater than 0 become 1. In this sequence,
states are subscripted as _s and total-summed-input vectors as_i. The result
of matrix multiplication W ・_s is shown as _s → _i, while the result of the
squashing (thresholding) function σ(_i) is shown as _i ⇒ _s. The time step
(iteration number) is shown as a superscript.
The same sequence of states are shown in the circle-and-stick diagrams
in Fig. 6.5. Units c and f are mutually excitatory, so activity starting in
c spreads to f after the first time step and then spreads back to c after
the second time step. In the absence of other inputs, the c ↔ f excitatory
loop allows activity to bounce back and forth between these two units. This
is seen from time 0 to 3 in this example. At time 2, activity also spreads
down from f to e and i. However, c inhibits i so the activity of c at time 2
100 6. Concept Neurons
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g
h
i
t=0
t=1
t=2
t=3
t=4
t=5
t=6
Fig. 6.5: Sequence of network states from time t = 0 to t = 6. 0 is
represented by a white unit and 1 by a black unit.
leads to i turning off again at time 3. It turns on again at time 4, due to
the activity in f at time 3. By time 4, activity has percolated throughout
the network, excluding only b, which receives only inhibitory inputs. In the
final state both c and f are active. Since c → i inhibition is greater than
f → i excitation, i is turned off at time 5. It’s pretty hard to keep track of
activity as it roams around in this tiny network. Imagine trying to follow
activity patterns in a network of 1000 or 10,000 units. To do this, it would
be helpful to discover rules that describe common propagation patterns. A
general taxonomy of network state evolution has not yet been developed.
Some networks will change indefinitely over time. In Fig. 6.5, the network
stops changing. A network that reaches a final constant state in this manner
is said to arrive at a steady state. The state of the network, as represented
6.4. From single unit to network of units 101
x
y
z
(0,0,1)
(0,1,1)
(1,0,1)
(1,1,1)
(1,1,0)
(1,0,0)
(0,1,0)
three−dimensional
x
y
(0,1) (1,1)
(0,0) (1,0)
two−dimensional
Fig. 6.6: A two-dimensional binary vector can be mapped as a
point at one of the vertices of a square. If a 2-D vector took on
analog values between 0 and 1, it could be mapped as a point
inside the square. Similarly, a 3-D binary vector is mapped on
a vertex of a cube; here (0,0,0) is on the hidden vertex. A 3-D
analog vector is a point inside the cube. The concept generalizes
to hypercubes in higher dimensions.
by the state vector, will not change further with time. In the case of the
network of Figs. 6.4 and 6.5, the final state shown at t = 6 in Fig. 6.5 is
not only the steady state for this particular initial condition, but also the
final state for any initial condition. This makes it the sole attractor for this
network. Since this attractor is a steady state, it is a point attractor. For
most initial conditions, the network will end up at this attractor. The other
attractor in this network is _s = _0 . . . 0_T . Wherever the network starts, it
will end up in one of these two attractor states.
The concept of an attractor comes from physics, where it refers to the
behavior of an object in an energy field, such as gravity. An object in a
gravitational field will tend to roll downhill until it reaches a state of lowest
potential energy. Then it will stay there. The fact that it goes toward a
goal makes that goal an attractor. If the dynamics made it run away from
a point, that point would be a repellor.
State space is the multidimensional space in which you can map the entire
state vector (Fig. 6.6). You can map a two-dimensional vector on a square,
102 6. Concept Neurons
a three-dimensional vector on a cube, and a nine-dimensional vector on
a nine-dimensional hypercube (not shown). An attractor that is a single
spot in state space is a point attractor. We will revisit point attractors
when we discuss memory (Chap. 10). If the system ends up switching state
repeatedly among two or more state vectors, the attractor is a limit cycle.
6.5 Network architecture
The term architecture refers to the connectivity of a network. We can
describe the architecture of Figs. 6.4 and 6.5 as being recurrent and
sparse. Recurrent means that there are loops in the connectivity, for example,
the loop between c ↔ f and longer loops like the circumferential
c → i → h → g → a → c. Note that a network would be considered recurrent
even if there were only inhibitory loops — the strength or sign of
the connection is not important. As for sparse, we noted above that the
network is sparse because only a few of the 81 possible interconnections are
present (17 – 21% connectivity). The term sparse is not strictly defined.
It is generally said that brain connectivity is sparse; although a single cell
may receive many thousands of connections, this is only an infinitesimal
fraction of all the cells in the brain or even in a small brain region.
A recurrent network is a general network architecture since all connections
are possible. In the brain, neurons that receive projections from one
brain area typically also project back in the other direction. Most cortical
function is probably dependent on these loops of recurrent connectivity.
In addition to recurrent networks, the other major architecture is feedforward.
Feedforward networks are layered. All information flows in one
direction from a presynaptic layer to a postsynaptic layer. Since there are
no back projections in the opposite direction and no connections within
either layer, there are no loops. A typical three-layer feedforward network
is shown in Fig. 6.7.
In general, the first layer of a multilayer feedforward network will be
referred to as an input layer, and the last layer will be considered an output
layer. Any layers in between are called hidden layers — they do not interact
with the environment, hence they are hidden from the environment. When
we translate this network into matrix form, we see large blocks of zeros
that would have entries if there were recurrence.
6.5. Network architecture 103
−.2
.1
−.9
.5 .4
−.4
.1
−.6
.6 .5 −.2
.7
a
IN IN
b
a
HID
c
HID
b
HID
a OUT b OUT
Fig. 6.7: A simple three-layer feedforward network. Activity percolates
from the input layer at bottom through the “hidden”
layer to the output layer at the top.
FROM ⇒ INa INb HIDa HIDb HIDc OUTa OUTb
TO ⇓ INa 0 0 0 0 0 0 0
INb 0 0 0 0 0 0 0
HIDa −0.2 0.7 0 0 0 0 0
HIDb 0.1 −0.9 0 0 0 0 0
HIDc 0.5 0.4 0 0 0 0 0
OUTa 0 0 0.6 0.1 −0.4 0 0
OUTb 0 0 −0.6 0.5 −0.2 0 0
The only blocks that are filled are those at center left, which represent the
projections from the input to the hidden (2nd) layer, and those at center
bottom, which are the projections from the hidden to the output layer.
The input units themselves have no inputs at all, and the connectivity
matrix is therefore all zeros in rows a and b. If we were implementing this
network on the computer, using the full connectivity matrix would be very
wasteful of memory. Therefore, we would most likely just use the non-zero
portions of the full weight matrix as two feedforward weight matrices. The
first would map the input layer to the hidden layer, and the second would
map the hidden layer to the output layer. These operations should be done
sequentially, giving the following sequential update rule:
104 6. Concept Neurons
step 1: _shid = σ (Win→hid ・ _sin)
step 2: _sout = σ (Whid→out ・ _shid)
Win→hid =⎛⎝
−.2 .1
.7 −.9
.5 .4⎞⎠
Whid→out = _ .6 .1 −.4
−.6 .5 −.2_
The two matrices are simply copied from the appropriate sections of
the complete connectivity matrix. Note that neither of these feedforward
matrices is square. Win→hid is a 3×2 matrix that maps the two input units
onto the three hidden units and Whid→out is a 2 × 3 matrix that maps the
three hidden units onto the two output units. There is no update rule for
the input units. In fact, these don’t have to be considered as units at all.
They are simply state values that are presented to the system, coming from
the environment.
Although the brain is certainly heavily recurrent, there are probably
systems in the central nervous system that are largely feedforward. For
example, in a reflex, a simple input (the tap of a knee) is quickly and
directly translated into an action (the kick).
6.6 Summary and thoughts
Despite enormous progress in neuroscience, the simple units utilized in artificial
neural networks are little evolved from the McCulloch-Pitts neurons
and vacuum tubes of 60 years ago. These arose in a hopeful era when understanding
neural processing seemed to be just around the corner. Although
archaic, sum-and-squash units are not obsolete. Most of the learning theory
of the past 20 years has been built upon them.
Not coincidentally, artificial neural network units fit neatly into the
mathematical tools that are typically used to handle large systems. Linear
algebra is a tool that provides a compact notation for describing and running
a large network of sum-and-squash units. The basic artificial neural
network update involves a multiplication of the weight matrix times the
state vector followed by application of an activation function to the result:
_s t+1 = σ (W ・ _s t). In addition to providing a compact notation, the linear
algebra formulation is readily translated into computer programs.
Neural networks are typically classified as either feedforward or recurrent.
As we see in Chaps. 9 and 10, different learning algorithms are used for
these different network architectures.
7
Neural Coding
7.1 Why learn this?
We discussed the idea of signals as codes in Chap. 4. We demonstrated
that we could take the same signals and interpret them as a picture, or a
sentence, or a set of commands. To interpret a one-dimensional bit string
as a picture, we had to determine how to properly lay out the bits in two
dimensions (Fig. 4.2). In general, both the code and the thing encoded
can be multidimensional. Part of the key for code interpretation involves a
redimensioning. The information to be encoded will have different defining
dimensions depending on the modality (touch, smell, sight, language), but
most if not all modalities will involve some sort of time-varying signal and
thus include the dimension of time. Similarly, in addition to its physical
dimensions, the brain also possesses time-varying signals, as indeed it must
in order to respond to a time-varying world with time-varying actions.
In Chaps. 4 and 5 I speculated about brain representations by drawing
analogies from standard computer-science representations. In this chapter,
I present more standard ideas based more directly on the brain. We want
to be able to answer the question: How does the brain store and process information?
Because we can’t answer this question unequivocally, this book
dances through metaphors, models, speculations, relevant details, possibly
relevant details, likely irrelevant details, etc. The 43rd edition of this
book is tentatively scheduled for publication in 2212. It will contain a more
thorough treatment.
106 7. Neural Coding
Information, like beauty, is in the eyes of the beholder. The information
content of a “message” depends on what you want to do with it. The
position of planets in the solar system is generally not thought of as a
message. However, an astrologer reads this as a code and an astronomer
uses this information to make predictions about future celestial events.
While wise men eternally wonder about the question, If a tree falls in the
forest, is there a sound?, I am prepared to answer. Yes, there is a sound (a
pressure wave), but is there a noise? No. A signal? No. Information? No.
Similarly, patterns of light in the world can be taken as a code when they
fall on a retina.
For our purposes, patterns of activity in the nervous system will generally
be regarded as codes, though some of these codes may not be involved in the
information processing tasks of the organism as a whole. For example, there
is a large amount of information processing going on at the cellular level in
order to handle the computational needs of genetic and metabolic processes.
This reproductive and metabolic information processing is generally kept
separate from the organism’s thought and behavior. However, if you have
low blood sugar (hypoglycemia), you suddenly discover that your cells’
metabolic misery is quickly made your own.
We divide the question of coding strategies into three categories: 1) ensemble
coding (how neurons work together to form a code), 2) neuron state
encoding (what aspects of the neuron’s chemical or electrical activity are
relevant for coding), and 3) temporal coding (how signals are interpreted
across time).
7.2 Coding in space: ensemble codes
Just as the pattern of pixels in Fig. 4.2 formed a picture, some areas of the
brain are organized into maps that effectively mirror the world. The visual
system has many maps, starting with the direct painting of the world by
light on the retinal photoreceptors and continuing with similarly organized
retinotopic maps in various areas of cortex. It is possible to measure the
activity of cells in cortex and then re-create a blurred version of what the
animal or person is looking at. One visual area projects to another in a
hierarchy of processing areas. In general, as we go to higher processing
areas in the visual systems, the responses of individual neurons become
more specialized. For example, there are higher visual areas with specialized
cells that detect motion, and other visual areas specialized for response to
shapes.
In addition to greater neuron specialization from lower to higher processing
centers, there is also a trend toward greater receptive field size. The
receptive field of a visual cell is the area of visual space where suitable activity
(movement, light, color, etc., depending on the processing area) will
7.2. Coding in space: ensemble codes 107
affect a response in that neuron. In higher visual processing areas, some
cells have enormous receptive fields and will respond to the entire hemifield
(everything off to the left, for example) or even the entire visual field. By
knowing the receptive fields of cells and looking at their responses, it is possible
to draw a picture in which activity in each cell represents a pixel. At
the photoreceptor level in the retina, this picture will be perfect (although
upside down), like the image on the film at the back of the camera. By the
time you get to ganglion cells in retina it will be a little more blurred and
so on up the processing pathways. In higher visual areas, the pixels would
overlap enormously and the picture would be blurred beyond recognition.
Similarly, different areas of the brain have skin maps for touch and movement
maps for motor activity. These maps can be superimposed on the
brain to indicate areas that are more or less heavily represented in cortex.
In the sensory realm, areas that are used for exploration by touch are heavily
represented. In a mouse, whiskers get the heaviest representation. In a
person, the hand, face, and tongue are most heavily represented. Fig. 7.1
shows a version of the humunculus, the distorted drawing of a human figure
according to the location of its representation on the cortex. The tongue,
face, and hand are big. The body and back of the head are small. The foot
is pretty big, too. In areas with large cortical representations, there are
many cells, and each cell will have a small receptive field, allowing it to
produce a very fine-grained map of that area. This can be tested by having
someone do two-point discrimination testing on your skin. This is done by
randomly poking either with two pins placed closed together, or with a
single pin. At some point, the two pins are so close together that you can’t
tell whether you’re being touched by two pins or by one. You will find that
you have much better two-point discrimination on your face or hand than
you do on your back.
The motor humunculus also devotes a lot of space to hands and lips and
relatively little to shoulders and foreheads. In general, left brain innervates
right body and right brain innervates left body in both the motor and sensory
systems (contralateral innervation). In the motor realm, areas that
are not heavily represented in cortex tend to be innervated by both sides
(bilateral innervation). This can be appreciated by comparing the brain’s
control of the forehead to its control of the mouth. Almost everyone can
move one side of the mouth without moving the other side. However, most
people cannot raise one eyebrow without raising the other eyebrow at least
a little. Because of the bilateral innervation, when your left cortex signals
a contraction of your right forehead muscles, it signals contraction of your
left forehead as well. Everyone’s brain is different. Some people have relatively
segregated facial innervation patterns and can raise one eyebrow
without difficulty. Also you can train your brain to increase the density of
representation for one area at the expense of another.
There are many other maps in the brain. Hearing has both frequency and
location maps. Hearing maps location by measuring differences in arrival
108 7. Neural Coding
head
hand
body
foot
tongue
Fig. 7.1: The humunculus — right cortex viewed in coronal
section.
time and intensity between the sounds received by the two ears. Bats and
dolphins, using echolocation, have much more sophisticated auditory maps
that involve both motor output and perception. Many aquatic animals,
including electric fish, sharks, and platypuses, have electric field maps and
can detect and apparently image electric fields.
Local vs. distributed ensemble coding
Ensemble codes can be classified as either local or distributed encoding
schemes. A local code is one in which an individual neuron can be identified
as representing a specific item (such an item could be an image, a sound,
a thought, or a word) out in the world. For example, some invertebrates
have command neurons that are triggered by particular stimuli and produce
7.2. Coding in space: ensemble codes 109
a stereotyped response. A local code is not necessarily one to one; there
can be many neurons, all of which represent the same location (or item or
thought or word). Local coding neurons are also called grandmother cells.
This phrase implies that there is a specific neuron whose activation would
indicate that grandma has been detected. If this cell was lost, one would
lose all knowledge of grandma. It does not appear that local coding is used
much in higher animals.
The alternative to a local code is a distributed code. In a distributed
code a bunch of neurons need to be coactive to represent item A. In this
respect, a distributed representation isn’t any different from a local representation
where each one of the bunch can represent A by itself. The trick
in a distributed representation is that we reuse the same units when we
are representing a bunch of items: A, B, C .... In a local representation,
any given neuron represents just one of these items. If you lose all of the
neurons that represent A, you no longer have any memory of A. In a distributed
representation, it is the pattern of activity that counts rather than
the activity in any single neuron. Different patterns, representing different
items, will involve some of the same neurons.
We generally think in terms of representations involving large numbers
of neurons. In a distributed encoding, the loss of a few neurons will degrade
the representation of several items. However, no single item will be
lost. As more and more neurons are lost, all of the representations will be
degraded more or less equally without losing any particular representation.
This property of distributed representation networks is known as graceful
degradation with damage. This is a valuable property and an advantage
over local representation networks, which are more prone to lose memories
if damaged.
A drawback to distributed coding is that there can be significant overlap
between representations of different items leading to confusion between
them. Although this is bad if you are trying to engineer reliable memory
systems, it is good if you are trying to model human memory, which is
notorious for its capacity for confusion. Overlap is also a potential advantage
where linking related items is beneficial, as in reasoning by example
or metaphor.
A technological example of a distributed representation is a hologram,
a three-dimensional picture made with lasers. The image in a hologram is
distributed, while the image in a photograph is local. In the photograph,
every item in the visual scene is represented at one spot on the photo. If
you cut out a piece of the photo, you lose an item. In a hologram, the
information about the scene is distributed across the image. As a result, it
needs to be decoded in order to be seen. However, if you cut a piece out
of a hologram, you do not lose individual items in the scene; rather, you
blur all of the items. Since information about each item is distributed, you
can’t cut out one item without removing information about other items as
well.
110 7. Neural Coding
7.3 Coding with volts and chemicals: neural state
code
In the previous chapter we introduced the notion of the state of a unit
or neuron. This hypothetical state comes from a top-down computer science
perspective, and represents a minimalist notion of neuron activity.
To reconcile this top-down notion with the bottom-up complexities of wet
biological measurements, we want to ask what we can measure in a real
neuron that corresponds to the state of the neuron.
The neuron is full of chemicals. Some are measurable by currently available
techniques; most are not. Nerve cells also have a large catalog of
electrical responses. A neuron is big — electrical potentials or chemical
concentrations in one spot will not generally correspond to their values
somewhere else in the neuron. Any and many of these potentials and concentrations
are likely to be involved in neural information processing. The
existence of so many possible neuron states makes it hard to see how
we can work within the context of the scalar-state single unit description
introduced in the previous chapter.
We return to the complex internal state of neuron when we consider the
neuron’s electrical and chemical infrastructure in more detail in Chap. 11.
For now, we simplify matters enormously by discussing only the output
of the neuron. Since this output is the only way that one neuron influences
another, it must represent a summary or synopsis of the complex
multidimensional state of the entire neuron.
Biologically, it is important to emphasize the distinction between the
neuron state, which is complex and high-dimensional, and the output state,
which is relatively simple and low-dimensional. This distinction is often
overlooked because there is no such difference in sum-and-squash units,
where unit state and output state are one and the same. By making this
state-output identity assumption for real neurons, we force the conclusion
that the neuron does not do major calculations internally. A simple neuron
can take only low-dimensional inputs, combine them, and produce a lowdimensional
output. On the other hand, if a neuron is running complex
programs on multiple internal states, it can take the input signals and do
substantial processing before producing another signal that is only distantly
related to the inputs. In the terms that I have previously used, the debate
is between the neuron as transistor and the neuron as CPU.
Having thrown most of the neuron out of our discussion, we can now
throw out the synapse as well. The synaptic mechanisms, very complex
as well, will be collapsed into a single dimensional weight. As we describe
later, this simplification collapses presynaptic, synaptic, and postsynaptic
mechanisms. Instead of worrying about all that, we assume, as before, that
the signal passes from one neuron to the next via a simple scalar weight. As
7.4. Coding in time: temporal and rate codes 111
with the oversimplification of the neuron state, I’m prepared to complain
about this, but I’m not really ready to do anything about it.
Having thrown away the neurons and the synapses, the only thing left
to discuss is the meaning of axonal spikes or action potentials. The spike
is the form of electrical activity responsible for long-distance signaling to
other neurons. The spike, or the pattern of spiking, can be taken to be the
output state of the neuron. As the single major output channel for many
neurons, this gives us a single place to measure state and a single thing to
measure. The action potential is a brief (about 1 ms) positive voltage that
travels as a wave down the axon to communicate via synapses with other
neurons. An action potential is a sudden upward deflection of membrane
voltage followed by an equally sudden return to the baseline — a brief
spike in voltage. Spikes are stereotyped; every spike looks pretty much like
every other. (Actually later spikes in a train tend to be smaller, but this
is not believed to make any difference in their ability to trigger synapses.)
This rules out two obvious possibilities for information transmission: spike
shape and spike size. Therefore, spike timing is the only attribute that is
thought to be important for information transmission.
7.4 Coding in time: temporal and rate codes
State in the sum-and-squash unit is instantaneous. The state value at any
time is dependent only on the previous step and independent of whatever
came before that. We can instead base state value on a train of spikes,
making unit state a reflection of that unit’s history. This is called temporal
coding, a code that utilizes sequences of interspike intervals. Because utilization
of temporal information bases the state values on what happened
in the past, it is a basic form of memory.
Temporal information is clearly used in human communication. To understand
speech, one has to process strings of phonemes (sounds) to
understand a word, and then process strings of words to arrive at a complete
idea. The fact that the brain is good at creating and understanding
speech demonstrates that temporal coding and decoding occur in the brain,
although not whether they are performed primarily at the neuron or at the
network level.
In signals and systems parlance, a signal processor is called a filter. Interpretation
of a temporal code requires that the filter store information
in some kind of memory to keep track of what came before. To interpret a
temporal code, it is necessary to chunk data, to group it. Consider Morse
code. Like the signal on an axon, an initial analysis of the telegraph signal
would suggest that it has two states: a high voltage and a low voltage. However,
since we know the code, we know that it is more accurate to regard
it as having at least three states: 0/dot/dash. To distinguish these three
112 7. Neural Coding
states we have to monitor activity on the line for a long enough period to
distinguish the dot from the dash.
Going further, we could consider the different letters of the alphabet as
distinct states (ignoring punctuation and special signs). By monitoring the
line for longer periods, we can distinguish 26 different states. Interpretation
of these 26 states requires that the filter store a series of high and low
voltage levels. This interpretation is made more difficult by the fact that the
different states are represented by signal sequences of differing durations.
If we didn’t already know Morse code, it would be difficult to extract
it from recordings made on a telegraph line. Usually Morse code is used
to transmit natural language. If natural language is being transmitted and
you know or suspect this to be the case, you could use statistical measures
to infer which symbol represented what letter. This is the domain
of cryptography. If the type of data being transmitted is completely unknown,
other techniques must be used. This is the domain of information
theory. To parse (divide up) an unknown code, information theory compares
various temporal integration times, using statistical measures to look
for recurring patterns. The information theoretical approach has been extensively
applied to spike trains of single neurons with some encouraging
results. However, it may not be possible to break the neuron code using
single neuron signals, if the code is dependent on coordinated firing across
multiple neurons. In that case, information theory would have to be applied
to multiple spike trains simultaneously, applying spatial as well as
temporal integration.
As in the case of Morse code, temporal code can generally be chunked in
various ways depending on the sophistication of the filter. Longer chunking
times require more sophisticated processing. To an extent, this will enhance
efficiency, but at the cost of slowing any responses to be based on the results
of the interpretation. Therefore, optimal chunking size will depend in part
on the requirements of sensorimotor integration.
Temporal integration
Temporal integration is a much simpler filtering algorithm that neurons
can readily perform. Integration is the calculus word for summation. The
mathematical symbol for summation is _; integration is _. Both are variations
on the letter S for summation. Temporal integration means adding
up all the signals that are received over a certain period. This is believed
to be a major form of filtering occurring at the single neuron level. Only
one number needs to be stored in memory, representing the signal sum at
that moment.
If we keep integrating incoming signals indefinitely, the sum that we are
storing just gets bigger and bigger. Therefore, we must define an integration
time τ (tau). At any time, the value of the temporally integrated signal is
the sum of all signals that have arrived during the last τ ms. Typically,
7.4. Coding in time: temporal and rate codes 113
we can imagine neurons as doing temporal integration by maintaining a
running average of activity. As signals come in they are added to this
average signal, but their influence cuts off (or more realistically wanes) after
period τ has passed. Note that temporal integration would not work for
Morse code. If we consider a dash to be twice as much signal as a dot and we
receive dash-dot (N), we could add up 2+1 = 3 as our temporally integrated
signal. However, dot-dash (A) would also give 3, as would dot-dot-dot (S).
Individual units in an artificial neural network don’t make use of temporal
information. Furthermore, the artificial neural network’s ensemble code
only uses the state of the network at the previous instant in time (for example,
t = 4 in Fig. 6.5) to determine network state at the next instant
(t = 5). Despite this, artificial neural networks can do temporal processing.
Events that happened in the past (t = 2, t = 3) percolate around a recurrent
network, providing implicit access to historical information. In this
way, artificial neural networks have been used to model trajectories that
unfold in time.
Clocking
In a typical artificial neural network implementation, updating is strictly
clocked: every unit is updated at the same time. This is called synchronous
updating. Artificial neural networks can also be designed to use asynchronous
updating: each unit updates itself on its own schedule. This
does not mean that clocking has been eliminated. Doing away with a central
clock requires that we delegate the clocking to the individual units.
Although units update at random times, they must remain grossly synchronized
in order to participate in the network. This generally requires
that units all have similar time constants. A network could be designed
with some units whose time constants are multiples of the time constants
of other units. These units would then update regularly after several cycles.
As we discuss in Chap. 11, real neurons do have a time constant and
can therefore do their own clocking. There are also various types of central
clocking apparent in the brain as well. The circadian (24-hour) rhythm is
the most obvious and best studied clock in the brain. The appearance of a
variety of frequencies in the electroencephalogram hints at the possibility
that there may be rapid clocking as well. Electroencephalogram frequencies
are grouped in ranges: delta, theta, alpha, beta, gamma (these are usually
written out like this, rather than abbreviated with the Greek letter). The
delta rhythms are the slowest, ranging from below 1 Hz (less than once per
second) to 3 Hz. These are seen mainly in deep sleep or deep coma. The
theta range is 4 to 7 Hz in humans (the same frequency names are used in
rats but the frequency ranges differ). Theta is mostly seen in drowsiness.
The alpha rhythm, 8 to 12 Hz, is a prominent occipital rhythm with eyes
closed that goes away when the eyes open. Beta, 13 to 26 Hz, and gamma,
27 Hz and above, are also seen in the awake state. High-frequency gamma
114 7. Neural Coding
rhythms don’t penetrate well through skull and scalp, so they can’t be seen
on the electroencephalogram. These high frequencies are better recorded on
the electrocorticogram, which uses electrodes placed directly on the brain.
The circadian rhythm is generated by identifiable cells in the suprachiasmatic
nucleus. By contrast, faster brain waves do not appear to be projected
from a central generator. They are instead an emergent property resulting
from the activity of large numbers of cortical neurons. Brain waves might
just be epiphenomenal, a consequence of activity but having no direct
functional relevance. On the other hand, it is possible that these waves,
emerging from one set of cells, are utilized by other cells as timing signals.
Simultaneously occurring waves at different frequencies could then be
playing different clocking roles in different brain areas.
If neurons in the brain are being clocked, neuron time constants and brain
wave frequencies can tell us something about how the clocking might work.
A spike lasts about 1 ms. Therefore, maximum spike rates are about 1 kHz
(1000 cycles per second). This would be the maximum rate that a clock
could run. Although this theoretical upper limit is probably unobtainable,
fast clocking at 100 to 200 Hz is possible in small brain regions. It has
been suggested that inhibitory interneurons, which have fast firing rates,
might provide such synchronization. Over larger regions, or perhaps over
the whole brain, gamma frequencies of 40 to 60 Hz have been suggested as
possible clocking signals. Synchronization of activity at these frequencies
across large areas of cortex has been found.
7.5 Frequency coding
The most widely accepted model of temporal coding in the central nervous
system assumes that temporal information is utilized through temporal
integration. As we saw in the Morse code example above, this means that
exactly when spikes are received will not matter — everything just gets
dumped into the lump sum. The only thing that matters is how many
spikes have arrived during the period (τ ), i.e., the rate of spike arrival.
This coding scheme is referred to as rate coding or frequency coding.
Frequency coding has several advantages when it comes to translating the
activity of real neurons into the nomenclature of artificial neural networks.
The primary advantage is that it translates the spike train into a scalar
state. A second advantage is that it gives us a way to express negative
states (Fig. 7.2). Artificial neural networks that use −1/1 as the range of
states are widely used in network memory models, as will be described
in Chap. 10. Spikes are always positive voltage deflections; there are no
negative spikes to use for representing −1. Using rate coding, we assume
that there is a natural spontaneous firing rate that represents a zero state
(Fig. 7.2 — 0). An increase from this spontaneous rate corresponds to a
7.5. Frequency coding 115
0
+
−
Fig. 7.2: Spontaneous rate represents a scalar state of 0. Increased
rate is positive state and decreased rate is a negative state. Spikes
are represented as short vertical lines atop the horizontal axons.
At bottom: negative state times negative weight equals positive
state.
positive state (Fig. 7.2 — +). A decrease from the spontaneous rate is a
negative state (Fig. 7.2 — −).
The baseline rate is produced by a balance between ongoing excitatory
inputs and ongoing inhibitory inputs. At the top of Fig. 7.3, I show excitatory
and inhibitory inputs producing the spontaneous rate in a follower
cell. There is a balance of excitation and inhibition, which favors excitation,
driving the follower neuron. I have illustrated the excitatory synapses as
little stubs and shown one inhibitory neuron (shaded). At the bottom of
Fig. 7.3, the rate of firing in the inhibitory cell has decreased (a negative
state). This negative state is transduced via a negative, inhibitory weight.
With the reduction in inhibitory input, the follower cell shows increased
firing. This demonstrates the rate coding of the following situation: negative
state times negative weight equals positive state. Similarly, it is easy
to see that both negative state (reduced rate) times positive weight (excitatory
input) produces negative state (reduced weight), and positive state
(increased rate) times negative weight (inhibitory input) produces negative
state (reduced rate).
A major appeal of the rate coding model is that it gives us the simplicity
and modeling tractability of a scalar state. However, there are neurobiological
as well as computational reasons to believe that rate coding is important
in the nervous system. There is clearly a frequency/strength relationship
in the efferent (motor) peripheral nervous system. Increased rate of firing
of a nerve that controls a muscle will lead to increased contraction strength
in that muscle. You can electrically stimulate a nerve from outside the skin
116 7. Neural Coding
Fig. 7.3: Spontaneous rate is produced by a balance of excitatory
inputs (line segment synapses) and a negative input (small circle
synapse, shaded cell body). Bottom: A negative state (reduced
rate) in the inhibitory neuron times a negative weight (inhibitory
synapse) produces a positive state (increased rate) in the follower
neuron.
and make its muscle contract harder as you increase the current, pushing
the firing rate higher.
There is also substantial evidence for stimulus specific increases in firing
rate in the afferent (sensory) central nervous system. In their studies of
occipital cortex, David Hubel and Torsten Weisel showed large increases in
the neuron firing rate in response to oriented line segments in the visual field
of a cat. Recently, Newsome and colleagues performed some remarkable
experiments in which they altered an animal’s visual perception by injecting
current into the brain. They studied area MT, a visual area with neurons
that respond to motion. They identified sets of neurons that fired when
most of the randomly moving spots on a display screen went from left to
right. They then showed the animal a display in which most of the spots
went from right to left. If they injected current so as to increase the firing
rate in the previously recorded left → right cells, they could deceive the
animal into thinking that the spots were moving left to right as before.
Paranoid schizophrenics sometimes complain that something (the CIA, the
TV networks, or a dog) is projecting thoughts into their mind. (Which of
these three will be the first to get their hands on the new thought-insertion
technology?)
In addition to the motor and sensory correlates, rate coding is suggested
by the fact that many cells appear to have spontaneous firing, meaning that
they fire even when no stimulus is present. Of course, unexplained firing
7.5. Frequency coding 117
500 ms
40 mV
Fig. 7.4: Trains of regular firing and bursting neurons from
thalamic cell simulations.
does not have to be spontaneous or meaningless. Instead, irregular firing
may be coding for things that are not directly connected to stimulation,
such as thought or memory access. We ask an experimental subject, or
provide rewards for an animal, to just pay attention to certain stimuli and
not think about anything else. I tried this in yoga; my mind didn’t want
to cooperate.
On the other hand, there are also neurobiological reasons to believe that
rate coding may not be the whole story. Many neurons fire bursts. In that
case one can define either a rate of spike firing within the burst or a rate
of bursting — there is no single scalar state for rate. Fig. 7.4 shows four
traces of simulated thalamic cells. The top two traces show fairly regular
firing at about 1.5 and 6 Hz (spikes per second), respectively. The third
trace shows bursting. The rate of bursting is about 2 Hz, but the spike
rate within the burst is much higher. The bottom trace is a complex mix
of bursting and regular firing with varying rates. Blowing up one of the
bursts in the trace at bottom demonstrates that it is not simply a burst of
spikes but is actually a burst of bursts. The bursts in the third trace are
similar; they are also bursts of bursts.
The top trace of Fig. 7.4 also illustrates that even “regular” neuron firing
is often somewhat irregular. In an experiment, irregular firing is typically
attributed to noise. As in Fig. 7.4, however, such irregularity is also seen
in a deterministic dynamical system. In both biology and simulation, apparent
noise may be evidence of deterministic chaos. Chaos is a relatively
118 7. Neural Coding
new field of mathematics that has exposed the tendency of nonlinear systems
to act weird. More precisely, chaotic systems appear to show random
behavior despite following precise mathematical rules (deterministic system).
Examination of chaotic behavior has shown that there is some slight
predictability within the chaos. Chaotic systems show sensitivity to initial
conditions. This means that two identical systems, started at slightly different
points in state space, will show completely different behaviors. This
is the source of the hypothetical “butterfly effect” — a butterfly flapping in
China can change the weather a year later in South America. Additionally,
a chaotic system will never revisit a point in state space; it’s always going
somewhere new. Despite all this wildness, chaotic systems do manage to
stay on strange attractors, distinct volumes of state space. If you use a
computer to graph the trajectory of a chaotic system gradually in time,
the scribbles will eventually fill in an area. This is the strange attractor.
Some researchers have suggested that chaos might be used by the nervous
system to store information. In this case, complex patterns of seemingly
random neuron firing would fit on one of many strange attractors. These
attractors would be mapped by following network state in n dimensions,
where n is the number of neurons.
A simpler source of firing rate changeability is adaptation. Neurons will
typically adapt to a constant input with a gradual reduction in firing rate.
Because of this, a constant stimulus is not precisely associated with a single
rate code. This does not necessarily contradict the rate coding model
because adapting neurons could underlie the psychophysical phenomenon
of habituation, the reduction in perceived stimulus strength with continued
exposure. It does illustrate that coding will not be a simple matter of
defining values that imply particular stimuli.
Problems in defining firing rate also arise when neurons fire rarely. Some
neurons may fire only a single spike when they fire and therefore have no
spontaneous rate. This kind of firing pattern is reminiscent of the all-ornone
binary coding originally hypothesized by McCulloch and Pitts. Some
neurons don’t spike at all; clearly these neurons are not candidates for any
kind of spike coding. These may be neurons that don’t have the channel
density needed to produce action potentials. Such neurons could communicate
signals through neurotransmitter release triggered by subthreshold
potentials, voltages that are too small to produce firing.
7.6 Summary and thoughts
In this chapter I have emphasized the most widely accepted coding models:
local and distributed ensemble coding, and rate coding. I have also taken
the opportunity to cast doubt on these models by pointing out how the
many complexities of real neurons don’t fit neatly into these simplified
7.6. Summary and thoughts 119
theories. In the following chapters, I try to stick with the basic theories
and hold my skepticism in check.
8
Our Friend the Limulus
8.1 Why learn this?
Artificial neural networks have primarily made their mark in engineering
and cognitive science. We look at one of the basic cognitive models in a
later chapter. The success of neural networks in these nonbiological domains
raised great hopes that artificial neural networks had captured the design
of the brain. A grand notion, hard to live up to, that unfortunately led
to artificial neural networks being regarded by many neurobiologists as all
bluster and no substance, at least with regard to biology.
In this chapter and the next, I present two applications of artificial neural
networks in biology. The example in this chapter describes the mechanisms
of vision in the eye of the horseshoe crab. This work was the research
of H.K. Hartline and Floyd Ratliff in the 1940s and 1950s. The horseshoe
crab, often called by its scientific name, limulus, has been around for a while
(about 500 million years) and has changed little over these thousands of
millennia. Its visual system is of course very simple compared to our own.
Despite this, there are aspects of human visual performance that can be
explained through our understanding of the limulus eye.
Because this chapter represents the first real modeling in the book,
I present here some general computer modeling concepts. These include
simplification, scaling, sampling, linearity, parameterization, and iteration.
122 8. Our Friend the Limulus
8.2 The biology
Sensation starts with a process of signal transduction — where a physical
signal from the environment is “led through” (the meaning of transduction)
a transforming step so that it can be processed by the nervous system. More
generally, this is only the first of a series of transductions as the signal is
repeatedly changed from one form to another. In visual transduction, for
example, light energy is converted into a chemical signal. This chemical
signal is in turn converted to an electrical signal, which is converted to
a chemical signal, which is converted to an electrical signal, and so on
through chains of synaptically connected cells. At each stage, we can speak
of a physical process that is encoding the original signal.
As we saw in Chap. 7, Section 7.5, rate coding provides one way to represent
a single scalar value in a neural signal. In the case of vision, an
important stimulus variable to be encoded is light intensity (signal amplitude).
Another major aspect of the visual scene that must be encoded
is spatial pattern (signal locations). These spatial locations are directly
present as locations in the retina and transferred, in higher animals, to
locations in cortex. This is called labeled-line coding since the meaning of
the message does not depend on the form or timing of the signal but rather
on the identity of the transmission line where the signal is received. For
example, in an old-time telegraph office in Denver, the identity of the line
would tell the operator whether the message came from Albuquerque or
San Francisco, even before any decoding took place.
The horseshoe crab has compound eyes—lots of little eyes work together
to make one big eye. One can think of the axons projecting from these as
so many labeled lines, each coming out of a separate eye segment. Each
of these little eye segments has its own cornea and photoreceptor. The
cornea collects light from a certain area of visual space, the photoreceptor
measures the amount of a light, and a neuron transmits this information
to a central visual processing station.
8.3 What we can ignore
The art of modeling consists of deciding what aspects of the object of interest
have to be included and what aspects can be ignored. This transforms
a real physical object into a simulacrum, which may be another physical
object, a set of mathematical equations, or a computer simulation. Generally,
you model something when it is too complex to understand by just
thinking about it. To do this, you must first simplify, deciding what to include
and what to omit. This winnowing process is the art of modeling. As
pointed out in Chap. 2, Section 2.3, reference to birds during development
of heavier-than-air flight proved both helpful and misleading.
8.4. Why the eye lies: the problem 123
cornea
light
Retinular cell Eccentric cell
lateral inhibition
Axons
Fig. 8.1: Ommatidium, illustrating lateral inhibition plus all the
stuff to be ignored.
As with most biological systems, the complexity of the limulus eye is
daunting. Each little photoreceptor complex, called an ommatidium (plural
ommatidia), consists of two cells (Fig. 8.1). The retinular cell contains
the photoreceptor chemical, rhodopsin, which changes shape when hit by
a photon of light. This chemical process, and the subsequent process of
transduction that changes the chemical change into an electrical signal on
the membrane of the cell, is itself complicated. The signal is then transferred
to the eccentric cell and from there out the axons to central processing
areas. From there, cells take the signal and transmit it to the side where
the signal reduces the output from other cells. This process is known as
lateral inhibition.
Any and all aspects of this signal handling might turn out to be critical
for understanding the process of vision in the limulus. We will see that
lateral inhibition is of particular interest. It will turn out that we can
strip away all of this complexity and just assume that the system somehow
takes light of a certain intensity and transforms it to an electrical signals
coding that same intensity. Simpler still, we’ll just represent light intensity
(luminance) as a number, and then represent the cell signal as the same
number.
8.4 Why the eye lies: the problem
Our long-distance senses use light, sound, or chemicals to transmit information
from the physical world around us. Each of these has its own properties
124 8. Our Friend the Limulus
that can produce illusions. For example, when we look up to the sky, a seemingly
real blue dome overhead, we are just seeing blue light from the sun
highly distorted. Even after we get through these physical distortions, our
information about the world passes through receptors and then through
the nerves that connect to these receptors. All of these intermediaries can
introduce further errors and distortions.
In an ideal world, it might be nice to have direct access to reality. It
is sometimes hard to appreciate that we do not, that we are only seeing
those platonic shadows on the cave wall. When Hermann von Helmholtz,
the 19th century pioneer of neurophysiology and medicine, discovered that
nerves conduct information at a finite rate, his father thought the idea
ridiculous. It is quite obvious, his father said, that we perceive the world
immediately and directly without delay. Actually, a variety of physical and
neural intermediaries create both delays and distortions. Some of these distortions
can be left alone, for example, the blue sky. Some distortions can
be useful — delay in sound propagation through air is used for sound localization
by perceiving the difference in arrival times in the two ears. Other
distortions must be fixed in order for the animal to compete and survive.
The limulus has been hanging around since long before the dinosaurs. It
is a survivor. One of the roles of the limulus eye and retina is to correct
receptor errors, sweep away illusions, and see life as it really is.
For the limulus, the world out there is an image of light and dark. In
an ideal situation of minimal distortion, the activity across the limulus eye
(or the back of a camera or the human retina) would accurately match
the image: brightness (the perception of strong light) would exactly match
luminance (the physical presence of strong light). In places where the image
reflects a lot of light (high luminance), the corresponding location on the
eye would have a highly active photoreceptor (appearing bright). Where the
image is dark, it would have a very inactive photoreceptor. However, two
physical factors get in the way (Fig. 8.2). One is scattering. As light passes
through water or air from the image to the eye, it spreads out, blurring the
representation of the image on the photoreceptors, just as blue light from
the sun is blurred out all over the sky. In Fig. 8.2, scattering of light is
seen affecting receptors a and c. Receptor a is positioned midway between
light and dark bands but receives more than half the maximum light due
to scatter. Receptor c is positioned over an area of darkness, but scatter
causes it to register the presence of some light.
The other problem is sampling. Even in the absence of scatter, accurate
detection of a sharp change in luminance requires that two receptors neatly
flank the contrast boundary, as do receptors b and c. If, as in the case of
a, a receptor sits in the middle, it will sample both the dark and the light
side and get an intermediate amount of light. This gives the perception
of an intermediate brightness, suggesting a gradual drop-off rather than
a sharp contrast boundary. Additionally, photoreceptor misalignment will
also add to the overlap between portions of the image received at different
8.4. Why the eye lies: the problem 125
bc
Image Luminance Brightness
Receptors a b c
a
cornea
axon
photo−
receptor
Fig. 8.2: Photoreceptor placement and light scatter turn the
sharp black and white image into a blurred image. Image at top
is depicted with a brightness graph, which is sampled by receptors
schematically made up of cornea, photoreceptor, and axon.
The resulting image can be reconstructed by drawing a smooth
line between sampling points. Lower left: simplified schematic of
receptor.
receptors. Although redundant sampling ensures that nothing is missed,
misalignment may cause it to increase blurring still further.
Nature has designed neural processing to compensate for the blurring due
to scattering and sampling. The solution is lateral inhibition. This solution
is used not only by limulus, but also by us and by most of the in-between
creatures of the great chain of being. Fuzziness is most troublesome in cases
where the original image is least fuzzy—at a sharp contrast boundary from
black to white, where contrast is maximal (Fig. 8.2). On one side of the
contrast boundary, the image is black; little light emerges from the surface.
126 8. Our Friend the Limulus
On the other side the image is white: maximum, saturating, light reflects
from the surface. If neural activation perfectly reflected this contrast, the
region facing the black part would be completely inactive, while the region
facing the white part would be maximally active. A major purpose of lateral
inhibition in an eye is to find this boundary location by deblurring.
As we will see, lateral inhibition restores and actually enhances contrast
boundaries. When a receptor detects light and transmits that information,
lateral inhibition reduces the strength of signals transmitted by adjacent
receptors. The amount of signal reduction on the flanks is proportional to
the strength of the signal at the center. This process allows the receptors
flanking the contrast boundary to reconstruct that boundary and determine
boundary location as precisely as possible.
8.5 Design issues
To construct a neural network, we start with the biology and then abstract
from it to make things as simple as possible. In doing so we typically make
a large number of compromises in order to make the model tractable. These
compromises reach into all aspects of modeling: translation from biology,
calculation, understanding the model, and translation back to biology.
In this section I describe some common modeling compromises: linearity,
size and dimensional reduction, wraparound, simple parameterizations, and
scaling. In the present case, the biggest compromises come from the translation
of biology into the model. We substitute a simple sum-and-squash unit
to represent a complex signal transduction process that is actually carried
out by a set of cells each utilizing multiple internal processes. The use of a
sum-and-squash unit also means that we assume that signals sum linearly.
This important approximation simplifies calculation and interpretation. To
further simplify calculation, we use scaling and dimensional reduction for
the network. To compensate for problems with scaling, we use wraparound.
We also use simple images that are easily parameterized instead of real pictures
of limulus predators. Finally, for the individual units, we use a scaling
factor that makes it easy to translate back to biology. As you can see, we
end up with interlocking compromises that make it hard to put together a
single list of assumptions that can be readily assessed, and then accepted
or rejected.
Making the model small — scaling
A common modeling simplification has to do with size. Neural systems
are compact but pack in a lot of neurons and synapses. Certainly, the
sheer number of units is a major factor in the brain’s extraordinary
information-processing abilities. In performing simulations, we cannot ef8.5.
Design issues 127
fectively emulate the large number of neurons in a given brain structure
such as the retina. We generally use two simplifications to reduce the computational
load: decrease the number of dimensions and reduce the absolute
number of neurons (scaling).
Scaling is pretty common in modeling of all sorts, hence the term scale
model. A sometimes overlooked aspect of the scaling process is the assumption
of linearity. A linear transformation preserves scale. If the wingspan
of an airplane is twice the length of its fuselage, and you create a scale
model, the wingspan will still be twice the length. By contrast, a nonlinear
transformation would not preserve ratios. You may have heard that, aerodynamically
speaking, bumblebees can’t fly. The confusion results from
the fact that a large scale model of a bumblebee would not be able to
fly because flight forces do not scale linearly. As pointed out in Chap. 2,
flapping does you less-and-less good once you get past the size of a duck.
Despite the fact that there’s actually not much linearity in biology, we often
make the assumption of linearity in modeling. Although it doesn’t work for
bumblebees, this mistaken assumption often works out well enough to get
something useful out of a model.
Linearity is of course the major feature of the linear algebra used in neural
networks. As described in Chap. 6, sum-and-squash units have both a
linear step (the sum) and a nonlinear step (the squash). The linear summation
step allows any number of excitatory or inhibitory inputs to be
added in together without distortion. In the limulus model, linearity offers
a major advantage in quickly illustrating the core issues we want to study.
However, any aspects of limulus eye function that depend on its nonlinearities
will not be represented in the model. This is a major compromise since
sensory systems generally show a highly nonlinear response with increasing
sensory stimulus strength. In many cases, this nonlinearity is logarithmic,
meaning that any doubling of stimulus intensity produces the same-size
increase in the perceived stimulus. This sensory response property is called
the Weber-Fechner law, which explains why light/dark contrasts look the
same whether viewed on a bright or cloudy day.
Making the model small — dimensional reduction
The retina is a sheet of photoreceptors and other nerve cells that lines the
back of the eye in the mammalian eyeball and lies right near the front in
limulus. As a sheet, it is a two-dimensional structure, which we could model
as a circle or square of interacting units. However, we can make things
much simpler and smaller by modeling it as a one-dimensional structure (a
line segment) instead of as a two-dimensional structure. Such a reduction of
dimensionality is a typical strategy in modeling that saves a lot of computer
time and space; compared to modeling a square, we need far fewer units.
The limulus has about 8000 ommatidia arranged in approximately a 6 by
12 mm ellipse. Modeling a line across the retina instead of a square retina
128 8. Our Friend the Limulus
a b c d e f g h
Constant Luminance
Luminance Brightness
peak peak
Fig. 8.3: Lateral connectivity of the limulus retinal network. Most
units receive a total of four projections: one from each of two
units on either side. However, unit a receives two, unit h two,
unit b three, and unit g three projections. Because they are on or
near the edge, these units behave differently. As a result there is
an illusion of brightening at the edge as perception differs from
luminance. This is an edge effect.
reduces the problem to √8000, which is about 90 units. Thus one main
effect here is to make the problem much smaller and therefore allow it to
be done with much less computer space and computer time. Working in
one dimension also makes it easier to set up the connectivity matrix and
easier to understand the results.
Eliminating edge effects — wraparound
In a two-dimensional network like the retina, there will be some neurons
that are situated far out at the periphery, at the rim or edge. Neurons of
the retina have a geometry, being connected preferentially to neighboring
neurons. Edge neurons will have a different connectivity, since they have
neighbors only on one side. This difference in connectivity will lead them
to respond differently to a stimulus. This difference in response will in turn
influence the activity of the neurons that they connect to, leading these
neurons to also behave a little differently. In this way, edge effects will
propagate a little way in toward the center of the network. Note that the
8.5. Design issues 129
a
e
d
f
g
c
b
h
Fig. 8.4: Wrapping the network around so that one end connects
to the other eliminates edges and thereby eliminates edge effects.
Compared to Fig. 8.3, units a and h have been wrapped around
and connected in a circle.
“edge” of edge effects refers to the edge of the retina or retina model. Above
we have been discussing contrast boundaries, which also can be called edges,
image edges between a light area and a dark area. Edge effects are probably
not much of a problem in the real retina. The vast size of the network means
that proportionally few cells are near the edge. However, edge effects do
become a problem in the model.
In a model network, the much smaller number of neurons accentuates the
edge effects. A model network will be simpler and easier to understand if it
is uniformly connected, with all units having the same number and types
of connections. Instead, Fig. 8.3 shows units at or near the edge with fewer
connections. In the limulus model, connections are all inhibitory. If the edge
units do not get as many connections, they will not get as much inhibition
and therefore will be turned on more than the other units. This will give an
illusory appearance of brightness at the edge of the image. The brightness
curve (shown smoothed out in Fig. 8.3 bottom) does not match the square
luminance curve above it. This is an example of perceptual enhancement
at a boundary. Here the boundary is not a contrast boundary out in the
world, but a histological boundary in the eye. Perceptually, an edge-effect
illusion would look like a segment of halo around any lighted object whose
image went off of the edge of the eye. If you looked at a wall or other
uniform surface, it would appear to have a brightening at the edge. The
brain presumably has mechanisms to get rid of edge-effect illusions.
In the model, we avoid the problem of edge effects by employing a common
trick called wraparound. This eliminates the edges by simply turning
the network into a circle so that unit a connects to unit h (Fig. 8.4). If we
were doing the same thing with a two-dimensional model, the wraparound
would give us a torus (also known as a donut). From the surface of a torus
130 8. Our Friend the Limulus
10 20 30 40 50
20
30
40
10
50
a=0 b=10
a=0.5 b=10
a=0.5 b=20
a=0.5 b=30
a=0.5 b=40
a=0.7 b=40
a=−0.5 b=10
x
f (x)
Fig. 8.5: Various parameter choices for a line: f(x) = a ・ x + b.
you can go around in a circle in one of two ways: through the hole or around
the entire donut.
Presenting the input — parameterization
Like scaling, parameterizing is one of the basic tasks in mathematical
or computer model construction. Modeling involves translation. Different
types of models require translation into different types of language: pictorial
models, metaphorical models, physical models. In mathematical (or
computer) modeling, we draw with equations. The art is to find the right
equation and to then parameterize that equation so that it has the right
form and size for the model. In general, parameterization refers to the
choice of constants that give a particular class of equation a particular
appearance. The form of an equation determines the general shape of the
curve, while the parameters define its specific location and appearance. For
example, the form of the function f(x) = a ・x+b, defines it as a line. (f(x)
is read “f of x” and is short for “function of x.”) a and b are the parameters.
a gives the slope of the line and b, the y-intercept, shifts the line up and
down (Fig. 8.5).
Things that are easy to draw pictures of are not always easy to parameterize
(e.g., natural scenes) and vice versa (e.g., things in four or more
dimensions). An important step in using the simulator will be to draw inputs,
images, to present. The image will be presented as an input vector
_p (p for picture) and therefore could be any arbitrary image. Nonetheless,
8.5. Design issues 131
a
b
x1 x2 N
Fig. 8.6: Sample pictures (_p).
it will be convenient to parameterize the input in order to produce images
easily without selecting individual values for tens or hundreds of inputs.
Since we are interested in how the eye handles contrasts, we want to
use functions with high contrast — big jumps from high to low and low
to high. The highest contrast is provided by a discontinuous function that
just pops from low to high at a specified point. A discontinuous function
cannot be described in a single functional form. Instead, it is made up
of line segments and described as a piece-wise linear function. A single
piece-wise linear function is written as three linear equations over three
different domains: f(x) = a (0 ≤ x < x1) ; f(x) = b (x1 ≤ x < x2) ; and
f(x) = a (x2 ≤ x < N) (Fig. 8.6 dashed line). The four picture parameters
are x1, x2, a, and b. N is a parameter of the network: the number of
units. This piece-wise linear function describes two contrast boundaries at
locations x1 and x2 jumping between levels a and b.
We also want to see how the eye handles blurring. We use a blurred
function that gives a flat top but sloping shoulder drop-off (Fig. 8.6, solid
lines). These lumps (not a technical term) are formed by raising a Gaussian
function (the familiar bell curve: f(x) = ex2) to an even power — the higher
the power, the broader the lump and the steeper the sides. However, shifting
and scaling our lump equation to get it to sit neatly on top of our little
square step function requires a bunch of parameters: f(x) = a + (b − a) ・
e{−(x−N/2)
w }2n, where a, b are again the vertical limits, w = x1−x2
2 is the halfwidth
of the lump, and n determines how flat the top is (larger n, broader
top). Although both the step function and lump function are defined for the
entire domain from 0 to N, we are only sampling the functions at discrete
points corresponding to the location of our photoreceptors (vertical hash
marks on x-axis in Fig. 8.6). The values at these points will be placed in
the input vector _p.
Because we are using wraparound, the leftmost part of the network (unit
0) is next to the rightmost part (unit number N). Therefore, it is convenient
132 8. Our Friend the Limulus
to have the value of our functions be the same at receptor 0 and receptor
N. This is why the step function in Fig. 8.6 has two steps instead of one.
Alternatively, we could have just defined one step and left the other step
as a discontinuity between receptors N and 0.
Parameterizing the activation function
As mentioned in Chap. 6, the activation or squashing function of a sumand-
squash unit usually uses a range of either 0 to 1 or −1 to 1. These values
then need to be scaled in order to relate them to biological measurements.
In the limulus model we assume rate coding and simply use the values from
actual limulus firing rates to make it easy to relate our results to limulus
experiments. By setting the model to suitable numbers at the onset, we
don’t have to do re-scaling for interpretation. Axons projecting from the
limulus ommatidium spike at rates of 10 to 50 Hz. Therefore, we use a
squashing function in our model that utilizes a range of 10 to 50.
We have chosen numbers for neuron state that relate to the biology. We
also need to choose input values for our image — the a and b of Fig. 8.6.
We are not concerned here about absolute light levels, so we won’t bother
with correctly defining the image intensity using units of lumens. Instead,
we simplify our arithmetic by just using the same range of numbers for
light intensity as we are using for neural state. This is highly artificial but
reflects the fact that we are interested not in modeling the physics of light
but only in modeling neural responses. Setting input range equal to state
range simplifies signal transduction. The transfer from input light to neural
response becomes a multiplication by 1 (identity function).
We use a piece-wise linear function for the squashing function. The simplest
procedure is to just let the value alone if it is in the appropriate
range and to move it to the nearest limit if it’s outside of that range:
f(x) = 0 (x < 10) ; f(x) = x (10 ≤ x ≤ 50) ; f(x) = 50 (x > 50)
(Fig. 8.7).
Parameterizing the weight matrix
It is also convenient to parameterize the weight matrix W so that we can
easily make minor changes without going through and setting each of N2
weights by hand. The weight matrix is going to be completely regular; the
connections from each unit to its neighbors is the same since we’re using
wraparound. It will be easiest to just parameterize a single column of the
weight matrix, a divergence weight vector for a single unit, and use this
to create all the columns of the weight matrix. Each column will have
the same weights shifted to correspond to the location of the postsynaptic
unit. Since this is a purely inhibitory network, the numbers will all be
negative. A graph of the divergence vector of a particular receptor reveals
how much that receptor inhibits itself and other receptors surrounding
8.5. Design issues 133
10
50
10 input 50
spike rate
Fig. 8.7: Limulus model squashing function.
(u−1,−s)
−s
−r
u
0 N
Fig. 8.8: Spatial decay of weight: exponential (solid line) and
linear (dashed line) functions
it. Nearest neighbors are inhibited most, those farther away are inhibited
progressively less. Two simple choices for fall-off functions are linear fall-off
and exponential fall-off (Fig. 8.8).
Again, the equations are basically very simple. However, we would need
to add a bunch of parameters to make it easy to scale the inhibitory drop-off
for different network sizes and then to place the drop-off correctly in each
row of the matrix. Rather than defining the whole kit-and-kaboodle in a
series of equations, as we did with the picture functions above, we can easily
do the needed mirroring and shifting algorithmically, using the computer.
Referring to the weight curves shown in Fig. 8.8, we see that the fall-off is
symmetrical on either side of a given unit u. Therefore, we just define one
134 8. Our Friend the Limulus
of these drop-off curves. We can easily parameterize a line segment running
between the points (0,0) and (u − 1, −s): w(x) = −r − s−r
u−1 ・ x. We can
use values from this equation to set values for the top half of the central
column of the weight matrix, corresponding to the divergence from unit u
at the center to units 0 through u − 1. The following shows the resulting
weight matrix for a 10-unit network with s = 0.99 and r = 0.95. Subsequent
simulations were run with an 80-unit network. The 80 × 80 matrix used is
too big to show.
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0 −.99 −.98 −.97 −.96 −.95 −.96 −.97 −.98 −.99
−.99 0 −.99 −.98 −.97 −.96 −.95 −.96 −.97 −.98
−.98 −.99 0 −.99 −.98 −.97 −.96 −.95 −.96 −.97
−.97 −.98 −.99 0 −.99 −.98 −.97 −.96 −.95 −.96
−.96 −.97 −.98 −.99 0 −.99 −.98 −.97 −.96 −.95
−.95 −.96 −.97 −.98 −.99 0 −.99 −.98 −.97 −.96
−.96 −.95 −.96 −.97 −.98 −.99 0 −.99 −.98 −.97
−.97 −.96 −.95 −.96 −.97 −.98 −.99 0 −.99 −.98
−.98 −.97 −.96 −.95 −.96 −.97 −.98 −.99 0 −.99
−.99 −.98 −.97 −.96 −.95 −.96 −.97 −.98 −.99 0
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
In each row, Wuu = 0, meaning that there is no self-connectivity. The
most negative values (strongest inhibition) flank the zeros. This is the
nearest-neighbor connectivity. From there, inhibition falls off linearly in
both directions until the furthest unit is reached with a −.95 strength inhibitory
weight projection. Each row (or each column) is identical to the
prior one except that it is rotated by one place. Because of the wraparound,
a number that falls off the right side of a row with row-to-row rotation shows
up back on the left side. A number that falls off the bottom of a column
with column-to-column rotation shows up back at the top.
A similar equation for exponential drop-off with distance x is w(x) =
−s・e−x−(u−1)
λ . The two parameters are maximal inhibition −s and a spatial
length constant λ, which indicates how far the inhibition will reach. Bigger
λ means that the inhibitory influence will extend further out. Here again,
wraparound is most easily done algorithmically by appropriate placing,
mirroring, and copying.
8.6 The limulus equation
Now that we’ve done all that scaling, redimensioning, and parameterizing,
we can actually put together the limulus equation itself. We need to make
only a small change in the basic sum-and-squash unit update equation
introduced in Chap. 6. This change is needed to account for the image
input. We have scaled the input, representing luminance, to match the
range of values used by the units to represent firing rate. Because of this,
8.7. State calculation 135
the input can be simply added in with the interneuronal connections as a
set of values all with weights set at 1. The limulus equation is
_s = σ (_p +W ・ _s)
where _p is a picture (image) vector (Fig. 8.6). As usual, _s is the state vector,
W is the weight matrix, and σ is the squashing function (Fig. 8.7).
Explicitly including the image vector _p in the equation provides ongoing
drive from the image to the system. This represents the fact that the input
(light falling on the photoreceptor) is continuous during the processing.
Rather than adding in the _p vector explicitly, it could be handled as an
external input represented as a separate layer of fixed (clamped) “units”
with feedforward connectivity projecting one to one onto the processing
layer.
To examine the meaning of an equation and get some insight into what
it means, it’s helpful to simplify the equation in various ways by setting
parts of it to zero and looking at what’s left. For example, if we set the
weight matrix to 0 (no connection between units), then _s = σ(_p). Because
σ is the identity function (σ(x) = 1 ・ x) in the 10 to 50 range, σ(_p) = _p.
This means that in the absence of lateral connectivity, the input vector
is simply mapped directly onto the unit state vector. This would be the
perfect solution in the absence of scattering and sampling error. If, on the
other hand, we suddenly turn off the lights, setting the input (_p values) to
numbers ≤10, the negative weight matrix will drive the states back down
to their minimum, 10, regardless of their starting point.
We scaled all the numbers to correspond to the range of rate coding
actually seen in the limulus neurons, namely 10 to 50 Hz. By using the
same values for both the input vector and the state vector, we can simply
add the input vector to the state vector at every time point simulated. To
start any system simulation, we need an initial condition, or starting point,
for the state vector. We would typically start the system with the image
already present on the retina by setting _s = _p. Alternatively, we could start
the system in darkness by setting _s = 0 — the zero here is actually a vector
of zeros: _000 ・ ・ ・ 0_ — and then flip the lights on suddenly by setting _s = _p
at time t = t1.
8.7 State calculation
The core operation of the limulus eye lateral inhibition model is the determination
of the values of each unit at each discrete time interval. This is
the process of simulating the limulus equation. In Fig. 8.9 we start with
no image on the retina so that at time t = 0 activity is zero throughout.
Each of the circles along the x-axis represents the state of one of the 80
photoreceptors along the line of retina. When the image is presented at
136 8. Our Friend the Limulus
0
1
1
2
2
3
retinal location
3
activity
Fig. 8.9: Limulus equation simulation. Sequential time steps are
shown by numbers 0–3 and by symbols: circle, triangle, square,
plus, ... Square wave image presented at t = 1. Mach bands
indicated by arrows.
t = 1, the states jump up to match the values of the image (triangles).
Note that there is no inhibition yet since the state vector at the previous
time step was uniformly zero so that all the weights were multiplied by 0 to
obtain the input for t = 1. At t = 2, however, inhibition kicks in, resulting
in a lowering of all values (squares) and the appearance of an exaggeration
of contrast at the contrast boundaries (small arrows). The difference in
activity between the receptors at the contrast boundaries is greater than
the activity difference between receptors elsewhere. Contrast enhancement
is a characteristic finding in all lateral inhibitory networks. In the visual
domain, this enhancement is referred to as Mach bands.
As we continue the simulation for a few more steps, the values converge
toward a final target (Fig. 8.9 — solid line under all of the piled-up symbols).
At the steps beyond t = 1, the values alternate from being above or
below this line, reflecting lesser or greater inhibition from the slightly lower
or higher values on the previous step. Mach bands are present at all steps
after t = 1.
In Fig. 8.10, we have started with a blurred image. The Mach bands
here are less pronounced — we do not succeed in completely reversing the
8.8. Life as a limulus 137
0
1
2
2
3 ...
retinal location
activity
Fig. 8.10: Blurred image presented at t = 1. Mach bands form but
are less pronounced.
blurring. If we play with the width of the inhibitory surround by changing
the fall-off in the weight matrix parameterization function, we can match
the width of the blurring and get improved contrast enhancement. However,
such a specially designed inhibitory filter would not be optimal for greater
or lesser blurring, as might occur when the limulus is in extra dirty or
extra clean water. To detect how much enhancement is needed and alter
the inhibition accordingly, we would need an adaptive filter. This brings us
to the topic of learning algorithms, which we take up in the next chapter.
8.8 Life as a limulus
People often ask me, “What’s it like to be a limulus?” This is the philosophical
problem of qualia. No matter how much we explain scientifically,
how can we know what experience is like for another person or another
creature? Another way of stating this is as the problem of subjectivity: Do
others perceive things the same way that I do?
One could argue, heck, I will argue, that if my neural responses to a
particular stimulus are organized similarly to those of other persons, or
animals, I will see things in the same way that they see things. Because
of lateral inhibition, the limulus model demonstrates Mach bands. Inso138
8. Our Friend the Limulus
Mach bands
Fig. 8.11: Mach bands and contrast illusion. The Mach bands are
seen at the central contrast boundary between dark and light
(arrows). A related illusion produces a difference in appearance
of the two identical central squares.
far as the model is right and insofar as the limulus experiences anything,
we can conclude that the limulus eye tells the limulus brain about these
Mach bands. Similarly, our retinas have lateral inhibition and our eyes tell
our brains about Mach bands. The Mach band illusion is not restricted to
limulus but is found throughout much of animalia. In this admittedly limited
sphere, we can understand what it is like to see things as a horseshoe
crab sees them. Fig. 8.11 shows the illusion. The central contrast between
black and white is exaggerated (arrows) — the black appears blacker and
the white appears whiter. Another consequence of lateral inhibition is also
demonstrated in the figure and may be easier to see. The two gray squares
are identical but the one surrounded by white appears darker than the one
surrounded by black. This is due to inhibition of response to the square
on the right in the presence of greater lateral inhibition due to increased
flanking activation associated with the white surround.
Of course the question of qualia gets more thorny when we ask whether
one person see things the same way that another person sees them. The
typical example given is color — How can I know that the red that you see
is the same as the red that I see? At the lowest level, we could measure the
8.9. Summary and thoughts 139
responses of the different cone subtypes in your eyes and my eyes (cones
are the type of retinal cells that respond to color; they differ slightly from
person to person). We could then adjust light frequency in order to give
your cones the same amount of stimulation that my cones are receiving at
the chosen “red” frequency. How about if we went up and around my cortex
and discovered that red vibrates certain ensembles more than others
due to my personal experience with red? We would attempt to produce
a similar experiential contrast in your cortex, relative to the ensembles
and wiring that you have available. This would give you an experiential
illusion, evoking comparable associations. As a simple example of this, consider
the condition of synesthesia, a benign condition that causes crosstalk
between sensory modalities, causing a person to experience sounds while
seeing shapes, or colors while listening to music. Mapping these complex
sensory combinations in the cortex of a synesthetic would allow us to reproduce
the same combinations in another person, giving the nonsynesthete
the experience of synesthesia.
8.9 Summary and thoughts
In this chapter, we have reviewed the decisions and compromises made in
designing any computer model: simplifying, scaling, parameterizing. We
also considered additional common compromises, such as linearizing and
wraparound. It may seem remarkable that one can learn anything after
stripping away so much detail. The lesson of the limulus model is just this,
however. One can contrast the limulus-model approach with its opposite:
throw everything that you know into a model, set it loose, and see whether
it tells you something you don’t already know. This everything-and-thekitchen-
sink strategy can be a useful starting point, helping one organize
information and ideas. However, such a mega-model is just a prelude to the
hard work of paring back, focusing on a problem and figuring out what to
take out and what to leave in.
Lateral inhibition is a common architectural feature of both natural and
artifical neural networks, often seen in a context when a neural sheet is
mapping some attribute of the world. In this context, the flanking neurons
mean something relative to the inhibiting central neuron. As in the limulus
example, this will produce sharpening, allowing a maximally activated neuron
to stand out more strongly from the background. Other examples could
be taken from the somatosensory (touch) system where lateral inhibition
would allow more precise localization of a poke on the skin.
A mechanism similar to lateral inhibition can be used to allow a subset
of neurons to emerge out of a background of many active neurons. Highly
active neurons inhibit less active neurons. As the activity of the inhibited
neurons decreases, they provide less inhibition onto the more active neu140
8. Our Friend the Limulus
rons, leading to a dynamic where the rich get richer and the poor poorer. In
the extreme, an inhibitory network can be organized so that only a single
neuron remains active in the steady state. Such a network is known as a
winner-take-all network.
9
Supervised Learning: Delta Rule and
Back-Propagation
9.1 Why learn this?
Learning and memory are favorite topics in both neuroscience and artificial
intelligence. They have thereby become a popular focus in computational
neuroscience as well. In this chapter and the next, we discuss two types of
learning: supervised and unsupervised.
In childhood, we are told what things are for and what they are called.
In the context of learning theory, this is considered supervised learning.
When we learn to write, there is a specific target for the formation of each
letter. As adults, we often learn things without ever clearly delineating
exactly what we are learning. Thrown into a new work environment, for
example, we simply absorb things and gradually sort out when to use which
strategy to accomplish which objective. This absorbing and sorting is a
process of unsupervised learning, where information is being organized in
our brain without our necessarily ever being aware of exactly what is what.
A still more common form of learning is reinforcement learning, in which
the feedback is only general success or failure (or approval or disapproval)
rather than an explicit target behavior to be replicated. For example, the
success of a tennis serve will gradually teach improved serving, while a
slightly raised eyebrow may be all of the feedback available to assist use in
fitting into a new social situation.
We have previously mentioned Hebb’s rule as the basis of one type of
learning algorithm. Hebb’s rule states that connection strength will increase
between simultaneously firing neurons. In the proper context, Hebb’s rule
142 9. Supervised Learning: Delta Rule and Back-Propagation
can be the basis for learning, as we illustrate in the next chapter. In its basic
form, Hebb’s rule is a part of unsupervised learning algorithms since it’s
just a rule for altering connections without regard to any goal or purpose.
It can be adapted to make it part of a reinforcement learning system by
only allowing the synaptic modification to take place when the positive
reinforcement conditions are in effect.
In this chapter, we discuss two supervised learning algorithms: the delta
rule, which is simple, and the backward-propagation algorithm, which is
complicated. We use back-prop (as it is affectionately called) to explain
how neurons work to move the eyes when looking at something and to
stabilize the eyes when the head moves.
9.2 Supervised learning
In supervised learning, we pair an input with an output and teach a network
to produce the output when it is presented with the input. Most supervised
learning algorithms use a feedforward network architecture, taking the input
at one end and processing it to produce the output at the other. For
example, we might want the network to associate faces with names. This
is difficult to do because different pictures of a face, even if all taken full
frontal, will differ in shading, photographic exposure, and facial expression.
Let’s say we have a database of 500 digitized names with 20 frontal facial
photographs for each one. Perhaps each photograph is stored as a 50 by 50
pixel array and each name is stored as 30 Ascii letters. Then we would construct
a feedforward artificial neural network with 2500 input units (50・50)
and 240 output units (30 times 8 bits per Ascii character). We would start
with random weights throughout the system. We would present the faces
to the system in random order by setting the input units to the pixel values
for each face and then forward propagating the values through the layers of
the network to obtain output values. Initially, each of the resulting output
vectors would be a meaningless bit stream. However, we would use a learning
algorithm to change the weight matrix so that the system would do
better next time. Over many thousands of presentations of all of the paired
names and faces, the network would get progressively better at accurately
producing the name when the face was presented. Finally, the network is
ready for sale to the FBI.
In discussing representations (Chap. 4) and parameterizations (Chap. 8,
Section 8.5), we noted that the worlds of mathematical and computer modeling
presume and prescribe the use of numbers for representing everything.
Therefore, when we discuss learning in this context, we are learning numbers.
When we discuss supervised learning, we are getting the number
wrong and correcting it to get the number right. Unlike many real-life
learning situations, errors in numbers are neatly defined and quantified.
9.2. Supervised learning 143
(0 1)
target
output (1 0)
input
vector
clamped
input
output
vector
hidden
units
0 1
1 1
1 1 0
Fig. 9.1: Learning in a feedforward network. An input is clamped
onto the network. Information feeds forward through a hidden
layer to produce a calculated output. This output is compared
to the target output. Weights are changed. Another input/target
pair is tried.
Feedforward networks take inputs and provide outputs. The teaching
is done by providing input/output pairs: a specific input is paired with
a specific target output (Fig. 9.1). Output is calculated from each input
provided. One or more processing layers between input and output are
referred to as hidden layers. In a real nervous system these units are in the
“black box”; they remain hidden from an experimenter who only presents
inputs and measures outputs. Calculated output must be distinguished
from the target output. For this reason the target output (the output of
the input/output pair) is called a target (_t) where it must be distinguished
from the calculated output (o_ut where outi is the state of unit i of the
output layer).
144 9. Supervised Learning: Delta Rule and Back-Propagation
On the input side, this use of an externally provided input for learning
leads to another confusion. In the limulus equation, we added in the picture
vector _p (the input) directly as part of the update rule. In the case of
feedforward learning networks, we handle the input vector as if the values
were the states of a set of input units. These input units are false units;
they are not processing units. It is useful to represent them as if they were
processing units, because we are interested in modifying weights between
these inputs and the next layer. The input units do not get updated by
an update rule. Instead they are successively clamped (set) to the values of
each input.
Similarly, in some algorithms, we associate a bias unit with each processing
unit. This is another false unit. The bias unit is always set to 1.
After learning is complete, the weight from the bias unit to its processing
unit produces a constant bias. This bias effectively repositions the totalsummed-
input on the squashing curve. Changing the weight for the bias
unit is equivalent to shifting the squashing function left or right. Therefore,
learning the bias weight is equivalent to learning the location of the
threshold for the squashing function.
In the example of Fig. 9.1, the input _0 1_ and output _1 0_ belong
to a training set for learning to reverse the order of the two bits. The
input/output pairs for the full training set are _0 1_ → _1 0_, _1 0_ →
_0 1_, _0 0_ → _0 0_, _1 1_ → _1 1_. The clamped values in the
input layer determine values in the hidden layer, which determine values in
the output layer. In this case, the output is shown to be in error. The error
would be_t −o_ut = _1 0_−_1 1_ = _0 −1_. This error would then lead
to weight changes as will be described below.
Presenting the input states as false units creates an additional nomenclature
confusion. This set of false units is called the input layer. The simplest
network is usually considered a two-layer network, having an input layer
and an output layer. As an example, Fig. 9.2 has a two-unit input layer
and a one-unit output layer. This would usually be called a two-layer network
since it’s drawn with two layers. This is what I’ll call it. However, it
only has one processing layer and could alternatively be called a one-layer
network. The two inputs are just numbers that are copied from the list of
input patterns and are not real units that update as sum-and-squash units.
In Fig. 9.2, only the single output unit is a sum-and-squash unit.
In feedforward networks, units are typically numbered within each layer
so that there is an input unit #1 and an output unit #1 in Fig. 9.2.
The same numbers are then used as subscripts for a weight to indicate
which units are connected by that weight. The update rule for Fig. 9.2 is
out1 = σ{w11 ・ in1 +w12 ・ in2}. For Fig. 9.2, σ is the thresholding function
shown inside the picture of the output unit. Below, we use subscripts i
and j as indices for units with si as the state of a postsynaptic unit (a
hidden or output unit) and sj as the state of a presynaptic unit (a hidden
9.3. The delta rule 145
inputs
0
1
output
unit 1
w12 w11
unit 1 unit 2
Fig. 9.2: Simple feedforward network: two inputs and one
processing unit. Sharp thresholding.
or input unit). In general, total-summed-input for a postsynaptic unit is
si = wi1 ・ s1 + wi2 ・ s2 + . . . + wiN ・ sN = _N
j=1 wij ・ sj . (The capital
sigma, _, standing for “sum,” says to add up all of the values obtained by
setting j = 1. . . N. Here, N is the number of presynaptic units.) The order
of subscripts in wij indicates the feedforward direction. This just gives us
another way of writing the sum of products that is represented by the dot
product: si = _w ・ _sj .
9.3 The delta rule
Neural network supervised learning algorithms change network weights little
by little until the input patterns used in the training are reliably mapped
onto their corresponding output targets. (Mapping is a general term for the
transformation of one representation to another, just as the physical representation
of a city is transformed into a graphical and textual map.)
When learning is complete, we should be able to present an input pattern
to the network, perform the update rules for all of the units, and have
the network produce output states that match those of the paired target
output. To get from a random network that will produce a random output
for each input to the correct network, we need to make appropriate
changes to each weight based on the errors the network makes. When an
error is made, which weight should be changed and by how much? This
is the credit-assignment problem — the basic problem that must be solved
146 9. Supervised Learning: Delta Rule and Back-Propagation
by supervised learning algorithms. Perhaps it would be better called the
blame-assignment problem. We determine which weights are most to blame
for the error and correct those weights most.
The Greek letter delta (capital Δ or small δ) used in math means
“change.” The delta rule and its variations are common supervised learning
rules that are used to change weights. There are a variety of similar rules
with different names that can be considered as variations of the delta rule.
These include the Widrow-Hoff rule, adaline rule, perceptron rule, least
mean squares (LMS) rule, and gradient descent rule. In general, the rules
differ by the use of different squashing or activation functions (sigmoid, linear,
or thresholding). They also use somewhat different factors to correct
the weights.
Consider a single output unit outi. As discussed in Chap. 6, the input to
that unit is the dot product of presynaptic weights and presynaptic states.
A learning rule should alter the weight vector in order to get the state of the
unit, outi, closer to a target state ti. The simplest rule would be to subtract
or add a small number α to all of the presynaptic weights depending on
whether ti −outi is negative (state too big so decrease weights) or positive
(state too small so increase weights). Let’s call this the α rule. The size of
α determines how fast learning takes place. The α rule doesn’t work very
well. The first problem with it is that it does not account for the magnitude
of the error.We can take account of this by multiplying α by the magnitude
of the error ti − outi to get a proportionate correction factor. This allows
learning to go fast when the error is big and to slow down as the actual
output state approaches the target.
The other problem with the α rule is that it doesn’t solve the creditassignment
problem. Multiplication of each weight by α・(ti−outi) changes
the weight whether or not it had a role in producing the error. Unnecessary
and counterproductive changes to weights can be prevented by considering
the presynaptic activity that is being communicated through that weight.
The delta rule includes this factor by using the value of the presynaptic
state, sj .
The delta rule is Δwij = α ・ (ti − outi) ・ sj . In the delta rule, α provides
the learning rate. The delta rule avoids weight changes where a presynaptic
state was very small or zero and therefore made no difference in the output.
As the credit/blame for the error increases, the weight change increases.
Using the delta rule, the update rule for weight wij is wij(new) =
wij +Δwij .
The energy analogy
When we talked about the evolution of network state through time, we
noted that this evolution could be mapped in a suitably high-dimensional
state space. In state space, the state of the entire network can be described
9.3. The delta rule 147
as a single point. Similarly, a learning network makes gradual changes in
weights that can be mapped in a high-dimensional weight space.
Within weight space, the movement of a single point defines the process
of learning. There will be a spot in weight space where the input/output
mappings are performed as well as possible. We want a learning algorithm
to be able to find this spot. Such an algorithm can be designed by defining
an energy field that assigns a potential energy for each point in weight
space. This potential energy is just an analogue of error. The energy function
for the delta rule uses the summed-square error: potential energy =
error = 1
2Σi(outi −ti)2, where the sum is over the number of output units.
The difference is squared so that we are always summing positive numbers,
regardless of whether the (outi − ti) difference is negative or positive.
The standard derivation for the delta rule utilizes this energy function.
By determining how the energy function changes with change in weight (a
derivative in calculus), one arrives at the same delta rule we determined
above. Using the delta rule and the energy function, learning is a process
whereby network weights are altered so that the weight vector gradually
flows downhill to reach the point of lowest potential energy. Big error is high
energy and small error is low energy. One looks at the learning progress as
a dynamical system, as if this were a bowling ball rolling downhill under
the influence of gravity (Fig. 9.3). The delta rule is guaranteed to find the
point attractor in the energy field that corresponds to the optimal mapping
of inputs to target outputs. Error getting smaller corresponds to energy
getting smaller with the reduction of potential energy down a gradient.
For this reason, algorithms of this mathematical form are called gradient
descent algorithms.
It is also useful to think of error in terms of this energy-minimization
analogy even when using error-minimization algorithms that do not use
gradient descent. For example, genetic algorithms jump around in weight
space. Although they do not walk down hills, it is helpful to think of them
as looking for low points on the error landscape.
The delta rule solves AND
Let’s take a simple example to see how the delta rule works. Typical test
examples for artificial neural networks are taken from Boolean algebra (see
Chap. 16, Section 16.3). We’ll look at the Boolean AND operation, which
maps 0,0→0; 0,1→0; 1,0→0; 1,1→1. (Using 0 for False and 1 for True,
this mapping says that statement A AND statement B are false if either
A or B or both are false but true if A and B are both true.) We’ll use the
architecture of Fig. 9.2 with its sharp threshold activation function with
a threshold of 0.5. The delta rule will be used to determine weights. Let’s
use α = 0.1. We’ll start with w11 = 1, w12 = −0.6 as our randomly chosen
initial weights. Fig. 9.4 shows the evolution of the weights both in time
(left graph) and in weight space (right graph).
148 9. Supervised Learning: Delta Rule and Back-Propagation
Error "rolls downhill"
Energy = 0
(No error)
weight change
Error
Fig. 9.3: Gradient descent. The parabola shown here: error =
1
2 (out − t)2 is the one-dimensional version (one weight dimension)
of the delta rule energy function.
We present each of the four patterns in the order given above. The first
pattern is trivial: with the input set to _0 0_ the values of the weights
are irrelevant; the output will always be 0. The error is 0 and credit assignment
tells us that neither weight would bear any blame even if there
was any error. Therefore, delta is 0 for each weight and the weights remain
at w11 = 1, w21 = −0.6. The next pattern presented is _0 1_.
0 ・ 1 + 1 ・ −0.6 = −0.6; σ(−0.6) = 0. Again, there is no error and no
weight change. However, the third pattern _1 0_ gives an incorrect value:
1 ・1+0・−0.6 = 1; σ(1) = 1. The error ti−outi = 0−1 = −1 is multiplied
by α to give a delta of −0.1, which changes w11 to 0.9. w12 is not changed
because its presynaptic state is 0, so it was not to blame. The fourth pattern
_1 1_ also gives an incorrect value: 1・0.9+1・−0.6 = 0.3; σ(0.3) = 0.
Now the error is +1 and the delta, +0.1 is applied to both weights since
both presynaptic states are 1. Note that w11, which was corrected down
from 1 to 0.9 for the _1 0_ pattern, is now pushed back up to 1 (arrow in
left graph). This type of competition in weight space is typical for neural
networks as they converge toward a solution. In this case, w11 gets pushed
down and then up repeatedly until w11 dips below 0.5 and w12 goes above
0.
Learning continues with each pattern presentation until each pattern
produces the right output and the summed error is 0 over all four patterns.
In weight space (right graph), we can follow the learning as it goes in the
9.4. Backward propagation 149
w11
w12
10 30 50
−0.5
0
0.5
1 w11
w12
−1
0
1
−1 0 1
Fig. 9.4: Left graph shows sequence of weights over 56 iterations
(14 presentations of four input/output pairs) during delta rule
learning. Right graph shows same progression in two-dimensional
weight space; arrow indicates direction with time.
direction shown by the arrow. Many pattern presentations don’t cause any
change in weights; therefore, the 56 points seen in the time graph can’t all
be seen in weight space—many lie on top of each other. The solution finally
arrived at makes sense. The output state for _1 1_ is σ(1 ・w11+1・w12) =
σ(w11 + w12). To get 1 as the output (σ(w11 + w12) = 1), the sum of the
weights have to be greater than 0.5 (w11+w12 > 0.5). However, each weight
alone has to be less than 0.5 so that inputs with a single one, _1 0_ and
_0 1_, will not be able to drive the output to 1.
9.4 Backward propagation
The solution to the credit-assignment problem for three-layer (and greater)
networks bedeviled investigators for many years. Neural networks were popular
in the early 1960s. There was much excitement about their potential
for understanding how the brain works until two researchers (Marvin Minsky
and Seymour Pappert) proved in 1969 that delta-rule type networks,
while fine for AND, couldn’t do more complicated Boolean expressions
like “exclusive OR” (XOR, mapping 0,0→1; 0,1→0; 1,0→0; 1,1→1). This
soured most people on neural networks for a decade or two. Additionally,
Minsky and Pappert guessed (and stated in their book) that three-layer networks
(two processing layers) couldn’t do anything special that two-layer
networks couldn’t do. They were wrong about that one. David Rumelhart,
Geoff Hinton, and R.J. Williams developed the back-propagation algorithm
in 1986 and showed that a three-layer network could solve XOR. Neural
networks rose from the grave. As with many such stories of excitement and
150 9. Supervised Learning: Delta Rule and Back-Propagation
remorse, neural networks were then again oversold as the solution to all
things beautiful or bright. Once again they suffered the scorn of skeptics,
as it was discovered that back-propagation probably was not happening in
the brain.
The solution to the credit-assignment problem for two-layer feedforward
networks is to look back at the input layer and see which units are contributing
to the output. Similarly, the solution to the problem for threeand
more-layer feedforward networks is to keep looking back down the
convergence tree through the layers. In Fig. 9.5, the dashed lines show the
convergence tree onto the shaded output. As we go backward down this
tree, we assign blame to weights at each level in order to decide how much
to change them. In Fig. 9.5, the error will propagate backward from the
shaded unit at upper right through three weights to all of the units of
the hidden layer and then through all of remaining weights between the
hidden and input layer. For example, if the error at the output unit was
0.2, this value could be multiplied by each of the back weights to obtain
an error value for each of the hidden units. As with the delta rule, weight
changes need to be proportional to presynaptic state values to assign proper
blame. In Fig. 9.5, the 0 in the left-most input means that the three weights
projecting from this unit cannot be blamed for an error for this training
pattern and will not be changed on this input presentation. Changing the
weights from the hidden to the output layer will also take into account the
presynaptic state values. In this case, the values will be the states of the
hidden units obtained during the feedforward phase.
The back-prop algorithm is derived by again considering the energy analogy.
As it turns out, solving the credit assignment problem is not sufficient
and the algorithm will not work as described above. An additional factor
based on the slope of the activation function is also required. This factor
allows us to make bigger changes where they will have more effect on solving
the error, that is, to make big changes in places where the postsynaptic
cell will pass on a lot of that change. To make three-layer feedforward networks
learn, back-prop not only solves the credit assignment problem but
also solves an “opportunity assessment” problem so that changes are made
in places where they make a difference. This assessment of the passage of
information at one layer allows the algorithm to appropriately credit lower
layers by seeing how much of their influence will pass through toward the
output.
By the way, back-propagation is sometimes referred to in current neuroscience
literature. However, most neuroscience references that use the term
are talking about “back-propagating spikes.” This has nothing to do with
the back-prop algorithm. We discuss back-propagating spikes in Chap. 13.
9.5. Distributed representations 151
input 0 1
output
hidden
Fig. 9.5: Back-propagation: if the output unit on the right is in
error (shaded), error propagates backward through the convergence
tree (dashed lines). However, an input unit with state 0
(left) will not get any credit/blame for the error, and the weights
from there to the hidden units will not be changed.
9.5 Distributed representations
The discovery of back-propagation made a big impact in cognitive science
and a lesser but still substantial impact in neuroscience. It was also newly
taken up by engineers, although ironically the same algorithm had been
discovered by the engineering community some years before but had not
been widely disseminated. The major conclusion from early studies with
back-prop was that information processing in the brain might be parallel
and distributed rather than serial and local. As has been mentioned
previously (Chap. 7, Section 7.2), this represented a shift away from the
prevailing artificial-intelligence paradigm to something more neural.
Study of parallel distributed processing networks showed that many features
of distributed representations were reminiscent of human learning and
memory. For example, the networks showed graceful degradation, falling
off slowly rather than abruptly with damage. Additionally, the back-prop
152 9. Supervised Learning: Delta Rule and Back-Propagation
paradigm offered incremental learning so that new information could be
added as it came in. However, this new information would tend to interfere
with older information, particularly related information. Again this is
a familiar attribute of human memory. In common with children, the networks
tended to initially overgeneralize before getting the problem right.
For example, both networks and kids first learn that “ed” makes a verb
past tense and overgeneralize, making errors like “goed” before learning
the correct “went.”
Early successes with cognitive applications made people want to believe
that back-prop was in fact how the brain worked. The only detail
remaining was to find back-prop in the brain. It couldn’t be found. Although
back-propagating action potentials were discovered, there is no
evidence that information will continue to proceed backward from the dendrite
since chemical synapses are largely one-way information conduits.
Although back-prop has not been found in the brain, there is evidence for
distributed representations. Back-prop is useful for producing distributed
representations and suggesting how they might work to solve biological
problems.
9.6 Distributed representation in eye movement
control
Back-propagation has been used as a tool to understand the complex
responses of interneurons (hidden units) in several sensory and motor systems.
Here we look at the mixture of sensory inputs that help control eye
movement. There are several things that cause your eyes to move. Fast
movements to look at something that has attracted your attention are
called saccades. Slow movements are used to follow a moving visual target,
for example a duck flying overhead. This is called visual pursuit. (Previously
I’ve been using the word target to mean the goal that output states
of the neural network have to match; now I’m talking about a moving visual
target that you track with your eyes. To keep this straight I’ll call the
pursuit target a “visual target” or a “duck.”) There is another type of slow
movement that is used for eye stabililization. This is called the vestibuloocular
reflex (VOR). Pursuit is based on the movement of an external visual
target. The VOR is based on the viewer’s own movement. Although apparent
movement of the entire visual field is also used to stabilize the eyes,
the major input for the VOR is an inertial sensing mechanism called the
semicircular canals, located together with the sound sensing organs in the
ears. The use of inertial sensing means that the VOR works in the dark.
(The closeness of visual and inertial sensation also accounts for the fact
that retinal slip, the movement of the entire visual field, produces the sensation
of movement, as for example when a train next to yours leaves the
9.6. Distributed representation in eye movement control 153
Fig. 9.6: Above, visual pursuit: guy watches duck. Below, VOR:
guy turns head side to side.
station.) The VOR compensates for head movement that could otherwise
cause the image to disappear off of the retina. Both the VOR and pursuit
operate simultaneously, as for example when you watch a duck fly by,
while sitting in a moving boat. Signals from both eye and ear (semicircular
canals) influence the muscles controlling eye movement (Fig. 9.6).
The neurons that mediate eye movements are in the brainstem, a place
where it is relatively hard to record neuronal activity. Before any such
recordings had been made, researchers had already been constructing models
of eye movement control, based largely on engineering principles. They
generally assumed that the neuronal systems for pursuit and VOR systems
would remain separate until they converged on the muscles of the
eye. When brainstem recordings were finally made, people were surprised
to find that many neurons could not be strictly defined as “pursuit cells” or
“VOR cells” but had a mixed response to both visual and movement input.
Even more surprising was that there were some cells where pursuit inputs
were pulling in one direction and VOR inputs were pushing in the oppo154
9. Supervised Learning: Delta Rule and Back-Propagation
site direction. This is not how an engineer would usually design a control
system.
Tom Anastasio and David Robinson were two researchers who looked
into this by applying the then-new back-prop algorithm to the problem of
how pursuit and VOR inputs might distribute the information through a
bunch of interneurons (hidden units) and still be able to produce the correct
effects on the eye. This represented a different approach to understanding a
neural circuit. Instead of using top-down ideas to figure out how the system
might work, their model was allowed to grow from the bottom up and was
then analyzed by the researchers and compared to the real thing. This had
some appeal since the nervous system has to develop from simpler rules
that cannot take account of how the finished brain ought to look.
Design of the model
Let’s look at a version of their model (Fig. 9.7). It’s a feedforward network
with four inputs, nine hidden units, and two output units. Unlike the binary
units of Fig. 9.2, the units here use analog coding so that they can take
on any value between 0 and 1 (activation function inset). Therefore, the
squashing function is a continuous rather than a thresholding function.
The training input/output pairs are simple in principle, but keeping track
of crossing influences pushing left and right makes it confusing. Left pursuit
(visual target moving to left) makes the eyes move left, while left VOR
(head move to the left) makes the eyes move right. Here are the basic
input/output mappings in the context of watching a duck flying by. L
stands for left and R for right.
INPUTS OUTPUTS
L L R R L R
VOR pursuit pursuit VOR eye eye
#1 head R 0 0.5 0.5 1 → 1 0
#2 head L 1 0.5 0.5 0 → 0 1
#3 duck L 0.5 1 0 0.5→ 1 0
#4 duck R 0.5 0 1 0.5 → 0 1
0.5 is the value of spontaneous activity for a unit; 1 means full activation;
0 means that the unit is being inhibited. Each of the training patterns
represents activation of only one input pathway, indicated by the boldfaced
1s in the table. By looking at the location of these 1s, one can read
off the meaning of each input. For example, the 1 in the first row indicates
that head turning right activates the R VOR input.
This mapping is easiest to understand if you imagine your own head
moving or your eyes tracking a duck. Pattern #1 is pure right VOR. This
means that the head moves right so that the right VOR unit is activated
(1) and the left VOR unit is inhibited (0). The pursuit units remain at
their resting activation levels (0.5). The output is full activation (1) of the
9.6. Distributed representation in eye movement control 155
− 0 +
1
0
State
Input activation
function
eye movement
L R
VOR
R
L R
pursuit
L
VOR
9 hidden
units
Fig. 9.7: Feedforward neural network takes information about
head movement and the visual pursuit target and transforms it to
eye movement. Sum-and-squash units are shaded. Inputs, which
do no processing, are shown in white.
muscles moving the eyes to the left and full inhibition (0) of the muscles
moving the eyes to the right. Head goes right; eyes go left. Similarly, pattern
#2 is pure left VOR. Pattern #3 is pure left pursuit. The duck goes left.
Left pursuit input is activated and right inhibited. VOR inputs both stay
at rest. Left eye muscles are activated and right inhibited. Duck goes left,
eyes go left. In pattern #4, duck goes right, eyes go right.
In this case we have an excess of hidden units and it’s easy to train the
system to produce these mappings. Easy is a relative term here. Training
takes thousands of iterations, as we repeatedly forward propagate inputs
and then back propagate error to get the total error across all patterns
down to a small number. The smaller the requested total error, the more
iterations required. In back-prop, the error measure for a single pattern
presentation is the same summed-square error that was used for the delta
function: 1
2ΣNi
(outi − ti)2 (i counted from 1 to N output units). As with
the delta function, this error function is used in the derivation of back156
9. Supervised Learning: Delta Rule and Back-Propagation
propagation. We can express the summed-square error in vector notation
by defining an error vector _e = o_ut−_t. Using the error vector, summedsquare
error is 1
2 (_e ・_e). Given P input/target pairs, total error is calculated
by testing each input in turn, comparing the output to the target, and
summing over pattern pairs: total error = 1
2ΣPi
=1(_ei ・ _ei) (i counted from
1 to P patterns). Mean error or average error is just total error over the
number of patterns: totalerror
P .
If you run the network exactly the way I’ve outlined it, it won’t work
very well. The typical squashing functions for analog units are sigmoidal
with asymptotes at 0 and 1. This means they never actually reach either
0 or 1. If you use 0 and 1 as targets, the network can never eliminate the
error. In trying to do so, the network will run for a long time to reduce
error. It may also create enormous weights in trying to push states toward
the unreachable zero or the unreachable one. To avoid this problem, it is
better to use less extreme targets (e.g., 0.4 and 0.6 instead of 0 and 1)
or to use a larger squashing function. In this example, I used a squashing
function that ranged from −1 to 2.
As training goes along, we repeatedly assess total error. Given the proper
conditions, the algorithm is likely to converge, meaning that the outputs
will get closer and closer to matching the target outputs. Unlike the delta
algorithm, however, the back-prop algorithm is not guaranteed to converge
to the best mapping or even to a good mapping. It may find a local minimum
of the energy function instead of the absolute minimum. In general,
for cases that use binary units, like the AND problem of Fig. 9.4, error
can be driven down to zero. However, with analog units, error is never
completely eliminated.
If you take a bunch of networks and start them with different random
weights, they will converge at different rates. After learning is stopped,
the resulting networks will have different weight values and different total
errors. In some cases, one of these networks will handle one set of patterns
well, while another network will handle another set well. Learning rate on
the VOR task varied enormously depending on the random numbers that
were used as initial weights. Using eight different weight initializations for
eight different networks, it took between 5800 and 61,100 iterations to
achieve a mean error less than 1 ・ 10−7. With this criterion, the worst
performance for a single unit was about 1.001 instead of 1. This is error
of 1 ・ 10−3, which gives a squared error of 1 ・ 10−6 (remember that this is
the error for one unit with one presented pattern; there are two units and
four patterns being averaged to give mean error). The enormous variability
in the number of iterations required to converge to the answer is typical
of neural networks with randomly chosen initial weights. In some cases, it
is possible to choose sensible initial weights and then use the back-prop
algorithm to improve the performance.
9.6. Distributed representation in eye movement control 157
Results from the model: generalization
Fig. 9.8 shows the performance of one example network on the VOR
problem. Each column of marks illustrates the response to one pattern
presentation. To simplify the graph, I summarized the two output values,
eye-right (output unit outR) and eye-left (output unit outL), as a single
value representing rightward eye movement. This is calculated by taking
the difference between the output states: outR−outL. In Fig. 9.8, rightward
eye movement is shown as a filled circle. Similarly, rightward head movement
is shown as a square and rightward duck (visual target) movement is
shown as a triangle.
The patterns in Fig. 9.8 are given in the same order as in the table
above. Pattern #1 is head right (square at +1) producing eyes left (circle
at −1 representing output states of _0 1_). However, while the pattern list
above gives target states, Fig. 9.8 shows actual outputs after training. The
training is good enough that the errors cannot be seen on the graph. The
output produced for pattern #1 was _0.0004 1.0004_ instead of _0 1_.
Therefore, the filled circle in the figure lies at −1.0008 instead of at −1.
A good test of the usefulness of a neural network is its ability to generalize.
This is a fundamental aspect of human intelligence. A child learns
what dogs are and then sees a St. Bernard for the first time. Even though
this dog is different from all other dogs, the child can still identify it as a
dog. He has generalized from experience so that new stimuli that are not
identical to learned stimuli can be recognized.
In a network, we would want the system to generalize and do sensible
things under conditions not in the training set. Any input will produce some
output. It only makes sense to test for generalization under conditions in
which the investigator (the teacher) has additional patterns that belong in
the set. In the context of neural networks, a common practice is to use only
some of a given input/output data set for training and reserve the rest of
it for testing. For example, if we had 50 dog and 50 cat pictures available
to train a network to distinguish dogs from cats, we might use 20 of each
for training and save 30 of each for testing. We might also want to try out
some pictures of cows and horses to see if the network was overgeneralizing,
as it will commonly do for unlearned categories. Again, this is reminiscent
of human learning. The city child who has never seen farm animals is likely
to assume “fat doggie” when first seeing a pig.
In Fig. 9.8, I show tests for several reasonable generalizations. In pattern
#5 the head is not moving and the duck is not moving. The network
produces a reasonable answer — the eyes do not move. In patterns 6 and
7, the head is going one way and the duck is going the other way. This
will require extra eye movement to keep up with the duck. In both cases,
the eye movement is in the correct direction and is increased, although not
doubled from the controls in patterns 1 and 2. In patterns 8 and 9, the
158 9. Supervised Learning: Delta Rule and Back-Propagation
−1
0
1
Learned Generalized
head
eye
duck
R
L
1 2 3 4 5 6 7 8 9
Head R
Head L
Duck L
Duck R
Nothing
Duck L
Head R
Duck R
Head L
Duck R
Head R
Duck L
Head L
Fig. 9.8: Inputs (triangle and square) and output (circles) from
neural network of Fig. 9.7. Near perfect responses are seen to four
learned patterns, and reasonable generalization is demonstrated
with four nonlearned patterns. (Value for each point is calculated
as right-side unit state minus left-side unit state.)
entire head follows the duck so no eye movement is needed. The network
again produces the correct response.
Overall, the network shows good but not perfect generalization. A limitation
of the network is seen in the response to patterns 6 and 7, where
the visual target is moving in one direction and the head in the opposite
direction. Linear summation of the opposing head and eye direction would
give doubling of the eye movement signal. Due to the nonlinearity of unit
responses, it is typical that the network will not produce this linear relationship.
Instead it gives somewhat less. The nonlinearity of the squashing
function is the limiting factor. Output values can only be pushed as far as
the sigmoid curve lets them go. For the sigmoid, the nonlinearity gets more
pronounced near the ends of the range. Therefore, expanding the sigmoid’s
range provides more linear responses.
9.6. Distributed representation in eye movement control 159
−1 1
−1
1
head movement
detection
duck movement
detection
L R
L
R
Fig. 9.9: Sensory detection profile for nine hidden units. Most
units show a mixture of influences, but two units (open circles)
are largely dedicated. Two units (open squares) exhibit paradoxical
combinations of pursuit and VOR influence. (Value for each
point represents a right weight minus a left weight.)
Exploration of the model: hidden unit analysis
Now we can look at the responses of the hidden units, shown in Fig. 9.9.
This is a virtual experiment: exploring the simulation to try to understand
how and why it works. The measurements here are the virtual equivalent of
the biological experiment of recording from interneurons in the brainstem.
If we were to do this type of recording, we would present the animal with
either head movement alone or visual target movement alone, and record
the response from different brainstem units. Similarly, in the virtual experiment,
we measure activity in each of the hidden units under conditions of
VOR activation alone or pursuit activation alone. To get everything neatly
onto a two-dimensional graph, we use total rightward pursuit input and
total rightward VOR input as we did in Fig. 9.8. In the network, this result
can be obtained by subtracting the values of two weights that come
into a hidden unit. To get the total rightward pursuit input we subtract
the weight connecting the left pursuit input to that hidden unit from the
weight connecting the right pursuit input to that hidden unit. We do the
same for VOR and can then graph one influence versus the other (Fig. 9.9).
From an engineering perspective, we expect that the points would either
lie along the axes or lie along the diagonal line from upper left to lower
160 9. Supervised Learning: Delta Rule and Back-Propagation
right. Hidden units that map to the x- or y-axis are dedicated to particular
sensory input. Hidden units on the y-axis are dedicated to communicating
VOR information alone. The open circle unit that lies near the y-axis in
Fig. 9.9 will transmit information about the head turning to the left. Hidden
units on the x-axis are dedicated to communicating pursuit information.
The open circle unit near the negative x-axis in Fig. 9.9 will mostly convey
information about visual targets moving left. These two units are dedicated
to particular types of inputs. By contrast, each unit near the down-sloping
diagonal are likely to be dedicated to a particular motor output. Units
near the upper left of the graph specialize in left pursuit and right head
movement and will presumably turn the eyes left. Units near the lower
right of the graph specialize in right pursuit and left head movement and
will presumably turn the eyes right. I say “presumably” since we are only
looking at the inputs to the hidden units. We could confirm the motor
influence by looking at the weights from hidden units onto output units.
By again subtracting left from right, we could produce a map of motor
influence from each hidden unit.
In this network, some of the hidden units are near the down-diagonal
but none are on it. Most of the units have a mix of influences from pursuit
and VOR. The two units at upper right (open squares) are particularly
peculiar. They code for pursuit to the right and for head movement to
the right. Since pursuit to the right makes the eyes move right and head
movement to the right makes the eyes move left, the units represent two
influences fighting to move the eye in different directions. It’s like pushing
on the brake and the accelerator at the same time. Units like this have
been found experimentally in animals. When these were first discovered,
physiologists thought they were too weird to be true and assumed that this
result must be an artifact. In the neural network, one can show how units
like this, although not useful in isolation, work together with other units
to convey information effectively.
One advantage of the parallel distributed processing of neural network
is redundancy. In this case, we are using nine units to do a job that can
actually be done by one unit alone. If we train a network with only one
hidden unit, the input weights always lie directly on the diagonal far from
the center, either at the upper left or lower right of the graph. If we train a
network with two hidden units, one of them invariably lies on this diagonal,
while the other one is free to wander off. As we train with more and more
units, there is less of a tendency for any single unit to sit on the diagonal. It’s
not clear whether this sprawling across the graph provides any advantage in
terms of reliability or tolerance to noise or damage. It would be interesting
to study this by comparing a system where all of the units lie on the
diagonal to one with the usual sprawl. One could then compare the two
systems by adding noise to the units or damage it by removing weights.
9.6. Distributed representation in eye movement control 161
Computer modeling vs. traditional mathematical modeling
The process that we used to evaluate this model is closer to an experimental
process than to the traditional mathematical modeling. Old-time mathematical
models were highly simplified versions of the real thing. These
old-time models were simple enough that the mathematician or physicist
could hold the whole model in his mind. As the modeler became more and
more familiar with his model, he could wander around in it as if he were in
his own home. He would come to appreciate how various parts fit together
and how the whole thing works. It’s impressive to see an old-time mathematician
stare for a minute at a set of equations on the blackboard and
then tell stories about what the system will do.
The good news is that, with computer models, it doesn’t take years of
training to begin to understand them. The computer provides an interactive
tool for doing the stuff the old-timer had to do in his head. The
bad news is that the complexity of some computer models places them out
of the reach of full mastery even after those years of training. The VOR
model is fairly simple. It can be mastered intellectually. But larger, more
complicated neural network models cannot be. These models are created
by the computer outside of the control of the human creator (shades of cyborg).
Despite being man-made artifacts, they cannot be held in the mind.
Instead they provide virtual preparations to be explored experimentally by
making the same measurements that one would make in the real thing.
The graphs in Fig. 9.8 are the same graphs one would make after doing
biological experiments.
So why bother with the computer model if you can do many of the same
experiments on the real thing? After all, the real thing is a better version
of itself. Well, you can do experiments on a computer model that are
impossible in the real thing for either ethical or technical reasons. In the
computer model, if it’s there, it can be measured. In the real thing, there
are lots of things that can’t be measured. As importantly, it is easy to do
“what-if” experiments on the computer model such as the one we proposed
above: what if the brainstem evolved so that all of the pursuit/VOR interneurons
lay on the down diagonal? Hey, maybe at some point animals
did evolve that way and they died out because their eye-movement system
wasn’t good enough and they couldn’t compete. If this is the case, then it
might eventually be possible to use genetic engineering to re-create such a
creature. However, many “what-if” situations will be biologically impossible
due to both developmental and ethical constraints; genetic engineering
won’t put neurocomputing out of business.
162 9. Supervised Learning: Delta Rule and Back-Propagation
9.7 Summary and thoughts
Back-propagation is an algorithm that was developed to solve the credit
assignment problem. This was a landmark in the history of artificial neural
networks in particular and computational neuroscience in general.
Back-propagation is still used extensively in the engineering community.
However, the biological community remains somewhat touchy about backpropagation
because it was originally marketed as the solution to how the
brain learns. It’s not. However, back-prop can be used to explain some
otherwise mysterious aspects of neural organization. Back-prop appears to
arrive at solutions similar to those found by biology using some different,
unknown, algorithm.
Back-prop solutions serve as an existence proof, demonstrating that it
would be possible for neurons to solve complicated problems by working
together in parallel, processing distributed information. The back-prop results
demonstrate that, in a population of neurons working together on a
task, some will be specialized, others will be doing a little of this and a little
of that, and some will be perverse, seeming to specialize in both going
right and going left. Alone, the action of an individual neuron may appear
inadequate or senseless, but the society of neurons gets the job done.
This analogy of neural networks and societies was first pointed out by
Friederich Hayek, who won the Nobel prize in economics for models based
on the progress of an economic system because of, rather than despite,
the contributions of individuals whose efforts are uncoordinated and undirected.
Hayek also wrote a theoretical neuroscience treatise entitled The
Sensory Order, presenting theories similar to those of Hebb. Neurons, like
people, just gotta be free.
10
Associative Memory Networks
10.1 Why learn this?
Hebb’s rule, an increase in synaptic strength between simultaneously active
neurons, is one of the most enduring and endearing concepts in computational
neuroscience. With the demonstration of long-term potentiation
(LTP) of synaptic strength between real neurons, Hebb’s stock rose still
higher. It is now generally believed that Hebbian learning does take place
in the nervous system. For Hebb’s rule to result in Hebbian learning, it is
necessary that the process provide the basis of some sort of information
storage (memory). Hebb suggested that this would occur through the formation
of cell assemblies. Cell assemblies form as cells that fire together
connect together (Hebb’s rule). Groups of connected cells represent memories.
Since individual cells will likely be involved in more than one memory,
this is a distributed memory mechanism. Various specific algorithms have
been proposed that will produce Hebb assemblies using Hebb’s rule. The
most popular of these is the Hopfield algorithm.
In the previous chapter I showed that the delta rule and back-propagation
can be used to produce neural networks that can do pattern matching. Although
the back-prop algorithm is nonbiological, it’s useful to create and
explore distributed representations. Distributed representations do appear
to be biological. The Hopfield network, in its original form, is also nonbiological.
The Hopfield network is another useful tool, allowing us to better
understand distributed representations and content-addressable memories.
164 10. Associative Memory Networks
10.2 Memories in an outer product
Hebb’s rule is easy to express arithmetically using multiplication of presynaptic
and postsynaptic weights: Δwij = spost ・ spre. If we use 0/1 binary
vectors, this gives the pure Hebb rule:
PRODUCT
PRE → 0 1
POST ↓ 0 0 0
1 0 1
Δ Weight
0 1
0 none none
1 none ⇑
This rule allows for increase in synaptic strength but there is no rule for
decrease. Without the possibility of decrease, the rule leads to synaptic
strengths increasing without bound. Eventually all states are pushed to
their maximum value all the time. We get a more useful rule by using
−1/1 binary values instead:
PRODUCT
PRE → −1 1
POST ↓ −1 1 −1
1 −1 1
Δ Weight
−1 1
−1 ⇑ ⇓ 1 ⇓ ⇑
This extension to the Hebb rule sounds reasonable: if one cell fires and
the other cell is silent, there is a decrease in synaptic strength. This has
also been described biologically as long-term depression (LTD). However,
this Hebb rule variant also states that synaptic strength will increase when
two cells are both inactive. This does not make much sense and does not
seem to be the case biologically.
We can bypass arithmetic and simply set up rules for synaptic changes
based on biological observations. For example, a more realistic rule might
be that presynaptic on (1) with postsynaptic on (1) would lead to synaptic
increase, and presynaptic off (0 or −1) with postsynaptic on (1) would lead
to synaptic decrease. However, the use of arithmetic-based rules has advantages.
Using math allows us to utilize mathematical reasoning in order
to understand what is going on. In some cases, we can even develop mathematical
proofs to guarantee that a particular neural network will store
and recall memories. It turns out that −1/1 vectors work particularly well
as a basis for a content-addressable memory. They are used in one of the
top-selling artificial neural networks, the Hopfield network.
Association across a single synapse
Conveniently, and not coincidentally, multiplicative associative memories
with −1/1 binary-valued units (Hopfield networks) work because of the
associative property of multiplication: (a・b) ・c = a・(b ・c). If we present preand
postsynaptic activity at a single synapse, the arithmetic form of Hebb’s
10.2. Memories in an outer product 165
rule sets a weight to spost ・spre. Using scalars (i.e., a two-neuron, one-weight
network), we can show that this learning rule produces a weight such that
a learned input, sin, maps onto the corresponding target output: sin → starget. We clamp the presynaptic unit onto the learned input: spre = sin.
We want to show that the learning and update rules produce an output on
the postsynaptic cell such that spost = starget.
The update rule is spost = σ{w ・ spre}. σ here is a sharp thresholding
at zero. The single weight, w, is determined by Hebb’s rule applied to
the learning of the particular input/target pair: w = (starget ・ sin). Substituting
for w in the update rule gives spost = σ{(starget ・ sin) ・ spre}.
Application of the associative property of multiplication gives spost =
σ{starget・(sin・spre)}.We now present our two-unit network with the learned
input, clamping spre = sin. Now we have spost = σ{starget ・ (sin ・ sin)}. In
this binary network, states are either −1 or 1. This is where the choice of
−1/1 binary-valued units becomes critical. Since −1・−1 = 1 and 1・1 = 1,
(sin ・sin) is always equal to 1. Therefore, spost = σ{starget ・1} = σ{starget}.
σ thresholds at zero and won’t have any effect here: σ{1} = 1 and
σ{−1} = −1. Therefore, the output, spost = starget, as desired. Quod erat
demonstrandum. Latin aside, it’s real simple: if the input and target of a
pair are the same (1 → 1 or −1 → −1), then the weight is 1, which as a
multiplier maps anything onto itself. If the input and target are different
(1→−1 or −1 → 1), then the weight is −1, which will produce an output
opposite in sign from the input.
To scale this scalar associative rule up from a two-neuron network to a
large network, we use vectors. We then need to use a linear algebra rule
that will match up the input and target states properly when we are using
vectors _sin and _starget instead of scalars. Such a rule is the outer product.
Recall that the inner product (dot product) matches up and multiplies
corresponding elements of two vectors and adds them up to give a scalar.
Instead of matching up corresponding elements of two vectors, the outer
product matches up every possible pair of elements. Instead of summing
these products to form a scalar, the outer product places these products
in a matrix. Instead of being the product of a row vector times a column
vector, the outer product is the product of a column vector times a row
vector. Instead of requiring that both vectors be of identical length, the
outer product can multiply two vectors of different size. The outer product
uses a column vector (size M) times a row vector (size N) to get a matrix
of size M × N.
The outer product of two vectors
To form the outer product, multiply the entire column vector in turn by
each element of the row vector, placing the new column vectors side by
side to make a new matrix of N columns. Alternatively, you can get the
same result if you multiply the entire row vector in turn by each element
166 10. Associative Memory Networks
of the column vector, placing the new row vectors one under the other to
make a new matrix of M rows. Using linear algebra, Hebb’s rule uses the
outer product W = _spost ×_spre. The update rule uses the product between
a matrix and a vector: _sout = σ{W ・ _sin}. This product is performed by
forming a vector out of the ordered dot products of each row of the W
matrix with _sin. _sin must be oriented as a column vector. Associativity
works, so the proof above is still valid. Unlike scalars, however, the outer
product is not commutative: _a×_b _=_b×_a. Therefore, the order _spost ×_spre
is important.
The many versions of linear algebra multiplication can be hard to keep
track of. Here’s a table:
dot (inner) product outer product
symbol a ・ b a × b or a ⊗ b
order row ・ column = scalar column × row = matrix
aT b = (・ ・ ・ ) .. ._ = number abT = .. ._(・ ・ ・) =⎛⎝
・ ・ ・
・ ・ ・
・ ・ ・
⎞⎠
lengths N ・ N any × any
process sum of products combined products
product scalar matrix
matrix-vector product
symbol A ・ b
order matrix ・ column = column
Ab =⎛⎝
・ ・ ・
・ ・ ・
・ ・ ・
⎞⎠
.. ._ = .. ._
lengths M × N matrix ・N vector
process matrix-row times column-vector dot products
product vector
So far we can “remember” a single memory. At the single synapse level,
this is spectacular only to the extent that you are prepared to be thrilled by
the properties of multiplication. However, at the matrix or network level,
we begin to enjoy some of the advantages of collective computation: the
network will correct mistakes in the input. This is because the total input
to a single neuron is represented mathematically by the dot product of a
row of the weight matrix and the input vector. This dot product only has
to have the right sign, positive or negative, and the squashing function will
give the right answer. The dot product is just a sum of multiplications. Each
element of _sx that’s the same as an element of the original _sin will push
this sum in the correct direction; each element that differs will push in the
wrong direction. As long as the correct matches outnumber the incorrect
matches, the thresholding will yield the correct output. So the input can
10.2. Memories in an outer product 167
have a few 1s where there should be −1s or −1s where there should be 1s.
The dot product will still have the right sign and the output will be right.
Making hetero- and autoassociative memories
We can use the outer-product rule to form either autoassociative (often
simply called associative) or heteroassociative memories. In an autoassociative
memory, memory for an item is prompted by something similar but
not identical to the remembered item, as if one got a glimpse of a friend
through a dirty window and could nonetheless recognize him. In a heteroassociative
memory, an unrelated image prompts recall; for example,
seeing the friend prompts recall of his name. For associative memories, the
outer-product multiplication is the learning or training operation. There
is no prolonged repetitive presentation of patterns as we used in the delta
rule or with back-prop. It is possible to get similar results by using gradual
Hebbian changes with repeated presentations during training. In either
case, I call this process “training,” even when it’s an immediate result of
multiplication and addition.
Here’s an example. Let’s learn the heteroassociative mapping
_sin = _−_________1 −1 −1 1 −1 1_T
→_starget = _−1 1 −1 −1 1_.T
The input and output vectors are of different sizes. The outer product gives
the 5×6 weight matrix shown below. We now test an input that’s a slightly
messed-up version of the original learned pattern. The test vector is
_sx = _−1 1∗ −1 1 −1 −1∗_T ,
differing from _sin at the starred locations. Presentation of the test input to
the memory is done by applying the update rule. This involves multiplying
the memory matrix times the test input vector and squashing:
σ{W ・ _sx} = σ_⎛
⎜⎜⎜⎜⎝
1 1 1 −1 1 −1
−1 −1 −1 1 −1 1
1 1 1 −1 1 −1
1 1 1 −1 1 −1
−1 −1 −1 1 −1 1
⎞⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎝
−1
1
−1
1
−1
−1
⎞⎟⎟⎟⎟⎟⎟⎠
_
= σ
⎛⎜⎜⎜⎜⎝
−2
2
−2
−2
2
⎞⎟⎟⎟⎟⎠
=⎛⎜⎜⎜⎜⎝
−1
1
−1
−1
1
⎞⎟
⎟⎟⎟⎠
= _sout = _starget
The overlap between the prompt (test) vector _sx and the _sin is four out of
six. The product for the four matches will add up to 4 and the two nonmatch
products will add up to −2 giving dot product _sx ・_sin = 4−2 = 2. The plus
168 10. Associative Memory Networks
or minus sign for each of the 2s comes from the sign of the corresponding
element in _sout.
Similarly, we can form an autoassociative memory by simply using
_sin ×_sin as the associative matrix. In either the auto- or heteroassociative
case, we find that these matrices have the qualities of content-addressable
memories: a partial input or an imperfectly formed input can be recognized
due to its similarity with previously memorized patterns. The original memory
can then be reconstructed or, in the case of a heteroassociative memory,
the association can be retrieved. The great thing about associative memory
matrices is that they can be used to store more than one memory simply by
adding up the matrices formed from the outer product of each individual
memory.
As one tries to stuff more and more patterns into an associative network,
adding more matrices together, the input patterns can begin to interfere
with one another, producing confusion between them. This problem will
be particularly severe if the learned inputs are too similar to one another
(it doesn’t matter if the outputs are similar or even the same). The same
confusion is seen in human memory or in any content-addressable memory.
If you try to remember events that are too similar, you will not be able to
separate them reliably — was that fried egg Friday’s breakfast or Thursday’s?
Unlike breakfasts, the similarity between vectors can be precisely
quantified using the dot product.
Vectors that are completely nonoverlapping are called orthogonal. Two
vectors are orthogonal if the dot product between them is 0. The higher
the dot product, the more the overlap, up to a maximum of n for two
identical n-length −1/1 binary vectors. If the input patterns for an associative
memory are orthogonal, then there is no risk of confusion between
them. Unfortunately, there are only a limited number of orthogonal vectors
of a given size. One can use nonorthogonal vectors to form associative
memories, realizing that if the dot product between memories is too high
there will be confusion between patterns. In general, one wants to train
the network with input patterns that have low overlap (small pair-wise dot
products). After the training phase is complete, the network will work best
when presented with unknown input patterns that have high overlap (large
dot-product) with one of the remembered patterns.
For example, the following three 4-bit vectors are pair-wise orthogonal:
_aT = _1 1 1 1_, _bT = _−1 1 1 −1_, _cT = _−1 −1 1 1_.
This is because _a ・_b = 0, _a ・ _c = 0, and _b ・ _c = 0. We can add up the three
10.2. Memories in an outer product 169
outer product matrices to get a full associative memory matrix:
⎛⎜⎜⎝
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
⎞⎟⎟⎠
+ ⎛⎜⎜⎝
1 −1 −1 1
−1 1 1 −1
−1 1 1 −1
1 −1 −1 1
⎞⎟⎟⎠
+⎛⎜⎜⎝
1 1 −1 −1
1 1 −1 −1
−1 −1 1 1
−1 −1 1 1
⎞⎟⎟⎠
= ⎛⎜⎜⎝
3 1 −1 1
1 3 1 −1
−1 1 3 1
1 −1 1 3
⎞⎟⎟⎠
Typically, we would normalize the network by dividing the final matrix
by the number of memories stored. In this case, we would divide by three
and get:⎛⎜⎜⎝
1 1
3 −1
3
1
3
1
3 1 1
3 −1
3
−1
3
1
3 1 1
3
1
3 −1
3
1
3 1
⎞⎟⎟⎠
This is not a critical step but is useful in the case of networks with large
numbers of remembered patterns that would otherwise give very large
weights.
This autoassociative memory will map each of the input vectors onto
itself, as it is guaranteed to do. Also it does a fairly good job of mapping
incomplete vectors, correctly mapping 100% of vectors with one bit set to
zero and 67% of vectors with two bits set to zero. (In the setting of −1/1
vectors, a zero can be viewed as an indeterminate value.) This is of course
a very small network with limited storage capacity and completion ability.
Since this is such a small network it is easy to illustrate in the stick-andball
form. Fig. 10.1 shows this network with the weights from the original
sum of matrices, without dividing by three. If you set the value of each unit
to a value and then update the other units step-by-step (as was done in
Fig. 6.5), it will converge on the closest remembered pattern. This is about
the smallest associative memory network that can do anything interesting
and about the biggest where you can draw a stick-and-ball picture.
Fig. 10.1 is an example of an autoassociative network with full connectivity
using −1/1 binary units. Networks with these features are called
Hopfield networks after John Hopfield, who explained why they work as
memories. Remember that state space is the multidimensional space in
which you map the entire state vector as a single point (Fig. 6.6). In this
case, the state space is four-dimensional, so it can’t be drawn. An autoassociative
mapping maps a pattern onto itself and therefore defines a fixed
point in state space — once you arrive there, you stay there. In the case
of an associative memory network, every fixed point is a point attractor.
This means that points near the fixed point will be sucked into it as into a
gravitational body. The set of points that will be sucked into a particular
170 10. Associative Memory Networks
a b
c d
3
1
−1
1 1
3
1
−1 −1
1
3
1
1
−1
1
3
Fig. 10.1: Stick and ball diagram of a four-unit neural network
from the summed matrix above. In its weights, this network
carries the bittersweet memories of three orthogonal vectors.
attractor is the basin of attraction for that attractor. From any starting
point in the basin the system will evolve to end up at the point attractors.
Hopfield’s influential contribution, and the reason that he got the network
named after him, was to prove that the fixed points (the memories)
of an autoassociative memories are point attractors. This guarantees that
inputs similar to remembered vectors will move toward these memories.
The process of memory recall in autoassociative memories could then be
thought of as gradient descent in state space. In Fig. 10.1, we have a fourdimensional
state space to map 4-bit vectors. The fixed points formed by
the three learned patterns lie at three of the 16 corners of the hypercube.
Each will have a corresponding basin of attraction. In addition to these
three desirable attractors, the matrix addition will also produce three spurious
attractors. Each of these “false memories” is the negative of one of
the trained memories. In Fig. 10.1, _−1 1 1 −1_ is an attractor and
_1 −1 −1 1_ is a spurious attractor. These attractors are also called
mirror attractors. It is characteristic of Hopfield networks that the negatives
of the memories (think in terms of a photographic negative for a large
bitmap image) will also be attractors.
These attractors and basins of attraction bring us back to the energy
metaphor. In Chap. 9, Section 9.3, we were looking at gradient descent
in weight space (Fig. 9.3). Now, as in Chap. 6, Section 6.4, we are back
to looking at gradient descent in state space. A Hopfield network with
many memories is a landscape defined by the Hopfield energy function:
E = −1
2ΣiΣjwij ・si ・sj , where the s are states, and both i and j count over
10.2. Memories in an outer product 171
all of the units. Although the two Σ’s make this equation look intimidating,
it is easily explained. We are talking about energy in state space after
learning, so all the weights are fixed. For a given connection wab between
units a and b, the product wab ・ sa ・ sb will be large if the weight wab
is large and if units a and b are both active. Notice that these are the
same conditions that were required to make wab large in the first place.
Since heavily connected coactive units will make the product large, state
vectors with lots of heavily connected coactive units will make the entire
summation large. Heavily connected coactive units are Hebbian assemblies.
The energy equation is just a way of associating a large number with each
Hebbian assembly. There is a negative sign at the front of the equation that
turns this large positive number into a large negative number, allowing us
to do gradient descent, climbing downhill to more negative values of energy.
Following pathways of decreasing energy the state vector will creep gradually
downward, descending through valleys into the depths of memory
(Fig. 10.2). Depending on the starting point (the initial conditions), the
network will progress to a different resting place. These resting places are
point attractors that correspond to each of the memories.
In the four-dimensional space of Fig. 10.1, an energy can be calculated
for each location in the space (each state vector). Since a five-dimensional
graph (four dimensions for the vector and one for the energy) can’t be
drawn, I illustrate the concept at a lower dimension in Fig. 10.2. The x, y
values are the states of a two-dimensional vector. The z value gives the
energy for each possible vector. If one starts with a vector near an energy
peak, the update rule will lead down to an energy nadir that is a memory
(arrow). This nadir or energy well is a fixed point attractor. The whole
system works because of two effects: 1) the learning algorithm (Hebbian
learning) produces point attractors, giving the system somewhere to go; and
2) the standard update rule gradually reduces energy so that the network
will get there.
Limit cycles
Autoassociative memories map an input onto itself. By definition they utilize
fixed points. These fixed points must be attractors if the memory
is going to be useful. Heteroassociative memories map one pattern onto
another. Therefore, they do not use fixed points. In general, a recurrent
network can show all kinds of dynamics, including, for analog-unit networks,
chaotic dynamics. A chaotic system will never visit the same point
in state space twice. A binary-unit network cannot be chaotic since there
are a fixed number of states available to the system and it will eventually
have to revisit places it’s already been.
Binary networks can settle into oscillations where the same set of states
are revisited repeatedly. In dynamical system nomenclature, these oscillations
are called limit cycles. At a fixed point the system stays in one place.
172 10. Associative Memory Networks
Fig. 10.2: A made-up memory landscape for a two-dimensional
system. Starting on top of a hill, the update rule will produce
a trajectory moving downhill to recover a memory (arrow).
Compare with one-dimensional landscape of Fig. 9.3.
On a limit cycle it “orbits” in state space, visiting two or more points in
the space. This notion of an orbit is of course abstracted from the concept
of a physical orbit, which visits all points over the course of a revolution.
For example, a simple associative network can be set up by using the
outer product rule to memorize the heteroassociative pair _−1 −1_ →
_1 1_ and the autoassociative pair _−1 1_ → _−1 1_. The resulting
memory matrix, after normalizing by 2, the number of memories, is:
_ 0 −1
−1 0_
Fig. 10.3 shows the simple stick and ball. Since the system is only
two-dimensional, we can graph state space. Because of the heteroassociative
mapping, the network flops back and forth in an oscillation between
_−1 −1_ and _1 1_. This is the limit cycle. The autoassociative mapping
maps _−1 1_ onto itself and produces a fixed point attractor. This
network has three basins of attraction. The first is the basin of attraction
for the remembered vector. This basin is the upper-left quadrant of state
space. The limit cycle has a basin of attraction consisting of the upperright
and lower-left quadrants of state space. Then there is the spurious
attractor at _1 −1_ with its basin in the lower-right quadrant.
Since we restrict ourselves to binary inputs to the network, these basins
don’t have much meaning. The only binary point in each basin is the attractor
itself. We can present the network with analog vectors lying elsewhere
in the basin and they will go to the appropriate attractor. This is partly a
trivial consequence of the squashing function, which will take any analog
vector and map it into the binary vector in the same quadrant. However,
10.2. Memories in an outer product 173
Point attractor at (−1,1)
−1 1
1
−1
Limit cycle between
(−1,−1) and (1,1)
Basin of
attraction
for (−1,1)
Mirror attractor
a b
−1
−1
Fig. 10.3: A simple two-unit mutual inhibition network and the
corresponding state space diagram. The point attractor represents
the autoassociative memory. Half of the limit cycle is the
heteroassociative memory. The network also has two spurious
memories: a mirror of the point attractor at (1,−1), and the other
half of the limit cycle, which maps _1 1_ → _−1 −1_. Only one
of the three basins of attraction is shown, for simplicity.
if we take a point on the basin of attraction for the limit cycle, it will get
sent into the opposite quadrant before returning. For example, the vector
_0.5 0.5_ will jump to _−1 −1_ on the first update.
174 10. Associative Memory Networks
Instantaneous vs. gradual learning and recall
With outer-product summation for learning and synchronous updating for
recall, both learning and recall occur instantaneously and globally. If we
had real neurons or real electronic components doing this, we would require
a central clock that told all the synapses to change at the same time
during learning and all of the units to update at the same time during
recall. This is probably unrealistic for the brain. Hopfield proposed that
the units in an associative network could operate independently. Instead
of all of the units being updated at the same time (synchronous updating),
individual units could update themselves at different times according
to some probability function (asynchronous updating). Alternatively, one
can get gradual recall by using continuous units and changing them slowly
(either synchronously or asynchronously) depending on the results of the
matrix multiplication. Gradual recall occurs slowly as network states gradually
head “downhill” toward one of the stored memories. Additionally, one
can do gradual learning by adjusting weights little by little as individual
patterns are presented.
The use of a Hopfield network for complex images is shown in Fig. 10.4.
This network is constructed using the same outer-product rule used in
Figs. 10.1 and 10.3. However, instead of four or two units, this network has
46,592 units, which are fully interconnected. Hence it has 46, 5922, which
is ∼2 billion synapses. The three stored memories are seen at the bottom
of the figure: Beardsley, bamboo, and Mozart. Each image is 182 × 256 =
46, 592 units, where each unit is shown as either white (−1) or black (+1).
With only three memories in this big network, the attractors are very
attractive. Simple matrix multiplication of any of the images at the top
will recover the memory in one step. To give a better picture of the gradual
downhill slide to the attractors, we have run the system so that it gradually
converges on the result over 10 iterations, of which three are shown. Such
gradual convergence would occur naturally if the attractor landscape were
more crowded, as units will initially be pulled toward different attractors
as they wander downhill. To show the gradual descent here, we had the
associative memory matrix gradually push continuous units (by changing
them a little bit on each iteration) instead of simply setting binary units.
Another way to achieve this gradual settling or relaxation of the network
to each attractor would be to use asynchronous updating.
Fig. 10.4 illustrates two aspects of the behavior of the contentaddressable
memories in general: pattern completion and pattern cleanup.
The center image is incomplete and the network gradually fills in the missing
region. The images on either side are complete but are obscured by
noise. The network removes the noise and restores the pristine image. Both
of these properties are present in the brain. The brain does pattern completion
in vision, filling in the information that is missing from the visual
field due to the blind spot. You are unaware of the blind spot even if you
10.2. Memories in an outer product 175
Iterations
Partial patterns
Final patterns
Fig. 10.4: Gradual settling of a Hopfield network into each of three
attractors (bottom). “Time” (number of iterations) progresses
from top to bottom.
close one eye, and you must test carefully to demonstrate that it is there at
all. An example of the brain’s extracting a signal from noise is the cocktail
party effect: your brain can filter out all the extraneous noise at a party in
order to focus on the relatively weak signal coming from the person you’re
talking to.
176 10. Associative Memory Networks
10.3 Critique of the Hopfield network
There are several problems that occur when we try to apply the Hopfield
network literally to the brain. The one that bothers me the most is that
point attractors are points where activity gets stuck and remains the same.
The electroencephalogram (brain wave recording) demonstrates that the
brain is oscillating continuously. Intuitively, our thoughts seem to flit easily
from one topic to another, sometimes having trouble staying focused long
enough to finish an exam or, later in life, an IRS form. On the other hand,
some people have obsessive-compulsive disorder and get stuck all the time.
In the Hopfield network, we can deal with the problem of getting stuck
by saying that after the attractor is reached (or just approached), the brain
heats up (produces more noise) and kicks the network out of the attractor.
This then allows the network to descend toward another attractor. (Heat,
energy, and noise are all closely related both mathematically and physically.)
Maybe attention-deficit disorder is a brain that’s too hot (too noisy)
and obsessive-compulsive disorder is a brain that’s too cold. Alternatively,
one can get away from the point attractor notion entirely and build similar
networks that depend on convergence onto limit cycles or even onto strange
attractors.
Another problem with the Hopfield network is that it works best with
−1/1 vectors. This means we need negative state values. In the context of
spiking neurons this is interpreted as firing rates below the spontaneous
rates of regular firing neurons. As we noted in Chap. 6, rate coding does
not seem like a valid model for neurons that fire rarely or fire in bursts. One
can build Hopfield networks using 0/1 vectors, but they have much lower
storage capacity due to crosstalk, the tendency of patterns to bleed into
one another as more memories are stuffed into a network. This, in turn,
can be solved with additional mathematical jiggering. However, at some
point all the mathematical manipulations begin to overwhelm the model.
In software marketing, when customers complain about a bug in the
program, the marketers check with the software engineers. The engineers
explain that though it seems like a bug, it’s really a feature, something good
that they put there purposely. Similarly, in neural networks, attributes that
seem like a problem from one perspective may have benefits from another.
For example, crosstalk is generally taken to be a bug. You stuff more and
more memories into an associative neural network and they start to get
mixed up. An image that should go to one attractor instead goes to another
or to a spurious attractor. On the other hand, crosstalk has some
appeal as the basis of mental association. One thing reminds you of another,
or becomes a metaphor for something. Such crosstalk linking would
be heteroassociative recall without explicit heteroassociative learning. So
crosstalk, as a feature, is a candidate for a form of cognition. It’s not clear
how to put together a neural network so as to get fruitful crosstalk without
fruitless confusion.
10.4. Summary and thoughts 177
In real life, memories are formed through experience, forward in time,
one on top of another. Another source of crosstalk arises when we attempt
to replace the unbiological instantaneous outer-product matrix summation
with Hebbian rules applied to incoming memories. Basically, a new memory
that has any overlap with an old memory will tend to activate the
old memory. The old memory will then be dragged into the learning and
a hodge-podge of old and new will be learned. Various algorithms have
been developed to prevent this mixed learning. The simplest is to suppress
all spread of activity during learning. Then activity spread has to be
turned on again for recall. There is some evidence that something like this
might actually occur in the brain. These issues will be explored further in
Chap. 14.
10.4 Summary and thoughts
Learning can be defined as the registering of associations. Memory can then
be defined as registering the same association after a delay. By utilizing
the convenient fact that the number 1 maps onto itself (i.e., 1 ・ 1 = 1),
arithmetic associativity provides a way of restoring binary vectors from
either partial versions of themselves (autoassociativity) or other vectors
(heteroassociativity). This yields what is currently the standard paradigm
for memory storage — storage is in the synapses.
This viewpoint is supported by the fact that a variant of Hebb’s rule can
be demonstrated biologically. Long-term potentiation (LTP) is a biological
process defined by a sustained synaptic strength increase following coactivation
of a presynaptic cell and a postsynaptic cell. In the original Hebb’s
rule, an increase in synaptic strength occurred only if the postsynaptic cell
fired. This is not a requirement in LTP and may not even be sufficient to
produce LTP in many cells. Additionally, there are many variants of LTP
that have somewhat different properties.
The cornerstone of the Hebb hypothesis is the Hebb assembly rather
than the Hebb rule. Cell assemblies cannot be readily demonstrated biologically
due to technical limitations. We have no way to measure activity
in thousands or tens of thousands of neurons in a behaving, learning animal
in order to show that certain subsets fire together in association with
particular stimuli or while thinking about those stimuli later. Evidence for
some small cell assemblies have been found in the early stages of smell
processing (olfaction), however.
An alternative to classical Hebbian assemblies that has been popular
lately is the synchronized-firing assembly. The classical Hebbian assembly
depends on a chain reaction where cells recruit other cells in a repetitive
firing mode until a large group of cells are all firing. A synchronized cell
assembly would involve a group of cells that fire a single spike at the same
178 10. Associative Memory Networks
time. In that case, there would be no time for one cell to activate another in
the assembly. It is not clear what role Hebb’s rule would have in permitting
synchronized assemblies. Such synchronization has been demonstrated in
the visual and motor systems.
Although synchronized-firing assemblies are a popular alternative, it’s
probably fair to say that Hebb’s rule and Hebb assemblies remain the dominant
paradigm at present, partly because they can be so well worked out
mathematically. However, there remains a huge divide between the elegant
simplicity of outer products and energy fields and the elegant complexity
of idiosyncratic dendritic trees, myriad neurotransmitters and second messengers,
and all that other stuff that creatures actually think with (before
it’s ground into slop to feed to other creatures). That is why I now finish
up with the nice neat reasonable stuff and dive into the slop in the next
chapter.
Part IV
Brains
Brains 181
An electrode is placed inside a neuron. It goes through a membrane to enter
the salt solution inside. On the outside of the cell the electrode showed the
same potential as the grounded reference electrode. Once inside, the voltage
drops suddenly to about −75 mV. If we record for a while we may see
signals: little 10-mV bumps going up or down and big, but brief, up-going
100-mV spikes. We also see a lot of higher frequency noisy stuff. Some of
this is an artifact of our recording equipment: it is easy to accidently pick
up nearby radio stations on an electrode. Some of it is real, showing the
opening and closing of small pores in the membrane or reflecting signals
from far-off neurons. An electrode placed outside of the neuron will pick up
more of these far-off units, eventually providing a mixed-up signal of many
neurons combined.
Imagine yourself as the prototypical green alien visitor guy. After a visit
to area 51 and an evening spent performing unspeakable procedures on
sleeping humans, you will naturally want to monitor electrical activity on
earth. You will assume, correctly, that much of this electrical activity has
to do with information processing and communication. But you will have
a heck of a hard time pulling apart any of the signals so as to interpret
them. One big problem is that there are multiple protocols in use, both the
electronic languages used to encode and decode signals and the underlying
human symbolic languages. Apparently simple signals may actually be
complex multiplexed signals, meaning that there are many different signals
mixed up inside what appears to be a single signal. Also on top of all of
this you have loads of noise, a mix of distant signals, and the background
radiation that comes from all electronic devices (and from outer space).
It’s hard being an alien.
From concept to data
In the foregoing chapters, I presented a view of the nervous system as a
well-built, rational, comprehensible machine. The theme of the following
chapters is that the nervous system is a hodge-podge, a hack, and a puzzle.
This perspective shift takes us from concept-based theory to data-based
theory, from what might be there to what is there. This shift is disconcerting
since it’s natural to feel that these reasonable theories of the prior
chapters really ought to be there, because they make so much sense. In
this way, the theories tend to take on a virtual life of their own, influencing
what is observed. To some extent, this is right and good since it is hopeless
to go mucking around in the brain without any plan or guidance, perhaps
counting red blood cells instead of counting neurons. When taken too far,
however, this reliance on theory leads to what has been called “the heartbreak
of premature definition.” If taken too seriously, the theory provides
blinders rather than guidance.
182 Brains
Current theories of neural information processing are probably inadequate.
We all await the grand paradigm shift, when we throw over the old
theories and enjoy the revolution. In some paradigm shifts, for example
the Copernican revolution (the paradigmatic paradigm shift), old theories
are completely discredited and discarded. In others, for example, the Bohr
atom to quantum mechanics, initial adolescent iconoclasm gives way to an
appreciation that the old theory maybe wasn’t so bad after all. This is
meant to be reassuring; maybe you didn’t entirely waste your time reading
the first half of the book. On the other hand, maybe you did. Sorry, I
should have mentioned that earlier.
Leaving high-level theorizing behind, we look at the guts and volts of
the brain. Unfortunately, most of these details can’t be understood at the
big-picture level.We replace the top-down conceptual world with the messy
bottom-up world. There are still concepts, but they are limited in scope
and don’t pretend to answer major questions. We give up the comfortable
heft of neatly encapsulated concepts well understood. We trade this for the
confidence of knowing about real things.
Isaiah Berlin, in a humanities context, described a corresponding contrast
in perspectives by quoting a Greek poet (Archilochus) who wrote: “The fox
knows many things, but the hedgehog knows one big thing.” In his famous
essay entitled “The Hedgehog and the Fox,” Berlin compares “those, on
one side, who relate everything to a single central vision” to “those who
pursue many ends, often unrelated and even contradictory.” The first half
of this book was meant to appeal to hedgehogs (and physicists), describing
coherent ideas. The second half is for foxes (and biologists), presenting lots
of little ideas that are intriguing in themselves but don’t really add up.
Neurons are cells
A central example of this discrepancy between happy fancy and hard reality
is found at the level of the single cell. The happy-go-lucky single cell we
met in neural networks has a primitive appeal reminiscent of some ancient
cave painting (Fig. 11.1). The real guy whom we are about to meet looks
hopelessly organic — tree rather than I-beam. The artificial guy is made
of numbers, while the real guy is soap, water, and salt.
A neuron is a cell, too. Before it can do any seeing or signaling or thinking,
a neuron has to take care of itself. This means it has the same enormous
machinery for energy metabolism, protection against toxins, and communication
both intracellular and extracellular as other cells in the body with
more prosaic job descriptions. When we find out about some remarkable signaling
mechanism available to neurons, it is generally the case that kidney
and liver cells have their own versions of the same mechanism.
Brains 183
v
Fig. 11.1: The mythical neuron meets its counterpart.
All of the machinery of the cell is entwined with neural information
processing technology. Chemicals passed around by the cell to obtain energy
during metabolism are reused for communication. Three of the basic
food groups for cells — glucose, adenosine triphosphate (ATP), and acetylcoenzyme
A (acetyl-CoA)—are either used directly or have close congeners
that are used for information processing. The major neurotransmitters,
glutamate and GABA (gamma-aminobutyric acid), are spin-offs from the
tricarboxylic acid cycle (Krebs cycle), the main sugar digestion route. Similarly,
ACh (acetylcholine) is half brother to acetyl-CoA, the main product
of the Krebs cycle. Glucose is processed in order to produce ATP, the
main energy storage medium. ATP is also a neurotransmitter. Various ATP
by-products — adenosine diphosphate (ADP), adenosine monophosphate
(AMP), cyclic AMP (cAMP) — are also used in signaling. Teleologically,
one can imagine that shortly after the simple single-cell creatures of long
ago learned to smell food, they began to use the same receptors to commune
among themselves. This would gradually evolve into the sophisticated
food-based data processing technology of our own brains. Are you thinking
about food or is your food thinking about you?
This sharing of resources is not confined to information processing. The
same compounds or their close relatives are also amino acids, the construction
materials of the cell and the body. Both glutamate and GABA are close
relatives of amino acids. Glycine, another neurotransmitter, is an amino
acid. Glycine is the major constituent of collagen, making it the building
block for skin and bones. Similarly, the nucleotides of DNA, the blueprint
of the cell whose code is used to construct proteins, is also shared with
184 Brains
both neurotransmission and metabolism (e.g., ATP). Cell maintenance also
requires additional intracellular communication between and among various
organelles (subcellular organs), as well as extracellular communication
with various supporting cells. This type of communication cannot always
be cleanly separated from classic neurotransmission. For example, during
development neurotransmitters and second messengers are being used to
grow neurons and coordinate their wiring and relations with various supporting
cells. While a child is using his transmitters to grow a brain, he
also has to think using the same transmitters.
Because of this close enmeshment of functions, it may never be possible
to cleanly separate fancy neural information processing from the boring
information processing of housekeeping chores. Even the electrical charge at
the cell membrane, the key attribute that permits action potential signaling
between cells, has generic cell maintenance tasks as well. Similar electrical
potentials are present in all body cells and in yeast and bacteria as well.
Information transmission (and related encoding and decoding) has to be
done in every organ. Bone cells communicate in order to adapt to changes
in stress patterns when you learn to rollerblade. The liver and endocrine
systems all have complex nonneural communication protocols in place. The
immune system has a remarkable interplay of cell types that chat with one
another and with other cells in the body, all of which have to continually
remind white blood cells that they belong there and should not be eaten.
What is the neuron state?
In previous chapters, a single number (a scalar) was used to define the
state of a neuron. An implicit assumption of the scalar model is that we
are dealing with the functional equivalent of a point neuron — a neuron
that has no geometry or spatial extent. Of course, real neurons have a
complex three-dimensional structure. Some of them are big enough to be
seen with the naked eye. Signals come in at dendrites that may extend as
much as a millimeter from the central soma (cell body). Signals then go
out through axon terminals that may be more than a meter away from
the cell body (e.g., the axon that goes to your big toe). However, from a
signal-processing standpoint, what is important is not the physical size of a
neuron but its electrical size — how far across the cell can a signal spread.
Given the many different morphologies and different electrical properties
of dendritic trees, some will turn out to be electrically large and others
electrically compact, concording with the role they have to play. Axons,
on the other hand, are all effectively compact; the action potential ensures
that a signal that starts at one end will get to the other.
In addition to being three-dimensional in shape, neurons are also multidimensional
in terms of the many different kinds of signals that are used.
Brains 185
Many types of electrical signals and a variety of chemicals participate in
neural signaling. These include the neurotransmitters that transmit information
across synapses as well as a variety of second (and third, and fourth)
messengers that transmit the signal further on inside the neuron. Many of
these chemical signals are likely to be important in information processing.
Thus, rather than think of neural state as a scalar, it might make
more sense to consider a vector of voltages and chemical concentrations to
describe the cell.
Although the chemical and electrical complexity of neurons makes it clear
that there is no single scalar state, it may be that multiple states occur in
series as a temporal chain. In this case, each state determines the next,
and any of the states could be used to represent the information processing
state. Serial states are assumed in rate-coding theory: presynaptic firing
rate determines synaptic potential determines soma potential determines
soma firing rate determines axon firing rate. We can treat these different
states as if they were the states of independent units and work out the signal
transduction (weights) between them. In the next two chapters, we do this
analysis; working out how synaptic inputs determine summed membrane
potential and how summed membrane potential determines spike rate.
On the other hand, if multiple states are operating in parallel, then the
neuron is more like a computer central processing unit (CPU) than like a
transistor. In that case, analysis of single-neuron information processing becomes
far more difficult. Many neurons have a single major output pathway,
the axon. Even if such cells are processing multiple information streams in
parallel, all of the information has to come through the axon. We can then
take this axon output to be neuron state. However, this output state would
just be the answer to whatever computational problem the neuron was
calculating. We would have missed all the information processing.
Still more complex are neurons that have multiple inputs and multiple
outputs. Thalamic cells are a prominent example of this in mammalian
brains. The thalamic cell is likely not only multiprocessing but also multiplexing.
Multiple signals come into the thalamic cell, multiple signals are
spit out. Some of these inputs combine, others remain separate. As long
as they are separate, the states can be understood separately. When the
inputs combine into a single measurable state, this is multiplexing. The
causes and results of such a signal can be hard to disentangle. If single
neuron processing turns out to be this complex, then techniques for analyzing
activity in the single neuron will be similar to the techniques that
we currently use to analyze a neural network.
No scalar state; how about a scalar weight?
In artificial neural networks, scalars were used to define neuron states.
Scalars were also used to define the weights between any two neurons.
The use of a scalar weight simplified learning theory, since learning was
186 Brains
Presynaptic Postsynaptic
axon
dendrite
Spine
Vesicles
Neurotransmitter
Receptors
spine
Fig. 11.2: Basic synapse design: presynaptic vesicles (big dots) release
neurotransmitters (little dots) into the synaptic cleft. They
float across the cleft to activate postsynaptic receptors (fat lines).
expressed as an increase or decrease in this single number. As you might
guess, the scalar-weight concept will also need some revision. Basic synapse
design looks pretty simple (Fig. 11.2). An electrical signal, an action potential,
invades the presynaptic axon terminal. This causes vesicles to release
neurotransmitter into the synaptic cleft between the neurons. The neurotransmitter
molecules float across and bind to receptors on the postsynaptic
neuron. These receptors are linked to channels that open up and create an
electrical signal, a postsynaptic potential, on the membrane of a spine or
dendrite of the postsynaptic neuron.
Although there are multiple processes in this description, the synapse can
still be assigned a single weight as long as these processes are occurring in
series. Just as with our description of neural state, we would then define
a scalar value for each process in sequence and determine transduction
between them. Again things gain in complexity if there is multiplexing
going on at the synapse. Just as there is a wide variety of neuron types,
there are also many different types of synapses, differing in their complexity.
There are various sources of synaptic complexity. Many synapses have
more than one kind of receptor postsynaptically. Some synapses release
more than one neurotransmitter. Some synapses are electrical rather than
chemical connections. Some synapses are not strictly one-way but have
retrograde transmission or are involved in perverse three-ways (thalamic
synaptic triads). Some axons synapse onto postsynaptic spines, as shown
in Fig. 11.2. The function of spines is a mystery; it seems likely that they
don’t provide a straight conduit to the main dendrite. Some researchers
Brains 187
suspect that significant chemical information processing may be happening
inside these spines. Many neurotransmitters don’t just head straight across
the synaptic cleft but instead spread far across neural tissue. This is called
volume transmission. Some nonclassical neurotransmitters are gases. Gases
can pass straight through neural tissue, expanding in a cloud from their
release site.
Different modeling tools
Another distinction between the first and second half of the book is the set
of tools used. The tools of the top-down approach were primarily those of
discrete mathematics. “Discrete” here refers to the use of distinct digital
numeric values instead of continuous analog values. Discrete math is the
math of computer science and digital electronics: binary numbers, Boolean
algebra, transistor-transistor logic.
By contrast, in the following chapters we mostly will be dealing with
continuous analog phenomenon. For continuous mathematics, calculus is
the primary tool. We discuss electronics and explain the mathematical descriptions
of capacitors, resistors, and batteries. As it happens, the math
of electronics is identical to the math of dynamics, the study of movement.
This is the original realm of calculus, beginning with Isaac and the
proverbial falling apple.
Whoops
The foregoing was meant to be an introduction to the realistic neural modeling
that will be discussed in the rest of the book. However, I have to
backpedal a bit here. I emphasized the importance of chemical signals. I
will now proceed to ignore them. The details of chemical signaling are not
well understood, and there hasn’t been very much modeling done on this.
Therefore, the emphasis in what follows will be on electrical modeling.
I also spent some of the foregoing maligning scalar state and scalar weight
theories. I take that back, too. These remain the standard theories because
they are still the best theories. Therefore, rather than abandoning these
theories, I will explore their implications.
11
From Soap to Volts
11.1 Why learn this?
In this chapter, I introduce the hardware of the brain. I develop some of the
concepts of neural membrane modeling. I show how currents, capacitors,
and resistors arise from the interactions of salt water with membranes. To
demonstrate this, I have to address the relatively hard concept of capacitance,
which must be handled with calculus. I explain some basic calculus
algebraically by using numerical, rather than analytic, methods. Using
numerical calculus as a simulation tool, I then explore a couple of fundamental
concepts of neural signal processing: temporal summation and
slow potential theory.
Moving quickly from soap to volt to signal processing requires that I
skip over some key concepts and issues that will be addressed in later
chapters. As an alternative approach, I could have introduced all of the
underlying material first and then moved on to discuss specific models and
what they mean. I chose the get-there-quick approach for two reasons. First,
delayed gratification is hard: the technical and algebraic details can get
pretty dull. Applications and ideas brighten this chapter up a bit. Second,
making the right simplifications is a major part of modeling. The use of
a stripped-down model in this chapter demonstrates how ignoring some
details can help focus attention on those details that are essential for the
issue at hand. Specifically, I demonstrate why time constants are critical
for understanding neural signal processing.
190 11. From Soap to Volts
In the following chapters, I fill in the details. In this chapter, the circuit
only has one resistor and one capacitor. In the following chapters, I put
in batteries and a few more resistors. In this chapter, I only use injected
currents as signals. In the following chapters, I show how biological signals
are largely conductance changes that cause current to flow secondarily.
In this chapter, I have also tried to avoid stumbling over units, the volts,
amperes, farads, siemens, hertz, and other famous guys immortalized as
stuff. Units are unloved and underappreciated (see Chap. 16, Section 16.2).
Sizes, durations, and magnitudes offer insights about the limits, capacities,
and capabilities of the hardware. Unfortunately, focusing on the units now
will entangle us in a nest of confusing conversions. I have largely avoided
this distraction.
11.2 Basic cell design
Cell design is based on the separating of inside from outside by means of
membranes. Prokaryotes, bacteria, are single cells with one big compartment.
In eukaryotes, like us, each cell has many separate subcompartments
doing separate tasks. Most of the body is salt water. Water and salt can’t
pass through fat since oil and water don’t mix. Soap is the compound that
connects oil and water, thereby allowing showers to wash oily stuff off your
skin. Soap works by having a fatty part that connects with the dirt and a
hydrophilic (water loving) part. Fat is hydrophobic (water fearing). Soap
thus provides a link that allows water to drag oil away.
Soap bubbles form with the fatty part of soap pointing inside and outside,
away from the water that stays on the interior of the bubble membrane
(Fig. 11.3). Cell membranes are called lipid bilayers. They are configured
in the opposite way: the fatty part is in the interior of the membrane, and
hydrophilic heads point both in toward the cytoplasm and out toward extracellular
space. Just as soap bubbles form spontaneously on the surface of
soapy water, lipid bilayers form spontaneously from a mixture of phospholipids
(biological soap) and water. The inside of the cell and the interior
of the membrane are not the same thing. The cell is filled with a volume of
intracellular solution (cytoplasm) that the membrane separates from the
extracellular solution outside of the cell. Although the membrane is only a
thin layer at the surface of the cell, there is enough room in the interior of
the membrane for chemicals to float around and react with one another.
Only hydrophobic compounds can exist in the interior of the membrane.
Ions and many proteins are hydrophilic. Ions are the charged versions of
certain elements, such as sodium, potassium, chloride, and calcium. They
are charged because they have either lost or gained an electron, which
carries a negative charge. The little plus or minus signs tell you how many
electrons were lost or gained: Na+, K+, Cl−, Ca++. Positive and negative
11.2. Basic cell design 191
Soap bubble:
water in membrane
Biological membrane
pore
extracellular: water outside
intracellular:
water inside
Soap:
hydrophobic tail
hydrophilic head
Fig. 11.3: Soap bubbles have fatty tails pointing in and out and
water (shaded) interior. Biological membranes keep the fatty tails
interior with hydrophilic heads sticking in and out. Proteins orient
themselves using both hydrophilic and hydrophobic amino
acids. They can form pores that allow ions to move between
intracellular and extracellular space.
ions stick together to make a salt; NaCl is table salt. Many proteins are
charged as well. Charge makes a compound hydrophilic since the hydrogen
of H2O will stick comfortably to a negative charge and the oxygen will
stick comfortably to a positive charge. Fat will stick only to uncharged
molecules. Therefore, ions and charged proteins can move around freely
192 11. From Soap to Volts
in either extracellular or intracellular space but can’t pass through the
fatty part of the membrane. Some proteins and various other compounds
are hydrophobic and can float around the interior of the membrane but
not move into the water on either side. There are also information-carrying
compounds that are gases. These don’t care whether they’re in fat or water.
They can pass through anything. The most well known of these is nitric
oxide (NO), a former molecule-of-the-year winner and the functional basis
of Viagra.
Proteins are the active components of cells, involved in metabolism, cell
reproduction, and practically every other function. Transporters and pores
are transmembrane proteins (e.g., the pore in Fig. 11.3). Although ions
cannot move directly across the fatty membrane, they can flow through
these pores. Ions will move passively from a high concentration to low
concentration, a process of diffusion down a concentration gradient. Other
proteins can provide ion pumps or transporters that push ions up against a
concentration gradient. Additionally, proteins provide transport and structure
within both the extracellular and intracellular spaces. Although the
extracellular and intracellular space are usually thought of as being salt
water, the large amount of protein gives it structure, making it more like
jello than seawater.
11.3 Morphing soap and salt to batteries and
resistors
Descriptions like the above are both model and metaphor: the extracellular
space is jello, the membrane is an inside-out soap bubble, the proteins form
holes. By switching to another representation, another language, we can describe
the same thing in different terms. A particular model/metaphor will
be better than another for a particular purpose. In this case, translation of
the model into the language of electronics (batteries, resistors, capacitors)
will allow us to describe neural signaling much more easily than we would
be able to do just chewing the salt and the fat.
Morphing from a membrane and salt representation to an electrical engineering
representation obscures the biology somewhat. Any representation
offers trade-offs and compromises. The advantage of the electrical representation
is that we can now use the many mathematical tools developed
for electrical circuit analysis. Ions moving through water carry charge. The
movement of charge is electrical current. Current can flow through the cytoplasm
and extracellular fluid freely. The pores in the membrane provide
conduits (conductors) for current to go through the membrane as well.
Protein pores of a particular type provide conductors through the membrane
that are selective for a particular ion. These pores are parallel
conductors providing parallel paths for current. A conductor is defined
11.3. Morphing soap and salt to batteries and resistors 193
by its tendency to permit the flow of current. A resistor is defined by its
tendency to resist the flow of current. A resistor and a conductor are the
same thing. It’s just one of those pessimist-optimist things: half-empty or
half-full; resisting or conducting. Resistance is expressed in ohms (Ω) and
is represented by R. Conductance is measured in siemens (S) and is represented
by g. Mathematically, conductance is the inverse of resistance,
g = 1
R. Because of this, siemens, the unit for conductance, is also called
“mho,” which is ohm spelled backward (seriously).
In the circuit diagram of Fig. 11.4, all of the pores of a single type are
lumped together as a single big conductor (resistor). This ubiquitous conductance
is known as the leak conductance (gleak or Rleak). Later, when we
add different types of pores to the circuit, we will have different conductors
in parallel with this one.
Made of fat, the membrane acts as an insulator. Current that does not
flow through one of the pores will just sit next to the membrane. Charge
acts at a distance to attract unlike charge or repel like charge. Charge that
is sitting on one side of the membrane will cause equal but opposite charge
to sit on the other side of the membrane. This phenomenon is known as
capacitance. In electronics, a capacitor is built by placing an insulating
material between two parallel metal plates that are attached to the wire
leads of the capacitor. Since these plates do not touch, electricity cannot
pass directly through the capacitor. However, electricity can flow indirectly
as one plate induces electrical flow in the other plate. The two parallel lines
in the standard symbol for the capacitor (Fig. 11.4) represent these two
plates. In the biological situation, the two plates are just thin accumulations
of ions in the water adjacent to either side of the membrane.
Capacitance has this name because it is a measure of the capacity of
the plates to hold charge. This capacity will have to do with the size of
the plates, the distance separating them, and the nature of the material
between them (its dielectric constant). A capacitor, or membrane, with high
capacitance has the ability to hold a lot of charge with only a small voltage
difference between the plates. Voltage times charge is a measure of energy.
A high capacitance means that charge can be stored easily, i.e., without
requiring as much energy. As you put more voltage across a capacitor, it will
hold more charge and more energy. A capacitor “expands” with voltage, as
a balloon expands with pressure. Capacitance tells how much the capacitor
can hold at a given voltage, just as a capacity measure for a balloon would
tell how big the balloon would be at a given pressure.
A resistor (R) and capacitor (C) placed next to each other (in parallel) is
called an RC circuit (Fig. 11.4). In electrophysiology, we insert an electrode
through the membrane and then inject current into the inside of the cell
(arrow in Fig. 11.4). This current will pass out of the cell through the
resistor or the capacitor (through the pores or via the capacitance). Once
the current reaches the outside of the cell it will disperse to the rest of the
body and the surrounding world. In the diagram this is represented by the
194 11. From Soap to Volts
capacitor
conductor
(resistor)
Inside
Iin
Outside
(ground)
Fig. 11.4: The membrane is represented as an RC (resistorcapacitor)
circuit. Current can be injected inside the cell (arrow)
by using a hollow glass electrode.
ground symbol. The human body is big enough and the currents are small
enough that the body can easily sink (conduct away) the current that cells
produce. When we put a cell in a dish and inject current, we place a wire
in the dish to conduct the current out to ground. This wire is attached to
something metal that leads to the physical ground. This is the idea that
Ben Franklin came up with for lightning — provide an electrode to lead
current harmlessly to suburbia.
11.4 Converting the RC circuit into an equation
Modeling the neuron as an electrical circuit involves predicting voltage
changes based on current flow. One can also model an electrical circuit the
11.4. Converting the RC circuit into an equation 195
other way around: given the voltages, predict the currents. In the case of
neurons we typically are injecting currents, or thinking about the effects of
extra pores that cause extra current to flow into the cells. Here are three
major electrical laws needed for neural modeling:
• Kirchhoff’s law: Conservation of charge. Charge does not disappear
or appear out of nowhere.
• Ohm’s law: V = I ・ R. The voltage (V ) drop across a resistor (R) is
proportional to the current (I).
• Capacitance: Q = C ・ V . The charge (Q) on a capacitor (C) is
proportional to voltage (V ).
In electronics, Ohm’s law is usually expressed in terms of resistance, while
in biology it is more often expressed in terms of conductance. Since g = 1
R
(not a law, just a definition), Ohm’s law can be expressed as V = I/g.
Turning this around gives I = g ・ V . This is the preferred formula for
biology. When we talk about pores opening up, it is most natural to think
of them as conductors of ions rather than as resistors to ion flow. The ability
to conduct is given by the magnitude of g. Ohm’s law states that a bigger
conductor will allow more current to flow down a given voltage gradient.
g will be a constant for a simple resistor but will vary for a rheostat, a
variable resistor.
We can’t relate these three laws directly in the forms given above. Ohm’s
law is expressed in terms of current, while the other two laws are expressed
in terms of charge. Current is the movement of charge. We can use this fact
to rewrite the latter two laws in terms of current. This is easy to understand
for Kirchhoff’s law: charge that is moving has to go somewhere. Therefore,
any current that comes out of one wire must go into some other wire. Where
one wire feeds two other wires, the sum of currents in the two will equal
the current that came out of the one. Conservation of charge translates
into conservation of current. This is the translation for Kirchhoff’s law.
Translating the capacitance law from charge into current is a little harder,
requiring that we use calculus.
Capacitance and current
In Chap. 16, Section 16.5, I introduce several notations commonly used
in calculus. For present purposes, I stick with a single numerical notation,
suited to numerical analysis by computer. The capacitance law given above
is Q = CV . Current, I, is the change in charge, Q, with time. Let’s look at
the charge on a capacitor plate at two times, time ta and a later time tb,
separated by a small duration. At time ta, the charge on the plate is Qa.
At time tb, it is Qb. The amount of charge that has moved off of the plate
during this time is Qb−Qa. The current, I, is Qb−Qa
tb−ta
, the change in charge
divided by the change in time. In mathematics, the Greek letter delta (Δ)
196 11. From Soap to Volts
is a short-hand notation used to mean “change.” ΔQ = Qb−Qa means the
change in charge on the capacitor. Δt = tb − ta is the time interval. So we
can write the expression for current as I = ΔQ
Δt . ΔQ divided by Δt is the
rate of change of charge — how fast charge changes.
Now we can consider the capacitance equation: Q = C ・ V . At time ta,
Qa = C ・ Va and at time tb, Qb = C ・ Vb. If we subtract these two equations
we get Qb −Qa = C ・ (Vb −Va). In between these two measurement points,
tb − ta of time has passed. We divide both sides by the time duration to
get Qb−Qa
tb−ta
= C ・ Vb−Va
tb−ta
. Using the Δ notation, this is ΔQ
Δt = C ・ ΔV
Δt . Since
I = ΔQ
Δt , the law of capacitance can be expressed in terms of current as
I = C ・ ΔV
Δt .
Current through a capacitor equals capacitance times the rate of change
of voltage. What does this equation mean? If voltage is changing fast, then
a lot of current is passing through the capacitor (high I). When the voltage
stops changing, the capacitor will stop carrying current. A capacitor can be
used to store charge that can then be released abruptly as a large current.
An example is the capacitor used for the flash bulb in modern cameras.
This capacitor has high capacitance. A little camera battery with its little
voltage can place a lot of charge on this capacitor. Capacitative charging
is accompanied by a humming noise, as the two plates vibrate slightly.
Once the capacitor reaches the battery’s voltage, the charge will sit on the
capacitor until needed. Pushing the shutter button suddenly connects the
capacitor to ground, meaning that voltage changes abruptly from battery
voltage to zero volts. This is a large ΔV so it causes a much larger current
than could be generated by the battery directly. This large current allows
the bulb filament to release a large amount of energy (Q2 ・ R/s where R is
the resistance of the filament) suddenly as a bright light.
Adding up the currents
We can now use our three laws to calculate what will happen in an RC
circuit when the current (Iin) is injected. Kirchhoff’s law says that charge
is conserved so the injected current has to go somewhere. It will go either
through the capacitor or though the resistor to get to ground. From there,
the current runs away to the rest of the body and the world beyond. We
can express this conservation of current arithmetically as an addition Iin =
IC + IR. Iin is a known number, let’s say 1 μA (microamp). IC and IR
can be re-expressed using the current-voltage equations for capacitance and
resistance. Therefore:
Iin = C ・
ΔV
Δt
+ g ・ V
With Δt set to an infinitesimal (infinitely small) value, this equation is
called a differential equation. Differential equations of various sorts are
one of the main topics in calculus. If you are familiar with calculus, you
11.4. Converting the RC circuit into an equation 197
may recognize this as a standard differential equation that can be solved
analytically in order to get the precise relationship between V and I. We
solve it using a numerical integration, getting very close to the analytic
solution by making Δt very small. By using finite numbers and finite time
steps, we create a discrete approximation to the differential equation.
We set Δt = 1 μs = 0.001 ms = 1 ・ 10−6 s. We can expand out ΔV as a
difference: V + − V . The negative sign in V + is not an exponential but is
used to denote voltage in the future. Plain V means voltage in the present.
V − would mean voltage in the past. The reason that we use a superscript
here instead of a subscript is that the subscript is typically used to indicate
the location in a neuron where the voltage is measured.
Substituting for ΔV leaves us with:
Iin = C ・
V + − V
Δt
+ g ・ V
Notice that the voltage multiplied by g is also V ; future voltage will be
based on present voltage. This is called the explicit Euler integration.
There is also an implicit Euler equation, which is described in Chap. 16,
Section 16.5.We can now gather up the present voltage terms, V , separating
them from the future voltage V +.
Iin = V + ・
C
Δt
+ V ・
C
Δt
+ g ・ V
and solve for current voltage in terms of past voltage:
V + = (1 − Δt ・
g
C
) ・ V +Δt ・
Iin
C
This is an update equation, similar to the ones we discussed in the context
of neural networks. The voltage (V +) is updated at each time step based
on the previous voltage V . At the same time, time, t, is updated by simply
adding Δt: t+ = t+Δt. The new voltage then becomes the V and the
new time the t for the next update step. We typically store all of the V s
as we calculate them. We can then graph voltage against time where time
increases by Δt at each update.
Let’s look at the numbers. Units and unit analysis are a big part of solving
these equations. I will use standard units, but avoid worrying about them
by making sure everything divides out neatly. The horror and beauty of unit
conversion is discussed in Chap. 16, Section 16.2. As is standard for neuron
simulation, I use units of millivolts (mV) for potential and milliseconds
(ms) for time. We set g = 1mS
cm2 and C = 1 μF
cm2 . (The common use of
square centimeter for these units contrasted with the usual use of microns
for neuron size is an example of why units are such a nuisance. There
are 100,000,000 square microns in a square centimeter.) Using the chosen
values, g
C = 1/ms. Similarly, we will set the injected current to 1 μA
cm2 so
that Iin
C = 1 mV/ms. We’ll set Δt = 0.001 ms. With these convenient
198 11. From Soap to Volts
choices, the update equation is
V + = (1 − 0.001) ・ V + 0.001
At every step, V + will be updated according to this rule and t+ will be
updated by Δt. So the update for the full simulation requires equations for
updating both t and V:
t = t + 0.001
V = 0.999 ・ V + 0.001
Notice that I have now left out the superscripts for V and t. This is common
practice since this is the form the update rules will take when entered into
a computer program.
We now just need a starting point, called the initial condition, and we
can simulate. We’ll start with V = 0 at t = 0. Then at t = 0.001, V = 0.001
and at t = 0.002, V = 0.999 ・ 0.001 + 0.001 = 0.001999. Values during the
first 10 μs are
t (ms) V (mV)
0 0
0.001 0.001
0.002 0.001999
0.003 0.002997
0.004 0.003994
0.005 0.00499001
0.006 0.00598502
0.007 0.00697903
0.008 0.00797206
0.009 0.00896408
0.010 0.00995512
as shown graphically over 5 ms of simulation in Fig. 11.5.
11.5 Parameter dependence
We are interested in how the properties of a neuron provide its information
processing capabilities. In Fig. 11.5, a continuously injected current
provides an input signal. The response of the membrane (the “system”
in signals and systems parlance) depends on fundamental properties:
membrane resistance and membrane capacitance. As we move to more complicated
models, we find that membrane conductance (g) can change with
different inputs or in response to voltage change. In a network of neurons,
input signals received by a particular neuron will also vary, as activity
moves around the network. For now, we use the highly simplified model
of Fig. 11.5 to see how varying conductance and injected current alters
11.5. Parameter dependence 199
0 1 2 3 4 5
0
1
0.5
V (mV)
t (ms)
Vmax
Fig. 11.5: Membrane charging curve simulation with C = 1
μF/cm2, g = 1 mS/cm2 , Iin = 1 μA/cm2 This gives time constant
τmemb = C
g = 1 ms and Vmax = Iin
g = 1 mV.
neuron response. Because of the simplicity of the differential equation in
this example, we could use the techniques of analytic calculus to solve the
differential equation. This would precisely define how the g and Iin parameters
determine voltage response. We instead use a computational numerical
analysis. This type of numerical technique will also be applicable to more
complex situations where analytic calculus cannot be used.
For starters, we need to understand how the update rule, V = 0.999・V +
0.001, generates the curve in Fig. 11.5. The value of the first coefficient,
0.999, reflects a tendency of V to stay put — its stickiness. The addend,
0.001, meanwhile, is the drive that pushes voltage up. The stickiness coefficient
is just a little less than 1. The drive addend is just a little more than
0. If the stickiness coefficient was 1 and the drive addend was 0, then the
voltage wouldn’t move at all. If we had used an update time step of 0.0001,
the stickiness coefficient would be 0.9999 and the drive addend would be
0.0001. This would give us a more precise approximation of the solution to
the differential equation. The curve in Fig. 11.5 would look identical since
in this example, the improvements would be tiny: i.e., V =0.001099 instead
of 0.001 after 1 μs.
In Fig. 11.5, we start with an initial condition of V = 0. At the beginning
of the curve, with V small, the membrane will charge quickly. The 0.999・V
stickiness is relatively small and the 0.001 drive dominates the update rule.
As V gets bigger, the stickiness begins to dominate and the rate of rise
gets slower until the curve reaches one. When V = 1, the update rule will
give V = 0.999 ・ 1 + 0.001 = 1, so V will stop increasing. In general, one
can find the maximum value (Vmax) by taking the update rule as a regular
equation and solving for V to get the steady state: Vmax − 0.999 ・ Vmax =
200 11. From Soap to Volts
0.001, 0.001・Vmax = 0.001, Vmax = 1 mV. We can similarly solve for Vmax
in the parameterized equation to show that in general Vmax ∼ Iin
g . We need
to say approximately equal (∼) rather than equal because the numerical
solution is just an approximation, as will be described further below.
Notice that Vmax = Iin
g = Iin ・ R. This is Ohm’s law. The value of this
maximum reflects the fact that at the end, the voltage is not increasing.
Therefore, ΔV
Δt = 0 and there is no capacitative current. All the current
flows through the resistor and the voltage is determined entirely by Ohm’s
law.
We referred above to the “stickiness” of the update equation, dependent
on how close the coefficient for voltage is to 1. This in turn depends on
the ratio of Δt to C
g . A bigger C
g (bigger R ・ C) will make the coefficient
closer to 1 and make the integration more sticky, causing the voltage to
rise slower. The measure of this stickiness is τmemb, the time constant:
τmemb = g
C = R・C. Using analytic calculus, τmemb can be shown to be the
time required for the voltage to reach about 63% (or more precisely 1−e−1)
of its final value (see Chap. 16, Section 16.5). In Fig. 11.5, τmemb = 1 ms.
In real life, we would get our parameters from measurements made in
neurons. Capacitance and membrane conductance cannot be easily measured.
However, both Vmax and τmemb can be measured using an electrode
that injects current into a cell and measures voltage (current clamp). Vmax
is the maximum voltage deviation reached during a prolonged current injection.
τmemb is the time required for the voltage to reach ∼ 63% of Vmax.
Although gmemb and Cmemb are the basic parameters used in the simulations,
these are only estimates based on gross neuronal properties that can
be directly measured.
Advantages and disadvantages of numerical integration
The differential equation of membrane charging is usually handled analytically,
as shown in Chap. 16, Section 16.5. This allows us to directly
calculate voltage as a function of time. In our specific example, the precise
solution is v(t) = 1 − e−t. When t = 0, e−t = 1 and 1 − e−t = 0. As t
gets bigger and bigger, e−t gets closer and closer to 0, and 1−e−t = 0 gets
closer and closer to 1. The precise solution, the real solution, never reaches
Vmax. It just approaches it asymptotically, meaning it will just get closer
and closer to Vmax without ever reaching it.
A numerical integration is just an approximation to the real solution. In
the present example, the numerical solution reaches Vmax, while the real
solution never does. Since the charging curve can be solved analytically, it
is easy to check the numerical solution against the correct solution. However,
most neural simulations can’t be solved analytically. The true, precise
solution of the equations is unknowable. This shouldn’t bother us too much
since the equations are themselves just an approximation of the physical
11.6. Time constant and temporal summations 201
reality. Approximating the approximation doesn’t take us too much further
away from the reality.
In the usual case, without an analytic solution to compare the numerical
solution to, one has to do computational controls to be confident about
the adequacy of the approximation. Usually this is done by rerunning the
simulation with smaller Δt’s to ensure that very similar curves result with
increasingly accurate approximations. Additionally, one can redo the simulation
with a different integration technique. Many such techniques are
available. One other integration method, the implicit Euler technique, is
given in Chap. 16, Section 16.5. Numerical integrations can and do fail. In
our example, if we choose Δt = 2, bigger than the time constant, then the
update equation is V = −V +2, which gives voltages that just bounce back
and forth between 0 and 2. This is an example of an unstable numerical
solution that doesn’t converge onto the correct solution. Instability often
occurs when Δt is too big. Generally, a good Δt is a small fraction (e.g.,
10%) of the time constant of the fastest process in a simulation. The choice
can be tested by demonstrating that further reduction in Δt produces little
change in the solution.
11.6 Time constant and temporal summations
In the first part of the book, we studied the simple signal transduction properties
of artificial neural network sum-and-squash units (Chap. 6): step 1)
multiplication of weight times state; step 2) summation; step 3) squashing
to final output. All of this is much more complicated in real neurons. We
now start to look at signal summation (step 2) in the context of a more
realistic model. In artificial neural networks, everything took place in discrete
time cycles so an input coming in at one time didn’t add to an input
coming in at another time. In real neurons, running in real time, a later
input can sum with an earlier one. This is called temporal summation.
Real synaptic signals are usually conductance changes that lead to
current flow. For the time being, we use current injections rather than
conductance changes in order to simplify the model and make it easier to
understand.We look at the summation of two brief current injection signals
to see how parameter changes alter temporal summation.
The slow charging and discharging of the membrane is a major factor in
temporal summation. The other major factor is the duration of the signal
itself. Long-duration signals are more likely to add together with other signals.
When the signal stops, the membrane does not suddenly drop back to
zero but instead discharges with an exponential discharging curve similar to
the charging curve, but upside down (Fig. 11.6). The time constant, which
determines charging time, also determines discharging time. In Fig. 11.5,
the current injection (the signal) lasted for the full 5 ms of the simulation.
202 11. From Soap to Volts
in I #2 in I #1
Fig. 11.6: Two brief current injections in a row. Membrane
charges exponentially and then discharges exponentially. 5 ms,
1 μ A/cm2 current injections start at t = 0 ms and t = 10 ms.
Scale bar: 5 ms, 0.5 mV.
In Fig. 11.6, the current stops at t = 5ms.At this time Iin = 0. The update
equation becomes V = 0.999 ・V , with an initial condition of V = Vmax = 1
mV. This gives an exponential discharging curve that has the same time
constant as the charging curve and drops to 0 mV (asymptotic to 0 for the
analytic solution).
In Fig. 11.6, the two current injections are separated in time so that
the first doesn’t appreciably affect the second. As in Fig. 11.5, the time
constant, τmemb, is 1 ms and Vmax is 1 mV. After 5 ms (5 ・ τmemb) the
membrane is pretty much fully charged: V = 0.9933 mV. Then it discharges
for 5 ms, ending up almost fully discharged: 0.0067 mV. At t = 10 ms, the
second current injection starts up, sending the membrane potential on the
same trajectory again. The simulation ends at t = 20 ms.
Each of the current injections in Fig. 11.6 is a signal. In this example,
there is no appreciable summation. Bringing the two signals closer together
may result in their adding up to a bigger signal. If we bring the two signals
closer together in time, this is temporal summation. If we bring them closer
together in space, this is spatial summation. Since this model (Fig. 11.4)
is a point neuron, a single compartment with no spatial extent, there is
no spatial summation. Using multicompartment models, we can get spatial
summation as two signals arrive at different locations in the neuron
(Chap. 13).
A variety of different measures can be considered as evidence of signal
summation. The simplest is to look at the peak voltage. The signals are
positively summating if the two signals together give a higher voltage than
either one alone. Another measure is the average voltage or the integral of
voltage with time (adding up the voltages at every time point). For now,
we just use peak voltage, since it’s easy to assess by eye, and is a better
predictor of whether a neuron will spike.
As a first parameter exploration, I started the second signal at earlier
and earlier times (Fig. 11.7). Because the second signal now starts before
11.6. Time constant and temporal summations 203
Fig. 11.7: Starting second signal during fall-off of first signal results
in no temporal summation by peak criterion. Scale bar: 5
ms, 0.5 mV.
Fig. 11.8: Starting second signal during the first signal results
in temporal summation. Different line types are used to tell the
lines apart. Scale bar: 5 ms, 0.5 mV.
the first signal ends, the second signal takes off from a higher voltage.
Despite this, it doesn’t end up going any higher than otherwise. This lack
of temporal summation occurs because the first signal is dropping off at the
same rate as the second signal is increasing. When the second signal starts
at t = 5 ms (dashed line), there is no drop-off between the signals. This
is not surprising since the two signals are now just one continuous signal
lasting 10 ms. The membrane charges over 10 ms instead of over 5 ms.
For a second exploration, I started the second signal still earlier so that
there is overlap between the two signals (Fig. 11.8). During the period of
overlap, the total current is twice what it would otherwise be and the membrane
is charging toward twice the single-signal Vmax (Vmax = 2 ・ Iin/g).
The greater the duration of signal overlap, the greater the membrane depolarization.
Since the membrane time constant is unchanged, the membrane
will fully charge over 4 to 5 ms as before. Temporal summation is maximal
for these two signals when the currents are coincident for that duration or
longer.
204 11. From Soap to Volts
Fig. 11.9: Reduction in conductance increases Vmax and temporal
summation. Second current injection starts at t = 6 ms; g = 1,
0.5, 0.2 mS/cm2. Scale bar: 5 ms, 0.5 mV.
Third, I explored the effect of conductance change (Fig. 11.9). By decreasing
g, we simultaneously increase the time constant and maximal
response. Lower conductance (higher resistance) gives a bigger, slower response.
For Fig. 11.9, I decreased g from 1 mS/cm2 (lower trace) to values
of 0.5 (middle) and 0.2 mS/cm2 (upper trace). The second signal starts at
t = 6 ms. The lower trace, identical to one of the simulations of Fig. 11.7,
shows no temporal summation. By decreasing g, we prolong the time constant
so that the membrane is not fully charged by the end of the first signal.
Now when the second signal kicks in, there is room to grow and temporal
summation occurs. The increased temporal summation is a direct consequence
of the increase in τmemb. We can demonstrate this by playing with
g and C so as to independently manipulate Vmax and τmemb. This is the
type of artificial exploration that cannot be done in real life, since there is
no way to independently control membrane capacitance experimentally. A
decrease in g with a proportional decrease in C will lead to a larger voltage
response (increased Vmax) but no increased temporal summation between
the two signals (same τmemb). An increase in C alone will increase τmemb
and lead to increased temporal summation, although the final voltage will
be lower due to less complete charging during the time of current injection.
In Fig. 11.10, I used a positive followed by a negative current injection.
Here the second signal was a current of −1 mA/cm2, which by itself would
give a down-going charging curve that would asymptote at −1 mV. This
current injection was started at t = 2 ms, at which time the trajectory
turns around and starts downward. The trajectory goes down still faster
after t = 5 ms when the first current injection ends and the second is
working unopposed. At t = 7 ms, the second current injection ends and the
11.7. Slow potential theory 205
Fig. 11.10: Temporal summation of a positive and a negative signal.
Negative signal is −1 mA/cm2 and starts at 2 ms. Dashed
line shows resting membrane potential. Scale bar: 5 ms, 0.5 mV.
negative membrane potential discharges back toward 0. This summation of
positive and negative currents is analogous to certain types of inhibition,
where a negative signal aborts or reduces the excitatory effect of a positive
signal.
Figures 11.5 through 11.10 demonstrate several critical properties of temporal
integration in passive neurons. The time constant determines how fast
the membrane will charge (Fig. 11.5). The membrane will discharge at the
same rate (Fig. 11.6). Even through the membrane has not fully discharged,
onset of a second signal will not lead to a higher final voltage because the
response to the first signal will fall off as fast as the response to the second
signal comes on (Fig. 11.7). If the second signal comes on while the membrane
is still charging, temporal summation will be seen (Fig. 11.8). The
contrast between Fig. 11.7 and Fig. 11.8 demonstrates that current injection
summation depends on the duration and overlap of the stimuli, not the
duration of the responses. Decreasing g (Fig. 11.9) will increase both Vmax
and τmemb, allowing inward currents to produce larger and longer depolarizations.
Only the time-constant effect will increase the degree of temporal
summation. Finally, hyperpolarizations can sum with depolarizations and
cancel them out (Fig. 11.10).
11.7 Slow potential theory
In Chap. 7 we discussed rate coding, also called frequency coding. The
rate-coding hypothesis states that the neural state can be measured at the
206 11. From Soap to Volts
axon as frequency of spiking. Rate coding uses the firing frequency of a
presynaptic neuron as that neuron’s output state. According to the basic
artificial neural network model, this output state will be multiplied by a
weight and added to the other state ・weight products to produce the totalsummed-
input for the postsynaptic unit’s squashing function (Chap. 6). To
look at the artificial neural network model in the context of neural membrane
theory, we need to find membrane-model parameters that correspond
to the weight in the artificial neural network model. This weighting process
will transduce presynaptic firing frequency to a postsynaptic value corresponding
to the state ・ weight product. The obvious postsynaptic value for
this purpose is membrane voltage. The voltage due to one input can add
to voltages due to other inputs to give the total-summed-input needed for
the squashing function. As we see in the next chapter, membrane voltage
can also be transduced into firing rate (postsynaptic neuron state) through
the dynamics of the Hodgkin-Huxley equation
To determine a firing frequency, one has to wait until at least two spikes
have arrived. If we measure the time between two spikes and invert the
interval between them, this gives us the instantaneous frequency. For example,
if two spikes occur 2 ms apart, then the instantaneous frequency is
1/(0.002 s) = 500 Hz (hertz, the unit of frequency, is equal to events per
second). In the context of rate-coding theory, instantaneous frequency is
not a very useful measure because instantaneous frequency values tend to
jump around a lot. Biological spike trains look noisy. Noise interferes with
frequency estimation and makes it necessary to smooth out the noise by
signal averaging.
Is this spiking irregularity really noise? What looks like noise may actually
be some kind of uninterpreted signal that is being used by neurons in
their calculations. This speculation is the province of an entirely different
class of models (e.g., synchronization models) and won’t be pursued further
here. Rate-coding theory postulates that the apparent noise really is noise.
So, for present purposes, noise is noise and that’s that.
Noise means that instantaneous frequencies vary and do not consistently
reflect the underlying average frequency. Frequency can only be reliably
assessed after the arrival of several spikes. Spikes are counted over a period
of time to determine average rate. This counting and adding up of spikes is
signal integration. Slow potential theory describes how signal integration
is performed as spikes trigger postsynaptic potentials (PSPs), which add
up to produce a voltage that estimates the average presynaptic frequency.
These PSPs are the slow potentials. They have to be relatively slow, long in
duration, in order to give a reliable estimate. Temporal summation occurs
as more spikes come in during the time while previous PSPs are still active
(e.g., Fig. 11.8).
From listening to the radio, you know that amplitude modulation (AM)
and frequency modulation (FM) are two methods of transmitting information
using an oscillatory signal. Since biological spikes do not vary
11.7. Slow potential theory 207
33.7 Hz 74.1 Hz 33.6 Hz
37.5 Hz 36.7 Hz
100 ms
41.8 Hz
54.8 Hz 23.1 Hz 50.2 Hz
Fig. 11.11: Different frequency estimates from a noisy FM spike
train. Top trace: estimates are fairly accurate reflections of the
underlying 50→25→50 frequencies. Second trace: Measurements
of instantaneous frequency can be highly inaccurate. Third trace:
Averaging with correct duration but wrong phase (timing of start
of average) also gives bad estimates. Fourth trace: averaging for
too long a duration misses the modulation entirely.
meaningfully in height (Chap. 3), the system is not using AM. Rate coding
theory assumes that the system is using FM. To describe an FM signal,
we need to discuss two different frequencies: the carrier frequency of the
spikes and the frequency at which the rate changes. In Fig. 11.11, I show a
noisy spike train. The carrier frequency varies between 25 and 50 Hz. This
frequency is modulated at a rate of 4 Hz, meaning that we have a shift
in frequency every quarter second (250 ms — the length of the brackets
at the top of Fig. 11.11). Clearly, the frequency of modulation must be
considerably lower than the carrier frequencies.
In Fig. 11.11, rates for a single artificial noisy frequency modulated spike
train are estimated correctly, and then incorrectly in several ways. Biologically,
it is not possible to be confident of whether a particular signal
estimation is correct or incorrect. In Fig. 11.11, I made the spike train, so
I know what was signal and what was noise. This spike train was produced
by a noisy frequency generator set to 50 Hz for 250 ms, followed by 25 Hz
for 250 ms, followed by 50 Hz again for the last 250 ms. The top trace is
208 11. From Soap to Volts
the most accurate estimate of the underlying signal that can be obtained.
Averaging is started and stopped at exactly the times of the generator’s frequency
shifts. However, instead of 50 Hz → 25 Hz → 50 Hz, the frequency
estimates are 54.8 Hz → 23.1 Hz → 50.2 Hz. This discrepancy is due to
the underlying noise and limited duration of sampling. If longer samples
were available, the estimates would be more accurate. In the second trace,
three instantaneous frequencies are chosen that give particularly inaccurate
estimates of average frequency. Long intervals are measured during
the fast firing at beginning and end. A short interval corresponding to 74.1
Hz is measured during the slow firing in the middle. In the third trace, reasonable
averaging intervals are used but averaging is started and stopped
at exactly the wrong time (180 degrees out of phase with the frequency
modulation). As a result, the estimates miss the frequency shift and average
across two different frequencies to give an intermediate frequency of
about 37 Hz. In the fourth trace, the estimate also misses the frequency
modulation by using too long an averaging time.
Averaging by adding PSPs
As you can see from the above example, accurate frequency interpretation
depends on averaging over a reasonable duration and starting the averaging
at the right time (right phase) to capture frequency shifts. Finding the
right averaging duration or time constant depends both on the range of
frequencies to be communicated and on the amount of noise that obscures
them. The ideal time constant can be calculated using communication theory.
Here, I’ll just make note of some straightforward conclusions: 1) signals
with low carrier frequencies require long averaging periods; 2) signals with a
lot of noise require long averaging periods; 3) signals with rapid frequency
modulation require short averaging periods. Noise can easily overwhelm
signal in cases where a low carrier frequency is modulated rapidly.
In Figs. 11.6 to 11.10, I showed temporal summation of square waves and
noted factors that would increase temporal summation. Artificial current
injections from electrodes are square. Real biological signals are more curvy.
The alpha function (Fig. 11.12) is a popular model for a PSP. The alpha
function is parameterized as
Iin = Imax ・
t
τα ・ e(−(t−τα)/τα)
This is a complicated equation that produces a simple curve. It rises quickly
for τα time and then falls slowly over about 5 τα (a little slower than exponentially).
Note that we now have a couple of time constants to discuss. We
call the alpha function time constant τα and the membrane time constant
τmemb.
The time constant of the slow potential must be chosen so that the PSPs
are long enough to average a reasonable number of spikes. Choice of too
11.7. Slow potential theory 209
10 20 30 40
5
time (ms)
current
Fig. 11.12: Alpha functions with onset at 5 ms and amplitude of
5 μA/cm2. Three curves with τα=2, 4, 6 ms. Note that τα equals
the time from onset to peak. τα also determines decay time. This
is used as a model of a postsynaptic potential (PSP).
short a τα will not allow integration to occur — the PSP will end after only
one or two impulses so that only instantaneous frequency can be measured
(e.g., Fig. 11.11, trace 2). An overly long τα will not allow enough PSP
decay during the arrival of many spikes, and will therefore average across
the frequency modulation (e.g., Fig. 11.11, trace 4).
For Fig. 11.13, I produced an FM signal with spike-train frequencies
varied randomly between 20 and 100 Hz. The frequency changed every
second (1 Hz modulation frequency) for 10 seconds. The carrier frequencies
are shown graphically in the top trace with the values given in Hertz.
I didn’t show all the spikes since at this scale they would just scrunch
together into an indistinguishable blob. However, at the bottom of the
figure I expanded an 800-ms period to show the spikes and membrane
response together.
With carrier frequency between 20 and 100 Hz, the interspike intervals
(ISIs) ranged from 50 to 10 ms (period is inverse of frequency: ISI = 1
f ).
A good choice of τα would be somewhere in this range; I used 30 ms.
Membrane time constant, τmemb, is 1 ms, which allows the membrane to
follow the PSP without substantial lag and without adding any additional
delay. Another way to do this would be to use a shorter τα but prolong the
signal with a long τmemb. It is more difficult to get this model to work well,
since the longer τmemb delays signal onset as well as signal offset.
Fig. 11.13 shows membrane potential in response to two spike trains
with the same underlying frequencies but without and with noise. There
is an initial charging period (arrow) when the membrane rises from resting
membrane potential by about 3 mV. Then the potential plateaus in
response to the constant-frequency input. When the input frequency shifts
upward from 47 to 81 Hz there is another charging period as the membrane
rises another 2 mV and plateaus. The correspondence between shifts
210 11. From Soap to Volts
2.5 mV
500 ms
64
FM signal 47 81 29 82 37 54 45 91 82
with noise
no noise
Fig. 11.13: Slow PSP response to FM square wave (carrier frequencies
shown in Hz). PSP parameters: τα = 30 ms; τmemb = 1
ms.
in frequency and shifts in membrane voltage can be easily appreciated. The
scaling is arbitrary: the change in millivolts as a number is not the same
as the change in frequency as a number. However, the relationship is linear:
a doubling of input frequency leads to a doubling of potential. During
the plateau there is a low-amplitude oscillation from the waveforms of the
constituent alpha functions. In the absence of noise, membrane potential
closely reflects frequency modulation, even reflecting small frequency shifts
such as the shift at right from 91 to 82 Hz.
Once noise is added to the FM signal (Fig. 11.13, with noise), frequency
estimation suffers. It is no longer possible to reliably identify frequency
shifts of under 10 Hz. Even the shift from 81 to 64 Hz is hard to see. The
expanded trace below shows the alpha function responses to individual
spikes up to the shift from 29 to 82 Hz.
If we increase τα to 100 ms (Fig. 11.14), averaging occurs over a greater
number of spikes and most of the noise is filtered out. As well as averaging
over a greater period, the longer τα also produces longer charging delays.
This means a longer wait for the voltage to stabilize on an estimate of the
incoming signal (arrows). With a charging delay of nearly half a second and
frequency modulation of 1 Hz (period of 1 second), the frequency estimate
barely registers as a plateau before the frequency shifts again.
Notice that slow-potential averaging solves the problem of phase choice.
As shown in Fig. 11.11, trace 3, one will obtain the wrong frequency average
if averaging starts and stops at the wrong times. With slow potentials,
averaging is going on continuously with activity in the remoter past gradually
wearing out. Instead of starting and stopping at certain times, the
sum of slow potentials is always most strongly influenced by the preceding
11.8. Summary and thoughts 211
2.5 mV
500 ms
Fig. 11.14: Slower PSP response to noisy FM signal of Fig. 11.13.
τα = 100 ms.
instantaneous frequency, less strongly by the one prior to that, and so on
backward in spike history for the duration of the slow potential.
However, this figure also illustrates a problem with slow potential theory
— it’s too damn slow. Depending on the degree of noise, which will vary
across different cell types in the brain, reliable frequency estimation can
take upward of 100 to 500 ms. The route between perception and action in
the spinal cord, thalamus, and brain encompasses many synapses. If each of
these synapses requires prolonged slow-potential processing, it could take
many seconds to react. This is plenty of time for a tiger to eat you (unless
you ask that your tiger use rate coding as well).
11.8 Summary and thoughts
My recurring theme is that hardware determines software. In this chapter, I
started with salt, water, and soap, the basic ingredients of the brain. Out of
these, the body builds capacitors and resistors. In the next chapter we’ll see
that it builds batteries as well. The physical limitations of these building
blocks makes neurons very slow compared to transistors. In particular, the
relatively large capacitance translates into slow signaling. Rather than be
dismayed by this slowness, I presented models that use it to advantage. The
slowness allows the membrane to hold onto a signal, permitting temporal
summation. The amount of summation has to do with the length of the
membrane time constant. Slow potential theory is a model that makes
slowness a feature, using long time constants to do signal averaging that
blurs out noise.
This chapter showed how the parameters of the membrane and of the
signal itself determine the influence a signal has on the neuron. These interactions
explained signal transduction from presynaptic spike rate to
postsynaptic membrane potential. They also explained the signal summation
required to arrive at a total-summed-input. The next step of an
artificial neural network update rule is signal transduction from totalsummed-
input to output state, in this case from membrane potential to
spike rate. In the next chapter, we explore how the size of the membrane
potential will determine the likelihood and frequency of neuron firing.
212 11. From Soap to Volts
Although rate coding and slow potential theory are about the best we
can do right now, I find them unsatisfying. As mentioned above, they are
too slow. Also as I’m sitting and thinking, it bugs me to think that most of
my substantial metabolic effort is just producing noise. Worse yet, I have to
wait around just so I can ignore most of what my brain is doing. Anyway,
I’m always complaining about other people using intuition to understand
the brain, and here I am doing it.
12
Hodgkin-Huxley Model
12.1 Why learn this?
In the 1950s Alan Hodgkin and Andrew Huxley worked out the ionic basis of
the action potential and developed a mathematical model that successfully
predicted the speed of spike propagation. Their work can be regarded in
retrospect as the beginning of computational neuroscience. It remains the
touchstone for much neural modeling today. The Hodgkin-Huxley model
demonstrates how computer models can reveal biological properties that
cannot be examined directly.
Hodgkin and Huxley described two ion channels. Since then hundreds
have been described and some of the basic parameterization has been updated.
Despite this, the modeling techniques that Hodgkin and Huxley
developed are still used and remain the standard model today.
12.2 From passive to active
The simulations in Chap. 11 used a simple RC circuit. This is a passive
membrane model because the conductance remains constant. In this chapter,
we use the same passive components (the R and the C), and add active
components as well. Active components are conductors that change their
conductance in response to changes in membrane voltage or due to activation
by a chemical. Voltage-sensitive channels are responsible for the action
potential. Chemical-sensitive channels are responsible for synaptic activa214
12. Hodgkin-Huxley Model
tion in response to the arrival of a ligand at a synapse. In general, voltage
or chemical-sensitive channels are considered active channels because they
are activated in response to some signal. Channels that remain at a fixed
conductance are called passive channels.
The resting membrane potential is about −70 mV
The simulations in the previous chapter were simplified by starting at a
resting potential of 0 mV. In real life, the membrane rests at a negative
potential of about −70 mV (different cells differ). This is the resting
membrane potential (RMP) of the cell. The resting potential is set up by
pumps that separate charge across the membrane, producing various fixed
potentials associated with different ions. This charge separation will be
represented by batteries in the electrical circuit diagram. They are denoted
the sodium battery, the potassium battery, etc. The polarity of the individual
batteries depends both on the polarity of the ion involved and on
the direction of the inside-outside concentration gradient for that ion. Each
battery will only affect membrane voltage when the ion channels open for
that particular ion. In the circuit diagram, this is represented by connecting
each battery through a variable conductor.
The membrane is insulator, capacitor, and battery
The addition of batteries and variable conductances to the circuit diagram
demonstrates several other roles played by the cell membrane and
by membrane proteins. As part of being an insulator and a capacitor, the
membrane also allows the charge separation that sets up the batteries, each
of which is associated with a different ion. Protein transporters pump the
ions to charge each of these batteries. Active protein channels form the
variable conductors (rheostats). These proteins will be sensitive to changes
in membrane voltage or to chemicals or both.
The resting (inactive) potential on the membrane is negative. Hence,
both negative-going inhibitory signals and many positive-going excitatory
signals will be negative relative to ground. There is a standard nomenclature
to describe voltage deviations from rest (Fig. 12.1). Negative
deviations, which make the membrane even more negative that at rest,
are called hyperpolarizing (hyper means more). Hyperpolarizing inputs are
generally inhibitory. Positive deviations, which make the membrane less
negative than it is at rest, reducing its polarization, are called depolarizing.
Depolarizing signals move the membrane potential toward or past 0
mV. Depolarizing inputs are generally excitatory.
Hyperpolarization and depolarization are not symmetrical. The membrane
can be naturally depolarized by about 120 mV. 120 mV above rest
is +50 mV relative to ground (−70 + 120), approximately the value of the
sodium battery. Natural activity will only hyperpolarize the cell by about
12.2. From passive to active 215
IPSP
AP
RMP
0 mV
EPSP
Positive Polarity
Negative Polarity
Depolarization
Hyperpolarization
Fig. 12.1: Resting membrane potential (RMP) is typically about
−70 mV (inside negative). Membrane can be depolarized as much
as 120 mV or hyperpolarized as much as 30 mV from rest. Excitatory
postsynaptic potentials (EPSPs) depolarize; many inhibitory
postsynaptic potentials (IPSPs) hyperpolarize. Action potentials
(APs) are depolarizations that can overshoot 0 mV, temporarily
reversing membrane polarity.
20 to 30 mV. 20 to 30 mV below rest is about −90 to −100 mV relative
to ground (−70 − 30 = −100). This is about the range of values for the
potassium battery, which may differ somewhat among different cell types.
Experimentally, one can hyperpolarize the membrane beyond −100 mV by
injecting negative current through an electrode inside of the cell. Artificial
depolarization with injected current is limited by the tendency of prolonged
depolarization to kill the cell.
Synaptic inputs aren’t current injections
In the simulations in Chap. 11, I started by using square wave current
injections as signals. These were similar to the current injections used by
physiologists. Then I moved on to the alpha function as a more natural signal.
I was still using current injections. Most synapses in the nervous system
are chemical synapses. The associated synaptic potentials, the natural input
signals for neurons, generally arise as conductance changes rather than
current injections. The current that flows is secondary to the conductance
change. (There are also electrical synapses where current flows through a
channel connecting two neurons.) As shown in Fig. 11.2, a chemical ligand
216 12. Hodgkin-Huxley Model
(neurotransmitter) is released presynaptically. It floats across the synaptic
cleft and binds to a receptor on the postsynaptic membrane. This receptor
is connected either directly or indirectly to one or more ion channels that
conduct current across the membrane. Depending on which ions the channels
allow through, the synaptic current can be inward (depolarizing) or
outward (hyperpolarizing). In our electric circuit diagrams, this will correspond
to connecting up a particular battery by activating a switch or a
controllable conductance (a rheostat).
12.3 History of the action potential
The story of the action potential starts with Luigi Galvani’s 1791 discovery
that electrical signals from lightning or primitive batteries could cause
contraction of the leg of an otherwise dead frog leg. This was an inspiration
to mad scientists, and researchers started performing similar experiments
on the heads of hanged criminals, attempting to bring back the dead with
electricity at around the time that Frankenstein was written. The National
Institutes of Health do not encourage this line of research nowadays. Perhaps
these demonstrations had some value in that they suggested that
mysterious human abilities such as motion, sensation, and thought could
be caused by a physical process. On the other hand, connecting thought to
electricity may not have been so disturbing at that time, since electricity
was itself another mysterious process that could be readily equated with
an unknowable “life force.”
A later finding that brought neural function further into the physical,
non-ethereal realm was the demonstration of neural delays by Hermann
Helmholtz, the famous 19th century scientist. Helmholtz showed that stimulation
of a nerve at different points led to contractions of the corresponding
muscle at measurably different times. Not only was the speed measurable,
but it was relatively slow (about 70 mph). Helmholtz’s father wrote him a
letter rejecting these findings as absurd. Helmholtz’s father was sure that
he could directly perceive the world, and that his thoughts were converted
directly into actions. If he moved his hand and watched it, perception didn’t
fall behind. Research into the seeming seamlessness of experience is now
actively pursued using careful timing of perception and brain electrical activity.
An understanding of how the illusion of simultaneity arises despite
varying delays of sensory signal processing would provide a partial solution
to the mind–body problem.
Hodgkin and Huxley
By the time Hodgkin and Huxley got to the problem, much progress had
been made. The resting potential was well described and the action po12.4.
The parallel-conductance model 217
tential had been described as a “negative variation” in this potential. (We
now describe the action potential from the inside as a positive variation:
positive inside, negative outside.) The squid axon had been picked out as
an ideal experimental preparation due to its enormous size compared to
other neurons. Squids, like other invertebrates, have unmyelinated axons
— wires without insulation. Their axons are leaky and prone to signal loss,
which has caused them to evolve extremely broad axons that conduct electricity
better. From the experimentalist’s point of view, this makes them
big enough to see and to stick wires into.
Hodgkin and Huxley threaded a fine silver wire inside the axon. With
this, they could measure the electrical potential inside and deliver enough
current so as to maintain a particular voltage despite the efforts of the
ion channels in the membrane to change it. This is called voltage clamp.
They measured how much current was required to keep the voltage from
changing. This told them how much current was being passed through the
axon membrane and in which direction. They could then figure out which
ions were responsible for which currents by doing the same experiments
in sodium-free or potassium-free solutions. By running these experiments
at many different voltages, they found out how the sodium and potassium
currents grew and shrank with changes in membrane potential. They used
these data to construct the parallel-conductance model (Fig. 12.2).
12.4 The parallel-conductance model
The parallel-conductance model is similar to the basic RC model of
Fig. 11.4. Once again all points on the inside of the membrane are electrically
connected via the cytoplasm (horizontal line at bottom). This is the
point where we measure potential. The outside of the membrane is connected
via the extracellular fluid (horizontal line at top) and is grounded,
keeping it at 0 mV.
The circuit
The inside and outside of the membrane are connected via four parallel
conducting pathways. On the left side are the membrane capacitor and
the fixed membrane conductance. These two passive components are similar
to those of the simple RC model. However, the resistor in this case
is connected to a battery. Batteries are represented by two parallel lines
of different lengths. The long line of the battery schematic indicates the
positive pole. The passive conductance in the parallel-conductance model
is known as the leak conductance. Because the channels carrying this current
are not voltage-sensitive, the leak conductance remains the same at
any voltage, providing a constant “leakiness” for current. The potential of
218 12. Hodgkin-Huxley Model
Inside
Outside
leak
sodium
potassium
+
+
capacitor
+
battery
conductor
(resistor)
rheostat
(variable
conductor)
+ EK ENa Eleak
leak g gK gNa
Fig. 12.2: Parallel-conductance model of the membrane. The various
variable conductors (variable resistors, rheostats) connect
the inside and outside. They are arrayed in parallel.
the battery associated with gleak is Eleak. (E for electrical potential and V
for voltage are synonymous but E is usually used for batteries.) The short
line of the leak battery is connected to the inside of the membrane making
the membrane inside-negative. Eleak is the major determinant of resting
membrane potential.
On the right side of the circuit diagram in Fig. 12.2 are the two active
branches: the sodium battery and conductance, and the potassium
battery and conductance. Note that these two batteries are pointed in opposite
directions. The potassium battery, like the leak battery, will make
the membrane negative. The sodium battery will make the membrane positive.
The conductance symbol under each battery has an arrow through
it. This means that it is a variable (or controllable) conductor, also called
12.4. The parallel-conductance model 219
a rheostat. The rheostat is the thing you turn to dim the room lights to
set a romantic mood. In this case, the rheostats will be controlled not by
the level of romance but by the level of the membrane voltage. Since the
rheostat influences membrane voltage and the membrane voltage influences
the rheostat, this will lead to positive or negative feedback loops, as will
be described below.
At rest, the sodium and potassium conductances are turned off so that
these two lines are not conducting. Under these circumstances, the two
associated batteries have no effect on membrane voltage. If one of these
conductors were to be turned all the way on (zero resistance), then the
associated battery would dominate the membrane potential. If both of these
conductors were turned on all the way at the same time, you would have
the situation that you get when you connect the leads wrong while jumping
a car battery — the battery will discharge massively, overheat, and blow
up. Luckily, this doesn’t happen in the brain.
Currents
Benjamin Franklin defined current as the flow of positive charge. In electronic
equipment, current through wires is carried by negative electrons.
Therefore, the direction of current in wires is opposite to the direction that
charge flows. In biology, current is carried by ions from dissolved salts that
move through water. Most of the ions involved are positive, like sodium
(Na+), potassium (K+), and calcium (Ca++). Chloride (Cl−), a negative
ion, is also important. Positive ions, with one or more superscript +, are
called cations; negative ions, with superscript −, are called anions. Because
calcium has twice the charge of sodium, movement of calcium ions will result
in twice as much current. When dealing with cation flux, the direction
of ion movement is the same as the direction of current.
Current is measured during voltage-clamp. To clamp a constant voltage
onto the membrane, current is injected or withdrawn from the inside of
the cell through an electrode. This current must exactly cancel out any
currents that are passing through the membrane in order to prevent these
membrane currents from changing the membrane potential. Current direction
is defined with respect to the membrane, not the electrode. Inward
current is positive charge going across the membrane from outside to inside.
Outward current is positive charge going from the inside of the membrane
to the outside.
Interpreting voltage-clamp results is a bit of an art. Sodium (positive
ions) passing into a cell or chloride (negative ions) passing out of a cell
would both be examples of inward current. Since sodium is at a higher
concentration outside of the cell, an increase in sodium flux will result in
an inward current. Potassium, being at higher concentration inside the cell,
would be a typical outward current. When the membrane is stable (whether
at RMP or at some other voltage imposed by voltage clamp), there is some
220 12. Hodgkin-Huxley Model
balance of ongoing outward and inward currents. In addition to an increase
in these baseline currents, it is possible for an experimental manipulation
(a drug or a voltage step) to produce a decrease in the baseline currents.
An increase in potassium current is an outward current, but a reduction in
the baseline potassium current is measured as an inward current. Chloride
reversal potential is near to RMP. Therefore, a chloride flux can be either
inward or outward. Because chloride carries a charge of −1, a decrease in
inward chloride flux is an inward current.
Calculations
The calculations for the parallel-conductance model are similar to those for
the RC model except that we have to add in the batteries. As in the RC
model (Fig. 11.4), Ohm’s law gives a resistive current equal to conductance
times membrane voltage. As before, this can be written as either IR =
g ・ Vmemb or Vmemb = IR ・ R. A battery in series with the conductor
gives a voltage boost on that line. For the leak conductance: Vmemb =
Eleak +Ileak ・Rleak. Eleak is the value of the leak current battery, usually
about −75 mV. We need to add up currents (Kirchhoff’s law), so we turn
the equation around to give Ileak = gleak ・(Vmemb−Eleak). Similarly, INa =
gNa ・ (Vmemb − ENa) and IK = gK ・ (Vmemb − EK).
All of the currents add up to zero: 0 = IC +INa +IK +Ileak. Therefore,
−IC = INa + IK + Ileak. Substituting for the currents gives the parallelconductance
equation:
−C
ΔV
Δt
= gleak ・ (Vmemb − Eleak)
+ gNa ・ (Vmemb − ENa)
+ gK ・ (Vmemb − EK)
Because IC = C・ ΔV
Δt , positive capacitative current is a depolarizing current
that makes the inside of the membrane more positive. In the parallelconductance
equation, as in the RC model, capacitative current is opposite
in sign from the conductive currents. Therefore, a negative conductive
current is a positive capacitative current and produces depolarization. Negative
conductive currents are inward currents, involving the flow of current
through ion channels from outside to inside. Notice that negative current
has nothing to do with the sign of the ion that is carrying the current. Instead,
it is an indication of the direction of current. Also notice that the sign
change is confusing: a negative membrane current produces a positive voltage
effect. The direction of negative current is an arithmetic consequence
of measuring membrane voltage on the inside rather than the outside. In
the literature, the phrase “membrane current” is used as a synonym for
conductive current. Therefore, negative current flows in and depolarizes;
positive current flows out and hyperpolarizes. However, it’s worth remem12.4.
The parallel-conductance model 221
bering that there is zero total current flow: the inward conductive current
is matched by outward capacitative current. It is actually the latter that is
most closely associated with the depolarization.
At steady state, there will be no capacitative current since voltage is not
changing: 0 = INa + IK + Ileak = gleak ・ (Vmemb − Eleak) + gNa ・ (Vmemb − ENa)+gK ・(Vmemb−EK). Solving this equation for Vmemb gives the resting
membrane potential:
Vmemb = gleak ・ Eleak + gNa ・ ENa + gK ・ EK
gleak + gNa + gK
This is a version of the Goldman-Hodgkin-Katz (GHK) equation. It looks
complicated but it just says that steady-state membrane voltage will be the
weighted sum of the batteries, with the weighting provided by the conductance
associated with that battery. Since gleak is the dominant conductance
at rest, it will have the greatest effect on determining RMP. If a conductance
is turned off completely (e.g., gNa = 0), the corresponding battery
has no influence. If, on the other hand, a conductance is very high, then
the other batteries will have very little influence, e.g., if gNa >> gK and
gNa >> gleak, then Vmemb ∼ gNa•ENa
gNa
, hence Vmemb ∼ ENa.
Where do the batteries come from?
The batteries are an indirect result of proteins that pump ions across the
membrane. These ions then try to flow back “downhill,” in the direction of
their chemical gradient from high concentration to low concentration. Only
a little current has to flow in order to set up an equal and opposite electrical
gradient. The electrical gradient, opposite in direction to the chemical
gradient, is the battery. This electrical potential is called the Nernst potential.
It can be precisely calculated by knowing the concentrations of a
particular ion inside and outside of the cell (see Glossary for definition).
Each ion has its own Nernst potential. The value in millivolts of the Nernst
potential is the strength of the battery that we use in the circuit diagram.
With its many ins and outs, the origin of the Nernst potential can be
confusing. In Fig. 12.3, I show the origin of the sodium battery. Sodium
is pumped from inside to outside (#1 in Fig. 12.3) by a protein that uses
energy from ATP. The pumping leaves sodium concentration outside of the
cell ([Na]o ∼ 140 millimoles) higher than it is in the cytoplasm ([Na]i ∼ 10
millimoles). The concentration difference across the membrane does not
in itself lead to any charge separation, since sodium ions on both sides
are appropriately matched with negatively charged proteins. Since there
is more sodium outside, it “wants” to flow inside due to diffusion (#2 in
Fig. 12.3). (Diffusion is what makes a drop of ink spread out in a glass of
water; it wants to go where no ink has gone before.) As long as the selective
channels for sodium remain closed, sodium cannot diffuse and the sodium
concentration gradient has no effect on membrane potential.
222 12. Hodgkin-Huxley Model
1. pumped
out
3. held back
by E field
[Na] i=10 mM
[Na] o =140 mM
chemical gradient
electrical gradient
−−−−−−
+++++
2. diffuses in
(when open)
−−−−−−
++++
+60 mV
Na (E )
0 mV
(ground)
Fig. 12.3: The origin of the Nernst potential for example of Na+.
Na+ is pumped out by an energy-consuming protein, flows back
in through Na+ channels, and is pushed back as it creates its own
electric field. The sodium flux eventually reaches electrochemical
equilibrium at a potential of about 60 mV. The superimposed
rheostat sign is meant to indicate that the channel can be open
or closed. This is a mixed metaphor: the symbol belongs to a
different model.
When the sodium channel opens, sodium rushes down its concentration
gradient. The negative proteins that are paired with the sodium ions
cannot follow; they are not allowed through the sodium channel. This diffusion
of sodium across the membrane leads to charge separation across
the membrane, with unmatched sodium ions on the inside and unmatched
negative protein molecules on the outside. The unmatched sodium ions inside
the membrane will stay near the membrane, in order to be close to
their lost negative brethren. This bunching of positives next to the inside
of the membrane, with a corresponding bunching of negatives next to the
outside, creates an electric field (#3 in Fig. 12.3) that opposes inward diffusion
through the ion channels. This outward electric field is the sodium
battery. The inward diffusive force and the outward electrical force reach a
steady state (Nernst equilibrium) so that there is no net flow of ions and
little need for continued pumping to maintain equilibrium.
12.5. Behavior of the active channels 223
The concentration difference between inside and outside can be directly
translated into an electrical potential by using the Nernst equation. ENa
is approximately +60 mV. The positive plate of the sodium battery in
Fig. 12.2 will be 60 mV relative to the negative plate. By contrast, potassium
is at high concentration inside and low concentration outside. The
potassium chemical gradient is outward so the electrical gradient is inward.
The positive inward electrical gradient would be +90 mV if measured from
the outside of the membrane, relative to a grounded inside. However, we
always measure the potential on the inside, relative to ground outside, so
the potassium potential (EK) is about −90 mV.
All of the reversal potentials can vary slightly in different cells (different
pumps) or under different conditions. For example, the sodium reversal potential
can be slightly lower if you’re sweating a lot: less sodium outside, less
inward chemical gradient, less outward electrical gradient. The potassium
reversal potential gets less negative under conditions where cells fire a lot.
The constant firing allows potassium to build up outside of the cells. The
accumulation of extracellular potassium reduces the outward chemical gradient,
reducing the inward electrical gradient. Because resting membrane
potential is largely determined by baseline potassium flux (the leak conductances
conduct primarily potassium), this change in EK changes Eleak
and depolarizes cells. Even cells that have not themselves been firing are
affected. In this way, extreme activity in one set of cells can depolarize
neighboring cells and make them more excitable. This is one factor that
can lead to the spread of uncontrolled activity in seizures.
12.5 Behavior of the active channels
To complete the Hodgkin-Huxley model, we have to describe the behavior
of the sodium and potassium channels. We continue to look at the model
from several different descriptive perspectives. First, we presented the electronics
perspective and considered the channels as rheostats. Second, we
talked about ion channels from a biophysical viewpoint, describing flux and
electrochemical equilibrium. Third, we will describe the interaction of the
channels in terms of positive and negative feedback systems. Fourth, we will
return to ion channel behavior, describing their conductance properties in
terms of somewhat mythical “particles.” Fifth, we will study the equations
that model the system. Sixth, we will run the simulation and demonstrate
the model graphically.
This mess of descriptions includes different levels of organization (single
channel vs. whole membrane), different approaches (electrical engineering
vs. biology), and different methods of presentation (numerical vs. graphical).
Doing modeling from so many directions would seem to make things
harder rather than easier. It means that we have to not only understand
224 12. Hodgkin-Huxley Model
concepts in biochemistry, biology, and electronics, but also be familiar with
various tools provided by math and computer science. Additionally, we will
see that not all of the descriptions are fully consistent with one another.
For example, electronically we described the sodium channel as a rheostat.
However, we also describe the sodium channel as having little on and off
switches, making the channel sound more like a tiny subcircuit than like a
rheostat.
The payoff for all of this hard work can be appreciated by considering the
famous parable of the blind men who meet an elephant (one feels the trunk,
one a leg, another a tusk, etc., and they give widely discrepant descriptions
of the beast). None of them knows anything useful about elephants but, if
they pool their knowledge, they may be able to create a passable picture.
Similarly, coming at a model from many angles permits us to come closer
and closer to understanding the thing itself. We have neither the concepts
nor the mental capacity to allow us to wrap our brains around all of this
complexity, and see nature as it really is. Instead we use these different
models as tools to pick up different clues to this underlying reality. As we
move back and forth between representations, we gain further insights.
There are other approaches to describing the Hodgkin and Huxley model,
in addition to five or six that I use in this chapter. For example, one can use
complex graphical representations that allow us to look at several dimensions
of the dynamics at once. This is called a phase-plane representation.
There are also other kinds of mathematical tricks that can help us understand
the system better. An example of this would be descriptions of
nullclines and of the space of solutions as a field. Going beyond the scope
of the Hodgkin and Huxley model, we could look at levels of detail that are
not directly considered in the model: we could describe the detailed molecular
conformation changes that determine channel opening and closing or
include the detailed physical chemistry of how salts and water interact.
Feedback systems
The Hodgkin and Huxley sodium and potassium channels are voltagesensitive
conductances. They go on and off with voltage change. As we
describe the cycle of the action potential, we can speak in terms of positive
feedback and negative feedback. Positive feedback occurs when the cycle
of influence causes a change to produce more change in the same direction.
If, for example, the faster you drive the more excited you get about driving
fast (a positive feedback loop), you will tend to drive faster and faster. Positive
feedback systems are not self-limiting; they are limited by something
outside of the positive feedback system (i.e., a crash) or the opposition
of some negative feedback system (e.g., prudence or the police). Negative
feedback systems are self-limiting, since the cycle of influence produces a
change that opposes the original change. Negative feedback is very common
in biological systems: hunger makes you eat, food reduces hunger.
12.5. Behavior of the active channels 225
positive
feedback
negative
feedback
Na on
Na off
K on
Fig. 12.4: Positive and negative feedback shape the action
potential.
The Hodgkin and Huxley model uses two types of ion channels that are
controlled by three types of switches. Turning on the sodium channel pushes
voltage up (depolarizes) and turning on the potassium channel pulls it back
down (hyperpolarizes). The channels are controlled by voltage. The sodium
channel is controlled by both on and off switches. Increasing voltage turns
on the sodium channel. This is a positive feedback loop. Increasing voltage
also turns off the sodium channel, by means of a different switch. This
is a negative feedback loop. The potassium channel only has one type of
switch. Increasing voltage turns on the potassium channel, hyperpolarizing
the membrane and providing a negative feedback loop.
The positive and negative feedback loops happen at different rates in
the Hodgkin-Huxley model (Fig. 12.4). Initially, the action potential is triggered
by a depolarization (movement of the potential in a positive direction,
toward 0 mV). This can be a result of a current injection or a synaptic potential.
This depolarization causes the sodium channel to switch on quickly.
Current through the sodium channel will cause more depolarization, which
will turn on the sodium channel more, which will cause more depolarization,
which will turn on the sodium channel more ... This is the positive
feedback that produces the upsweep of the spike as voltage rises rapidly.
The sodium channel’s on-switch provides the positive feedback by activating
the channel with depolarization. The off-switch provides the negative
feedback by inactivating the channel during depolarization. While the
sodium channel is being switched on, it is being more slowly switched off.
The delay in negative feedback allows potential to rise about 100 mV over
about a millisecond. Then negative feedback kicks in. The action potential
226 12. Hodgkin-Huxley Model
reaches its peak as the sodium channel switches off and the potential heads
back down toward resting potential. Further negative feedback is provided
by the turning on of the potassium conductance. This connects the membrane
to the negative potassium battery, which opposes the positive sodium
battery.
Particle duality
Hodgkin and Huxley emphasized that their model was empirical, meaning
that they matched the behavior of the action potential without needing to
know what exactly was going on at the molecular level. They had no reason
to expect that the details would correspond to what actually exists in the
membrane. However, the model did provide a set of successful predictions
at the membrane level. Most notably, the model predicted the existence
of ion channels turning on and off independently. There was no way to
demonstrate these ion channels until patch clamping was developed decades
later.
Hodgkin and Huxley also had an implicit view of the functionality of
switches controlling the ion channels. They called these switches particles.
At the single-channel level, each particle was binary, taking on a value of
0 for blocking and 1 for nonblocking. The implicit concept portrayed the
particles as physical objects that could block a single channel and prevent
flow of current. Both the sodium and the potassium channel were described
as having four particles. In the case of the potassium channel, each of the
four particles behaves identically. Each of these potassium particles is called
n. Mathematically, the presence of four particles per channel is denoted by
a multiplication: n ・ n ・ n ・ n = n4. In the case of the sodium channel there
are 3 m particles and one h particle, giving m3 ・ h.
When considering the single-channel level, each particle is binary-valued.
However, at the population level, each particle takes on analog values between
0 and 1. This binary-analog particle duality is the kind of anomaly
that can crop up as we move from one view of a model to another. Looked
at from the bottom-up view, the particles appeared as little independent
objects that could either block the channel and prevent conduction or get
out of the way and allow conduction. If any particle is blocking a channel,
that channel is closed and there is no flow through it. This view of the
model gives the binary values: 1 for not blocking and 0 for blocking.
Looking at the model from a higher level (perhaps not high enough to
be top-down, maybe middle-sideways), the particles in the Hodgkin-Huxley
model represent a population and are represented by analog values. At the
single-channel level, a channel is either opened or closed, 1 or 0. When we
consider a large population of channels, some percent of this population will
be open and the rest closed. We use an analog value from 0 to 1 to denote
the percent of channels that are open. Similarly, particles take on a binary
value, blocking or not blocking, when considered at the single-channel
12.5. Behavior of the active channels 227
level but an analog value when considering the entire channel population.
This analog value for the particle represents an unblocking probability or,
equivalently, the percent of particles that are not blocking at a given time.
Hence, to model a population of channels, we end up modeling a “population”
of m particles, a “population” of h particles, and a “population”
of n particles. This doesn’t really make much sense because an m particle
belongs to a channel. There is physically a population of channels, but not
an isolable population of m particles.
This binary-analog particle duality is an example of a common problem
in modeling. An aspect of a model that makes literal sense at one level
is manipulated for computational or organizational convenience at another
level. Or, as in this case, an aspect of a model that was arrived at empirically
reveals interesting predictions when its detailed implications are looked at.
Either way, the back and forth between levels alternately obscures and
illuminates. In the present case, binary particles are conceptually helpful
at the single-channel level, making predictions about what may be going
on in individual channel molecules. Although the prediction of a physical
blocker has not been borne out for the particular channels that Hodgkin
and Huxley studied, other potassium channels have been described in which
a piece of a protein physically occludes the channel.
At the membrane level, individually modeling a vast population of individual
channels with individual particles was hopeless in Hodgkin and
Huxley’s day. It is computationally feasible nowadays but would not be
practical for large simulations. The m population makes little sense at the
channel level but describes the action potential extremely well. Thus, this
compromise is both useful and used.
Particle dynamics
The population of m particle moves gradually from near 0 toward 1 during
depolarization. This turns the sodium channel on. At the same time as
the sodium channel is being turned on through the movement of the m
particle from 0 to 1, it is also starting to turn off through the movement
of the h particle from 1 to 0. h changes more slowly than m. Eventually h
approaches 0 and sodium channel conductance drops to a small value. The
n particle goes from 0 to 1 at about the same rate at which h goes from
1 to 0. This increases the potassium conductance, giving the potassium
battery more influence over the circuit. The potassium current opposes
and eventually reverses the spike depolarization.
m moves quickly to unblock the sodium channel, while h moves slowly
to block it and n moves slowly to unblock the potassium channel. Since m
moves so much faster than anyone else, sodium channel unblocking dominates
at the beginning. m increases, unblocking the channel and providing
the positive feedback. At the same time, the slower n movement is lead228
12. Hodgkin-Huxley Model
ing to some unblocking of the potassium channels, increasing potassium
current and reducing this depolarization through negative feedback.
During the action potential downswing, everything reverses. m heads
toward 0. n heads toward 0 more slowly. h goes back toward 1.
12.6 The particle equations
Now let’s look at the equations that describe m, h, and n. I show the
underlying functions graphically here; the full set of functions is given under
“Hodgkin-Huxley equations” in the Glossary. Conveniently, the same basic
form is used for all of the particles. The value of the sodium and potassium
conductances are products of a maximal conductance value and the values
of their associated particles. Conductance is represented by g and maximum
conductance by g, which is called either “g-bar” or “g-max.” For sodium,
conductance gNa = gNa ・ m3 ・ h. For potassium, gK = gK ・ n4. If all the
particles were set to 0, then gNa = gK = 0 and we would be left with only
the passive membrane. If all the n particles were set to 1, then potassium
current would be maximal: gK = gK. If all the m particles were set to 1,
but the h particle was 0, then gNa would still equal 0. As we will now see,
the parameterizations for these particles involve asymptotes that do not
allow them to ever reach 0 or 1, but they can get close.
Each particle is indirectly parameterized by voltage. At the negative
resting membrane potential, the m particle of the sodium channel is near 0
(off) and the h particle is near 1 (on). During the action potential m will go
from 0 toward 1 and h will go from 1 toward 0. m will change faster than
h will. We are dealing with rates of change, so once again we use calculus
and differential equations to describe the process.
The discrete form of the differential equation for m is τm・ Δm
Δt = m∞−m,
a standard form for a first-order differential equation. The passive membrane
equation, introduced in Chap. 11, can also be written using this
standard form: τmemb ・ ΔV
Δt = Vmax −V . In the charging curve of Fig. 11.5,
we noted that V approaches Vmax as the steady state. Similarly, m will
approach m∞. (The “∞” denotes steady-state — the value that would be
reached after an infinite amount of time.) In the case of the charging curve,
we noted that τmemb determines how fast the membrane will approach
Vmax. Similarly, τm determines how fast m will approach m∞.
As with the charging curve, the solution to the differential equation for
m will be an exponential. However, in contrast with the fixed parameters
of the passive membrane equation, the parameters of the particle equations
are not constant. They are functions of voltage (Fig. 12.5) and the voltage
keeps changing. Let’s take m as an example. Since voltage is changing, m is
chasing a moving target, m∞. If the membrane is being depolarized, m∞ is
getting bigger and m is moving toward a bigger number. τm is also changing
12.6. The particle equations 229
−100 −50
1
mV
8m
8n
8h
−100 −50 0
10 ms
mV
m
h
n
Fig. 12.5: Control parameters m∞, h∞, n∞, τm, τh, τn are themselves
parameterized by voltage. (τ values are given at a squidish 6.3o
C.)
with voltage, speeding up or slowing down the rate at which m is moving
toward m∞. τm triples with the first 30 mV of depolarization from rest,
slowing down the rise of m and thereby providing another, relatively minor,
negative feedback. The equations for h and n are described by equations
of the same form. m, n, and h are all simultaneously following their own
moving targets at their own changing rates.
The time constant curves show that τm is much smaller than τn or τh at
all voltage levels. This means that m will always move faster than h or n.
Specifically, τm is less than a half millisecond at all voltages. This allows
m to follow m∞ with a time lag of about a millisecond (two to three time
constants). τn and τh are in the range of 2 to 10 ms. Grossly, these particles
will lag behind their infinity values by about 10 to 20 ms. The rates for
all of these processes, like other chemical reactions, are dependent on temperature.
Action potentials at ocean temperature (e.g., in a cold-blooded
squid) take place much slower than they do in a warm squid or a warm
person. The τ ’s all have to be adjusted to take account of temperature.
The adjustment factor for channels and for other active proteins is called
Q10 (pronounced “q-ten”).
State variables define a state
Having introduced all the components, it is apparent that this is a highly
complex system of interacting parts. It is worth stepping back and summarizing
the interactions. There are four differential equations, one each
for V, m, h, and n. We have represented these differential equations in their
discrete form in order to handle them numerically. V, m, h, and n are the
four state variables of the system.
We previously discussed the concept of state in the context of neural
activity and neural representation. In the case of linked differential equations,
one defines the state of a dynamical system by noting the values of
230 12. Hodgkin-Huxley Model
all of the state variables. The Hodgkin-Huxley system is a four-dimensional
system. The state at any time can be given as a point in four-dimensional
space. You could draw that point on a hypersheet of four-dimensional graph
paper. Another well-studied system, the earth and the moon, is also a fourdimensional
system. The earth–moon system’s dimensions are position of
the moon, velocity of the moon, position of the earth, and velocity of the
earth. The fact that the earth and moon exist in three-dimensional space
is unrelated to its four dynamical dimensions. By the way, if we add in
the sun, we have a six-dimensional dynamical system with the positions
and velocity of each of the three objects. This is the famous three-body
problem.
The dynamics of astronomy and axons becomes complicated and interesting
due to the interactions of the state variables. The differential equations
are not independent. They interact. Something that happens to one state
variable will eventually affect all of them. If a meteor hits the moon, it
will affect the earth. Depending on the equations, the time constants, the
feedback loops, interactions with other bodies, etc., this effect may become
apparent in several weeks or only after many millennia.
Interlocking or linked differential equations are hard to grasp. All of
these feedback systems can make it impossible to figure out who is doing
what to whom when. The behavior of each state variable is influenced by
its history and by the history of the other state variables. When state
variable A changes, it influences state variables B, C,... These changes
then feed back and change element A in turn. Chasing these state variable
interactions around multiple feedback cycles produces proverbial chickenand-
egg predicaments. On the bright side, as linked ODEs go (ODE =
ordinary differential equation), the Hodgkin and Huxley ODEs aren’t so
bad. Everything is linked through only one state variable, voltage. This
means that we can sometimes view events as if voltage were “controlling”
everything, even though voltage is itself controlled by the current through
the sodium and potassium channels.
Taken in these terms, the genesis of the action potential can be viewed
as the story of four state variables, each chasing its steady-state values. As
an example, we can start with a current injection, Iin. V will start to chase
Vmax = Iin/gmemb. As V rises, m∞ rises and m will chase m∞ with a slight
delay. m rising causes rising gNa causes rising INa. The additional current
and conductance will push Vmax = Itotal/gtotal. Notice that the increase
in sodium conductance is actually opposing the increase in sodium current,
another minor negative feedback. Meanwhile h is chasing h∞ toward
smaller-and-smaller numbers. This is slowly turning off the sodium channel
even as the m particle is turning it on. Additionally, n is chasing n∞ and
turning on the potassium channel, pulling the potential down toward the
negative value of the potassium battery.
12.7. Simulation 231
5 ms
50 mV
0
1
8m
8h
m
h
Fig. 12.6: Simplification of Hodgkin-Huxley model, without
potassium channel. m follows m∞ and h follows h∞. Voltage rises
during sodium activation and falls during inactivation.
12.7 Simulation
The proof of the pudding is in the simulation. To simplify the description,
I start with a partial model, initially leaving out the potassium channel.
Without n, we are dealing with a three-dimensional system that would
allow us to map the state in our regular three-dimensional space if we
wanted to. Instead, I use two-dimensional plots to map each state variable
against time. Fig. 12.6 also shows the steady state values m∞ and h∞ as
they change with time. Note that m follows m∞ with a lag of about 1 ms,
while h follows h∞ with a longer lag. At the end of the simulation m has
caught up with m∞. h still lags h∞ since τh is considerably greater at this
V value. The peak of the τh curve, about 8.6 ms at 6.3oC, is at −67 mV,
which places it near resting membrane potential. Therefore, the relaxation
of h to h∞ is relatively slow at this voltage.
The vocabulary for describing changes in active channels employs a set of
confusingly similar words. Using V as a control variable that turns the other
state variables on or off, we can construct the following table of descriptors
for sodium channel control:
232 12. Hodgkin-Huxley Model
0
1 5 ms
50 mV V
m
h
n
Fig. 12.7: The full Hodgkin and Huxley simulation showing all
state variables. Arrows indicate afterhyperpolarization (AHP)
and its cause.
⇑ V ⇓ V
m activation (↑ m) deactivation (↓ m)
h inactivation (↓ h) deinactivation (↑ h)
Activation (of m) and deinactivation (of h, note the double negative)
are both needed in order to open the channel. Activation occurs with
depolarization, and deinactivation occurs with repolarization. Without
deinactivation, a second spike cannot occur. Similarly, either deactivation
(of m) or inactivation (of h) can close the channel. Inactivation terminates
the sodium influx near the peak of the action potential, while deactivation
is simply the resetting of m during repolarization. Using this vocabulary we
can provide yet another description of the cycle of Fig. 12.6. During the upswing
m activates and h inactivates. During the downswing, m deactivates
and h deinactivates.
Putting the potassium channel back into the model, the behavior changes
slightly (Fig. 12.7). Activation of the potassium channel turns on an outward
current that helps pull the voltage back down toward rest after the
peak of the spike (repolarization). The lag of n behind n∞ causes the potassium
current to remain on a little longer so that the voltage overshoots
12.8. Implications for signaling 233
5 mV
2 ms
Fig. 12.8: Identifying the spiking threshold for a Hodgkin-Huxley
model. The area around threshold is expanded below (rectangle).
the resting potential, producing an afterhyperpolarization (AHP, arrow in
Fig. 12.7) as the potassium channel slowly deactivates (arrow on n). (Note
that the potassium channel shows deactivation but no inactivation — there
is an activation particle n but no inactivation particle.)
12.8 Implications for signaling
The genesis of the action potential gives it several important properties that
have implications for data transmission. These can be illustrated using the
Hodgkin and Huxley model. These properties of neuron signal generation
can provide bottom-up clues for constructing network models.
The threshold and channel memory
The action potential has a threshold (Fig. 12.8). A current injection that
does not reach the threshold does not generate a spike. At the threshold,
234 12. Hodgkin-Huxley Model
inward (sodium) current exceeds outward (potassium) current and positive
feedback kicks in. In Fig. 12.8, the threshold for firing is about −51 mV.
From the perspective of neural network theory, this threshold could be
taken to be the sharp threshold of a binary activation function. This would
allow the neuron to add up its inputs and then provide a rapid signal
indicating whether or not sufficient excitation had been received. However,
in contrast to standard neural network theory, the Hodgkin and Huxley
threshold is not a fixed value.
The three channel particles, m, h, and n, all respond with a lag. This
lag provides a simple form of memory. Something that happened in the
past can be “remembered,” while the m, h, or n state variables catch up
with their steady-state values. The afterhyperpolarization (Fig. 12.7) is an
example of this. The AHP reflects firing history — it’s only present after
the neuron has fired.
This history is not always immediately reflected in the membrane potential
but can be held hidden in the state variables, inaccessible to
experimental detection. For example, a hyperpolarizing input provides immediate
inhibition. The hyperpolarization opposes any depolarization that
would push the potential up to threshold. However, after the hyperpolarization
ends, h is left at a relatively high and n at a relatively low value
for a brief period of time. This pushes the effective threshold down closer
to rest, making it easier to fire the cell. A subsequent depolarization will
open the sodium channel more, and the potassium channel less, than it
otherwise would (Fig. 12.9). Similarly, a preceding depolarization, which is
immediately excitatory, will have a late effect that is inhibitory.
From a neural network perspective, this membrane memory could be
tuned to allow the neuron to respond preferentially to certain sequences of
inputs. In this simple case, an optimal stimulation would involve an IPSP
followed by an EPSP after an interval of two to three times τn at RMP.
A neuron has dozens of channel types, allowing the construction of more
complex responses that can build up over relatively long periods of time.
A novel firing pattern could be the result of some combination of inputs
occurring over several seconds. This would allow the use of very complex,
hard-to-interpret coding schemes.
Rate coding redux
Having speculated about complex history-dependent coding schemes, I now
wish to return to the comforting simplicity of rate coding. In Fig. 11.13
we showed that slow potential theory explains the transduction from a
presynaptic rate code to a postsynaptic depolarization plateau; increasing
input rate gave an increased depolarization, due to increasing current flow.
Using the Hodgkin-Huxley model, we can complete the sequence of signal
transductions by showing that a depolarizing current injection converts to
increasing firing rate within a certain range (Fig. 12.10). The greater the
12.8. Implications for signaling 235
10 mV
5 ms
0
1
h
n
Fig. 12.9: With preceding hyperpolarization (solid line), h is elevated
and n is depressed allowing a small depolarization to fire
the cell 10 ms later. In the absence of the hyperpolarization, the
same small stimulus is subthreshold (dashed line).
current, the greater the firing rate. Outside this range, there is no spiking.
Below, there is a threshold for repetitive spiking. Above, spiking is blocked.
The trace at the bottom of Fig. 12.10 (0.88 nA) illustrates activity just
below the threshold for continuous repetitive spiking. Below 0.84 nA, this
Hodgkin-Huxley model produces only one spike. (By this Hodgkin-Huxley
model, I mean a parallel-conductance model using sodium and potassium
channels parameterized using the Hodgkin-Huxley equations, but with different
specific parameters describing the infinity and tau curves.) This
model produces one spike all the way down to the spiking threshold, about
0.2 nA. On the other end, high current injections produce higher rates.
236 12. Hodgkin-Huxley Model
Fig. 12.10: Increasing firing frequency with increasing current
injection. At and below 0.88 nA, there is no continuous repetitive
spiking. At and above 7.7 nA depolarization blockade is seen.
The spikes become smaller and smaller. This contradicts what I said earlier
about spikes being stereotyped and of constant amplitude. In fact, spike
size does carry information about spike rate. It does not appear that this
amplitude information is used however. In Fig. 12.10, a 4-nA current injection
gives a measurable 137-Hz spike frequency. The spikes at this rate are
only about half the size of the spikes produced by a 1-nA current injection.
As we go to higher and higher injections, the spikes get less and less
spike-like as we gradually pass over to the low-amplitude oscillation that
is characteristic of depolarization blockade. Depolarization blockade occurs
when the voltage gets so high that the h particle remains near 0. This
means that the sodium channel does not deinactivate. Since the sodium
channel is continuously inactivated, it is not possible to generate spikes.
For example, in the top trace, with 7.7 nA of injected current, the tiny
oscillation has an amplitude of about 4 mV and frequency of about 165
Hz. Examination of state variables demonstrates that this oscillation is
based on an interaction between V and m without substantial contribution
from n and h. This is the dynamics of depolarization blockade, not the
dynamics of neural spiking.
Using the Hodgkin and Huxley model of neuron spiking, we can compare
this realistic input/output (I/O) curve with the sigmoid (squashing) curve,
the idealized input/output curve used in artificial neural network modeling
(Fig. 12.11). Both curves are monotonically increasing, meaning they only
20 mV 50 ms
Hz
74
105
123
137
1
2
3
4
nA
0.88
7.7
2 mV 5 ms
12.9. Summary and thoughts 237
0 2 4 6
40
80
120
Hz
nA
Fig. 12.11: An I-f curve from a Hodgkin-Huxley model compared
to the standard sigmoid I/O curve of artificial neural networks.
go up. Although it does not asymptote, the realistic I-f (current-frequency)
curve in Fig. 12.11, like the sigmoid curve, does show some reduction in
slope with higher input values. However, the sigmoid curve covers all input
values, while the realistic I-f curve (current-frequency curve) only has
outputs for a certain range of inputs. In math-speak, this means it has
a limited domain (x-axis) and limited range (y-axis). Below this domain,
there is a floor effect; above, there is a ceiling effect. By altering the Hodgkin
and Huxley parameters we can move the ceiling, the floor, and the precise
relationship between current and frequency. However, these measures are
not independent, so that if you try to move the floor down, the ceiling and
slope of the I-f relation (the gain) will change as well. This means that it is
not possible to precisely tune a Hodgkin-Huxley model to produce exactly
the response one might want for a particular network model.
12.9 Summary and thoughts
The Hodgkin-Huxley model of the action potential is the most influential
computer model in neuroscience and as such remains a touchstone of
computational neuroscience. It’s a dynamical model that arises from the interaction
of four time-dependent state variables — V, m, h, and n. Of these
only V , voltage, is directly measurable. The others are putative populations
of switches that turn sodium and potassium channels on and off.
Electrically, the Hodgkin-Huxley model is the basic membrane RC circuit
with two conductances added in parallel. Hence the circuit is called
the parallel-conductance model. The two added conductances are the active
sodium and potassium conductances. These conductances are active
238 12. Hodgkin-Huxley Model
because they change with change in voltage. A controllable resistance (conductance)
is called a rheostat. Each of the conductances, including the
passive “leak” conductance, is attached to a battery. The battery potential
(voltage) depends on the distribution of the particular ion that flows
through its own selective conductance. This Nernst potential is the electrical
field that holds back the chemical flow of the ion across the membrane
down its concentration gradient.
The spike is the result of a set of interacting feedback loops. Depolarization
activates sodium channels (↑ m) producing positive feedback with
further depolarization. This is the upswing of the spike. Following this, two
negative feedback influences kick in. The sodium channel starts to inactivate
(↓ h). Additionally, activation of the potassium channel actively pulls
the potential back toward and past the resting membrane potential.
The Hodgkin-Huxley model can be used to see how action potential behavior
will influence neural signal processing and signal transduction. For
example, the neuron has a threshold for action potential generation that
can be altered by preceding inputs in a paradoxical way. An earlier excitatory
input will raise the threshold, producing a late inhibitory influence. A
preceding inhibitory input will lower the threshold, resulting in a relatively
excitable state.
Repetitive action potential firing is possible over only a limited range of
inputs. Too little input produces no spikes or only a few spikes. Too much
input produces depolarization blockade with a low amplitude oscillation.
This limited range makes it difficult to use standard Hodgkin-Huxley model
dynamics for rate coding in neural network models. Adding in the dynamics
of other channels that are present in neurons makes it possible to get a wider
range of firing frequency.
13
Compartment Modeling
13.1 Why learn this?
In the previous chapters, I took two steps away from artificial neural
network simplifications toward more complex neural realities. First, I
demonstrated how membrane properties allow a neuron to do temporal
integration. Then I showed how active channels produce the action potential,
a complex signal. In this chapter, I build on these concepts in order
to produce a full modern model neuron.
Most neurons have sodium and potassium channels similar to those that
Hodgkin and Huxley found in squid axon. Neurons also have many other ion
channel types. In the case of both potassium and calcium channels, there
are large families of channels with different thresholds and time courses
for activation and inactivation. Some of these channels are activated by
the presence of ligands. These ligands may be other ions (e.g., a calciumsensitive
potassium channel) or chemicals of various types. The ligands may
bind the channel from within the cytoplasm or, in the case of synapses,
from outside of the cell. The ligand sensitivity may be accompanied by
voltage sensitivity as well. Sophisticated multistate Markov models have
been developed to describe these channels in great detail. For simplicity,
these various channels types are often modeled using modifications of the
Hodgkin-Huxley formalism.
Complex branching dendritic trees introduce additional complexity
compared to the simple parallel-conductance model. Dendritic trees are
modeled using compartment modeling. Each compartment is an RC or
240 13. Compartment Modeling
parallel conductance circuit. Individual compartments are connected by
resistors. Different inputs (synapses) can be located in different compartments,
placing them at different locations in the dendritic tree. In general,
inputs that are further away from the spike-generating zone will have less
influence on output than those that are closer. However, active ion channels
can also be located at different points in the dendritic tree. Inward currents,
such as those produced by sodium or calcium channels, would give excitatory
synaptic inputs a boost. Outward or shunting conductances, produced
by potassium or chloride channels, would tend to decrement excitatory
inputs.
There are various types of synapses that provide inputs into neurons.
Not surprisingly, they are far more complicated that the simple weight
times state multiplication connection that we used in artificial neural network
models. Most synapses are chemical and introduce complicated issues
of chemical kinetics, diffusion, and breakdown. We will not attempt to address
this level of modeling here but will utilize a simple variation on the
Hodgkin-Huxley model to describe synapses.
By the end of this chapter, we will be dealing with a complex singleneuron
model. From a numerical processing point of view, these single
neuron models are actually bigger than many artificial neural network models.
As I’ve previously suggested, it is entirely possible that the real single
neuron might also be bigger from an information processing perspective
— single neuron as CPU rather than transistor. However, with all of this
complexity, such models of single neuron electrical activity still only scratch
the surface of the information processing potential of the cell. Specifically,
there remains a big gap between simulation and reality due to the lack of
data and modeling techniques for handling chemical interactions within the
cell.
The neuron is a giant chemical plant (well, a tiny chemical plant with
lots of different chemical reactions). Much of this chemical activity is standard
cell metabolism that looks the same in the liver or kidney. However,
other chemical messenger networks are involved in neural information processing.
Neurotransmitters are chemicals that transmit information across
chemical synapses between neurons. Some of these transmitters cause the
release of other chemicals that pass the message along inside of the cell.
These secondary chemicals are called second messengers. Second messengers
may then cause the release of third and fourth messengers. The various
higher-level messengers may include positive and negative feedback loops,
producing chemical networks of great complexity. Such chemical networks
can be modeled. However, the complexity of these chemical models makes
it difficult to incorporate them into standard electrical models of single
cells, much less into networks of these cells.
13.2. Dividing into compartments 241
13.2 Dividing into compartments
Compartment modeling is used widely in biology, typically to follow concentrations
of chemicals. In general, one divides some biological entity (e.g.,
a person) into compartments. In the model, the chemical we’re measuring
is at the same level everywhere within a single compartment. The different
compartments exchange the chemical among themselves depending on their
connectivity. A typical example comes from pharmacology.We can describe
the distribution of a drug in terms of the compartments of the human body:
blood, bone, fat, muscle, urine, etc. If we are treating a bone infection, we
place penicillin intravenously in the blood compartment. Since “blood” is
a single compartment, we assume, somewhat unrealistically, that the drug
instantly mixes with all of the blood so that the drug concentration is the
same everywhere in that compartment. We then look at how quickly this
compartment exchanges penicillin with bone, urine, and muscle, the first
being the target location, the second being the primary route of excretion
(ground in the electrical analogy), and the third being an alternative location
where drug may be shunted away. We can use the compartment model
to find out how much penicillin will get to the bone, how fast it will get
there, and how fast it will leave. This enables us to calculate a good dosage
and schedule in order to achieve adequate penicillin concentration to kill
the germs.
In neural modeling, we are dealing with voltages instead of concentrations
and with current instead of chemical flux. The single compartment is
equipotential (syn. isopotential) — there is one voltage everywhere in that
compartment. This is equivalent to the instantaneous mixing referred to
in the previous example. If the compartments are too large, equipotentiality
cannot be assumed and the numerical simulation will produce a poor
approximation. Each compartment corresponds to a section of membrane.
The single compartment model must include at least capacitance and leak
conductance.
In addition to the obligatory capacitance and leak conductance, active
components will also be added to some compartments. This requires additional
rheostat/battery branches, similar to the sodium and potassium
branches of the parallel-conductance model (Fig. 12.2). Some of these
rheostats will be synapses as explained below. Others will be similar to
Hodgkin-Huxley sodium and potassium channels, usually readjusted to
more closely match the kinetics of the ion channels found in a particular
cell type of a particular non-squid animal. Other types of sodium and
potassium channels, as well as calcium channels, may be included. Each
of these will have the ion-appropriate battery. Each will have a rheostat
with some kind of dynamics. Rheostat dynamics may be parameterized
by equations similar to the Hodgkin and Huxley equations or by some
other kinetic scheme. Rheostat sensitivity to second messengers may be included
in addition to, or instead of, voltage sensitivity. Potassium channels
242 13. Compartment Modeling
that are sensitive to calcium are commonly found in neurons. Although
complex chemical modeling is not usually done in compartment models,
internal calcium concentrations are sometimes included in order to model
the activation of these calcium-sensitive potassium channels.
An individual compartment is equipotential but will typically differ in
voltage from neighboring compartments. In this way, different parts of the
neuron can be doing different things at the same time. For example, one
part of the neuron may be spiking while another part is hyperpolarized.
Building the model
To build a model of a real neuron, we measure the lengths and diameters of
dendrites and measure the soma. A made-up picture of a neuron is shown
in Fig. 13.1 (left). Neurons are filled with dye to make their dendrites easily
visible; the Golgi staining method colors neurons black. Using a microscope
linked to a computer, one can trace a neuron. This is hard work since
neurons are not typically flat and therefore must be traced through multiple
sections that don’t always match up well, due to differential shrinkage of
the tissue during processing.
Using these measurements, we can assign cylindrical compartments
(Fig. 13.1 center). The soma or cell body is in reality a roughly spherical
or pyramidal structure. However, it’s just going to end up as a bunch
of conductances and a capacitance in the circuit model. Therefore, we can
represent it as a cylinder in the intermediate model as long as we preserve
its surface area. (More surface area will translate into more capacitance
and more conductance.)
In the context of numerical integration, the spatial division of dendrites
into small cylinders of length Δl is analogous to the temporal division of
duration into subdivisions of Δt. The quality of any numerical integration
depends on making the time steps Δt short enough so that the change in
state variables during a single time step is insignificant. Similarly, the quality
of a compartment model depends on making the spatial steps (cylinder
length) small enough that the change in voltage over the length of the
cylinder is insignificant. This is the assumption of equipotentiality. Choosing
smaller cylinders or smaller time steps will give a better approximation
but will increase simulation time.
Within the dendrite, current flows through the cytoplasm inside. Cytoplasm
is glop with mobile ions that carry current. Cytoplasmic resistance
is a significant impediment to flow of current along the dendrite. Wider
cylinders allow more ions to pass, giving less resistance, less signal dropoff,
and faster signal transmission. This is the reason that squid axons are
so wide. If a dendritic cylinder is narrow, it will have relatively low conductance
(high resistance) and a greater voltage drop-off along its length.
The voltage drop-off can be calculated using Ohm’s law. The difference in
voltage from end to end equals longitudinal current times longitudinal re13.2.
Dividing into compartments 243
anatomy cylinders
circuit
Fig. 13.1: Three representations of a neuron model: faux Golgi
anatomy (kind of what a cell looks like under the microscope);
cylindrical compartmentalization; equivalent circuit model.
sistance: V+−V− = Il ・Rl. (Notice the use of subscripts for spatial intervals
compared to the use of superscripts for temporal intervals.) Narrow cylinders
have high longitudinal resistance and therefore must be divided into
a larger number of compartments. This is illustrated in the two branches
in the cylindrical representation in Fig. 13.1.
Having chosen the cylindrical compartmentalization of a dendrite, we can
put together an equivalent circuit model for each compartment and connect
them with resistors. Each compartment will have capacitance and leak
conductance. Some may have additional conductances representing voltage244
13. Compartment Modeling
Passive Active/Synaptic Synaptic
ECa EK Esyn Esyn
Fig. 13.2: Three example compartments from a dendrite model.
On the circuit diagram, different rheostat branches are distinguished
only by the directions of their batteries; I have added
labels to identify them further.
sensitive channels and synapses. In the circuit of Fig. 13.1, the ground
symbol indicates the extracellular side of each compartment. For simplicity,
I have illustrated each compartment as a standard parallel-conductance
circuit.
When putting together a model of a particular neuron, neurophysiological
data would likely suggest the inclusion of additional conductances
in some compartments. Other parallel rheostats would be included to
represent synapses. Active conductances might be omitted from some compartments
in the interest of computational speed and efficiency. In Fig. 13.2,
I show three compartments from a neuron model. The passive compartment
to the left has capacitance and leak resistance but no rheostats. The second
compartment is labeled to indicate that it has active potassium and
calcium channels as well as an inhibitory synapse. The synapse can be identified
as inhibitory by noting that the negative pole of the battery connects
to the inside of the membrane. The third compartment has an excitatory
synapse. Without the labels, there is nothing at the circuit diagram level to
distinguish a synaptic (ligand-gated) channel from a voltage-gated channel.
All of that detail is in the parameterizations, the details of the equations
describing the rheostats.
Individual compartments in a multicompartment model are connected
to neighboring compartments by a resistor. This resistor represents the
cytoplasmic axial resistance, also called longitudinal resistance, between
the center of the two connected compartments. As mentioned above, wider
dendrites will have less resistance. However, wider dendrites will typically
be represented with longer cylinders so that total distance, and total resistance,
between centers will be similar. Where two compartments represent
cylinders of different widths, resistances sum from cylinder center to the
connecting point on either side of the junction. A branch point, where two
dendrites come off a single parent dendrite, is represented with two resistors
as shown in Fig. 13.1.
The whole massive circuit in Fig. 13.1 or Fig. 13.2 can be simulated using
the same methods we previously detailed: 1) Kirchhoff’s law, to sum
13.3. Chemical synapse modeling 245
up all currents at intersections; 2) Ohm’s and capacitance laws, to determine
currents across passive components; 3) Hodgkin-Huxley equations
and parameters, to determine behavior of particles for active components;
4) numerical integration, to solve the differential equations. This is a big
project in itself, likely to delay any simulation project for months to years.
For this reason, it is generally best to use a software package that is designed
to handle solutions of differential equations. Since these packages
exist, I will not further detail the mathematical and programming complexities
of compartment-model calculations. These details aside, a basic
knowledge of compartment modeling remains useful for understanding how
to set parameters and handle simulations.
13.3 Chemical synapse modeling
We previously utilized square wave (Fig. 11.8) and alpha function
(Fig. 11.12) current injections as inputs to our models. The former was a
realistic model of current injected through an electrode by an experimentalist.
The latter was an intermediate model that had the approximate form of
a synaptic current. Chemical synaptic responses are actually conductance
changes rather than current injections. A synaptic conductance change will
permit current flow. As I show, sometimes it is the synaptic conductance
change, rather than the synaptic current, that affects postsynaptic activity.
The Hodgkin-Huxley model describes membrane rheostats controlled by
voltage. Chemical synapses are membrane rheostats controlled by neurotransmitters.
When modeling a network, these neurotransmitters are not
explicitly included. Instead, a spike in a presynaptic neuron is used to activate
a rheostat in the postsynaptic neuron. The postsynaptic rheostat
represents an ion channel triggered by a postsynaptic receptor. Depending
on the polarity of the battery associated with that rheostat (inward positive
or inward negative), that synapse could be made excitatory or inhibitory
(Fig. 13.2).
As with most things in nature, chemical synaptic transmission is terribly
complex. Synaptic simulation can be made terribly complex as well. A
full synaptic model would have to consider chemical diffusion, reuptake,
breakdown, and interaction with proteins before one even gets to a receptor.
Then, at the receptor level, one would consider the fact that receptors
are proteins that typically have multiple states (each of which could be
included in a more complex model, e.g., a Markov model). This gives them
properties such as activation, deactivation, inactivation, and deinactivation
as well as desensitization (channel turns off if the chemical ligand has been
sitting there too long), allosteric interactions (other chemicals binding at
different sites can alter the response to the main neurotransmitter), and
multiple binding sites (more than one molecule must bind for the channel
246 13. Compartment Modeling
to open). Then, there’s the fact that the synapse can grow and change.
This depends not only on classical learning effects but also on more lowlevel
adaptation to the chemical environment. For example, receptors will
typically upregulate (more receptors will be inserted into the membrane)
if they do not see much neurotransmitter for a while and downregulate in
the converse condition. They can also change shape and shift the location
of receptors.
Of course, we want to ignore all of that and focus on the basics: a chemical
is released and binds, the associated channel opens (activates) and then
closes (deactivates). For simplicity, we will have the channel open instantly.
This means that the conductance will go instantly from 0 to gmax and the
current will also shoot up instantly. However, the voltage will not shoot
up since the membrane capacitance provides a brake. We will then use an
exponential decay for conductance inactivation. This corresponds to the
simple differential equation (discrete version): τg ・ Δg
Δt = −g.
In Fig. 13.3, excitatory and inhibitory synaptic responses are compared.
The same conductance is used in both cases. However, the synaptic reversal
potentials (Esyn) differ. For this reason the excitatory postsynaptic current
(EPSC) is negative (inward and depolarizing), while the inhibitory postsynaptic
current (IPSC) is positive (outward and hyperpolarizing). This
is the result of the expression for current: Isyn = gsyn ・ (V − Esyn). In
the case of the excitatory synapse, the reversal potential (Esyn) is more
positive than resting potential. For the inhibitory synapse, the reversal
potential is more negative than the resting membrane potential (RMP).
Although the conductances in Fig. 13.3 are the same in both the excitatory
and inhibitory cases, the magnitudes of both postsynaptic currents
and postsynaptic potentials differ. The differing size of EPSC and IPSC
reflects the difference between membrane voltage and Esyn in each case:
V −Esyn. Eexcite is far from rest, while Einhib is close to rest. This potential
difference is called the driving force for the current. In Fig. 13.3, the
initial driving force (RMP −Esyn) is −70−−20 = −50 mV for the excitatory
case, and −70−−90 = 20 mV for the inhibitory case. As membrane
voltage moves toward Esyn, driving force gradually decreases. A very large
synaptic conductance can drag membrane voltage all the way to Esyn.
Shunting inhibition
Classically, inhibitory synapses produce a hyperpolarizing IPSP. However,
some inhibitory synapses do not change the membrane potential. This is
known as shunting inhibition. Shunting inhibition occurs when Esyn =
RMP. Activation of a shunting synapse alone will not cause any current
to flow: Isyn = gsyn ・(RMP − Esyn) = 0. Therefore, there will be no change
in membrane potential. However, in the presence of an excitatory input,
the shunting influence reduces depolarization. Inward current that would
otherwise pass outward through the capacitor and depolarize the membrane
13.3. Chemical synapse modeling 247
10 ms
10 mV
.25 nA
10 nS
Conductance
Current
Voltage
IPSC
EPSC
EPSP
IPSP
Fig. 13.3: Simple model of a chemical synapse: Conductance
activates instantly, then inactivates with time constant τg. Current
flows inward (negative) for excitation (dashed line), outward
(positive) for inhibition (solid line). Inward current produces a
depolarizing EPSP (dashed line); outward current produces a
hyperpolarizing IPSP (solid line).
is instead passed outward (shunted) through this synaptic short circuit and
made ineffectual.
Consider the two examples in Fig. 13.4. A hyperpolarizing IPSP and
a depolarizing EPSP roughly add up. In this case they produce a small
compound PSP (compound since it is the summation of postsynaptic potentials),
consistent with the fact that a large EPSP was being added to a
smaller IPSP. This synaptic interaction may be regarded as being approximately
additive, comparable with the linear summation of inputs used in
artificial neural network models. However, a check of the numbers reveals
that these PSPs are not strictly additive. In the top traces of Fig. 13.4,
the peak values are 23.4 mV depolarization for the EPSP and −9.3 mV
hyperpolarization for the IPSP. The numbers sum to +14.1 mV, but the
248 13. Compartment Modeling
=
=
10 mV
10 ms
Hyperpolarizing
Shunting
Fig. 13.4: A large EPSP and small IPSP add together to produce
a smaller EPSP. Each demonstration is the result of three separate
simulations: excitation alone (left top), inhibition alone (left
bottom), compound PSP (right).
compound postsynaptic potential is only +9.7 mV. The lost millivolts are
due to the shunting that accompanies all synaptic activation.
Shunting is present in association with any conductance change, whether
synaptic or due to voltage-sensitive channels. Shunting is most dramatic in
cases where the voltage shift is minimal. In the lower trace of Fig. 13.4, I
used a shunting conductance 10 times greater than the excitatory conductance.
The +23.4 mV EPSP is reduced to a 4.2-mV compound PSP. By
increasing the shunting conductance, one can make the resulting compound
PSP smaller and smaller. Shunting inhibitory synapses are sometimes called
multiplicative synapses to illustrate the notion that a zero IPSP “times”
an EPSP gives a zero result. Actually, the compound PSP will never go
to zero, no matter how large one makes the shunting inhibition. Without
the depolarization from rest, there is nothing to shunt. The concepts of
additive and multiplicative inhibition can be used to map the neurobiology
onto the simple arithmetic of artificial neural networks. However, they are
only crude approximations.
GABA and glutamate
There are many neurotransmitters. Each neurotransmitter is associated
with many different receptors. Each receptor may mediate many different
actions. A description of what is know about synaptic responses fills books.
I will be brief.
Neurotransmitters are somewhat arbitrarily classified as either a classical
neurotransmitter or a neuromodulator, based on their time scale of
action. The implicit hypothesis is that neuromodulators act on relatively
13.3. Chemical synapse modeling 249
long time scales, while classical neurotransmitters are quick and are involved
in moment-to-moment information processing. However, given the
myriad actions of most neurotransmitters, it is likely that many have some
modulatory and some classical functionality. Furthermore, it may be that
the time scales of neurotransmitter actions are a spectrum, and cannot be
readily classified into two distinct groups. In Chap. 14, I discuss acetylcholine,
a neuromodulator that is believed to influence learning. For now, I
focus on the two major classical neurotransmitters, GABA and glutamate.
The major inhibitory neurotransmitter in the central nervous system is
GABA. The major excitatory neurotransmitter is glutamate. There are
two types of GABA receptors, conveniently called GABAA and GABAB.
GABAA is directly connected to an ion channel selective for chloride.
Chloride is distributed so that its Nernst potential is near to the resting
potential. Therefore, the GABAA effect is primarily shunting. It is also
a relatively fast PSP, going on and off over about 5 to 10 ms. The GABAB
receptor, by contrast, is complexly coupled via second messengers that have
a variety of effects. A hyperpolarizing effect on the membrane is produced
via a link to a potassium-selective channel. Because of the time needed for
the second-messenger linkage and the properties of the associated channel,
GABAB is relatively slow, lasting about 100 ms.
On the excitatory side, with glutamate as the neurotransmitter, the receptors
have somewhat more obscure names: AMPA and NMDA (defined
in the Glossary). As with many receptor names, these are based on sensitivity
to alternative agonists — chemical ligands that are particularly good at
activating the receptor. As in most cases, these are artificial drugs and are
not present in the body, where glutamate is the natural ligand. The AMPA
receptor responds rapidly. The NMDA receptor is slower. These two receptors
are well studied since together they are responsible for the biological
analogue of Hebbian learning, long-term potentiation (LTP). The properties
of Hebbian learning through LTP are largely based on the response
profile of the NMDA receptor.
Both AMPA and NMDA receptors are linked directly to ion channels,
rather than to second messengers. Both have a reversal potential (Esyn)
of approximately 0 millivolts (about 70 mV depolarized from rest). The
Hebbian rule states that a synapse is strengthened when it simultaneously
registers pre- and postsynaptic activity. The NMDA receptor can make this
simultaneous measurement because it has both ligand and voltage sensitivity.
The sensitivity to an external chemical ligand allows it to register
presynaptic activity. Presence of glutamate indicates presynaptic activity
since the glutamate is released with the arrival of a presynaptic action
potential. The NMDA channel’s voltage sensitivity gives it a way of measuring
postsynaptic excitation. However, postsynaptic depolarization does
not necessarily lead to postsynaptic spiking, so LTP-mediated synaptic enhancement
can take place without increased postsynaptic activity. This
suggests that LTP may not be a classical Hebbian mechanism.
250 13. Compartment Modeling
Apical dendrite
Distal dendrites
Proximal
dendrites
Basilar dendrites
1 mm
Fig. 13.5: A neocortical pyramidal neuron.
The voltage sensitivity of the NMDA channel is a little different than
that of the Hodgkin-Huxley channels. In the Hodgkin-Huxley channels,
the voltage-sensitive mechanism is an intrinsic part of the channel protein
that shifts with voltage change. By contrast, NMDA voltage sensitivity
is due to blockade of the pore by a magnesium ion (Mg++) coming from
outside of the cell. This ion will sit in the pore unless it is kicked out
due to depolarization. With the magnesium out of the way, the pore will
conduct ions if it has also been activated synaptically by glutamate, the
neurotransmitter.
LTP is much studied. It has been found that the expression of LTP is due
to a change in AMPA conductance rather than NMDA conductance. Thus
the NMDA channel acts as the Hebbian trigger, while Hebbian expression
is handled by AMPA. In addition to depolarization, local calcium entry is
important in producing the Hebbian change. There has been much debate
as to whether the Hebbian strengthening takes place at the postsynaptic
receptor level or at presynaptic neurotransmitter release sites or both.
13.4 Passive neuron model
Fig. 13.5 shows a tracing of a cortical pyramidal neuron. Such neurons are
called “pyramidal” because the soma is somewhat pyramid-shaped (not
shown in figure). The long dendrite at the top, emerging from the apex of
13.4. Passive neuron model 251
the pyramid, is called the apical dendrite. The hairy dendrites at the base
are called basilar dendrites. Locations are referenced with respect to the
soma: a distal dendrite is far from the soma and a proximal dendrite is
close to the soma.
Passive compartment models have long been the standard model for
assessment of response properties. In a passive model, each individual
compartment has only capacitance and leak conductance, with no active
voltage-sensitive conductances. Synaptic conductances may be included as
inputs to passive models. Synapses are active elements, in that they involve
variable responses to a ligand. Despite this, the model is still considered
passive. The response of a passive model does not depend on prior activity
history.
Passive neuron properties have been a subject of debate for decades. The
leakiness of neuronal membranes can’t be measured directly. It can only
be inferred from responses to current injection. Unfortunately, connecting
an electrode to a cell damages the cell, making it hard to know whether
subsequent measurements are accurate. Some electrodes are stuck right
through the membrane (cell impalement). This makes a hole. Current leaks
through the hole. Other electrodes solve this problem by sealing tightly on
the membrane (whole-cell patch). However, this actually produces a bigger
hole that goes from the cell into the electrode itself. Current leakage isn’t
a problem, but chemical leakage is. This type of recording changes cell
membrane function by altering the second messengers inside the cell. Both
approaches introduce artifact. This is the classic experimental dilemma —
probes that mess up the thing that you’re probing.
The debate about passive properties has been remarkably passionate.
Perhaps this is because a conclusion would be a clue in a larger debate
about the processing role of the neuron: simple or complex, transistor or
CPU? If neuronal membranes are very tight (high resistance), then the
neuron is electrotonically compact. This is also expressed by saying that
the cell has a long length-constant. (This can be confusing: a cell with a
long length-constant is compact. The length-constant is long so the cell is
electrotonically small by comparison.) A long length-constant means that
signals coming in from anywhere will easily reach the soma and be able to
trigger action potentials. In the case of a compact cell, even large neurons
might act like the simple processing units of the artificial neural network
models. Conversely, if neurons are leaky (high conductance, short lengthconstant),
then signals will be handled differently depending on where they
arrive in the cell. In a situation where single inputs are ineffective, complicated
spatiotemporal input combinations might be required to produce
cell activity. This would suggest complex coding and decoding strategems.
Recent studies have suggested that many, perhaps all, neurons have
active membranes with voltage-sensitive channels. Some active dendrites
that have been studied appear to generate inward currents that increase
depolarizing potentials, boosting the EPSP down to the soma and help252
13. Compartment Modeling
25 mV
50 ms
Fig. 13.6: Identical synaptic inputs at four locations demonstrate
variation in signal spread. Position of trace shows recording location.
EPSPs are activated from bottom to top. Arrow shows
which EPSP in the sequence was initiated at that recording
location.
ing the action potentials head back up. Conversely, some dendrites may
have depolarization-activated channels that mediate outward current. Such
channels would reduce excitatory potentials and thereby decrease the excitability
of the cell. It is also possible that some neurons may have varying
mixtures of boosters and reducers located near particular synapses as an
adjunct synaptic-weighting scheme.
13.4. Passive neuron model 253
Synaptic responses
Distal synaptic inputs will generally have less effect on the soma than will
proximal inputs. If we initiate an EPSP at different points in this neuron,
we can see how much signal spreads to other parts of the neuron (Fig. 13.6).
In the passive neuron model, membrane conductance is the major parameter
that will determine signal spread. If membrane conductance is high
(membrane resistance low), then the membrane is leaky, hence the lengthconstant
is short, hence the cell is electrically large. Signals will not get
very far. Conversely, if the conductance is low (membrane resistance high),
then the membrane is tight, the length-constant is long, and the cell is
compact. In Fig. 13.6, I used a low membrane conductance so that signals
could propagate from one end of the neuron to the other.
As expected, there is a fall-off in the signal along the dendritic tree.
Notice, however, that the EPSPs initiated at the ends are larger than those
triggered at the soma or halfway up the basal dendrite. Remember that
the same synaptic conductance was used at each location. The difference
in response is due to the fact that current has only one way to go at a
sealed end and therefore tends to “pile up” instead of leaking away in both
directions down the dendrite. In electrical terms, these terminal locations
have relatively high input impedance. Impedance refers to any obstacle to
current flow. In a dendrite, resistance provides a constant impedance, while
capacitance provides a frequency-dependent impedance.
In addition to increasing the size of an EPSP at a distal dendrite, high
terminal input impedance also reduces the drop-off of the EPSP going
away from the soma, compared to the drop-off of an EPSP going toward
the soma. These two effects, the increased size of a distal EPSP and the
increased drop-off toward the soma, tend to cancel out. Therefore, a distal
input arriving at the soma is about the same size as a somatic input arriving
at the distal dendrite. High distal input impedance produces a large distal
EPSP (top trace, 4th EPSP). This large EPSP is substantially decremented
when seen at the soma (3rd trace, 4th EPSP). Going in the other direction,
the somatic EPSP is relatively small due to lower input impedance (3rd
trace, 2nd EPSP) but signal fall-off from the soma to the distal dendrite is
relatively small (top trace, 2nd EPSP). In this way, input-impedance differences
are balanced by transfer-impedance asymmetry in passive models.
David Jaffe and Ted Carnevale have described this as “passive normalization,”
a possible mechanism to equalize synaptic inputs arriving at different
locations in the dendritic tree.
As we saw in our discussion of shunting synapses, leakiness of the membrane
can increase due to synaptic activation. It may be that neurons that
have very tight membranes when measured in vitro are much leakier in vivo
due to the constant barrage of synaptic signals. A pyramidal cell can have
tens of thousands of synapses. Large numbers of excitatory and inhibitory
synaptic signals could balance each other so as to produce no net change in
254 13. Compartment Modeling
membrane potential, yet make the neuron leaky. This would substantially
decrease signal spread in the dendrites.
13.5 Back-propagating spikes and the Hebb synapse
According to the classical Cajal neuron doctrine, the action potential, generated
in the soma, spreads forward to the axon but not back into the
dendrites. Lately, there has been considerable interest in the spread of
spikes back up the dendrite. This is referred to as a back-propagating
action potential. Although it is named so partly in reference to the
back-propagation artificial neural network algorithm (Chap. 9), the backpropagating
spike and back-prop algorithm have nothing in common other
than their backwardness.
It is generally assumed that back-propagating action potentials require
active dendrites.We can test this assumption in a model by comparing spike
back-propagation with passive dendrites to back-propagation with active
dendrites. In Fig. 13.7, the soma is active (Hodgkin-Huxley channels) but
the dendrites are passive. A synapse is activated at the soma (rightward
arrow). An action potential, generated in the soma, propagates up the
dendrite, broadening and decrementing as it goes. At the distal dendrite
the action potential is barely apparent (oblique arrow). Even if one sets
Rmemb to a very high value, increasing the length-constant so as to make
the neuron as compact as possible, the spike only dents the membrane
potential in the distal dendrite.
Repeating the simulation of Fig. 13.7 with active dendrites (Hodgkin-
Huxley channels) produces a far more robust back-propagating action
potential (Fig. 13.8). In this case I placed the synapse in the distal dendrites
(downward arrow) rather than in the soma. While inhibitory synapses are
often on or near the cell soma, excitatory synapses tend to be located more
distally. With the passive dendrites of Fig. 13.7, a distal synapse would not
be able to fire the cell. In Fig. 13.8, the EPSP is augmented by the active
channels and easily reaches the soma. After the neuron fires, the action potential
heads back up the dendritic tree. The earlier passage of the EPSP
activated and began to inactivate the sodium channels in the dendrites.
This partial inactivation slightly distorts and delays the back-propagating
spike (oblique arrow).
Recall that the back-prop algorithm propagates error signals backward
across synapses. Spike back-propagation propagates an outgoing signal
from the soma back into the synaptic arrival zone. Although back-prop,
the spike, is not back-prop, the algorithm, it does have implications for
learning. Hebb’s rule states that synaptic strength increases when there
is a coincidence of presynaptic activity with postsynaptic activity. Postsynaptic
activity can be interpreted in different ways, but Hebb’s original
13.5. Back-propagating spikes and the Hebb synapse 255
10 ms
25 mV
Fig. 13.7: Synaptic activation at the soma (rightward arrow) triggers
an action potential that then passes up the dendritic tree.
Each trace is positioned next to its location on the dendritic tree.
Only the soma compartment is active; all dendrite compartments
are passive.
idea was that postsynaptic activity is the activity coming out of the postsynaptic
neuron. Back-propagating action potentials provide a way for this
output signal to get back to the synaptic zone so that pre- and postsynaptic
activity can be simultaneously monitored.
As mentioned above, the phenomenon of LTP and the NMDA receptor
have been intensively studied as a biological example of the Hebb synapse.
256 13. Compartment Modeling
10 ms
25 mV
Fig. 13.8: Back-propagating action potential. Down arrow: EPSP
location. Oblique arrow: delay in back-propagating spike.
We can now put together models that connect input and output in the
way Hebb originally envisioned. As described above, NMDA serves as a
coincidence detector by being sensitive to both intracellular voltage and
extracellular ligand. The ligand is glutamate and the voltage sensitivity
comes from channel blockade by magnesium. Active dendritic channels
permit the action potential to propagate back up the apical dendrite. The
depolarization from the back-propagating action potential relieves the magnesium
block. In the presence of glutamate, NMDA will signal for synaptic
strength augmentation via the AMPA receptor.
If we assume that augmentation will be proportional to total NMDA current,
we can look at how the concentration of dendritic active channels will
affect this NMDA signal. We would expect that increased sodium-channel
density will lead to a bigger back-propagating spike, which will produce
more NMDA current. Modeling shows this to be true for a limited range
of values. However, when the density starts to get too high, the NMDA
current starts to decline. This is due to two factors. The high-density case
13.6. Summary and thoughts 257
gives a larger but shorter back-propagated action potential. The shorter
duration of the action potential means less time for the incoming current.
Remember that the incoming current is dependent not only on the conductance
but also on a driving force dependent on the distance between
the voltage and the reversal potential for the channel. Therefore, a higher
amplitude spike reduces driving force and also reduces incoming current.
13.6 Summary and thoughts
Compartment modeling combined with Hodgkin-Huxley-like parameterizations
of ion channels and synapses is the state of the art for detailed neural
simulation. Each compartment is a parallel-conductance model. Different
rheostats can be used in different compartments to allow representation of
excitatory and inhibitory synapses, as well as a wide variety of ion sensitive
channels with different dynamics. In general, chemical processing in
neurons is omitted from these models, although some will include internal
calcium concentrations in order to accurately model calcium-sensitive
potassium channels. Synapses are modeled by triggering a rheostat based
on activity in a presynaptic cell. There are a variety of common neurotransmitter
receptors that are modeled, included GABAA, GABAB, AMPA, and
NMDA.
There remains considerable uncertainty regarding the passive properties
of neurons. This may have implications for the processing power
of the single neuron. If neurons are tight (high resistance, long lengthconstant,
electrotonically compact), then the neuron will tend to treat all
signals similarly and sum them in the manner of an artificial neural network
sum-and-squash unit. If neurons are leaky (high conductance, short
length-constant, electrotonically large), then different signals will have very
different effects on the neuron, suggesting more processing complexity. It is
likely that some neurons are compact and that others are not. Of course,
the physical size of the neuron will also play a role: little neurons are likely
to be electrotonically compact. The amount of synaptic input will also
be an important factor because synaptic activation increases conductance.
Neurons that receive constant synaptic bombardment will be leakier and
electrotonically larger. There are also neuromodulators that will alter the
electrotonic properties of a neuron.
This active debate about passive properties may be irrelevant for the
many large neurons that have active dendritic channels that boost depolarizing
signals. There is still relatively little known about these dendritic
channels, some of which may actually mute, rather than boost, signals. Depolarizing
currents have been shown not only to boost EPSPs on their way
down to the soma but also to permit action potentials to back-propagate
up the dendrites from the soma. This has implications for the biological
258 13. Compartment Modeling
instantiation of the Hebb rule, since back-propagating spikes relay the neuron’s
output signal back to the synapse, allowing input and output to be
simultaneously assessed at the synapse.
There is a remarkable conservation of pyramidal cell dendrite length
across different mammalian species — about 1 mm. Pyramidal cells are
the major projection cells of cortex. Some researchers have hypothesized
that pyramidal cells have kept their particular size due to some particular
signal integration advantage involving temporal and spatial summation.
Knowledge of the detailed topography of active channels and synaptics in
the apical dendrite might finally reveal what neurons really do.
14
From Artificial Neural Network to
Realistic Neural Network
14.1 Why learn this?
As noted toward the beginning of this book, the field of computational
neuroscience, still bound by its hybrid conception, has yet to really integrate
computation and neuroscience. The computer science side, represented by
artificial neural networks, continues to come up with the big concepts. The
neuroscience side continues to generate often confusing, if not confounding,
data. Reconciliation would require that the concepts of artificial neural
networks be directly applied to realistic neural network modeling and from
there to direct measurement in brain circuits.
In this chapter, I give an example of artificial neural network learning
theory being brought to the biological level. Similar examples could be
taken from work in the visual system. I chose this particular example because
it also helps illustrate some issues and problems in computational
neuroscience. The focus of this chapter is as much on the theory of theory
as it is on the theory of learning and the particular model discussed.
14.2 Hopfield revisited
Hebb’s rule for learning through activity-dependent changes in synaptic
strength is one of the few concepts commonly accepted all the way from
artificial modeling through realistic modeling to experimental neuroscience.
Hebb’s rule is used to form cell assemblies. The cell assembly hypothesis is
260 14. From Artificial Neural Network to Realistic Neural Network
also widely accepted. To directly demonstrate cell assemblies experimentally,
it would be necessary to record from very many cells simultaneously
in an awake, behaving animal. This is not technologically feasible. Development
of a biologically realistic Hebb assembly model would make it possible
to predict and then search for experimentally testable biological corollaries
of Hebb’s theory. Because the Hopfield model is the best demonstration of
how cell assemblies would form, Hasselmo and other researchers have been
trying to port the Hopfield artificial neural network into a realistic neural
network implementation.
The standard implementation of Hopfield’s algorithm for learning in a
content-addressable memory is given in Chap. 10. We start with N vectors,
each representing a different memory. Each vector is multiplied by
itself using the outer-product rule. This produces a separate matrix for
each memory. Each matrix can map an incomplete memory, a part of the
original vector used to form that matrix, onto the whole memory — making
it content-addressable. Next, the N outer-product matrices are added
together to form a final summed matrix. This summed matrix is the connectivity
matrix for the Hopfield network. The product of this connectivity
matrix and a partial or degraded copy of any of the N memories will
produce the full, corrected memory.
Implemented in this way, the Hopfield network is nonbiological. Outerproduct
and matrix multiplications require global processing algorithms
to pool and coordinate information over arrays of values. Biologically, this
would require that populations of neurons share and exchange information
instantaneously. Real neurons are local processing elements that operate
sequentially in real time.
Focusing on learning, the problem with this standard artificial neural network
implementation is that learning in the Hopfield network does not take
place over time, but instead just happens instantaneously as the vectors are
multiplied and matrices summed. Actually, Hopfield’s original description
of his network did not emphasize this implementation. Instead he talked
about asynchronous updating: individual synapses would be updated at
different times in response to activity in the network. This interpretation
of local interactions is more biological than the standard global implementation.
However, when we try to implement the network in this way, we
run into a problem.
That problem is cross-talk: presentation of new memories activates old
memories. The network learns the combination of old and new instead of
learning the new pattern independently. In Fig. 14.1, two overlapping patterns
are presented sequentially: AB and AC. Ideally, Hebbian learning
should potentiate the A ↔ B and A ↔ C synapses so that activation of
unit B will recall AB and activation of unit C will recall AC. Instead, we
end up with a mix. First, AB is learned. Then AC is presented. This activates
B via the potentiated AB synapse. Because both B and C are now
active at the same time, the B ↔ C synapses are also potentiated. How14.3.
Suppression model for reducing interference 261
A
B
C A
B
C
A
B
C
AB
AC
A
B
A C
B
C
Pattern:
Fig. 14.1: Cross-talk spreads activity during learning. Top: activation
of AB (arrows) potentiates A – B synapse (bold line).
Bottom: subsequent activation of AC potentiates A – B synapse
but B triggers C and B – C synapse is potentiated. (Dashed lines:
naive synapses; cross-hatched circles: active cells).
ever, B and C are not found together in either of the memories. They were
mistakenly coactivated because of cross-talk during learning. Now, when B
is activated during recall, it will activate the entire network instead of just
the pattern of which it was a part.
Instead of having constituent assemblies, the whole network has become
one big cell assembly. The different patterns have not remained separate,
but instead blend into each other. This makes sense when we consider
that a cell assembly is a content-addressable memory that is designed to
complete a partial input. If there is any overlap between a new memory
and an old memory, the new memory will trigger the old memory and
the new memory will not be learned independently. Instead, the system
will learn a combined memory that contains all the features (the union)
of both memories together. As another example, let’s say the system was
being trained on faces. Instead of learning a new face distinct from other
faces, the system would instead reinforce its knowledge of anatomy — the
nose lies below the eyes, the mouth lies below the nose. Not only that, but
if it were presented with a beard it would overgeneralize, remembering that
all faces have beards. This would not be a very useful memory system.
14.3 Suppression model for reducing interference
The solution to the interference problem is to make learning and recall
distinct phases so that they don’t get in each other’s way. Turning off
262 14. From Artificial Neural Network to Realistic Neural Network
A
B
A C
B
C
AB
AC
A
B
A C
B
C
turn off
stay on
Fig. 14.2: Suppression model: turning off associative synapses
during learning prevents cross-talk. Symbols and patterns as in
Fig. 14.1.
synaptic connections during learning would prevent the new memory from
bleeding over into the old one. However, turning off all synapses would
also prevent the initial activations of units by the external presentation
of the memory. Therefore, it is necessary to turn off or reduce the intrinsic,
associative synaptic connections while preserving the incoming afferent
connections (Fig. 14.2). Although the turned-off associative synapse will
not activate the postsynaptic unit, it can still be potentiated. The evidence
of this synaptic potentiation cannot be seen until the synapse is turned on
again.
Hasselmo and his colleagues recognized this problem and tracked down
mechanisms that could turn off these associative synapses during learning.
They identified acetylcholine (adjective: cholinergic) as a modulatory
neurotransmitter that was known to reduce the amount of glutamate
released at some synapses. The finding of damaged cholinergic cells in
Alzheimer dementia had previously suggested that acetylcholine might have
an important role in learning and memory. The researchers suggested that
acetylcholine was necessary to avoid cross-talk during learning. The critical
experimental observation, now confirmed in more than one brain area,
was that cholinergic activation primarily suppresses associative synapses,
leaving afferent synapses intact.
14.4 A digression into philosophy
Modern philosophy of science hypothesizes that science progresses by
paradigm shifts, major changes in theory such as occurred when the earth14.4.
A digression into philosophy 263
centered system of Ptolemy was replaced by the sun-centered system of
Copernicus. A corollary of this theory of theories is that theory should
lead experiment: all experiments should be driven by specific hypotheses.
I’m certainly in favor of developing explicit hypotheses. However, there
is an interplay between theory and data that is not entirely captured by
the notion of hypothesis-driven research. We can see in the present example
how theory (cross-talk in Hebb assemblies) suggested a search for
data (neuromodulatory suppression of associative synapses). This was an
hypothesis-driven search for facts.
Classical theory of science is derived from physics. In physics, a new observation
can cleanly disprove a governing hypothesis and perhaps suggest
a new hypothesis. Biology ain’t physics. In biology, if a new fact doesn’t
fit into the hypothesis, the hypothesis will be bent and shaped in order to
make it fit. If the fact still doesn’t fit, that fact will be ignored. This may
be awfully sloppy intellectually, but unfortunately experiments are rarely
conclusive in biology. The results in vitro may differ from the results in
vivo; the results in cat may be different from the results in rat; the results
under urethane anesthesia may not look like the results under halothane
anesthesia; the results from electron microscopy may appear to contradict
the results from light microscopy. This partly reflects the many artifacts
that can pop up in the complex technology of data acquisition. It is also
related to the notion, introduced early in this book, that biology is a hack.
Things are cobbled together from bits and pieces of old stuff in order to
get the job done. This means that a particular experiment may uncover
vestigial mechanisms that are not currently in use. It also means that two
different species may be doing the same thing in different ways through
homologous evolution (e.g., the independent development of wings in the
birds and the bees, and the bats).
Because of this intrinsic sloppiness, biological theorizing must constantly
battle the odd couplet of excessive sloppiness and excessive neatness. You
do not want to be so attached to your theory that your dead fingers cannot
be pried loose no matter how much evidence is adduced against you.
However, you need to be confident enough so that you aren’t immediately
dislodged from defense of your poster by hecklers at the trade fair.
In neuroscience, the search for facts does not always require that someone
do an experiment. In many cases, needed data will turn up after a search
through the literature. There are plenty of orphan facts out there waiting
for their theory to come in. Similarly, once the fact is fit into the hypothesis,
more orphan facts raise their hands and demand to be let in as well. When
they were originally hatched, these orphan facts were probably associated
with some kind of hypothesis. In many cases, however, this old hypothesis
has been long forgotten. The orphan facts now live out their lives in journal
limbo, hypothesis-free, waiting for a brave theorist to give them meaning
and purpose.
264 14. From Artificial Neural Network to Realistic Neural Network
In this case, the Hopfield network provided a hypothetical need for synaptic
suppression. A search turned up acetylcholine as a candidate mechanism.
Following this hypothesis-driven fact-finding, we can now do fact-driven
hypothesis-building. Specifically, we can look at the many additional orphan
facts about acetylcholine and use these orphan facts to extend the
Hopfield hypothesis.
14.5 Acetylcholine has multiple effects
In the jargon of pharmacology, drugs can be clean or dirty. Clean drugs
have a single effect. They bind exclusively at one spot and leave everything
else alone. Dirty drugs are all over the place. They bind at multiple places
and have multiple effects. From the perspective of pharmacology, clean is
good and dirty is bad. You want specificity: your penicillin should kill the
bacteria and leave your own cells alone.
From the perspective of biology, dirtiness appears to be the rule rather
than the exception, a part of the aforementioned biological sloppiness. Biologically,
endogenous “drugs” are ligands that float from one location in the
body to another. Such ligands include not only neurotransmitters but also
hormones and a variety of immune agents. Such endogenous compounds
invariably turn out to have multiple receptors. In some cases this multiplicity
of receptors just reflects the reuse of the same ligands in a variety of
unrelated roles. As mentioned previously, biological chemicals are reused
endlessly — various metabolic and genetic agents are recycled as neurotransmitters.
However, it seems probable that multiple neurotransmitter
actions will often turn out to be functionally significant, that the many cellular
changes effected by release of one neurotransmitter will all contribute
to a single effect at the network level. A particular neurotransmitter in a
particular part of the brain would then have a particular functional role.
This would not preclude its having a different role in a different part of the
brain or in different states, such as sleep and wakefulness.
In the present example, acetylcholine is hypothesized to place the network
into learning mode. One way that it does this is by depressing
intrinsic, while preserving afferent, synapses. We then assume that other
cholinergic effects also contribute to learning. These effects include 1) suppression
of inhibitory synapses, 2) suppression of spike adaptation, 3)
augmentation of the NMDA component of all synapses, and 4) increased
propensity to long-term potentiation (LTP).
Fig. 14.3 shows some of acetylcholine’s effects graphically. Impinging on
one cell, these various effects will be expected to interact in complex ways.
There appears to be a direct effect on Hebbian learning: acetylcholine lowers
the threshold for producing LTP and can even produce LTP by itself when
given in larger doses. This could be a direct action of acetylcholine on
14.5. Acetylcholine has multiple effects 265
Afferent Fiber
Inhibitory
Interneuron
Excitatory
Potential
Inhibitory
Potential
Intrinsic
Fiber
Potential
Adaptation
Excitatory
Afferent Fiber
Inhibitory
Interneuron
Adaptation
Intrinsic
Fiber
suppressed
by ACh
Not suppressed
by ACh
Suppression
by ACh
Suppression
by ACh
ACh
Fig. 14.3: Cellular effects of acetylcholine.
NMDA synapses or might be a secondary effect. For example, acetylcholinemediated
increased postsynaptic firing might secondarily enhance LTP.
Ideally, we would want to examine all of the known acetylcholine effects
in concert using both the experiment and the model at the cellular level.
This would tell us how the cells’ overall input/output functions will be
altered. We could then place the altered cells into the network and test
learning. Typically, modeling projects have a more modest scope. In this
case, it is easier to study just one of the cholinergic effects in isolation.
This single-effect modeling may fail if the learning effect is an emergent
property due to several individual acetylcholine effects acting together.
Nonetheless, the pursuit of single effects is simpler and may provide incremental
improvements in network learning and in our learning about the
network.
The dual-matrix hypothesis
One of the cholinergic effect mentioned above, augmentation of the NMDA
component of synapses, is one that my colleagues and I have looked at. I
retell the story in order to contrast the relative clarity of a completed story
with the preliminary confusion of modeling research.
The biological version of the Hebbian synapse utilizes two receptors.
The NMDA receptor acts as monitor of pre- and postsynaptic activation
through its dual ligand and voltage sensitivity. The AMPA receptor serves
as the output device. Activity across AMPA is augmented following Hebbian
activation. If NMDA were solely a monitor and a triggering device,
it would communicate only with the AMPA receptor or with presynaptic
mechanisms, leading to greater transmitter release. However, NMDA is pri266
14. From Artificial Neural Network to Realistic Neural Network
marily connected to an ion channel that depolarizes the postsynaptic cell.
Thus it is not only a monitor and trigger, but also a synaptic communicator.
The dual role of NMDA has dual implications. First, it sets up a conflict
of interest. NMDA both depolarizes the postsynaptic cell and is responsible
for monitoring depolarization. This creates the potential for positive
feedback at the synapse: NMDA could depolarize and then call for further
depolarization through AMPA augmentation. The further depolarization
would permit greater NMDA current. However, the time constants for
synaptic augmentation are relatively long so that this positive feedback
is not explosive. Additionally, other mechanisms reduce synaptic strength
and control runaway depolarization.
The second implication has to do with the dual activation through AMPA
and NMDA triggered by glutamate release. The network may be viewed
as having two connectivity matrices. AMPA strength is augmented and
decremented though Hebbian mechanisms. NMDA strength remains unchanged.
Even if they start out the same, the effective connectivity of the
two networks would end up being different. Additionally, AMPA strength
may be zero at some synapses. These are called silent synapses since, absent
postsynaptic depolarization, the synapse will not do anything. Such connections
would be represented in the NMDA network alone unless synaptic
potentiation took place.
In addition to the dual synaptic receptors, there is another major discrepancy
between the Hopfield network and experimental reality. The Hopfield
network is fully connected (Fig. 14.4, top). The biological network is not; it
is sparsely connected (Fig. 14.4, bottom). In the Hopfield network, any two
neurons that fire together will become connected. In biology, a single neuron
connects with relatively few of the many other neurons in a particular
brain area. Neurons that are not connected anatomically will not become
newly connected as a result of their activity. (Axonal sprouting can occur,
but it is not believed to be Hebbian.) The sparse cell-to-cell connectivity
in the brain limits the formation of cell assemblies.
Now we can connect the dots. The matrix is sparse, making it harder to
store information in the network. Each connection is precious since there
are so few of them. However, each connection is effectively two connections.
In our study, we hypothesized that the second, unaugmented, NMDA
connection would preserve this precious connectivity by allowing activity
during learning when the AMPA connectivity is suppressed. We did modeling
to test this hypothesis. In line with our hypothesis, a network with
dual connectivity worked better than the network with mono connectivity.
True confessions
I’ve told a lovely story about the dual-matrix hypothesis. Alas, it is not the
true story of the modeling. As with most science, we did not really set out
to discover what we did discover. I’ve told the story in reverse, with the
14.6. Summary and thoughts 267
Fig. 14.4: Top: Connectivity in the Hopfield network is full and
reciprocal (thin lines). Bottom: In the brain the cell-assembly
layer is sparsely connected.
explanation first and the model last. Actually, the simulations were done
as explorations. Various facts, opinions, and phenomena congealed along
the way.
The real story is as follows. We took a Hopfield network. We put in
some biological features. The network worked better with them than it did
without them. We then went back and figured out why it worked better.
Finally, the story made some sort of sense to us. After suitable writing
and rewriting, it made some sense to non-us individuals as well. Above, I
pointed out that biology is not physics. To that, I will add that biological
modeling (and science in general) is not the same as engineering. As you
set out to do one project, you may end up with a different one.
14.6 Summary and thoughts
In this chapter, I tried to show how we can now begin to close the circle
of theory and experiment. Artificial neural network theory suggests
how a content-addressable memory can be built. Efforts to implement this
268 14. From Artificial Neural Network to Realistic Neural Network
artificial neural network in a realistic network reveal problems with the algorithm.
A work-around is developed that fixes the algorithm but requires
an additional mechanism. The additional mechanism is found experimentally.
It is the neurotransmitter acetylcholine. Other effects of acetylcholine
are then explored for their possible contributions to learning. One such effect
(NMDA augmentation) helps solve another problem with the biological
version of the original artificial neural network algorithm (sparseness).
One advantage of realistic neural networks compared to artificial neural
networks is that they include detailed results that can allow pharmacological
predictions. For example, a model of acetylcholine in learning could
be used to determine the relative importance of the various acetylcholine
effects. This information could suggest which acetylcholine agonists would
be more likely to preserve learning in patients with early Alzheimer disease.
I favor using artificial neural network ideas to develop realistic neural
networks that make experimental predictions. An alternative approach is
to skip the realistic neural network step and go straight from artificial neural
network to brain measurements. This requires that we accept literally
the assumptions of artificial neural network modeling, most notably the assumption
of rate coding. fMRI is a physiological method that can measure
correlates of metabolic activity in the awake behaving person or animal.
Metabolic activity will increase with increased firing rate. As fMRI resolution
improves toward the single neuron level, it may be possible to look
directly into the brain for evidence of artificial neural network algorithms.
Having now stretched neural modeling about as far as I am able to take
it, I would like to step back and again consider the brain more directly. All
of our models are barely rubbing the edges of this enormous terra incognita.
Columbus and crew set foot on an unknown island at the middle of the end
of the earth. Maybe it looks like India. Maybe it smells like India. Is this
really India?
15
Neural Circuits
15.1 Why learn this?
In prior chapters, I’ve mostly generalized about brain areas and brain cells
— talking about typical neurons and generic network architectures. Most
of the brain doesn’t look anything like these generic versions. Each brain
area has its own peculiar cells that have idiosyncratic dendritic morphology
and differing complements of voltage-sensitive ion channels. Various brain
areas also have very different circuitry. Some brain areas have odd types
of connections (e.g., dendrodendritic or axoaxonic) that aren’t covered in
standard brain theory. In this chapter, I describe a few brain circuits. For
the most part, I only give gross connectivity without giving any specifics
about the forms or spiking patterns of the cells involved.
15.2 The basic layout
The hip bone is connected to the knee bone and the knee bone is connected
to the hip bone. This basic principle of biological reciprocity is often respected
in the brain as well. Neocortical areas typically project back to
areas from which they receive projections. Principal cells of a brain area
are generally taken to be those cells that are largest and that project to
other brain areas. The principal cells of most areas are excitatory. There are
two major brain areas, cerebellum and basal ganglia, where the principal
cells are inhibitory.
270 15. Neural Circuits
Microscopically, the brain is made up of neurons and glia. In anatomy
and pathology, the opposite of “microscopic” is “gross,” meaning able to be
seen with the unaided eye. Grossly, the brain is made up of white matter
and gray matter. The white matter is made up of myelinated axons, the
wires. The gray matter contains the neuronal cell bodies and the neuropil,
the tangle of dendrites and axons that makes up local circuitry.
Grossly, mammalian brains have an outer gray matter cortex that forms
a rind around the brain. Below that is white matter made up of projections
to and from cortex. Below that are the deep nuclei or ganglia of the brain,
irregular balls of gray matter. Cortex and these various nuclei and ganglia
are the major circuits for modeling.
15.3 Hippocampus
The hippocampus is the seat of episodic memory. It is one of the most
heavily investigated areas of the brain. There is a famous hippocampusless
man who is known only by his initials — H.M. Hippocampus was
removed from both sides of his brain to cure his epilepsy. This left him
with some odd memory problems that have been intensively studied over
the past several decades. He can remember things that happened before
his hippocampectomy (removal of his hippocampus) but cannot form new
complex memories. However, this does not mean that he cannot learn. He
can learn complex games and unusual tasks like mirror writing (writing
while watching your hand in a mirror). Afterward, he cannot remember
having learned these things. When he is later asked to mirror write, he is
surprised to find that he can do it easily. He cannot learn to find his way
around the place where he lives or to recognize people whom he sees every
day. However, if a person is consistently nasty to him, he will avoid that
person. However, he will be unable to identify the person in a lineup or say
why he doesn’t like him.
It appears that the hippocampus is responsible for things that are newly
learned, but that information is then gradually passed to neocortical areas.
LTP, the biological form of the Hebb synapse, is prominent in the hippocampus
and has been particularly well studied there. The hippocampus
shows distinct states of activity, ranging from slow theta oscillations to high
amplitude sharp waves. It may be that this propensity to produce sharp
waves explains why the hippocampus is a common site of seizure initiation.
Activity in the hippocampus and nearby areas appears to proceed around
a loop: entorhinal cortex → dentate gyrus → CA3 → CA1 → subiculum → entorhinal cortex (Fig. 15.1). (CA stands for cornu ammonis but it is rarely
written or said in full.) All of these projections are excitatory. Entorhinal
cortex receives information from much of the neocortex and projects back
to the neocortex in turn. The most intensively studied section of this loop is
15.3. Hippocampus 271
CA3
CA1
dentate gyrus
subiculum
entorhinal
cortex
Fig. 15.1: Hippocampus. Dashed lines represent location of cell
bodies (cell body layers); solid lines are connections. The aggregation
of cells making up the sharply angled “blades” of dentate
gyrus and the loop of cornus ammonis (CA) are easily seen with
a low-power microscope.
called the trisynaptic pathway. It runs from the perforant path output axons
of entorhinal cortex via the mossy fibers to CA3 and thence via Schaeffer
collaterals to CA1.
The perforant path is the major input pathway from entorhinal cortex
to hippocampus proper. Perforant path projects most strongly to dentate
gyrus but also projects directly downstream to CA3 and CA1. Dentate
gyrus granule cells are infrequently firing neurons that receive perforant
path inputs and then project via mossy fibers. The mossy fibers form powerful
inputs onto the proximal apical dendrites of pyramidal cells of area
CA3. CA3 neurons are heavily interconnected. The relatively dense connectivity
matrix of this area has led some to speculate that CA3 might form
an attractor memory network similar to the fully interconnected Hopfield
network.
CA3 projects in turn via Schaeffer collaterals to CA1. These are collaterals
(branches) of axons that connect the CA3 neurons to each other. (There
is also a CA2 between CA3 and CA1 but it appears to just be a transition
zone.) CA1 has sparse interconnections. The Schaeffer collateral synapses
are the site where NMDA-dependent LTP has been most fully described.
CA1 projects to the subiculum, which projects to the entorhinal cortex,
completing the loop.
272 15. Neural Circuits
15.4 Thalamus
The thalamus lies at the center of the brain and likely has a central function
as well. It is made up of a variety of small nuclei. Most of these have
reciprocal (back-and-forth) connections with a particular area of overlying
cortex. The ones that are connected with a sensory area are considered relay
nuclei because they connect a peripheral sensory area to primary sensory
cortex. For example, the retina projects to a thalamic nucleus called the
lateral geniculate nucleus, which projects in turn to the primary visual area
of occipital cortex. Other thalamic nuclei are reciprocally connected with
a motor area or are connected to hippocampus and other memory areas.
The moniker “relay nucleus” is probably an oversimplification. Presumably,
the thalamus is doing something to incoming messages, not just receiving
them and sending them on to the cortex. One theory, made popular by
Francis Crick, is that the thalamus might play a role in directing attention
by activating a particular area of cortex when immediate processing is
required.
When a person or animal is awake, thalamic cells produce fast spikes.
In certain stages of sleep, the thalamus produces infrequent bursts that
add up to slow oscillations. In addition to being seen in sleep, similar slow
oscillations also turn up in absence epilepsy, a peculiar epileptic syndrome
of children that causes brief periods of unconsciousness that look like simple
daydreaming. This slow oscillatory pattern of thalamic activity has been
much more extensively modeled than has the fast firing of the awake state.
An oscillation can be produced through the interactions of thalamocortical
cells and reticularis cells. Thalamocortical cells are the principal
excitatory cells of thalamus. They have reciprocal projections with cortex.
Thalamocortical cells are the relay cells in sensory nuclei. Relay cells
receive information from the sensory periphery and send it up to cortex.
Reticularis cells are inhibitory cells that lie next to the main nuclei in a thin
layer called the thalamic reticular nucleus. Reticularis cells have reciprocal
connections with thalamocortical cells. There is also a population of local
inhibitory interneurons in most thalamic nuclei.
Fig. 15.2 is an approximate circuit diagram of a generic sensory thalamic
nucleus. Information comes in from the outside to the interneurons and the
relay cells. It goes up from the relay cells to the cortex. The relay cells are
also influenced by descending projections from cortex and by inhibitory
connections from reticularis cells and interneurons. There are no direct
connections between relay cells. It is unclear whether the back-projection
from cortex involves cell to cell or area to area reciprocal connectivity.
One of the most peculiar aspects of thalamic circuitry is the existence of
synaptic triads, shown in the inset in Fig. 15.2. The triads are found where
the sensory input arrives in the thalamus and forms an excitatory connection
with a dendrite of a relay cell and with a dendrite of an interneuron.
The interneuron then makes an inhibitory connection with the relay cell
15.5. Cerebellum 273
CORTEX
Relay
Cell
Interneuron
Reticularis Cell
Sensory
Input
Triad
Input
Relay
Cell
Interneuron
Fig. 15.2: Thalamus. Synaptic triad (rectangle and inset) involves
interaction of thalamocortical and interneuron dendrites with the
sensory afferent. It is unknown whether an axon from that interneuron
would project to the same thalamocortical neuron, as
shown here.
at the same location. This is a dendrodendritic synapse from interneuron
to relay cell, which would be expected to provide a very rapid, highly localized,
feedforward inhibition. It is unclear whether this immediate triad
inhibition is the sole effect of interneuron activation at this dendrite. It is
possible that the interneuron excitation could propagate from the dendrite
down to the interneuron soma. In this case, this same input might also
activate an action potential that would travel to the axon and inhibit relay
cells (perhaps even the same relay cell) via axodendritic synapses.
15.5 Cerebellum
The cerebellum is the little brain that is piggy-backed on the back of the
big brain. It has its own cortex and its own deep nuclei just like the big
brain does. However, the major projecting cells of cerebellum, called Purkinje
cells, are inhibitory rather than excitatory. The cerebellum is involved
in coordinating movement. People with damage to the cerebellum cannot
reach and grab things — they are likely to either miss the thing that
they are reaching for or knock it over. They may also have problems walking,
tending to weave as if drunk. In fact, reversible cerebellar damage is
274 15. Neural Circuits
Olive
Climbing
Fibers
Parallel
Fibers
Mossy
Fibers
Basket
cell
Purkinje
cells
Golgi
cell
Granule
cells
Spinal
Cord
Fig. 15.3: Cerebellum. Inputs and parallel fibers are excitatory.
Other intrinsic connections are inhibitory.
the reason for the gait, speech, and nose-touching abnormalities of alcohol
intoxication.
In the human, the cerebellum has about 10 million Purkinje cells that
collectively receive more than a trillion synapses. That’s a lot of wiring.
The most numerous inputs to the Purkinje cells come from granule cells,
whose excitatory projections look like telephone lines as they run across
connecting to about 100 Purkinje cells in sequence (Fig. 15.3). These telephone
lines are called parallel fibers. Major sensory inputs to the cerebellum
arrive from the limbs via the spinal cord and thence the mossy fibers to
produce excitatory synapses on granule cells. Additionally, there is a pow15.6.
Basal ganglia 275
erful excitatory input from a brainstem structure called the inferior olive.
The excitatory inputs from the olive are called climbing fibers because they
climb up and wrap around the Purkinje cell main axon like ivy around a
tree trunk.
There are also a variety of inhibitory cells involved in this circuit. The
most prominent of these is the Purkinje cell itself, which provides an inhibitory
input to the deep cerebellar nuclei. There are also basket cells,
which are activated by parallel fibers and produce lateral inhibition on
Purkinje cells. Another set of inhibitory cells are the Golgi cells, which get
inputs from parallel, mossy, and climbing fibers, and feedback to inhibit
granule cells.
15.6 Basal ganglia
The basal ganglia are a group of deep nuclei anterior to the thalamus.
This nuclear group is involved in motor execution and planning. Like the
cerebellum, the principal cells of basal ganglia are inhibitory. The basal
ganglia accept input from much of the cortex and funnel this input down
into relatively few cells. In the rat, for example, about one billion cortical
inputs converge onto only about one million striatal cells (the striatum is
another name for the putamen and caudate). These spiny stellate cells of
striatum project in turn onto only about 100,000 cells of globus pallidus
pars externa. The convergence from order 1 ・ 109 cortical cells to order
1 ・ 105 globus pallidus cells represents a convergence ratio of about 10,000
over these two synaptic steps.
The basal ganglia circuitry is generally described as having two major
pathways, called the direct pathway and the indirect pathway. In Fig. 15.4,
I have shown the direct pathway as solid lines and the indirect as dashed
lines. Cortex projects to putamen and caudate in both cases. The direct
pathway leads from there to substantia nigra pars interna and thence to
thalamus. There are projections from thalamus to cortex, closing the loop.
The direct pathway involves excitation (cortex to striatum) → inhibition
(striatum to globus pallidus) → inhibition (globus pallidus to thalamus) → to excitation (thalamus to cortex), making it overall excitatory around the
loop.
Again starting in caudate/putamen, the indirect pathway gets to the
thalamus via a somewhat more circuitous route through three way-stations:
globus pallidus pars externa, subthalamic nucleus, and globus pallidus pars
interna. The indirect pathway from cortex to cortex involves excitation
→ inhibition → inhibition (globus pallidus pars externa to subthalamic
nucleus)→excitation (subthalamic nucleus to globus pallidus pars interna)
→ inhibition → excitation. The three inhibitory projections would make it
overall inhibitory around the loop.
276 15. Neural Circuits
Subthalamic
nucleus
Globus
pallidus
externa
Globus
pallidus
interna
Cortex
Thalamus
striatum=
caudate+
putamen
Fig. 15.4: Basal ganglia. Solid lines: direct pathway; dashed lines:
indirect pathway.
Feedback loops can only work through inhibitory links if the cells being
projected to are active. In the rate-coding paradigm, the cells would be
tonically active and the inhibitory projections would reduce this rate of
firing. In fact, both globus pallidus and thalamic cells, the two targets of
the inhibitory projections, do have relatively high firing rates. Spiny stellate
cells of caudate and putamen, on the other hand, fire rarely.
As with all such simplifications, the division of basal ganglia circuitry into
two pathways leaves out several other pathways and subpathways. There is
intrinsic connectivity in the striatum, where both GABA and acetylcholine
serve as neurotransmitters. There is also a well-described and important
interaction with the substantia nigra. The substantia nigra has two parts.
Substantia nigra pars reticulata is just an extension of globus pallidus and
has the same connectivity. Substantia nigra pars compacta, on the other
hand, has dopaminergic cells that project up to the striatum. These are
the cells that are lost in Parkinson disease, a disease that can be reversed,
at least initially, by replacing the lost dopamine with drugs.
15.7. Neocortex 277
15.7 Neocortex
When people think about the brain, they usually first consider cortex, that
magnificent edifice that separates us from the lowly beasts of field and forest,
or at least from the nonmammalian beasts of field and forest. Another
result of this corticocentrism is that neocortex is perhaps the most modeled
areas of brain, although many such models are just generic artificial neural
networks padded with a brief mention of cortex in the introduction to a
paper.
Despite all this attention, a single basic cortical circuit has not been delineated.
This failure is largely due to the complexity of cortex. It may also
be due to the fact that different areas of cortex appear to have somewhat
different wiring suited to their function. Brodmann, a famous late-19th
century anatomist, was able to separate out 52 areas of cortex based on
morphological criteria alone. There are likely many more areas if one considers
functional specialization, although it is not clear to what extent
functional differences require different wiring.
Neocortex has six layers. Layers 3 and 5 house large projecting pyramidal
cells. Layer 4 is generally an input layer. Other layers have intersecting
dendrites and axons. Such cell-sparse areas are called neuropil. There are
other, smaller, excitatory cells in addition to the large pyramidal cells.
There are also a variety of inhibitory interneurons of varying sizes and
shapes. Some of these inhibitory cells have large dendrites that reach up to
higher layers just as the pyramidal cells do. There is an unusual interneuron,
the chandelier cell, that synapses on pyramidal cell axons. It has generally
been assumed that chandelier cells can shut down pyramidal cell output
but it has not been possible to demonstrate this.
Visual cortex is the most thoroughly studied neocortical area. Primary
visual cortex is called V1 in cat and Brodmann area 17 in monkey and
human. It is also called striate cortex because of the prominent white stripe
(stria of Gennari) in layer 4. This stripe is white due to the myelinated
fibers that arrive there from lateral geniculate nucleus, the visual area of
thalamus. Many models of visual cortex have been developed. Most of these
models seek to explain the responses of cortical neurons to particular forms
of visual stimulation. They generally do not take account of the detailed
circuitry of cortex or the particular firing patterns of different cell types.
Fig. 15.5 is my take on a basic neocortical circuit diagram. It is loosely
based on a model put forth by Kevan Martin and Rodney Douglas. Input
from the thalamus arrives in layer 4 and projects onto basal dendrites
of layer 3 cells and apical dendrites of layer 5 cells. Excitatory cells in
each layer interact with each other and with the cells of the other layer.
Inhibitory neurons form networks of mutually connected cells within each
layer and also interact with the pyramidal cells of that layer. Outputs
project from layer 5 to the thalamus and beyond. There are also outputs
from low-lying pyramidal cells in layer 6.
278 15. Neural Circuits
thalamus
layer 5
layer 3
Fig. 15.5: Neocortex.
15.8 Summary and thoughts
Although I’ve tried to be fairly specific about connectivity in this chapter,
I’ve only scratched the surface of the enormous variability of brain areas.
For example, I presented a generic thalamic circuit. Actually, the thalamus
is made up of different nuclei, and these different nuclei have somewhat different
circuitry. In some nuclei of some animal species there are no thalamic
interneurons, so the circuitry necessarily differs from the generic norm that
I’ve shown.
I’m concluding on a note of mystery. The variety of circuits and cells
in the brain is remarkable. From an engineering perspective, as a brain
designer, one would prefer to build this complex system in a hierarchical
manner, starting with a small set of circuit elements and organizing these in
an only slightly larger set of fundamental subcircuits. This way things can
be kept neatly organized and understandable. Conflicts between different
circuit elements are kept to a minimum because the elements understand
each other, and they share common assumptions about input, output, and
processing methods.
15.8. Summary and thoughts 279
The brain, like the eukaryotic cell, was built more in the spirit of a
junkyard scavenger than of a space shuttle designer. The different pieces
were originally developed for different purposes and gradually evolved so as
to work together in a productive manner. Our knowledge of the brain has
progressed to the point where different brain areas can be assigned general
functions. A few things can be said about connections and their strength
and importance. So how does it all work?
16
The Basics
16.1 Why learn this?
It’s hard to be comfortable learning a new field whose routine chores are
intimidating or mysterious. Although some aspects of computational neuroscience
are based on fairly high-level math, most of what is needed can
be handled with algebra. Most of algebra can be explained in plain words.
William Clifford, an algebra reformer of a century and a half ago, said that
“algebra, which cannot be translated into good English and sound common
sense, is bad algebra.” He was actually complaining about the linear algebra
that is presented below. Despite its shortcomings, I’ve tried to present
even linear algebra in good English.
As mentioned previously, the other saving grace of computational neuroscience
is the computer itself. It’s always been the case that a picture
or demonstration is worth approximately 1 ・ 103 words. The computer can
easily generate 1 ・ 103 demonstrations, allowing it to quickly outpace the
paltry 1 ・ 104 words of this chapter. Most of the sections below have companion
programs. The presentation here can also be readily supplemented
by standard programs such as Octave, Matlab, Maple, or Mathematica.
In the following, I cover a variety of useful applied math skills. I start
with the handling of units since these express the distinction between math
and applied math — applied math is about specific things. I then discuss
the binary number system, used to work with computers at the machine
level. I cover some of the basic rules of linear algebra, frequently used in
artificial neural network implementations. From there, I switch from the
282 16. The Basics
discrete to the continuous and talk about calculus, reviewing methods of
numerical integration and briefly talking about analytic calculus as well. Finally,
I consider some essential electricity concepts, particularly the effects
of resistance and capacitance.
This chapter is made to be skimmed rather than read. Selective sections
can then be read in more detail as needed to supplement information
contained in the preceding chapters.
16.2 Units
Units and unit conversion is a nuisance. Different neural simulators use
different units. Different physiology papers give results in different units. A
great program for playing with units is the Unix “units” program, which I
used to check all of the calculations here.
Scientific notation
In science, we encounter lots of numbers that are too big or too small to represent
in the usual way. For example, 752400000000 and 0.00000000007524
are not easy to read. Instead, we separate significant digits from magnitude
by using scientific notation. The significant digits can be written as
a number between 1 and 10. The magnitude is written as a power of 10.
Then, 752400000000 is written 7.524 ・ 1011 and 0.00000000007524 is written
7.524 ・ 10−11. 7.524 is the mantissa in both cases, but the exponents
differ. For numbers greater than 1, the exponent can be readily determined
by counting the digits to the right of the most significant digit (7 here).
For numbers less than 1, count the zeros from the decimal point and add
one.
Starting with Fortran, computer languages, and now computer lingo, has
used the letter “e” (or “E”) as a shorthand to mean exponent of 10. Instead
of 7.524 ・1011, we can write 7.524e11; for 7.524 ・10−11, we write 7.524e−11.
The “e” here, pronounced “e,” has nothing to do with e ∼ 2.71828, the
base of the natural logarithm.
Numerical prefixes
An alternative to scientific notation, sometimes called engineering notation,
is the use of numerical prefixes in the metric system. These numerical
prefixes express the power of 10 that would be the second part of the
scientific notation. They are used just before some type of unit, such as
meter or amp. Some of these are widely used, for example, kilo and milli in
kilometer and millimeter. With the exception of centi (10−2), the commonly
used prefixes express exponents that are multiples of 3: e.g., 103, 106, 109.
16.2. Units 283
Therefore, the significant digits will be given as numbers between 1 and
1000. Above 1000, you go to the next prefix up. For example, 32432.5 meters
would be written as 32.4325 km (kilometers), while .0324325 meters would
be written as 32.4325 mm (millimeters). These conversions are easy to do if
you just think about shifting the decimal point left or right by three places.
prefix name magnitude original meaning
T tera 1012 monster
G giga 109 giant
M mega 106 large
k kilo 103 thousand
c centi 10−2 hundred
m milli 10−3 thousand
μ micro 10−6 small
n nano 10−9 dwarf
p pico 10−12 a little bit
Here’s a table of prefixes used in computer science and neuroscience.
There are more of them, all the way up to 1024 (yotta; I guess 1072 is
“yotta, yotta, yotta”) and down to 10−24 (yocto). Except for μ for micro,
they are generally easy to remember as the first letter of the full prefix.
Small numbers are small letters and big ones (beyond kilo) are capitals:
note the difference between “m” (milli) and “M” (mega).
The meanings of these prefixes is usually changed when dealing with
the bits and bytes of computers. For the prefixes from kilo and above,
each is taken to be a power of 1024 (210) instead of a power of 1000.
Therefore, a megabyte (MB) is actually 1,048,576 (10242) bytes. More awful
yet, this special computer rule is used only for bits and bytes measured
directly, but is not used for information transfer rates. Specifically, bits per
second (bps) does not use this convention. 12 kbps is 12,000 bps, which
would actually transfer only 12000
1024 kb ∼ 11.7 kb in 1 second. 12 kbps is not
equal to 12 kb per second! This is all such a mess that people screw it up
all the time. When 3.5-inch floppy disks doubled in density so as to hold
1440 kB, this was marketed as a “1.44 MB disk.” However, it actually has
1440 ・ 1024 = 1, 474, 560 bytes, which is correctly written as 1,474,560
1,048,576 ∼ 1.4
MB.
Units and abbreviations
Because computational neuroscience makes use of many fields of study,
we end up using lots of different units that can be hard to keep track of.
The abbreviation for a phenomenon will differ from the abbreviation for
that phenomenon’s units, but is sometimes the same as the abbreviation of
something else’s units. For example, farads (F) are the units of capacitance
(C). However, C as a unit is short for coulombs, the unit of charge, whose
284 16. The Basics
symbol is Q. Current, another C word, uses I as a symbol and is measured
in amperes (A). Case matters: s is seconds, but S is siemens, the measure
of conductance. Synonyms add to the confusion. Voltage, V, is also called
electrical potential, E.
Below is a table of abbreviations. I’ve ordered them to put together
similar or identical abbreviations that could be mixed up. I’ve focused
on units commonly used in computational neuroscience, so I don’t bother
mentioning that “c” is also the speed of light. I give units as they are
commonly used in computational neuroscience rather than giving the basic
SI unit. For example, cell diameters are generally measured in micrometers
(microns), not in meters.
In realistic neural modeling, subscripts are frequently needed to qualify
symbols. For example, membrane resistance (Rm or Rmemb) must be distinguished
from longitudinal cytoplasmic resistance, which is represented
by Ra or Rl. When referring to membrane resistance and capacitance, a
lower-case letter is used to represent the resistance or capacitance per unit
length: rm in ohm-cm or cm in farads per cm. An upper-case letter is used
to refer to specific resistance or capacitance, which is given per a unit area
of membrane: Rm in ohms times square centimeter and Cm in farads per
square centimeter.
phenomenon abbrev. unit unit
abbrev.
computer memory RAM byte B
data bit b
charge Q coulomb C
capacitance C farad F
current I or i ampere, amp A
voltage, potential V or E volt V
energy E joule J
volume V milliliter, . . . ml, cc
area A squared length μm2, cm2
velocity v meter per second m/s
resistance R ohm Ω
conductance g siemens, mho S
time t second s
frequency f hertz Hz, /s
period T second s
temperature T degree oC, oK
diameter, length d, L micron μm
chemical amount n mole mol
concentration [x] molarity M
16.2. Units 285
Another set of confusing abbreviations comes from common numbers and
constants. For example, i is used in mathematics to represent the √−1, the
basic imaginary number. However, i is needed in electronics for current, so
electrical engineering uses j for √−1. Here are some number symbols and
constants.
name abbrev. value comments & examples
imaginary
number
i or j √−1 j in electronics,
i in math
circle ratio π 3.1415926. . . circumference/diameter
base of
natural log
e 2.718281828. . . ln(e) = 1
base 10
exponent
e or E none 1e3=1E3=
1 ・ 103 = 1000
Faraday F 96485.341 A s a mole of charge
gas constant R 8.314 J/(mol K) PV=nRT, R=k・Av
Boltzmann’s
constant
k 1.38 ・ 10−23 J/K similar to gas constant
Avogadro
number
Av 6.022 ・ 1023 molecules in a mole
Unit conversions
Unit conversions can be hard to do in your head, but they are easy to do on
paper if you remember one rule: always multiply by 1. Given an equivalent
measure, you can form a ratio that equals 1. For example, 5280 feet = 1
mile. Therefore, 1 mile
5280 feet = 5280 feet
1 mile = 1. Each time we multiply by 1, a
unit that is in the numerator at one step is in the denominator in the next
or vice versa. These units cancel each other and can be crossed out.
Knowing that 1 inch is approximately 2.54 cm, we can convert 50 million
millimeters into miles as follows:
5 ・ 107mm
1 ・
1 cm
10 mm ・
1 in
2.54 cm ・
1 ft
12 in ・
1 mile
5280 ft
=
5 ・ 107 ・ 1 ・ 1 ・ 1 ・ 1 mile
10 ・ 2.54 ・ 12 ・ 5280 ∼ 31 miles
Here the mm in the numerator in the first fraction cancels with the mm
in the denominator of the second fraction and so on. We are left with only
miles.
We frequently have some measure that’s given as a ratio (e.g., miles per
hour) and have to determine one of the measures given a value for the other
one. For example, how long does it take to drive 1 inch at 60 mph?
286 16. The Basics
1 inch
1 ・
1 hr
60 miles ・
1 mile
5280 ft ・
1 ft
12 in ・
60 min
1 hr ・
60 sec
1 min ・
106μs
1sec
=
1 ・ 1 ・ 1 ・ 1 ・ 1 ・ 60 ・ 60 ・ 106
1 ・ 60 ・ 5280 ・ 12 ・ 1 ・ 1 ・ 1 ∼ 947μs
Notice that I turned the fraction for 60 mph upside down to provide the
second fraction. I also put all the given values down first and then converted
each of them, rather than insisting that unit matching should be done at
each neighboring fraction. The answer is just short of 1 ms, about the
duration of an action potential.
Another example, common in neuroscience, is to calculate the total
membrane capacitance (cm, in farads) of a cylindrical compartment given
specific capacitance (Cm, in farads per unit area) and the size of the compartment.
The commonly accepted value for specific capacitance is 1.0
μF/cm2. We will use a compartment 30 microns long and 20 microns wide.
The surface area of a cylinder is π・ length ・ diameter (π ・ L ・ d), so in this
case area = π ・ 30 μm ・ 20 μm ∼ 1885 μm2. A ratio for a squared or cubed
volume can be obtained by simply raising the ratio for the unit conversion
to the appropriate power (12 = 13 = . . . = 1). In this example, since 1 cm
10 mm
provides a ratio for cm to mm distance conversion, we can square this to
get the ratio 12 cm2
102 mm2 as a conversion for area. Similarly, we would cube it
to get a conversion factor for volume.
1885 μm2
1 ・
1 mm2
(103)2 μm2 ・
12 cm2
102 mm2 ・
1 μF
1 cm2 ・
1 F
106 μF
=
1.885 ・ 10−11F =
18.85 ・ 10−12F = 18.85 pF
Here, as in many cases, we didn’t know ahead of time the overall magnitude
of the answer, so we originally expressed it in farads. We then change the
exponent from −11 to −12, dividing by 10, and multiply the mantissa
by 10, in order to get an exponent that corresponds to a standard prefix
of engineering notation. If, for some reason, we wanted to work with mF
instead, we could easily convert to get 18.8 ・ 10−9mF.
Dimensional analysis
Dimensional or unit analysis is an assessment as to whether units that
are being used together really belong together. If you try to convert miles
to feet, that’s fine; but if you try to convert miles to seconds, you can
immediately see that you have an error in units. Such conversion error
are less obvious when you deal with complicated conglomerations of units.
For example, it turns out that a calorie per second can be converted into
16.3. Binary 287
volt-amps (it’s about 4.2 of them) since they are both units of power. (Voltamps
means volts times amps, not volts minus amps.) In Chap. 11 we add
C
Δt +g. It’s only possible to add numbers if the units for both addends are
the same. In fact, 1F
1s = 1S.
To make working with units a little easier, SI units are neatly matched
up so that the major electrical and mechanical laws work out when you use
standard units. Ohm’s law states that V = IR; 1 volt (V) equals 1 amp-ohm
(1 A-Ω). These can then be used for unit conversions by producing a ratio
of 1: 1 = V
A•Ω. R = 1
g : 1 ohm equals 1
siemens; 1 = S ・ Ω. Q = CV so 1
coulomb (C) is 1 farad-volt (F-V); 1 = C
F•V . I = C ・ ΔV
Δt so 1 amp equals
1 farad-volt/second; 1 = A•s
F•V . Of course, all these units divide out equally
well if you make them all into milliunits or microunits or kilounits.
16.3 Binary
The details of care and handling for binary numbers seem trivial and may
remind some readers of bad days in third grade. However, comfort with
binary is useful for understanding the computer as a machine. In addition
to their use in regular algebra (addition and subtraction mostly), binary
numbers are also used in Boolean algebra (combinations of true and false).
To handle binary numbers, it is also helpful to know octal (base 8) and
hexadecimal (base 16).
Translating back and forth
Numbers can be represented in different forms depending on the value
used as the basis for the number system. The noncomputer world generally
uses base 10, decimal. A base specifies how many different symbols
are available. Base 10 has 10 symbols: 0,1,2,3,4,5,6,7,8,9. Base 8 (octal)
has eight symbols: 0,1,2,3,4,5,6,7; base 16 (hexadecimal) has 16:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F; base 2 (binary) has two: 0,1. Binary is hard
to work with when doing arithmetic by hand because the numbers get so
long. Translation from binary into octal or hexadecimal (also called “hex”)
is easy, so these are commonly used by programmers.
Modern number systems all use place notation. This means that the
location of a numeral specifies its value. With any of the bases, the position
of a numeral relative to the radix point determines a power of the base to
be used as a multiplicative factor. In the decimal system, the radix point is
called the decimal point. Every position to the left of the point is a power
of the base counting up from zero. Every position to the right of the point
is a power of the base counting down from negative one.
The base is sometimes indicated with a subscript: e.g., 108 = 810. To
identify the value of a number, you simply raise the base to the power
288 16. The Basics
appropriate to the place and then multiply by the numeral that’s in that
place. The number 25710 = 2・102+5・101+7・100. 2578 = 2・82+5・81+
7 ・ 80 = 12810 + 4010 + 710 = 17510. 25716 = 2・ 162 + 5 ・ 161 + 7 ・ 160 =
51210+8010+710 = 59910. Similarly, 1001.1012 = 1・24+1・20+1・2−1+
1 ・ 2−3 = 9.62510.
Translation of numbers between bases is relatively straightforward. For
example, let’s translate 532 into base 2. It’s easiest to do if one refers to a
table of powers of 2 in base 10. Translation into base 2 is easier than other
bases since we only have to multiply each place by 1 or 0.
Powers of 2
power value power value
0 1 6 64
1 2 7 128
2 4 8 256
3 8 9 512
4 16 10 1024
5 32 11 2048
The first step is to determine how many places will be needed. Looking
at the table of powers, 532 is between 512 and 1024, so the base 2 representation
will have a 1 in the 9th place (counting from a 0th place) and
must have 10 places. 512 goes into 532 one time, so we put a 1 in the 9th
place of the base 2 number. With 512 out of the way there is only 20 left
(532−512). The biggest power of 2 that can go into this is 16, so there is a
1 in the 4th place. This leaves 4, which is taken care of with a 1 in the 2nd
place. All of the rest of the places get zeros. The number is 10000101002.
Another algorithm for converting to base 2 uses repeated division by 2
to determine place values in reverse order, from right to left. If a number
is even, then the right-most (0th) place will have a 0 in it, and if it is odd,
there will be a 1 there. This generalizes so that if we keep dividing by 2 we
can use the remainder as the value for that place. Using 532 as an example
again and working right to left:
532 0
266 0 0
133 1 0 0
66 0 1 0 0
33 1 0 1 0 0
16 0 1 0 1 0 0
8 0 0 1 0 1 0 0
4 0 0 0 1 0 1 0 0
2 0 0 0 0 1 0 1 0 0
1 1 0 0 0 0 1 0 1 0 0
After completing this process the number can be read from left to right at
the bottom: 10000101002. This same process works for translation to other
bases as well — repeatedly divide and list remainders from right to left.
16.3. Binary 289
Addition and subtraction
The trick in adding or subtracting in a base other than base 10 is to notice
when you’ll have to carry (in addition) or borrow (in subtraction). In
binary, there’s no room to grow within a place so you’re always carrying
and borrowing.
1 10 1111 10101
+1 −1 +1011 −1010
10 1 11010 1011
Octal and hex
When working with computers it’s very convenient to work in octal (base
8) or hexadecimal (base 16) instead of decimal, since it is easy to translate
from these into and out of binary. Hexadecimal utilizes 16 unique digits
(radix 16): the 10 standard arabic numerals followed by the letters A to
F. Hex is easily translated into binary since every hex numeral (called a
hexit maybe?) corresponds to a sequence of 4 bits. Similarly every octal
digit (0–7) corresponds to a sequence of 3 bits. Therefore, you can easily
translate into and out of octal or hex on a digit-by-digit basis by using this
table.
Octal to binary and hex to binary conversion
octal binary hex binary hex binary
digit value digit value digit value
0 000 0 0000 8 1000
1 001 1 0001 9 1001
2 010 2 0010 A 1010
3 011 3 0011 B 1011
4 100 4 0100 C 1100
5 101 5 0101 D 1101
6 110 6 0110 E 1110
7 111 7 0111 F 1111
To translate AF16 to binary just copy the digits off of the table:
101011112. Translating this binary number into octal requires rearranging
the bits in groups of three starting at the right: 010 101 111. The
left-most bits have to be left padded with zeros to complete the triplet.
Then the octal number can be read off of the table: 2578. Translating from
any of these into decimal cannot be done with a simple table readout but
must be multiplied out by hand: AF16 = 10 ・ 161 + 15 ・ 160 = 17510 or
2578 = 2 ・ 82 + 5 ・ 81 + 7 ・ 80 = 17510. Nowadays, octal is less commonly
used than hex because the modern byte (the elemental unit of computer
memory) is typically 8 bits long, hence 2 hex numerals.
290 16. The Basics
Boolean algebra
Boolean algebra uses bits as truth values. T, true, is 1, and F, false, is 0.
The three major operators are AND (∧), OR (∨), and NOT (∼). NOT
is also represented by placing a bar over a value: T = F. In C, and some
other computer languages, AND is ‘&,’ OR is ‘|,’ and NOT is ‘!’.
These are logical operators and the results can generally be understood
by using logic. For the result of an AND to be true, both of the arguments
must be true. For the result of an OR to be false, both of the arguments
must be false. The following is the truth table for AND and OR.
A B A ∧ B A ∨ B
T T T T
T F F T
F T F T
F F F F
This same table can be generated by using standard arithmetic by defining
any positive number as true and only zero as false. Then AND can be
calculated using multiplication, and OR can be calculated using addition.
A B A B A ・ B A + B
T T 1 1 1 ⇒ T 2 ⇒ T
T F 1 0 0 ⇒ F 1 ⇒ T
F T 0 1 0 ⇒ F 1 ⇒ T
F F 0 0 0 ⇒ F 0 ⇒ F
Boolean algebra shares many of the familiar rules of arithmetic. The commutative
and associative laws are both valid for AND and OR. Boolean
algebra has other properties that differ from those of arithmetic. One
of the most useful of these is De Morgan’s theorem, which states that
A ∧B = A ∨ B and that A ∨B = A ∧ B.
A popular Boolean operation in the world of neural networks is XOR,
exclusive or, symbolized by ∨. Here’s the truth table for XOR:
A B A ∨ B
T T F
T F T
F T T
F F F
16.4 Linear algebra
Linear algebra is a basic tool for studying large systems. In this section, I
present just the tools directly applicable to simulating neural networks and
thereby omit most major concepts. For the newcomer to matrices, even the
16.4. Linear algebra 291
basics will seem confusing at first. However, the notions we need consist of
a few rules that will become familiar after playing with some examples.
What is algebra? Why linear?
Algebra is the study of equations with one or more unknowns. In linear algebra
we restrict ourselves to equations for straight lines (e.g., y = 2・x+1).
This means that none of the variables will be raised to a power or multiplied
with one another, and that we won’t use any of the transcendental
functions such as sin, cos, exponential, or logarithm.
From this perspective, linear algebra would seem to be very simple. Complexity
arises because we don’t only want to handle a single linear equation
at a time. Instead, we want to handle many linear equations simultaneously.
To do this, we use matrices (plural of matrix) and vectors. In a linear algebra
course, one first learns how to use linear algebra to solve simultaneous
equations. This does not concern us here, since we are interested only in
using vectors and matrices as a tool for artificial neural networks.
As explained in Chap. 6, a vector is an ordered set of numbers. A vector
is defined by its length N: the number of numbers in the vector. A standard
vector is a column vector, drawn from top to bottom. A vector drawn from
left to right is a row vector. A column or row vector is like a column or row
of a spreadsheet; a matrix is like the whole spreadsheet. A matrix is a twodimensional
array of numbers with every row having the same length as
every other row and every column the same length as every other column.
A matrix is defined by its dimensions, M × N, where M is the number of
rows and N is the number of columns (important: rows then columns). A
matrix can be formed by putting together M row vectors of length N or
equivalently by putting together N column vectors of length M. If M=N,
the matrix is called a square matrix. M and N can take on any integer
values from 1 on up.
A sparse matrix refers to a matrix with a lot of zeros in it. Sparse weight
matrices are often encountered in neural networks. The weight matrix represents
all possible connections. If most neurons are only connected to a
small subset of the entire population, then the weight matrix will be sparse.
The transpose of a matrix is the result of flipping it over the diagonal
that runs from the top left to the bottom right corner. The symbol for
transpose is a superscripted ‘T.’ For example:
⎛⎝
−5 −7
−25 −8
10 14⎞⎠
T
= _−5 −25 10
−7 −8 14_
A vector is really just a special case of a matrix for which one of the two
dimensions is 1. A column vector is an N × 1 matrix and a row vector is a
1 × N matrix. Therefore, the transpose of a column vector is a row vector
292 16. The Basics
and vice versa. By convention, _x is assumed to represent a column vector.
_x T would be the same values as a row vector. A scalar can be considered
a 1 × 1 matrix. Since vector and scalars can be considered members of
the matrix family, the rules of arithmetic for matrices can also be used for
handling vectors and scalars.
Elements of a matrix are typically referenced by an addressing scheme
based on their row, column position. For example, in a matrix A, the number
in the first row and fourth column of A is known as a14. Note that
the order for the numbering is row, then column, corresponding to M ×N
for the definition of matrix size. Since we often use a computer language
for manipulating matrices, it is worth noting that different computer languages
use different conventions for array offsets. In some languages arrays
enumerate locations from zero. In C, for example, the first location in a
matrix would be A[0][0]. In some other languages arrays start counting
locations from 1. The first location in a Fortran matrix would be denoted
A(1,1). An additional difference between these languages is that Fortran
stores arrays by columns, while C and most other languages store by rows.
Addition and subtraction
Vector and matrix addition is very simple. To add two matrices, both matrices
must be of identical dimensions. If not, the operation is undefined.
Each element of the matrix is added to the corresponding element in the
other matrix. The element in the first row, first column of matrix A is added
to the element in the first row, first column of matrix B, and the result is
placed in the first row, first column of the solution matrix. Consider the
two matrices below. Both are 3 × 4 matrices, so we can add them.
⎛⎝
5 1 1 10
2 2 4 18
1 1 4 15 ⎞⎠
+⎛⎝
2 7 0 12
3 6 1 1
2 3 4 5⎞⎠
=⎛⎝
7 8 1 22
5 8 5 19
3 4 8 20 ⎞⎠
Subtraction is carried out in an identical manner: each element in the
second matrix is subtracted from the corresponding element in the first
matrix.
Dot product
The dot or inner product of two vectors is a scalar (a number). The two
vectors must be of identical length. The vector on the left must be a row
vector and the one on the right a column vector. When placed the other
way around, column on the left and row on the right, the result is the outer
product, described below. To find the inner product, multiply each entry
of the first (row) vector by the corresponding entry of the second (column)
vector and then add up all of these products. For example:
16.4. Linear algebra 293
_5 3 1 10 −4 15_・
⎛⎜⎜⎜⎜⎜⎜⎝
2
−6
4821
⎞⎟⎟⎟⎟⎟⎟⎠
_______=
5 ・2 + 3・ −6 + 1・4 + 10・ 8 + −4 ・2 + 15・ 1 =83
Orthogonality
Orthogonal vectors are vectors that are at right angles to one another when
graphed. On a two-dimensional graph, you graph vectors by drawing a line
from the origin (0,0) to the x, y coordinates given by the two numbers in
the vector. In general, the easiest way to determine whether two vectors
are orthogonal is to see if their dot product is zero. This works for higher
dimensional vectors where drawing a picture is either impractical (i.e., three
dimensions) or impossible (i.e., four or more). Here are two examples of
orthogonal vectors. The first example is two-dimensional and can be readily
seen to be orthogonal from a graph (Fig. 16.1); the second example is
six-dimensional and cannot.
_5 2_・ _−3
7.5_ = 0
_5 1 −1 10 −40 15_・
⎛⎜⎜⎜⎜⎜⎜⎝
−2
−1
4821
⎞⎟⎟⎟⎟⎟⎟⎠
= 0
Outer product
For the inner (dot) product, we take a row vector times a column vector and
get a scalar as the result. For the outer product, we take a column vector
times a row vector and get a matrix as a result. For the inner product the
two vectors must be the same size. For the outer product they do not need
to be. Using a column vector of size M and a row vector of size N, we can
form the outer product by putting M row vectors on top of one another
and multiplying each one by the corresponding value in the column vector,
thus producing an M × N sized matrix. We can equally well put N of the
294 16. The Basics
−4 5
(−3 7.5) 8
(5 2)
Fig. 16.1: Orthogonal vectors (5,2) and (−3,7.5) are at right angles
to each other in a two-dimensional graph. Each vector is graphed
by drawing a line segment from point (0,0) — the origin — to
the point represented by the vector used as (x,y) coordinates.
column vectors next to each other and multiply each by the corresponding
value in the row vector.
⎛⎝
−2
−1
4 ⎞⎠
_5 1 −1 10_ =⎛⎝
−10 −2 2 −20
−5 −1 1 −10
20 4 −4 40 ⎞⎠
This can be done by writing out a pile of rows:
−2 ・ _5 1 −1 10−1 ・ _ _ 5 1 −1 104 ・ _ _ 5 1 −1 10_ or a series of columns:
⎛⎝
−2
−1
4 ⎞⎠
・ 5, ⎛⎝
−2
−1
4 ⎞⎠
・ 1, ⎛⎝
−2
−1
4 ⎞⎠・ −1, ⎛⎝
−2
−1
4 ⎞⎠
・ 10
Matrix multiplication
To add matrices, the matrices have to be of identical dimension. The rule
for multiplication is more complicated. To multiply matrices, the number
of columns for the matrix on the left must be the same as the number of
rows for the matrix on the right. This means that matrix multiplication
is not directly commutative — often the commuted operation is undefined
because the dimensions don’t match up.
Remember that a row vector of length N is a 1 × N matrix. A column
matrix of length N is an N ×1 matrix. To multiply a vector times a matrix
(vector on left side), the vector must be a row vector. To multiply a matrix
times a vector (vector on right side), the vector must be a column vector.
In neural networks, we generally just use right-side column vectors. Rightside
multiplication is done by taking the dot product of each row of the
16.5. Numerical calculus 295
matrix with the entire column vector. Each dot product result becomes a
single element in the resultant column vector:
⎛⎝
−1 −2 3 −2
−5 −1 1 −1
2 4 −7 4⎞⎠
・⎛
⎜⎜⎝
5
−1
23
⎞⎟⎟⎠
=
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
_−1 −2 3 −2_•⎛
⎜⎜⎝
5
−1
2
3
⎞⎟⎟⎠
_−5 −1 1 −1_•⎛
⎜⎜⎝
5
−1
2
3
⎞⎟⎟⎠
_2 4 −7 4_•⎛
⎜⎜⎝
5
−1
2
3
⎞⎟⎟⎠
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
=⎛⎝
−3
−25
4⎞⎠
For any two matrices, the element in row a and column b of a resultant
matrix is the dot product of row a of the left-side matrix with column b of
the right-side matrix. This is why the left-side matrix row-length (number of
columns) must be the same as the right-side matrix column-length (number
of rows). It works out easily if you look at first at matrix sizes using the “row
× column” convention. The sizes for an arbitrary matrix multiplication are
M × N ・ N × P = M × P. Any element Xab of the new M × P-sized
matrix is the dot product of row a from the left matrix and column b from
the right matrix. Similarly, a dot product is multiplication of matrices size
1 × N ・ N × 1 = 1 ×1 — the 1 × 1 matrix is a scalar.
Here’s a full matrix multiplication for its entertainment value:
⎛⎝
−1 −2 2 −2
−5 −1 1 −1
2 4 −4 4⎞⎠
・⎛⎜⎜⎝
5 1
−1 1
2 3
3 5
⎞⎟⎟⎠
=⎛⎝
−5 −7
−25 −8
10 14⎞⎠
16.5 Numerical calculus
“A calculus” is a small stone that forms in the gallbladder or urinary tract;
“the calculus” is a technique for solving problems involving change. A
calculus is often painful; the calculus may be as well. A calculus can be
treated medically or surgically; the calculus can be treated analytically or
numerically. In this chapter, I treat it numerically and painlessly.
Biological relevance aside, it would probably be fair to say that calculus is
the foundation of modern mathematics. The funny thing about modernity
in mathematics is that it’s been going on for about 300 years.
296 16. The Basics
Infinitesimals
Since we are interested in computer modeling, we don’t need all of the
fancy calculus symbols. This makes things a lot easier, enabling us to use
basic algebra to explain the concepts and solve the problems. Nonetheless,
I introduce some of the symbols in order to make it easier to read other
books where calculus is used.
Calculus is used to describe change. The Greek letter delta means change.
Δ is the capital letter; δ is the small letter. Our letter “d” is also used.
Usually, we will be talking about change over time. Time is represented as t,
so change in time is Δt = tnow−tbefore. The original application of calculus
was mechanics, the physics of how objects move in time: falling apples,
cannonballs, the moon, and so on. We use the variable x to represent the
position of an object. So the change in position from location a to location b
is Δx = xb−xa. Once we have measured both change in position (Δx), and
change in time (Δt) we just divide to get average velocity: v = Δx
Δt . (Notice
that v is used to represent velocity, while V is used to represent volts.) For
example, to check the accuracy of your speedometer, you measure the time
it takes you to drive 1 mile. Let’s say it takes 1 minute. Then Δx = 1 mile;
Δt = 60 seconds; Δx
Δt = 1 mile
1 minute . We now just have to change units to show
that a mile per minute is 60 miles per hour.
This procedure works well if you maintain a constant speed while driving
the measured mile. However, you could also cover the mile in 1 minute
by driving 100 mph for 20 seconds and 40 mph for 40 seconds. Again you
would be traveling an average of 60 mph. If the police want to know if you
were speeding, timing a measured mile may not tell them. The calculus
and the police are interested in instantaneous rather than average velocity.
To estimate instantaneous velocity, you have to make your measurements
over a very short time, as the police do with lasers and radar guns. If we
reduce the measured distance to 100 meters, this distance will be covered
in about 3.7 seconds at 60 mph (sorry about mixing meters and miles).
This still would not be short enough to be certain of getting an accurate
instantaneous velocity since the driver could brake and change his velocity
considerably during this time. However, if your reduced the measured
distance to 1 meter, then the transit time would be about 37 ms (milliseconds)
and the measurement will give a pretty good estimate of actual
instantaneous velocity. If you go down to 1 mm, it will take about 37 μs
(microsecond) and this will be a very good estimate.
In the abstract world of mathematics, analytic calculus uses the symbol
“dt” to represent an infinitesimal duration in time. This allows the
definition of a true instantaneous velocity. In the physical world every measurement
takes some time. One always averages velocity over some period.
Similarly, in the computer simulation world of numerical calculus, one uses
a number for Δt. If you needed to simulate a car driving at high speed for
a video game, 10 ms would be a reasonable choice for Δt.
16.5. Numerical calculus 297
Numerical solutions
In Chap. 11, Section 11.4, I showed a numerical solution for the membrane
charging equation:
Iin = C ・
ΔV
Δt
+ g ・ V
Using the explicit Euler method, the update rule for voltage worked out to
be
V + = (1 − Δt ・
g
C
) ・ V +Δt ・
Iin
C
There are many different methods for solving differential equations numerically.
Different methods will have different advantages and different
drawbacks, both in terms of the quality of the solution and in terms of the
computational demands. The major issues in solution quality have to do
with accuracy and stability. In Chap. 11, I showed how the explicit Euler
equation would become unstable if the time step was made too large. By
contrast, the implicit Euler integration has the advantage of remaining stable
for large time steps. The explicit Euler solution has the advantage of
being much simpler to program for sets of linked differential equations.
While the explicit Euler solution bases its calculation of the present voltage
on the past, the implicit Euler solution bases its calculation on the
future. This is what makes it implicit. The concept of basing a calculation
on a future value seems counterintuitive. It is possible because calculus is
about prediction of future events in deterministic systems.
Just as we did with the explicit solution, we start by expanding out
ΔV = V +−V
Δt . However, now we use future voltage instead of present voltage
elsewhere in the equation:
Iin = C ・
V + − V
Δt
+ g ・ V +
The box highlights the difference from the derivation of the explicit equation:
now we use the future conductance current g・V + instead of the present
conductance current g ・V . The rest of the algebra is pretty much as before:
Δt ・ Iin = C ・ V + − C ・ V +Δt ・ g ・ V +
Δt ・ Iin = (C +Δt ・ g) ・ V + − C ・ V
Δt ・ Iin + C ・ V = (C +Δt ・ g) ・ V +
V + =
Δt ・ Iin + C ・ V
C +Δt ・ g
V + = C
C +Δt ・ g ・ V +
Δt
C +Δt ・ g ・ Iin
298 16. The Basics
Remembering that Vmax = Iin ・R = Iin
g and τ = τmemb = R・C = g
C, we
can simplify the expression and compare it to the explicit version:
Implicit : V + = τ
Δt + τ ・ V +
Δt
Δt + τ ・ Vmax
Explicit : V += (1−
Δt
τ
) ・ V +
Δt
τ ・ Vmax
In the example of Chap. 11, we used Δt = 0.001 ms, τ = 1 ms, and
Vmax = 1 mV. We can plug the numbers into the implicit solution and
compare the numerical update rules.
Implicit : V = 0.999000999 ・ V + 0.000999000999
Explicit : V = 0.999 ・ V + 0.001
These produce slightly different values over the course of five steps.
t (ms) mV Implicit mV Explicit
0.000 0 0
0.001 0.000999 0.001
0.002 0.001997 0.001999
0.003 0.002994 0.002997
0.004 0.003990 0.003994
0.005 0.004985 0.004990
A major advantage of the implicit equation has to do with stability. As
previously noted, a large Δt of 2 ms will produce an unstable result when
using the explicit solution. By contrast, the implicit solution will still give
the correct result:
t (ms) mV Implicit mV Explicit
0 0 0
2 0.666667 2
4 0.888889 0
6 0.962963 2
8 0.987654 0
10 0.995885 2
Because the implicit solution is stable, it is possible to go straight to the
solution of an integration in one step by using an even larger Δt. If we
use Δt = 100 ms, then V = 0.99 after only one time step. This will not
work with sets of nonlinear interlocking differential equations — we cannot
generally do whole neural simulations in one time step.
16.5. Numerical calculus 299
Mathematical symbols
A barrier to understanding biology is the complex jargon that has developed
to describe organic things—Latin words for creatures, long chemical names
for compounds, whole geographies of cells and organ systems. In math, the
problem is understanding the symbols.
We have introduced Δx
Δt as the numerical representation of change with
time. The simplest calculus representation just involves using a regular “d”
instead of the Δ : dx
dt , this is “the first derivative of x with respect to t,”
or simply, by reading the letters, “d x d t.” It is written as a fraction
since it can be treated algebraically as a fraction. Since change with time
is the most common application of calculus, there is a special shorthand
for representing it, a dot over the variable: dx
dt = ˙x. Spoken, this is “x dot.”
In the case of a function f(x), a first derivative with respect to x can be
represented as f_(x).
The above is the first derivative. There are also higher derivatives: second
derivative, third derivative, etc. The first derivative of distance with respect
to time is velocity. The first derivative of velocity with respect to time is
acceleration. Therefore, acceleration is the second derivative of distance
with respect to time. This is represented as d2x
dt2 or ¨x. Newton’s first law, F
= ma, is also written as F = mv˙ or F = mx¨. The second derivative of a
function of x with respect to x is written f__(x).
Analytic solution to the charging curve
Above, we numerically solved the charging curve Iin = C・ ΔV
Δt +g・V . Here’s
the analytic solution for charging the membrane with a constant current.
This probably won’t make much sense if you’ve never studied calculus.
The charging of the membrane RC circuit is described by the equation
Iin = C ・ ΔV
Δt +g ・ V . The finite numerical ΔV
Δt is represented in calculus as
dV
dt . Rearranging the terms gives the canonical form:
C
g ・
dV
dt
= −V + Iin
g
dV
dt is not really a normal fraction. However, it is written as a fraction
because the terms can be handled separately as if it were. This makes sense
when you consider that the expression comes from taking a limit of ΔV
Δt as
Δt approaches zero. We separate terms involving t from terms involving
V , and then integrate both sides as follows. In the following expressions,
K1 and K2 are constants that will need to be determined based on initial
conditions.
300 16. The Basics
_ −dV/(−V + Iin
g
) = −_ g
C ・ dt
ln (−V + Iin
g
) = g
C ・ t+K1
V −
Iin
g
= K2e−g•t/C
V = K2 ・ e−g•t/C + Iin
g
We’ll use V (t) = V0 as the initial condition for setting K2:
V0 = K2 + Iin
g
K2 = V0 −
Iin
g
So the solution is
V = (V0 −
Iin
g
) ・ e− g
C •t + Iin
g
Substituting in the terms for τ = g
C and Vmax = Iin
g we get
V = (V0 − Vmax) ・ e−t/τ + Vmax
We can confirm what we illustrated with the numerical solution. The curve
is exponential. As time gets very large, V asymptotes at Vmax. The time
constant τ is rise time to 1 − e−1 ∼ 63% of total excursion.
16.6 Electrical engineering
Digital has now beat out analog in almost every realm: CDs instead of LPs
for music, digital telephones, and now digital TV. Study of analog electronics
has become almost quaint, with its foreign names from the heroic age of
electrical discovery. Whether or not the neuron turns out to use some sort
of digital code, neuronal signaling will still need to be explained by analogy
with the batteries, resistors, and capacitors of folks like Volta, Amp`ere,
Ohm, Galvani, Franklin (he’s not foreign), Kirchhoff, and Faraday.
16.6. Electrical engineering 301
The three big laws: Ohm, Kirchhoff, and the other one
Ohm’s law deals with resistors. Kirchhoff’s law deals with currents. The
law of capacitance has no name. In electronics, capacitance tends to slow
things down and becomes a nuisance. Maybe that’s why no one wanted to
claim it. In neurons, the capacitor’s ability to hang on to charge permits
signals to add up.
Kirchhoff’s law is conservation of current. Like the laws of conservation
of mass and energy in their respective domains, this law says that
all stuff comes from somewhere and goes somewhere. Nothing appears out
of nowhere or disappears into nothingness. (These laws don’t hold in the
quantum realm.) Kirchhoff’s law says that recycling of current is required
as long as you stay within the circuit. However, there is an inexhaustible
landfill, called ground, where all current eventually goes. Going to ground,
current doesn’t disappear — it just dissipates.
Fig. 16.2 shows how Kirchhoff’s law is typically applied at nodes where
wires are connected. From the positive voltage (or potential) at the bottom
of the figure, positive current will flow “downhill” to ground. Ground always
remains at 0 volts. If the side of the circuit away from ground is held at
a negative voltage, positive current will flow out of the ground down to
this more negative potential. Current (I) is measured in amps (A), which
represents flow of charge (Q) per unit time (t). In Fig. 16.2 the two branches
are the same, so the current will split equally at the point where the two
wires separate. Half of the 3 amps goes down one branch and half goes
down the other. When the wires come back together, the currents add
back up. Charge is conserved; current is conserved. The charge then goes
into the ground and might show up next in a lightning bolt or just creep
into someone’s radio as static.
Ohm’s law
Current can flow through wires, through salt water, through anything that
has mobile charge carriers. Anything that will carry current is a conductor
of electricity. The quality of a conductor is called its conductance (g).
No conductor is perfect, however (superconductors come close). Any time
that current flows it will encounter some resistance to its flow. Thus any
conductor can equally well be described in terms of resistance (R). A good
conductor will have a low resistance. Conductance and resistance are just
two ways of describing the same thing: g = 1
R;R = 1
g . Resistance is measured
in ohms (Ω). The unit of conductance, inverse ohm, is sometimes
called mho (ohm spelled backward). It is also called siemens (S, not s).
Ohm’s law states that current times resistance equals voltage (V = IR).
It therefore also states that conductances times voltage equals current (I =
gV ). V = IR is more commonly used in electrical engineering; I = gV is
more commonly used in neuroscience. Given a choice, current will follow
302 16. The Basics
ground (0 V)
positive voltage
3 amps
3 amps
1.5 amps
1.5 amps
Fig. 16.2: Kirchhoff’s law and ground.
ground (0 V)
3 amp
3 amp
1 amp
2 amp
5 volt
5 ohm
(0.2 mho)
2.5 ohm
(0.4 mho)
Fig. 16.3: Current follows the path of least resistance. The
resistors are the sawtooth symbols.
16.6. Electrical engineering 303
ground (0 V)
5 volt
0.67 amp
3.33 V
ground (0 V)
5 volt
2.5 ohm
(0.4 mho)
5 ohm
(0.2 mho)
.67 amp
1.67 V
2.5 ohm
(0.4 mho)
5 ohm
(0.2 mho)
Fig. 16.4: Resistors in series.
the path of least resistance. If we modify Fig. 16.2 to include resistors in
the two branches, most of the current will go into the better conductor
(Fig. 16.3). If conductance is twice as good in one branch, that branch will
get twice as much current. With one side of the circuit at 5 volts and the
other at ground, the voltage drop across each resistor is 5 volts. The two
resistors are parallel routes to ground so each resistor has this 5-volt drop
across it. For the smaller resistor (larger conductor), the current will be 2
amps = V/R = 5 V/2.5 Ω = g ・ V = 0.4 S ・ 5 V . For the larger resistor
(smaller conductor), the current will be 1 amp = V/R = 5 V/5 Ω = g ・V =
0.2 S ・ 5 V .
Normally, we build a circuit out of known components and have the
values for voltage (from a battery) and the values of the two resistors. We
then calculate the current through each branch and add them up to get
the total current. We can then use Ohm’s law in the other direction and
calculate the total resistance of the circuit. In this case, total resistance
R = V/I = 5 Volts/3 amps = 1.6667 Ω. Equivalently, g = I/V = 0.6 S.
Using symbols for the resistance values and working through a bunch of
algebra gives the rule for adding resistances in parallel branches: 1
Rparallel
=
1
R1
+ 1
R2
. The equivalent rule for conductance is easier: gparallel = g1 + g2.
In this case, 0.6 S = 0.2 S + 0.4 S.
Resistors can also be arrayed in series. The serial arrangement is called a
voltage divider. Perhaps it would be better to call it a voltage subtractor.
304 16. The Basics
Each resistor drops the voltage a certain amount, providing a portion of the
voltage in between. Fig. 16.4 shows two different serial resistance circuits.
These two circuits are not connected to each other — each has its own
separate battery. The easiest way to calculate the current is to know that
resistance adds in series Rseries = R1 + R2, just as conductance adds in
parallel. Hence the total resistance in both circuits is 5 Ω + 2.5 Ω = 7.5 Ω.
The current in both cases is I = V/R = 5 V/7.5 Ω = 2
3 A. In the left
circuit, the small resistor is encountered first, providing a voltage drop of
V = IR = 2
3 A・2.5 Ω = 12
3 V . This is the drop in voltage from the starting
potential of 5 V so the voltage at this intermediate location is 5 V −12
3 V =
31
3 V . The same current flowing through the next resistor drops the voltage
down to zero at ground V = IR = 2
3 A ・5 Ω = 31
3 V ; 31
3 V − 31
3 V = 0 V .
This circuit drops off one-third of the voltage in the first resistor and the
other two-thirds in the second resistor. Switching the resistors, as at right,
drops off two-thirds of the voltage first and the other one-third second from
the divide to ground.
Capacitance
Anthropomorphically speaking, electricity just wants to flow downhill to
the ground. Anything that impedes this flow is called impedance. Resistance
is one kind of impedance. The two other kinds of impedance are
capacitance and inductance. Inductance isn’t much of a factor in neurons.
The three kinds of impedance differ in their response to time-varying signals.
Resistors don’t care about the speed of change: V = IR whether
voltage is changing quickly or slowly.
A capacitor will pass current easily in the presence of a quickly changing
voltage but will not pass any current in the presence of constant voltage
(Fig. 16.5). It is therefore a high impedance for slow-changing voltage signals
and a low impedance for fast-changing voltage signals. Capacitance
means the capacity to hold charge. The equation Q = CV says that more
charge Q can be held at a given voltage V if the capacitor has high capacitance
C. In Chap. 11, Section 11.4 and in Section 16.5, I explained
how to go from charge to current by calculating the change in charge with
time (first derivative with respect to time): ΔQ
Δt = I. By taking the first
derivative of both sides we get the impedance relation: I = C ・ ΔV
Δt .
Current flow through capacitors is dependent on how fast voltage is
changing. Capacitance tends to take current away from fast-changing voltages
— shunting the current off to ground. Fig. 16.5 shows the passage of a
fast-moving voltage signal through a wire that is connected through capacitors
to the ground. This is the situation in transmission lines, which always
have some capacitative coupling to grounded material. As long as the voltage
is oscillating slowly, the signal will not leak out of the wire through the
capacitors (top). If the voltage is oscillating at a high frequency, however,
16.6. Electrical engineering 305
Vin Vout
Fig. 16.5: Capacitative drop-off in an alternating current (AC)
increases with increasing signal frequency. No drop-off would be
seen with a constant direct current (DC).
there will be a substantial drop-off (bottom). Capacitance also produces a
drag on the signal, slowing it down. For these reasons, capacitance is typically
a nuisance in electronics — preventing equipment from running as
fast as one might want. This is the problem that computer chip designers
are always fighting. They want to run the chips faster so that they can do
faster logic. But if they try to send signals through too fast, the signals get
sucked away by parasitic capacitance and don’t get to their destination.
In neurons, however, capacitance is a good thing. Since the brain is not
strictly clocked like a computer is, one cannot depend on signals coming in
at exactly the same time. Capacitance broadens and slows signals, allowing
for the temporal summation of signals that come in at slightly different
times.
References
Here’s a list of some references that are interesting, fun, useful, or of historical
interest (sometimes all four). Some of these are textbooks that
are complementary to the present work. Others are general references. I
also give some primary literature references that describe research that I
presented or alluded to in the text.
I’ve divided the references into sections based on the chapters of the
book, although in many cases a reference is pertinent to more than one
section.
General
• MA Arbib. The Handbook of Brain Theory and Neural Networks. MIT
Press, Cambridge, MA, 1998. This miniature encyclopedia is a good general
references that covers most major topics in computational neuroscience.
• JA Anderson and E Rosenfeld. Neurocomputing: Foundations of Research.
MIT Press, Cambridge, MA, 1988. This collection of papers contains several
of the artificial neural network papers cited below.
Perspectives
• PS Churchland. Neurophilosophy: Toward a Unified Science of the Mind-
Brain. MIT Press, Cambridge, MA, 1986. One of the first books that looked
at what philosophy could learn from neuroscience. Considers the mind–
body problem and the question of subjectivity (qualia).
308 References
• PS Churchland and TJ Sejnowski. The Computational Brain. MIT
Press, Cambridge MA, 1994. A broad review of issues in computational
neuroscience.
• A Damasio. The Feeling of What Happens: Body and Emotion in the
Making of Consciousness. Harcourt Brace, New York, 1999. About as close
as anyone’s gotten to getting a bead on the mind-body problem.
• DO Hebb. Organization of Behavior. John Wiley & Sons, New York,
1949. The classic Hebbian book. Honestly, I’ve never actually read it, but
the little parts I’ve looked at are good.
• D Marr. Vision. Freeman, San Francisco, 1982. Description of a modeling
approach to vision that presents Marr’s ideas of levels of investigation.
Basic neuroscience
• ER Kandel, JH Schwartz, and TM Jessell. Principles of Neural Science.
McGraw-Hill, New York, 4th edition, 2000. The grandaddy of neuroscience
textbooks. An excellent introduction. The only downside is that it has
become very long – it was shorter back when I read it four editions ago.
• JG Nicholls, AR Martin, PA Fuchs, and BG Wallace. From Neuron to
Brain. Sinauer Associates, Sunderland, MA, 2001. Another classic text,
from which I stole the title.
Representations
• PE Ceruzzi. A History of Modern Computing. MIT Press, Cambridge
MA, 2000. A computer science history that gives an introduction to the
history of representations.
• D Knuth. The Art of Computer Programming. Addison-Wesley, New
York, 1973. The classic multi-volume work on data structures and programming
techniques. However, if you wish to learn to program, this is not
the place to start. Instead, figure out what language you want to learn and
then pick up a tutorial book with exercises.
Computers
• M Campbell-Kelly and W Aspray. Computer: A History of the Information
Machine. Harper-Collins, New York, 1997. A well-written history of
computer science.
• J Von Neumann, PM Churchland, PS Churchland, and K Von Neumann.
The Computer and the Brain. Yale University Press, New Haven, CT, 2000.
A quite readable book based on von Neumann’s lectures on the concept of
brain function that led him to help invent the computer.
References 309
Cybernetics
• N Wiener. Cybernetics. MIT Press, Cambridge, MA, 2nd edition, 1965.
This is the book that introduced ideas and concepts of cybernetics, as well
as coining the term. It’s a hard read.
Concept neurons
• J Hertz, A Krogh, and RG Palmer. Introduction to the Theory of Neural
Computation. Addison-Wesley, Redwood City, CA, 1991. This is a thorough
and well-written treatment of artificial neural networks. It’s rigorous
mathematically and offers a good introduction for physicists and engineers.
• LV Fausett. Fundamentals of Neural Networks. Prentice Hall, New York,
1994. A more widely accessible introduction to artificial neural networks.
• W McCulloch and W Pitts. A logical calculus of ideas immanent in
nervous activity. Bulletin of Mathematical Biophysics 5:115–133, 1943. This
is the McCulloch and Pitts work that helped inspire von Neumann. It’s
quite mathematical.
• D Hubel and T Wiesel. Receptive fields, binocular interaction and functional
architecture in the cat’s visual cortex. Journal of Physiology –
London 160:106–154, 1962. Classic paper identifying response patterns in
visual cortex.
• CD Salzman and WT Newsome. Neural mechanisms for forming a
perceptual decision. Science 264:231–237, 1994. Paper describing the
thought-insertion experiment.
• F Rieke, D Warland, R de Ruyter, and V Steveninck. Spikes. MIT Press,
Cambridge, MA, 1999. A mathematical treatment of neural coding by spike
trains and spike timing.
• J Gleick. Chaos: Making a New Science. Penguin, New York, 1988. A
wonderful history of the discovery of the mathematics and physics of chaos.
There are no direct applications to computational neuroscience in this book,
but it offers a very readable introduction to ideas in dynamical systems and
complex systems.
• S Singh. The Code Book: The Evolution of Secrecy from Mary, Queen
of Scots to Quantum Cryptography. Doubleday, New York, 1999. A fun,
readable history of codes and decoding. Nothing directly about nervous systems
but the description of the methods used to disentangle lost languages
provides some parallels.
• JA Anderson, editor. An Introduction to Neural Networks. MIT Press,
Cambridge, 1995. This textbook has a chapter on limulus that gives more
details than I have given. It is also one of the few texts in the field that are
readily accessible to undergraduates. Compared to the present work, this
book has less emphasis on biology and more on cognitive neuroscience.
310 References
Supervised learning: the delta rule and back-propagation
• DE Rumelhart and JL McClelland. Parallel Distributed Processing: Explorations
in the Microstructure of Cognition. Vol. 1: Foundations. MIT
Press, Cambridge, MA, 1986. JL McClelland and DE Rumelhart. Parallel
Distributed Processing: Explorations in the Microstructure of Cognition.
Vol. 2: Psychological and Biological Models. MIT Press, Cambridge, MA,
1986. Highly readable companion volumes that formed the manifesto of the
PDP movement. Still very worthwhile reading.
• R O’Reilly, Y Munakata, and JL McClelland. Computational Explorations
in Cognitive Neuroscience. MIT Press, Cambridge, MA, 2000. An
introduction to the use of artificial neural networks in a cognitive context.
• D Rumelhart, G Hinton, and R Williams. Learning representations
by back-propagating errors. Nature 323:533–536, 1986. The paper that
introduced back-propagation.
• TJ Anastasio and DA Robinson. Distributed parallel processing in
the vestibulo-oculomotor system. Neural Computation 1:230–241, 1989.
T Anastasio. Modeling vestibulo-ocular reflex dynamics: from classical
analysis to neural networks. In Neural Systems: Analysis and Modeling.
Kluwer, 1993. Two original papers about the application of
back-propagation to the VOR.
• JA Anderson. A simple neural network generating an interactive memory.
Mathematical Biosciences 14:197–220, 1972. T Kohonen. Correlation
matrix memories. IEEE Transactions on Computers C-21:353–359, 1972.
Two papers that independently introduced the concept of outer-product
formation of associative memory networks.
• JJ Hopfield. Neural networks and physical systems with emergent collective
computational abilities. Proceedings of the National Academy of
Sciences 79:2554–2558, 1982. Hopfield’s classic paper in which he showed
why autoassociative networks would have point attractor dynamics.
From soap to volts
• C Koch and I Segev. Methods in Neuronal Modeling: From Synapse to
Networks. MIT Press, Cambridge, MA, 2nd edition, 1998. C Koch. Biophysics
of Computation: Information Processing in Single Neurons. MIT
Press, Cambridge, MA, 1998. Two books by leading biophysicists who
describe how cells might be able to do complex processing.
• D Johnston and SMS Wu. Foundations of Cellular Neurophysiology. MIT
Press, Cambridge, MA, 1994. A neuroscience textbook with an emphasis
on neurophysiology.
• CF Stevens. Neurophysiology: A Primer. J. Wiley, New York, 1966. A
classic brief description of neurophysiology, which is now out of print. It
presents the original treatment of slow potential theory.
References 311
Hodgkin-Huxley model
• D Junge. Nerve and Muscle Excitation. Sinauer, Sunderland MA, 1992.
An introduction to basic neurophysiology.
• B Hille. Ionic Channels of Excitable Membranes. Sinauer Associates, Sunderland,
MA, 3rd edition, 2001. A very thorough treatment of ion channels
with an excellent chapter on the Hodgkin-Huxley model.
• AK Hodgkin and AF Huxley. A quantitative description of membrane
current and its application to conduction and excitation in nerve. Journal
of Physiology – London 117:500–544, 1952. The original modeling paper,
now 50 years old. It is quite readable.
Compartment modeling
• I Segev, J Rinzel, and GM Shepherd, editors. The Theoretical Foundation
of Dendritic Function: Selected Papers of Wilfrid Rall with Commentaries.
MIT Press, Cambridge, MA, 1995. A collection of the classic papers of
compartment modeling.
• JJB Jack, D Noble, and RW Tsien. Electric Current Flow in Excitable
Cells. Oxford, New York, 1983. An analytic treatment of passive dendritic
processing. Now out of print.
• DB Jaffe and NT Carnevale. Passive normalization of synaptic integration
influenced by dendritic architecture. Journal of Neurophysiology 82:3268–
3285, 1999. A nice simulation paper presenting passive normalization.
• K Neville and WW Lytton. Potentiation of Ca++ influx through NMDA
channels by action potentials: a computer model. Neuroreport 10:3711–
3716, 1999. A simulation paper of mine (with colleague) describing possible
effects of active dendritic channels on Hebbian learning.
From artificial neural network to realistic neural network
• RD Traub, JGR Jefferys, and MA Whittington. Fast Oscillations in
Cortical Circuits. MIT Press, Cambridge, MA, 1999. Oscillations in neural
networks presented by a pioneer in the development of realistic neural
modeling with physiology colleagues.
• ME Hasselmo and JM Bower. Cholinergic suppression specific to intrinsic
not afferent fiber synapses in rat piriform (olfactory) cortex. Journal of
Neurophysiology 67:1222–1229, 1992. Barkai E and Hasselmo ME. Modulation
of the input/output function of rat piriform cortex pyramidal cells.
Journal of Neurophysiology 72:644–658, 1994. Two papers explaining the
need for synaptic suppression when porting the Hopfield model to realistic
neural networks.
• J Chover, L Haberly, and WW Lytton. Alternating dominance of NMDA
and AMPA for learning and recall: a computer model. Neuroreport 12:2503–
312 References
2507, 2001. A paper written by my colleagues and me that presents the dual
matrix hypothesis.
Neural circuits
• G Shepherd, editor. The Synaptic Organization of the Brain. Oxford University
Press, Oxford, 1997. A multiauthor work that describes connectivity
in different regions of the brain.
• RJ Douglas, KAC Martin, and D Whitteridge. A canonical microcircuit
for neocortex. Neural Computation 1:480–488, 1989. A suggestion for a
basic cortical circuit.
Units
• Units program — http://www.gnu.org/software/units/units.html (free
software)
Linear algebra
These two programs are specialized for performing linear algebra operations
(and yes, much more):
• Matlab: http://www.mathworks.com/
• Octave: http://www.gnu.org/software/octave/octave.html (free software)
Numerical calculus
• Press WH, SA Teukolsky, BP Flannert, and WT Vetterling. Numerical
Recipes in C++: The Art of Scientific Computing. Cambridge University
Press, Cambridge, 2002. A wonderful introduction to all kinds of numerical
computing. In addition to the algorithms and programs, individual chapters
give good introductions to various math topics. Other books in the series
give algorithms for C, Fortran, and Pascal.
• S Wolfram. The Mathematica Book. Cambridge University Press, New
York, 4th edition, 1999. This book describes the use of Mathematica, a
program for doing both numerical and symbolic mathematics. Mathematica
is an excellent computer tool for learning math.
Electrical engineering
• P Horowitz and W Hill. The Art of Electronics. Cambridge University
Press, Cambridge, 2nd edition, 1989. The introductory chapter is a great
description of RC circuits and impedance relations. The rest of the book
give lots of practical information for building circuits.
Glossary
ablation: In neurology, an area of destruction or damage in a region of
brain. Verb: ablate.
ablative disease: Disease that involve loss of brain matter such as stroke
or head trauma.
absence epilepsy: Seizure disorder of childhood characterized by brief
(several seconds) periods of loss of consciousness with staring but without
collapse or limb movements. This disorder can be confused with daydreaming.
However, electroencephalogram shows characteristic abnormal three
per second spike-and-wave discharges.
absolute refractory period: In neurophysiology, the time after firing
of an action potential during which another action potential cannot be
initiated.
accommodation: In neurophysiology, decrease in spike firing rate with
time (also called adaptation). In neuropsychology, reduction in response
to sensory stimulation with time (also called habituation). In ocular
physiology, relaxation of the lens.
accumulator: In computer science, a single specialized register where
arithmetic results are stored. Abbrev.: ACC. The term is no longer current;
modern computers have many such registers.
acetylcholine: A neurotransmitter that serves as a neuromodulator
in the central nervous system and is the principal transmitter at the
neuromuscular junction. Adj.: cholinergic.
acquired disease: A disease that is picked up during life. Contrasted with
congenital.
314 Glossary
action potential: Spike in membrane voltage conducted down an axon to
transmit information to other neurons. Action potentials can also occur in
dendrites.
activation function: See squashing function.
active channels: Ion channels that change permeability (conductance)
with changes in voltage. Contrasted with passive channels.
active membrane: Membranes that have voltage-sensitive channels. Also
called excitable membrane.
adaline: An early version of the sum-and-squash unit used in artificial
neural networks. It is an acronym for ADAptive LInear NEuron.
adaptation: In neurophysiology, reduction in firing rate over time with
continuing stimulation.
adaptive filter: In signals and systems, a signal processing technique that
uses a learning algorithm to change the filtering properties depending on
aspects of the signal.
afferent: An axon, neuron, or tract that brings information in toward a
neuron or a neural structure.
afterhyperpolarization: Negative voltage that overshoots resting membrane
potential following an action potential. Abbrev.: AHP.
agonist: A drug or ligand that activates a receptor. Opposite of antagonist.
Algol: An early computer language that was precursor to Pascal.
algorithm: In computer science, a sequential series of discrete steps used
to solve a problem. Comparable to the steps of a recipe used in cooking.
alien-hand syndrome: In neurology, a disconnection syndrome resulting
from cutting of the corpus collosum. The two cerebral hemispheres then operate
independently, resulting in two largely separate consciousnesses. The
left typically remains dominant because of its control of speech. The “self”
of the left hemisphere regards the left hand as alien since it is controlled
by the right hemisphere.
all-or-none: A binary phenomenon without any intermediate state. Term
used to describe the action potential.
allele: A particular form of a gene; e.g., different alleles of a single gene
produce different eye colors.
allosteric: In biochemistry, alteration of the activity of an enzyme or ion
channel produced by binding of a chemical onto a site separate from the
main binding site.
alpha waves: Waves of 8 to 12 Hz in the electroencephalogram.
alternating current: Electrical current delivered as a sinusoidal wave of
changing voltage amplitude. This is the form of current delivered at a wall
outlet. Abbrev.: A.C.
amino acid: A chemical containing an amino group (nitrogen and hydrogen)
and a carboxyl acid (carbon, oxygen, and hydrogen). Amino acids
are the basic building blocks of proteins. Some amino acids serve as
neurotransmitters.
Glossary 315
AMPA receptor: A synaptic receptor for glutamate that mediates
excitatory (depolarizing) signals. Abbrev. for α-amino-3-hydroxy-5-methyl-
4-isoxazole propionic acid, an artificial agonist at the receptor.
ampere: Measure of current. 1 ampere=1 coulomb/second. Abbrev.: amp,
A.
amplitude modulation: In signals and systems, transmission of a signal
by change in the size of a carrier wave. Abbrev.: A.M.
amygdala: A deep-brain nucleus involved in emotional and appetitive
behavior. Part of the limbic system.
analog: A signal characterized by a continuous quantity. Contrasted with
discrete, digital, or binary; e.g., temperature is an analog property.
analog computer: A computer that organizes RC circuits so as to emulate
physical phenomenon, using the fact that the differential equations of
electronics are identical to those of mechanical dynamics.
analytic: Capable of being solved exactly using algebra, calculus, or other
methods of mathematics. Analytic solutions are contrasted with numerical
solutions, which are always approximations.
anatomy: The study of structures in biology. Contrasted with physiology.
AND: A Boolean operator. Symbol: ∧. See Chap. 16, Section 16.3.
animalia: The animal kingdom.
anion: A negatively charged ion, e.g., Cl−.
annealing: In physics, cooling so as to solidify in a lower energy state.
In neural networks, using analogous mathematics to solve problems of
minimization by reducing a high-energy network to an attractor state.
anomalous rectification: In electrophysiology, a channel that mediates
inward current with membrane hyperpolarization.
antagonist: A drug or ligand that blocks a receptor. Opposite of agonist.
antidromic: In electrophysiology, signal flow against the normal signaling
direction. Contrasted with orthodromic.
apamin: A potassium channel blocker that is a constituent of bee toxin.
aphasia: Loss of language following brain damage to the dominant
hemisphere.
apical dendrite: A large dendrite (∼ 1 mm) that typically protrudes from
the apex of the soma in a pyramidal cell.
Archilochus: A dead white male who said something clever about small
brown mammals (page 182).
architecture: In computer science, the details of the hardware design of a
particular machine. In neural network theory, the connectivity of a network.
area MT: A cortical area that mediates visual perception of motion as
well as other visual functions.
array: A data structure that places information in locations addressed
by a consecutive numerical index. These are commonly used in numerical
programming languages like FORTRAN and C. On a computer, a vector
is stored as a one-dimensional array, and a matrix as a two-dimensional
array.
316 Glossary
artifact: An experimental finding that arises from the experimental design
or experimental equipment and not from the thing being studied.
artificial intelligence: The development of computer programs to
replicate complex intellectual skills. Abbrev.: AI.
artificial life: A science dedicated to building computer data structures
to simulate evolving organisms.
artificial neural network: A system of interconnected computing units
with simple properties loosely based on the behavior of neurons. Abbrev.:
ANN.
Ascii: An international computer standard for binary encoding of letters,
numbers, symbols, as well as certain screen and keyboard commands.
asphyxiation: Death due to lack of oxygen.
assembler language: Human-readable version of machine language. Assembler
uses brief strings like MUL (for multiply) as a stand-in for
the numerical representation of machine language. Verb: assemble —
conversion from assembler to machine language. Also called assembly
language.
assignment operator: In computer science, a symbol that represents the
procedure whereby a variable (e.g., x) takes on a particular value (e.g., 5).
In C: x = 5; in Algol x := 5; in Lisp (set ’x 5).
association cortex: An area of cortex that is not connected to primary
afferents (sensation) or efferents (motor output). These areas are believed
to coordinate or associate information from lower levels of cortex.
associative long-term potentiation: Increase in strength in a weakly
stimulated synapse in the presence of strong activation in a nearby synapse.
associative memory: In neural networks, a network that associates one
pattern with another. Sometimes used as a shorthand for autoassociative
memory. Also associative network.
associative property: An arithmetic property that allows regrouping of
terms: e.g., (a+b)+c = a+(b+c); (a∧b)∧c = a∧(b∧c) (Boolean AND).
astrocyte: A glial cell believed to be involved in maintaining the chemical
composition of extracellular space.
asymptote: A value that is approached but never reached by a function.
asynchronous updating: A neural network technique where unit states
are reset in random order.
ATP: Adenosine triphosphate. A compound used as energy storage for
cells. ATP and its metabolites (ADP, AMP, cAMP) are also used as
transmitters and second messengers.
attention: A neural mechanism to focus brain function on a particular
sensory input.
attractor: A stable state or stable sequence of states in a dynamical system.
A point attractor is a single point, e.g., the low point in a landscape
where water will gather in puddles. A limit cycle is a repeated oscillation
that remains stable over time — like the oscillation of a pendulum. A
strange attractor is seen in chaotic dynamics.
Glossary 317
attractor network: A neural network whose dynamics causes the state
of the network to move to a single state (point attractor) or a sequence of
states (limit cycle) representing the stored patterns.
autoassociative memory: A neural network memory that maps inputs
onto themselves. Because of pattern completion, this form of memory can
be used to identify a pattern despite an incomplete or degraded input.
autoimmune: Diseases produced by the body’s own immune system
attacking other body parts.
autonomic nervous system: The section of the nervous system that
innervates the body’s organs. Abbrev.: ANS.
Avogadro number: ∼6.02 ・ 1023. Defined by the number of atoms in 1
gram of carbon-12.
axial resistance: In neurophysiology, the resistance along the length of a
dendrite. Also called longitudinal resistance. Abbrev.: Ra, Rl, or Ri.
axoaxonic: Describing an axon that connects to another axon.
axon: Long, thin projections that typically carry action potential signals
long distances in the nervous system.
axon collateral: A branch of an axon.
axon hillock: An initial expanded area of axon. Believed to be a spike
initiation zone.
axon terminal: The end of an axon or axon collateral, usually presynaptic
to another cell.
back-propagating spikes: In neurophysiology, action potentials that go
backward from the soma up the dendrite.
back-propagation: An algorithm to permit reduction of error in multilayered
networks by sending the error signal back through the layers. Also
called back-prop.
band-pass filter: In signal and systems theory, a system that allows only
a specific range of frequencies through, removing all frequencies below and
above.
basal ganglia: A large complex of nuclei rostral (anterior) to the thalamus.
Believed to be involved in motor activity, planning, and initiative.
base: In electrical engineering, the controlling input to a transistor. In
mathematics, the value of the number system being used, e.g., base 8, base
10.
BASIC: An early interpreted computer language developed at Dartmouth
College.
basin of attraction: In dynamical systems, area of state space from where
system state will proceed to a particular attractor.
battery: A source of constant voltage.
bauble: A small pretty object that has no value.
belladonna: A drug that blocks the action of acetylcholine at the
muscarinic receptor (a muscarinic antagonist).
beta waves: Waves of 13 to 26 Hz in the electroencephalogram.
318 Glossary
bias: In electrical engineering, the voltage that must be applied to the base
to get current flow from collector to emitter. In artificial neural networks, a
false input that is always set in order to effectively shift activation function
threshold during learning.
bilateral innervation: Projections to or from a brain area to both sides
of the body. Contrasted with the more common contralateral innervation.
binary: The base two number system (Chap. 16, Section 16.3).
binary operator: In computer science, an operator that takes two arguments;
e.g., ‘+’ takes two arguments and produces the sum. Note that the
binary in “binary operator” refers to two arguments and has nothing to do
with binary numbers.
binding problem: Question of how the brain figures out which parts and
attributes of a perception can be assigned to the same object. One aspect is
how object attributes can be reconnected after being processed in different
areas of the cortex. Must be solved by the brain in order to produce the
perception of a single object rather than separate perceptions of color,
motion, shape (see illusory conjunction).
bit: A binary digit — one place in the base 2 system.
bit-wise complement: Binary operation that changes every 0 to 1 and
every 1 to 0.
bitmap: A data structure that stores an image using bits that represent
single spots (pixels) of a picture.
black box: In engineering, term used to describe a device whose internal
functioning is unknown.
blind spot: The area of visual space that projects onto the optic disk
where the optic nerve exits. There are no photoreceptors at this location.
Boolean algebra: Mathematical system for calculation of truth values.
Symbols for true (T) and false (F) are used with operators such as ∨ (OR)
and ∧ (AND).
bouton: Synonym for synaptic bouton — a form of presynaptic specialization.
brain: The part of the mammalian central nervous system within the skull.
brainstem: The lower part of the brain that connects to the spinal cord.
Divides into three parts: mesencephalon, pons, and medulla.
Broca’s area: A frontal area involved in producing language.
Brodmann area: A cytoarchitecturally defined area of cortex. These areas
were originally mapped by Brodmann in the late 19th century.
bug: An error in a computer program.
bus: In computer science, the central wiring through which computer
components communicate.
butterfly effect: A hypothesis in dynamical system theory: a butterfly
flapping in China can change the weather a year later in South America.
This is an evocative example of the sensitivity to initial conditions seen in
chaotic systems. Weather is a chaotic dynamical system.
Glossary 319
byte: In computer science, a unit of data or memory. Nowadays usually
equal to 8 bits.
C: A popular programming language. C++ is a more modern variation on
the language.
CA1, CA3: CA1 and CA3 are two areas of the hippocampus. CA stands
for cornu ammonis.
cable equation: A partial differential equation describing the attenuation
of voltage along an undersea cable or a passive dendrite. Attenuation is
due to resistance along the cable as well as leakage through capacitance
and resistance through the sides.
calculus: The area of mathematics dealing with change (differential calculus)
and finding areas or volumes (integral calculus). See Chap. 16,
Section 16.5.
capacitance: Ability of separated conductors to store electrical charge.
See Chap. 16, Section 16.6.
capacitative coupling: Tendency of any transmitted oscillatory signal to
induce charge buildup in neighboring conductors through field effects. For
example, alternating current sent through a transmission line will induce
charge in nearby people, creating temporary capacitors, with the line as
one plate and the person as the other plate.
capacitor: In electronics, a device with parallel conducting places
separated by an insulator.
carbon-based: Used to refer to living things, built primarily from carbon,
hydrogen, oxygen, and nitrogen.
carrier frequency: In signals and systems, a constant or central frequency
of a signal that does not itself convey information but is modulated in some
way in order to transmit data.
catalyst: In chemistry, a material that increases the reaction rate of two
chemicals by bringing them together. Enzymes serve as biological catalysts.
cation: A positively charged ion, e.g., Na+.
cauda equina: The “horses tail” of nerves at the end of the spinal cord.
caudal: In anatomy, orientation toward the tail of the body. Due to the
curvature of the neuraxis in the human, this direction is posterior for the
brain and inferior for the spinal cord.
caudate: An input area of the basal ganglia. Part of the striatum.
cell assembly: Groups of simultaneously active neurons active together
due to their mutual connections. Also called Hebb assembly.
cell body: See soma.
cell membrane: Lipid bilayer separating outside from inside of a cell.
central nervous system: The brain (including retina and brainstem) and
spinal cord of higher organisms. Abbrev.: CNS.
central processing unit: The major data processing chip of the computer.
In modern computers this can be a single silicon wafer with millions
of transistors etched into it. Abbrev.: CPU.
320 Glossary
cerebellum: A little cortex located behind the brainstem believed to be a
movement or sensorimotor coordination center.
cerebrospinal fluid: The fluid that lies outside the brain and within the
ventricles. Abbrev.: CSF.
cerebrum: The large outer lobes (cerebral hemispheres) of the mammalian
brain. Adj.: cerebral.
chandelier cell: Inhibitory interneuron of cortex that synapses on
pyramidal cell axons.
chaos: Unpredictable behavior in a deterministic dynamical systems with
no random factors. In a chaotic system, small changes in initial conditions
will completely alter the subsequent evolution of the system.
charge: Excess or deficiency in electrons in a substance. Measured in
coulombs.
charge-couple device: Electronic array of photodiodes that captures the
image in a digital camera. Abbrev.: CCD.
charging curve: In electronics or membrane physiology, the exponential
increase in voltage with constant current injection.
charybdotoxin: A potassium channel blocker that is a constituent of
scorpion toxin.
checksum: In data transmission, a data field that represents a sum of
previously transmitted data used to check that no bits were lost.
chemical cascade: Signal amplification through a series of enzymatic
transformations of second- and higher-order molecular messengers. At each
stage, a single enzyme catalyzes production, resulting in amplification from
a small number to a large number of signaling molecules.
chemical synapse: A synapse that uses neurotransmitters that diffuse
across the synaptic cleft to bind receptors postsynaptically.
chromosome: DNA containing structure in the cell nucleus.
chunking: Organizing multipart information into a single datum for ease
of remembering; e.g., remembering a telephone number as a three-digit
followed by a four-digit number or as a single spelled word instead of as
seven independent numerals.
circadian rhythm: In physiology, the daily cycles of body and brain.
CISC: Complex instruction set chip. In computer architecture, a central
processing unit design that executes complicated instructions. With
growing CPU complexity, the contrast with RISC is now historical.
clamp: In neurobiology, voltage or current may be clamped (see voltage
clamp, current clamp). In neural networks, input units are clamped, meaning
that their values are held fixed and not reset according to an update
rule. In some cases, output units are also clamped — this is called teacher
forcing.
clear bit: Binary bit value corresponding to 0 or False.
climbing fibers: A projection in the cerebellum. See Fig. 15.3.
clock: In computer science, a central oscillator that provides timing signals
to coordinate signal transmission across the bus.
Glossary 321
CNS: See central nervous system.
cocktail party effect: In psychophysics, the ability of the brain to selectively
filter out large numbers of signals in order to attend to a single signal
— as is used to carry on a conversation in a loud cocktail party.
code: In computer science, a program or the process of programming. In
cryptology, a communication that has been altered in order to disguise its
contents. In neuroscience, means by which information is passed in the
nervous system.
coefficient: A number multiplying a variable in an equation. For example,
5 is the coefficient of x in the equation 5 ・ x = 10.
collaterals: Branches off the main trunk of an axon.
collector: In electrical engineering, one of the connections of a transistor.
column: In neuroanatomy, an internally connected volume of cortex often
defined functionally by the presence of similarly responding cells.
command neuron: A neuron whose activity produces a stereotyped motor
response. Found in some invertebrates.
communication theory: Mathematical theory for calculating the quantity
of information that can be transmitted using particular types of signals.
Closely related to information theory.
commutative property: An arithmetic property that allows swapping of
terms: e.g., a + b = b + a; a ∧ b = b ∧ a (Boolean AND).
compact: In electrophysiology, an adjective describing a neuron in which
signals propagate throughout the dendritic tree with little decrement (long
length constant).
compartment modeling: A standard biological modeling technique that
divides an organism or part of an organism into homogeneous compartments
that exchange some substance of interest. In electrophysiological
modeling, the compartments are equipotential sections of dendrite that exchange
current with neighboring compartments depending on differences
in voltage.
compiler: Computer program that converts a high-level computer language
such as C or Fortran into machine language. Verb: compile. Process:
compilation.
complex number: A number that has both a real and imaginary part.
The imaginary part is a multiple of i, the square root of −1. Complex
numbers are used to describe oscillatory signals.
compound eye: Common eye design in invertebrates. A large eye is
made up of many small eyes, each of which has its own cornea, lens, and
photoreceptor.
computer: A term originally used to describe people who were employed
to do calculations. Now used to describe machines that do calculations.
conditional statement: In computer science, a programming command
testing truth to decide on the branching of control. In many computer
languages, this takes the form: if true? then do A else do B.
conductance: See conductor.
322 Glossary
conduction failure: In neurophysiology, failure of an action potential to
transmit to the end of the axon.
conduction velocity: In neurophysiology, the speed of action potential
propagation along an axon.
conductor: Something that conducts electricity — e.g., a wire or salt
water. Symbol g, units siemens (S) or mhos. Conductance is the inverse of
resistance: g = 1/R. See Chap. 16, Section 16.6.
cone: A type of retinal photoreceptor that detects color, primarily found
in central retina.
confabulation: Pathological making up of stories to fill in inconsistencies
in perception or memory, typically due to brain illness associated with
dementia, delirium, or psychosis.
confectionery: Candy store.
congenital: Something present from birth.
connection matrix: A two-dimensional array of numbers give connection
strengths between units of a neural network. Synonym: weight matrix.
connection strength: A parameter that determines the ability of a
presynaptic neuron to drive a postsynaptic neuron. Also called weight.
consolidation: In neuropsychology, hypothetical process whereby memories
are moved from temporary storage in the hippocampus to more
permanent storage in cerebral cortex.
content-addressable memory: A memory system that allows retrieval
of information based on a part of the information itself. Contrasted with
the pointer-based addressing of a computer or a file cabinet.
continuous network: A network in which unit states change continuously
in time instead of being updated at discrete time step increments. Of course,
when a simulation is done on a computer, continuous time is approximated
with discrete time steps anyway.
contralateral: Referring to the other side of the body. Opposite of
ipsilateral. The cortex innervates contralateral body.
convergence: The number of inputs coming into a neuron. In mathematics,
the ability of a numerical calculation or infinite series to reach a finite
result.
core memory: In computer science, an old name for random-access memory
(RAM). This term dates from an old technology that built RAM from
magnetized iron cores.
cornu ammonis: Hippocampus, abbreviated CA in area names CA3, CA2,
CA1.
corollary: In mathematics, a side proof that reveals another aspect or
conclusion of a theorem.
coronal plane: A forward-facing vertical plane for sectioning the body
(see Fig. 3.3).
corpus callosum: A large white matter tract that connects the two
cerebral hemispheres.
Glossary 323
correlation is not causality: Science buzz phrase indicating that just
because two things have been observed to occur one after the other doesn’t
mean that the first one caused the second one to happen.
cortex: The outer gray matter of the cerebrum (cerebral cortex) or
cerebellum (cerebellar cortex). Plural: cortices.
corticospinal tract: Motor pathway that runs from motor cortex to spinal
cord. Also called pyramidal tract.
co-transmitters: Neurotransmitters released together at a single synapse.
coulomb: The measure of charge: 1 coulomb ∼ 6.2 ・ 1018 electrons.
CPU: See central processing unit.
credit-assignment problem: In neural network memories, the difficulty
of figuring out which unit or which connection was responsible for an error
in a pattern. Neural network algorithms such as back-propagation have
been developed to solve this.
critical period: Limited period of development of an organism when a
particular brain area (e.g., vision, language) is plastic and can be molded
by the environment.
cryptography: The art and science of coding and decoding.
CT scan: Computed tomography scan. An imaging method that re-creates
two-dimensional cross sections by adding up x-ray conductances measured
from different angles. Also called CAT scan.
current: Movement of charge: ΔQ
Δt . Symbol I, measured in amperes (amps,
A).
current clamp: Injection of continuous current to a neuron. Voltage is
measured.
CVA: Cerebrovascular accident. See stroke.
cytoarchitectonics: Distinguishing brain areas by differences in neuron
types and organization.
cytoplasm: The fluid inside a cell but outside of organelles.
Dale’s principle: The hypothesis that a single neuron expresses only one
transmitter. Although this has turned out to be wrong in detail (see cotransmitters),
it’s still used as a general rule to indicate that a single cell
will not be both excitatory and inhibitory.
data structure: Representation format for organizing information in
computer software.
database: In computer science, data structures for storing large amounts
of information so as to make them compact and readily accessible.
De Morgan’s theorem: A property of Boolean algebra that relates AND
to OR and vice versa. See Chap. 16, Section 16.3.
DEC: Digital Equipment Corporation. A company that was an early innovator
in producing small computers that could be run and used by an
individual rather than a large team. See Parallel Data Processor.
decimal: Base 10.
declarative memory: Synonym for episodic memory.
324 Glossary
degenerative disease: Diseases that involve progressive loss of neurons
or neural functioning, e.g., Parkinson and Alzheimer disease.
degree: Unit of angular measure. There are 360 degrees in a circle. Also,
unit of temperature measurement.
delayed rectifier: Synonym for the Hodgkin-Huxley potassium channel.
delta rule: In neural network theory, an algorithm for updating weights
so as to correct errors and solve the credit-assignment problem.
delta waves: Slow waves (≤ 3 Hz) in the electroencephalogram. Mostly
seen in sleep.
dendrite: Cellular extensions of the neuron with postsynaptic specializations.
Distinguished from axons by being generally fatter, shorter, more
tapered, and unmyelinated.
dendritic tree: The full set of dendrites.
dendrodendritic: Describing a dendrite that connects to another dendrite.
denominator: The bottom of a fraction, e.g., the 3 in 2
3 .
dentate gyrus: An area of hippocampus with cells that project to area
CA3.
depersonalization: Mental or neurological condition producing loss of a
person’s sense of his own identity.
depolarization: Positive deviation from the negative resting membrane
potential.
depolarization blockade: In electrophysiology, inactivation of sodium
channels by prolonged depolarization causing failure of spiking.
derealization: Mental or neurological condition producing sense of
disconnection from surrounding reality.
derivative: A basic operation of calculus. Ratio of amount of change in a
function (y or f(x)) to the amount of change in an independent variable
(x). This equals the slope of the tangent to the curve of y graphed against
x. The derivative is represented by dy
dx. If x is position, then dx
dt , the change
in position with respect to time, is velocity.
description length: In computer science, a technique for assessing the
size of an algorithm by considering both the amount of memory devoted
to the program and the memory devoted to data.
desensitization: Reduced responsiveness of a receptor to a ligand after
prolonged exposure.
deterministic: In mathematics, a system whose behavior is entirely defined
by its present state and its governing equations with no random
inputs. Contrasted with stochastic.
dielectric constant: In electronics, describes the polarizability of a compound
in an electric field. Capacitors use substances with a high dielectric
constant between the plates in order to increase capacitance.
diencephalon: Part of the brain connecting the cerebral hemispheres with
the brainstem. Includes thalamus and hypothalamus.
Glossary 325
difference equation: A functional form that describes the differences
between values in a sequence.
differential equation: An equation where derivatives appear on one or
both sides of the equal sign.
differentiation: The taking of derivatives to determine the rate of change
of one quantity with respect to another (e.g., time or space).
diffusion: The movement of particles in solution out of a high concentration
toward lower concentration.
dimensional analysis: Determination as to whether the units being used
in two aspects of an equation or conversion are compatible. Also called unit
analysis.
dimensional reduction: In dynamical systems, a mathematical technique
for analyzing a high-dimensional system by collapsing into a
lower-dimensional system.
dimensionality: In dynamical system theory, the number of state variables
in a system. The state of the system is a point and the full dynamics
is a curve in state space with this number of dimensions.
diode: In electronics, a component that passes current in only one
direction.
direct current: Electrical current delivered at a constant voltage. This is
the form of current delivered by a battery. Abbrev.: D.C.
discrete: A measure or enumeration that only takes on a finite number of
specified values. Opposed to continuous or analog.
discrete mathematics: Calculations that utilize specific numbers rather
than symbols representing continuous values. Numerical analysis done by
computers is discrete mathematics.
disinhibition: Removal or reduction of inhibitory effect in a neuron or
circuit.
distributed representation: A representation coding information diffusely
through multiple units. Opposed to local representation or grandmother
cells. Also called distributed code.
divalent: Describing an ion with two excess electrons or protons (e.g.,
Ca++).
divergence: The number of outputs coming from a neuron. Failure of a
numerical calculation to converge to a finite result.
DNA: Deoxyribonucleic acid. The double helix that serves as primary
genetic material.
domain: In mathematics, the set of values that can serve as inputs to a
function. See range.
dominant hemisphere: The cerebral hemisphere that controls language,
usually the left hemisphere.
dopamine: A neurotransmitter. Loss of dopamine cells causes Parkinson
disease.
dorsal: Toward the back side of the neuraxis: posterior in the spinal cord
and superior in the brain.
326 Glossary
dot product: A linear algebra operation that takes two equal-length
vectors and creates a scalar that is the sum of pair-wise products from
corresponding positions in the two vectors. Also called inner product.
downregulate: In biology, reducing the amount or activity of an enzyme,
receptor or other active protein due to negative feedback from the products
or results of that enzyme or receptor.
driving force: In electrophysiology, the difference between the Nernst or
reversal potential for a particular ion and the membrane potential: V −Erev.
duty cycle: In signals and systems, the proportion of the period of a binary
periodic signal that is spent at the high value.
dynamical system: Originally, a dynamical system was a set of any objects
that moved and influenced each other through gravitation (e.g., the
solar system). The term has generalized to include any system with multiple
state variables that interact in a time-dependent fashion. In practice
this means any system that can be described by a set of differential or
difference equations.
dyslexia: Difficulty in reading due to brain disorder. Can be congenital or
acquired.
dystopia: A vision of future society as an awful place. Opposite of utopia.
edge effects: Artifacts in neural network activity occurring in units at
or near the boundaries of the network due to the reduced number of
connections at the edge.
efferent: Leading or projecting away from neural structure.
electric eye: A photodiode paired with a constant beam of light that is
used to detect the passage of a person that interrupts the beam. The electric
eye has been partly supplanted by infrared, microwave, and ultrasonic door
monitors.
electrical self-stimulation: A mechanism for allowing an animal to
deliver small electric currents to activate a particular spot in its own brain.
electrical synapse: A gap junction between cells that allows current to
flow directly from the interior of one cell to the interior of another cell.
electrochemical equilibrium: The balance point between diffusive force
in one direction and membrane potential gradient in the other direction.
The membrane potential where this occurs is the Nernst potential.
electrocorticogram: Brain waves recorded using electrodes placed directly
on the brain.
electrode: A conductor used to probe an electrically active material. See
microelectrode.
electroencephalogram: A recording of electrical signals from the brain
made on the skin of the head. Abbrev.: EEG.
electrolyte: A substance that will carry electricity when dissolved in
water.
electron microscopy: An imaging technique that uses electrons to get
far better resolution than is possible with light microscopy.
Glossary 327
electrophysiology: Set of techniques for measuring the electrical properties
(voltages and currents) of neurons, of other electrically excitable cells,
and of cellular connections.
electrotonic: In electrophysiology, concerning the response of a cell to a
constant current injection.
emergent properties: Properties resulting from the combined effects of
many elements in complex systems that cannot be explained by knowing
the properties of these elements.
emitter: In electrical engineering, one of the connections of a transistor.
emulation: Use of a physical, as opposed to a virtual, model system.
Contrasted with simulation.
emulator: In computer science, a program written to reproduce the
behavior of a different computer system or component (like a CPU).
en passant synapse: A synapse at a site where an axon passes by a
dendrite.
endogenous: Arising from or belonging to an organism or other system.
Contrasted with exogenous.
endorphin: An endogenous peptide neurotransmitter that is active at the
same location where opium or morphine acts.
energy function: A mathematical equation that assigns a potential energy
value to every point in a multidimensional space.
engineering notation: A standard for representation of numbers that
uses prefixes (e.g., kilo, mega, giga) for powers of 1000. See Chap. 16,
Section 16.2.
enteric nervous system: The nervous system that runs the gut. It can
function on its own even after being completely disconnected from other
parts of the nervous system.
enzyme: A protein that catalyzes a chemical reaction, causing it to occur
at a rate far faster than would occur in the absence of the enzyme.
ephaptic: Extrasynaptic interneuronal communication mediated by the
effects of electric field generated by one neuron on another.
epilepsy: A brain disorder in which repeated seizures occur.
epiphenomenon: A measurable side effect of a phenomenon that has no
functional relevance.
episodic memory: Human memory of events in life.
epsilon: In mathematics, a very small number.
equilibrium potential: The Nernst potential that produces electrochemical
equilibrium for a particular ion.
equivalent: In chemistry, a unit corresponding to the quantity of ions
needed to carry 1 faraday.
equivalent cylinder model: A reduction of a branched dendrite model
into a single cylinder that preserves the electrical properties of the original.
error-correcting code: A redundant representation of information that
includes an internal check of accuracy that allows slight transmission errors
to be corrected by the receiver.
328 Glossary
error minimization: In artificial neural network supervised learning theory,
the goal of reducing the difference between network output and desired
target to a minimum.
eukaryote: In biology, organisms that have cells with a nucleus. This
includes animals, plants, and yeast.
excitatory: Serving to excite or turn on a neuron — often said about
synapses, weights, or connections.
excitatory postsynaptic potential: Depolarization of the postsynaptic
membrane potential due to synaptic action. Abbrev.: EPSP.
exclusive OR: See XOR.
existence proof: In mathematics, a proof that demonstrates that a certain
thing (e.g., a type of function) must exist, even if we cannot yet, or can
never, determine what it actually is. It is said that the brain provides the
existence proof for the ability of a machine to do language, vision, etc.
exocytosis: Release of material from a cell due to fusion of a vesicle with
the cell membrane. This is how neurotransmitters are released into the
synaptic cleft.
exogenous: Compound or material introduced to a system from outside.
Contrasted with endogenous.
expert system: Computer program that attempts to replicate human
reasoning in a particular domain such as medicine or law.
explicit Euler integration: A numerical integration technique that uses
the past value of state variables in the approximation. See Chap. 11,
Section 11.4.
exponent: The power a in xa.
extracellular: The volume outside of cells in the body.
facilitation: In neuroscience, transient increase in excitability. Generally,
used for effects that are short in duration as compared to potentiation.
false unit: Unit of a neural network that does not do any processing but
simply serves as an access point for presenting inputs or bias.
farad: Measure of capacitance. 1 farad=1 coulomb/volt. Abbrev.: F.
faraday: A mole of electric charge. Equals approximately 96,485 coulombs.
feature: In computer science, desirable functionality in software, whether
deliberately programmed in or not. Contrasted with bug. In image processing,
an elemental attribute to be extracted from an image during initial
processing.
feedback: Information or signal that is communicated back to a component
from a later processing component.
feedforward: In artificial neural networks, a network architecture that
sends information in one direction only. Contrasted with recurrent. In neuroscience,
refers to projections (particularly inhibition) that arrives before
or together with a major projection.
field potential: In electrophysiology, an electrical measurement representing
the summed activity of many neurons.
Glossary 329
filter: In signal and systems theory, a program or physical device that
transforms a signal. There are a variety of named filter types that handle
signals differently: low-pass, high-pass, band-pass, etc.
fitness function: Function used in genetic algorithms to describe how well
a result fits a target.
fixed point: In the context of a dynamical system, a point in state space
where the system stops and stays, the state vector remaining the same
thereafter.
fluid mosaic model: The standard conceptual model of the cell membrane:
the lipid bilayer is in a fluid state somewhere between a liquid oil
and a solid fat. Proteins and other uncharged molecules can float around
in the membrane.
flux: Flow. In neurophysiology, used to describe the flow of an ion across
the membrane.
FORTRAN: FORmula TRANslation language, the earliest compiled
computer language.
fovea: The central part of the retina that has the most acute vision.
frequency: The number of oscillation in a fixed period of time. Measured
in hertz = oscillations/second.
frequency coding: The notion that increases in spike frequency are an
important part of the code of the nervous system. Also called rate coding.
frequency modulation: In signals and systems, transmission of a signal
by changes in the frequency of a carrier wave. Abbrev.: F.M.
frontal lobe: The most anterior lobe of the brain. Believed to play a role
in initiative and planning.
functional magnetic resonance imaging: A technique for imaging
physiological activity using magnets and radio waves. Abbrev.: fMRI.
GABA: γ-aminobutyric acid. A major inhibitory neurotransmitter in the
central nervous system.
GABAA: A GABA receptor that mediates rapid inhibitory postsynaptic
potentials.
GABAB: A second-messenger linked GABA receptor that mediates slow
inhibitory postsynaptic potentials.
gain: In signals and systems, the amplification factor provided by a system.
gamma waves: Waves of 20 Hz and above in the electroencephalogram.
ganglion: A collections of neurons. Usually refers to neurons lying outside
the central nervous system in vertebrates (except in case of the basal ganglia).
In invertebrates, ganglia are the central processing structures (the
brains). Plural: ganglia.
gap junction: An ion channel that connects two cells together.
gating: In neurophysiology, the ability of channels to be opened or closed
either by membrane potential (voltage gated) or by the arrival of a
neurotransmitter or other chemical (ligand gated).
Gaussian function: The bell curve used in statistics; the basic form is
f(x) = e−x2 .
330 Glossary
gene: The unit of information storage in the coding nucleic acid (usually
DNA) of a cell. Classically a single gene coded for a single protein; this is
an oversimplification.
generalization: Flexibility in learning that allows a system to correctly
classify an object not previously learned.
genetic algorithm: A computer data-fitting algorithm modeled after the
process of evolution through changes in DNA. Many sets of parameters are
chosen randomly and tested against a fitness function. Those that are most
fit are then combined with each other through crossing-over and mutated
to produce a next generation to be tested.
genetic code: Representations of amino acids as well as stop and start
signals by triplet sequences of four nucleotides (A,T,G,C) in DNA and
RNA.
genome: All of the genes of an organism.
glial cell: The most common type of cell in the nervous system. These
cells play roles in support, hormone release, and balancing extracellular
concentrations. Two important types are astrocytes and oligodendroglia.
globus pallidus: An area of the basal ganglia. It has two parts: pars
externa and pars interna.
glucose: The sugar used as food by cells.
glutamate: An amino acid that is the major excitatory neurotransmitter
in the central nervous system. Also called glutamic acid.
glycine: An amino acid that is also an inhibitory neurotransmitter.
Goldman-Hodgkin-Katz equation: Equation describing the contribution
of multiple ions to membrane potential based on their independent
permeabilities and concentrations.
graceful degradation: In neural networks, the ability of a network to
show graded reduction in function with damage. Contrasted with the
catastrophic breakdown property of computers.
gradient: A slope.
gradient descent algorithm: In artificial neural network theory, process
for reducing the energy level of network state by heading gradually
“downhill” from higher to lower potential energy.
grandmother cell: An individual neuron that represents a specific thing,
place, or person (such as one’s grandmother). An evocative term for local
representation.
granule cell: Name for small, densely packed neurons found in dentate
gyrus and cerebellum.
grapheme: The written representation of a sound in a language. See
phoneme.
gray matter: Brain volumes made up of cell bodies. Cerebral cortex, basal
ganglia, and thalamus are gray matter.
ground: In electronics and electrophysiology, a connection to a standard
zero electrical potential, provided by attaching to a large conductor (typGlossary
331
ically the ground under your feet) that can readily source or sink charge.
When used as a verb: attach to ground or to a grounding wire.
gyrus: The cerebral hemispheres are folded; a gyrus is one of the ridges.
H.M.: Initials of a man who had bilateral hippocampectomy (removal of
the hippocampus on both sides) to cure his epilepsy. Over the past several
decades, he has been relentlessly studied by neuropsychologists trying to
infer the function of the hippocampus from his many memory dysfunctions.
habituation: Reduced response to a stimulus after repeated stimulation.
A model of simple learning. See accommodation.
hack: In computer science, a programming trick.
Hamiltonian: The total energy (kinetic plus potential) of a system.
Extended from use in statistical mechanics to information theory.
hardware: In computer science, the underlying machine that does data
processing.
Heaviside function: A discontinuous step function with discontinuity at
0. If x < 0, h(x) = 0 else h(x) = 1.
Hebb rule: The hypothesis that synapses will be strengthened when the
pre- and postsynaptic neurons are active at the same time. This is referred
to in multiple expressions: Hebb’s law, Hebb’s postulate, Hebb synapse,
Hebbian synapse, Hebbian learning, etc.
hertz: The unit of cycles per second for a sinusoidal oscillation. Abbrev.:
Hz.
heteroassociative: Referring to a memory system that maps one thing
onto another, e.g., names onto faces.
hexadecimal: Base 16; also called hex.
hidden layer: A layer of units in a feedforward neural network that are
not exposed to either the input or the output.
hidden unit: A unit in a feedforward network that does not receive inputs
or produce outputs.
high-pass filter: In signal and systems theory, a system that removes the
low frequencies from a signal and only lets the high frequencies get through
to the output.
hippocampus: A cortical area in the temporal lobe involved in episodic
memory (memory for events).
histology: The study of the organization of living cells into tissues.
Hodgkin-Huxley equations: Differential equations describing the origin
of the action potential. Each of three particles is described by differential
equation of the form τx ・ dx
dt = x∞ − x, where x is m, h, or n. The infinity
and tau curves (Fig. 12.5) are defined in terms of α(V ) and β(V ) as follows:
x∞(V ) = αx(V )
αx(V ) + βx(V ) τx(V ) =
1
αx(V ) + βx(V )
332 Glossary
The α and β curves are defined differently for each of the three particles:
αm(V ) =
0.1 (V + 40)
1 − e−0.1(V +40) βm(V) =4・ e−0.0556(V +65)
αh(V ) = 0.07 ・ e−0.05(V +65) βh(V) =
1
1 + e−0.1(V +35)
αn(V ) = −0.01 (V + 55)
e−0.1(V +55) − 1 βn(V) =0.125e−0.0125V +65
Voltage is given by the parallel conductance model (Fig. 12.2):
C ・ ˙V = gNam3h (V − ENa) + gKn4 (V − EK) + gleak (V − Eleak)
where each g is an individual maximal conductance, each E the potential
for a battery, and C the membrane capacitance.
holding potential: In electrophysiology, a voltage being maintained at
the beginning of an experiment. Voltages other than resting membrane
potential are maintained by injecting positive or negative current.
hologram: An imaging method that records frequency and phase rather
than intensity of light.
homologous evolution: In biology, the production of similar structures
in different creatures through Darwinian evolution. The existence of similar
wings in birds and bats is an example.
Hopfield network: A fully connected Hebbian recurrent network that
shows attractor dynamics and works as a content-addressable memory.
horizontal plane: The horizontal plane for sectioning the body (see
Fig. 3.3).
horseshoe crab: Common name for Limulus.
humunculus: In neuroscience, distorted representation of the body
mapped from receptive fields of neurons on cortical areas. See Fig. 7.1. Also
used to refer to the putative little guy in the brain who watches everything
that goes on.
Huntington disease: An inherited degenerative brain disease involving
the basal ganglia. It is a movement disorder characterized by dance-like
movements.
hydrophilic: In chemistry, a charged or polar compound that will exist
comfortably in water but not in fat. Opposite of hydrophobic.
hydrophobic: In chemistry, a compound that doesn’t mix with water but
will mix (partition) into fat. Same as lipophilic, opposite of hydrophilic.
hyperpolarization: Negative deviation from resting membrane potential.
hypoglycemia: Low blood sugar.
hypothalamus: A collection of brain nuclei responsible for central control
of various hormones and of appetite and thirst.
hypothesis-driven: Ideal that scientific research should generally (perhaps
always) be motivated by the search for confirmation or refutation of
a specific theory.
Glossary 333
I-beam: In civil engineering, a steel structural unit shaped like a capital I
in cross section. Commonly used to build skyscrapers and bridges.
I-f curve: In electrophysiology, the current-frequency curve. The firing rate
of a neuron graphed as a function of amount of injected current.
I/O: In engineering, an abbreviation for input/output. Describes the
output that will be produced by a system in response to a particular input.
IBM: International Business Machines, one of the first major manufacturers
of computers. It dominated the industry for several decades.
if and only if: Mathematics phrase for sufficient and necessary. Symbolized
↔ or “iff.”
illusory conjunction: A perceptual error that can occur with brief visual
presentations of multiple objects. For example, many different colored
shapes are presented briefly, and the visual system mistakenly binds some
shapes with the wrong color.
imaginary number: In mathematics, a number that is a multiple of i,
the square root of −1.
impalement: In electrophysiology, a technique for making measurements
inside of a cell by sticking an electrode through the membrane.
impedance: Opposition to current flow. See Chap. 16, Section 16.6.
impedance matching: In electronics, providing similar resistance at a
junction between circuits so as to optimize flow by preventing reflection of
current back into the source circuit.
implicit Euler integration: A numerical integration technique that uses
the future value of state variables in the approximation. See Chap. 16,
Section 16.5.
in vitro: Study of biological tissue after removal from the organism; means
“in glass.”
in vivo: Study of biological tissue in the intact organism; means “in life.”
inactivation: Closing of ion channels with time.
inertial guidance system: A navigation device that detects motion in
order to calculate position.
inferior olive: A brainstem nucleus involved in movement.
infinite loop: In computer science, a programming error (bug) that results
in the computer endlessly executing the same commands.
infinitesimal: In mathematics, an infinitely small duration or size.
information theory: A mathematical framework for describing the
information content of messages.
inhibitory: Serving to turn off a neuron.
inhibitory postsynaptic potential: A change in membrane voltage due
to synaptic activation that tends to reduce the activity of a postsynaptic
neuron. Usually hyperpolarizing. Abbrev.: IPSP.
initial condition: A starting point for a dynamical system.
inner product: See dot product.
innervate: Form neural connections onto some structure.
334 Glossary
input impedance: In neurophysiology, a measure of the effective resistance
(V/I) encountered by a current injected into a cell. Similarly in
electronics for any circuit.
instantaneous frequency: A minimal frequency measure, made between
two spikes in a spike train or by taking the inverse of a single period of any
oscillatory signal.
instantiation: A physical implementation of a hypothesis or concept.
instruction register: In computer science, a location on the central
processing unit where a command is stored immediately before execution.
instruction set: In computer science, the set of commands built into a
central processing unit.
integral: A basic operation of calculus. Represented by _, a symbolic “S,”
it gives the sum of all the values of a curve — the area under the curve.
The reverse operation of derivative.
integrate-and-fire model: A simple neural model that sums inputs and
produces a spike when the model reaches a predetermined threshold.
integration: In mathematics, a calculus operation that sums up the area
under a curve. In neuroscience, the tendency of neurons to add up synaptic
signals.
interneuron: A neuron with axons that project locally. In the cortex many
of these appear to be inhibitory.
interpreter: A computer program that reads a program line by line and
performs the requested operation. Contrasted with compiler.
interspike interval: The period of time between two action potentials.
intracellular: The inside of biological cells.
intrinsic disease: Diseases that strike directly at the functioning of neural
systems.
invertebrate: An animal without a spine. Can be divided into squishies
(octopus, squid, leech) and crunchies (insect, limulus, lobster).
inward current: In electrophysiology, current that flows through a conductor
from extracellular fluid to cytoplasm. Inward conductance current
produces outward capacitative current, which will depolarize the cell.
inward rectifier: In electrophysiology, a channel that mediates inward
current with membrane depolarization.
ion: A charged atom. The most important ions in the nervous system are
Na+,K+,Cl−,Ca++. The superscript identifies the charge.
ion channel: In neuroscience, an integral membrane protein that allows
the passage of ions down their chemical gradient. Ion channels are often
selective and may be voltage-sensitive.
ipsilateral: Referring to the same side of the body. Opposite of
contralateral.
Ising model: A classical physics model describing the submicroscopic organization
of magnets. The way that the individual magnetic domains
line up with each other has been taken as an analogue of cooperative
organization in neural networks.
Glossary 335
isolable: Something that can be isolated.
isopotential: Having the same electrical potential everywhere. Same as
equipotential.
JAVA: A modern computer language optimized for portability.
joule: Measure of energy.
Kelvin: A temperature scale with its zero at absolute zero. Degree size is
the same as in the Celsius (centigrade) scale.
kinesthesia: The sensation of one’s own movements. This will include joint
position sense and tactile clues.
kinetic energy: The energy associated with motion.
Kirchhoff’s law: Conservation of charge, conservation of current.
kit-and-kaboodle: A silly old phrase meaning “the whole thing bundled
together.”
kludge: In computer science, a programming trick that is very hard
for others to understand, making a program unreliable or impossible to
maintain.
knock-out: In genetics, an animal born without certain proteins due to
the removal of certain genes before fetal development.
Krebs cycle: The series of enzymatic reactions used to extract energy by
oxidizing (burning) the 3-carbon products of glucose breakdown.
labeled-line coding: Coding that uses the identity of a specific wire or
active neuron to identify a message. For example, activity in one set of
neurons indicates pain, while activity in another set signals vibration.
lateral: In anatomy, toward the side. Opposed to medial.
lateral geniculate nucleus: An area of the thalamus that relays
information from the retina. Abbrev.: LGN
lateral inhibition: In neurophysiology and neural networks, inhibition of
flanking units.
learning: Storage of new information in a biological or artificial system.
learning rate: In neural networks, a coefficient that can be adjusted to
speed up or slow down learning.
least mean squares rule: See delta rule. Abbrev.: LMS rule.
length constant: The distance in a passive dendrite over which a steadystate
voltage drops to about 37% (e−1) of its initial value. Also called space
constant.
ligand: A chemical that binds (ligates) to some receptor.
ligand-gated channel: An ion channel that opens and closes depending
on the arrival of some chemical — usually either a neurotransmitter or
second messenger.
limbic system: A set of interconnected brain structures that may be
involved in emotion (see Papez circuit).
limit cycle: See attractor.
Limulus: The short Latin name for the horseshoe crab.
limulus equation: An update rule for modeling lateral inhibition in the
limulus eye: _s = σ (_p +W ・ _s).
336 Glossary
linear: In signals and systems, having a response or output that is directly
proportional to the input. In mathematics, functions without powers higher
than 1.
linear algebra: A form of mathematics that uses vectors and matrices
(see Chap. 16, Section 16.4).
linearly dependent: In linear algebra, denotes a vector can be represented
by a sum of multiples of other vectors.
linearly independent: In linear algebra, denotes a set of vectors for which
no vector can be represented as a sum of multiples of any or all of the other
vectors.
lipid bilayer: The architecture of membranes in living things. See
Chap. 11, Section 11.2.
LISP: A list processing programming language popular in artificial
intelligence.
lissencephaly: Smooth cortex lacking normal gyri and sulci found in some
primates and in some children with abnormal brain development.
local code: See local representation.
local minimum: In attractor dynamics, a low-energy point that is not the
lowest-energy point in the field.
local representation: A representation where each individual unit
represents a specific thing. Opposed to distributed representation.
logarithm: Exponent needed to raise a base to produce a given number;
e.g., log10(100) = 2 because 102 = 100.
logistic function: A function of the form 1/(1+e−x). This function ranges
from 0 to 1 and is sometimes used as a squashing function.
long-term depression: A persistent weakening of synaptic strength.
Abbrev.: LTD.
long-term potentiation: Persistent strengthening of synaptic strength
after paired pre- and postsynaptic activity. Considered a neural analogue
of the Hebb synapse. Abbrev.: LTP.
longitudinal resistance: See axial resistance.
low-pass filter: In signal and systems theory, a system that removes the
high frequencies from a signal and lets only the low frequencies get through
to the output.
LTP: See long-term potentiation.
lumen: In physics, unit of light intensity. In biology, the space at the center
of a tube, e.g., the intestine.
Mach bands: A visual illusion seen at the edge of a contrast boundary.
The bright side of the edge looks extra bright and the dark side of the edge
looks extra dark.
machine language: In computer science, the set of binary operations that
are understood and executed by the central processing unit (CPU).
macroscopic: Big enough to be seen with the naked eye.
mantissa: The significant digits in scientific notation, e.g., 6.022 in 6.022 ・ 1023.
Glossary 337
map: In neuroscience, an area of the nervous system that represents some
aspect of sensory or motor space in a topographic manner. In mathematics,
transformation of one set of variables to another.
mapping: Mathematically the transformation of one representation into
another. For example, a matrix multiplication can be used to map vectors
in one space to vectors in another.
Markov model: A modeling technique used in describing active proteins,
such as ion channels or enzymes. A finite number of conformational states
of the protein are defined. Transition maps and rates among the states
describe how the model passes from one configuration to another. In the
case of an ion channel, some configurations will be open and others closed
to ion conductance.
matrix: Array of numbers arranged in rows and columns. See Chap. 16,
Section 16.4.
Maxwell equations: The four differential equations that describe the
interaction between electricity and magnetism.
McCulloch-Pitts model: An early artificial neural network model using
summation of excitation with veto inhibition.
medial: In anatomy, toward the middle. Opposed to lateral.
medulla: The lower part of the brainstem connecting to the spinal cord.
membrane potential: Voltage measured at the inside of a membrane
compared to ground outside.
memory: In computer science, usually refers to random-access memory
(main memory). In general, any form of information or signal storage.
mesencephalon: The upper part of the brainstem, connecting to the
diencephalon.
metabolism: Biological process of extracting and utilizing energy.
metabolite: A substrate or product of metabolism.
mhos: Siemens.
microelectrode: Tiny electrodes used to measure electrical activity in or
around neurons or other cells.
microglia: A type of glial cell involved in protecting the central nervous
system from infection.
micron: A micrometer: 10−6 meters.
micropipette: A small glass tube filled with conducting solution that can
be used as a microelectrode.
microtubule: A cell support protein found in cells.
mirror attractors: In memory networks, additional attractors that form
during learning that are the negatives of the learned patterns.
mitochondria: In cell biology, the organelles responsible for energy production.
Mitochondria have their own genome, suggesting that they were
originally free-living organisms before banding together with others to form
the eukaryotic cell.
molar: Measure of concentration in solution equal to 1 mole of solute per
liter.
338 Glossary
mole: Standard unit of material quantity equal to approximately 6.02・1023
atoms or molecules (the Avogadro number).
monotonic: In mathematics, property of a function that continues
ever upward (monotonically increasing) or ever downward (monotonically
decreasing).
monovalent: Describing an ion with a single excess electron (e.g., Cl−) or
proton (e.g., Na+).
mossy fibers: A descriptive term for an axonal pathway. There are pathways
by this name in the hippocampus (from dentate gyrus to CA3) and
cerebellum (from spinal cord to granule cells).
motor: In neuroscience, descriptor for systems involved in producing
movement of the organism.
motor cortex: Cortical areas that control movement.
MRI: Magnetic resonance imaging. A computed tomography technique
that produces an image using nuclear magnetic resonance (NMR), the interaction
of atoms with radio waves and magnets. This was called NMR
until a savvy marketer realized that it would be difficult to sell a medical
procedure that used the word “nuclear.”
multiple sclerosis: A neurological disease involving autoimmune attack
on myelin in the central nervous system, which produces axonal conduction
failures and ephaptic transmission in tracts.
multiplex: In data transmission, sending more than one signal or type of
signal on a transmission line at the same time.
multiunit recording: In electrophysiology, extracellular recording of action
potentials from multiple neurons simultaneously. Contrasted with field
potential or single-unit recording.
muscarinic: Adjective referring to a set of acetylcholine receptors in the
central nervous system.
myelin: An insulating material wrapped around axons to permit faster
conduction.
nadir: A low point. Opposite of zenith.
natural log: Logarithm using e ∼ 2.18281828 . . . as the base. Symbol: ln;
e.g., ln(10) ∼ 2.303 because e2.303 ∼ 10.
negative feedback: A signal communicated back to a prior system
component that decreases the output or activity of that component.
neglect syndrome: A disorder of perception seen after damage to the
nondominant hemisphere that involves a failure to perceive contralateral
space, sometimes including the person’s own body.
neocortex: Major area of cortex in higher mammals. Contrasted with
more primitive archicortex.
Nernst potential: The membrane potential associated with an ion at electrochemical
equilibrium accross the membrane. It can be calculated using
the Nernst equation, which relates the membrane potential (battery) associated
with an ion to its concentrations inside and outside. E = RT
zF ln( [A]out
[A]in
)
Glossary 339
where R is the gas constant, T the temperature, F the faraday, and z is
the valence of ion A.
nerve: A bundle of axons in the peripheral nervous system. Nerves run to
muscles and from sensory organs in the periphery.
neuraxis: In anatomy, the curve defined by the extent of the central
nervous system.
neuroanatomy: Study of brain and other nervous tissue using observation
of fixed specimens.
neuromodulator: A neurotransmitter with a relatively prolonged effect.
neuromuscular junction: The synaptic connection between nerve
terminal and muscle.
neuron: The principal information processing cells of the nervous system.
neuron doctrine: Cajal’s theory that neurons are separate cells. Contrasted
with Golgi’s reticular hypothesis suggesting that the neural cells
formed one large syncytium.
neuropeptide: Short amino acid sequence that serves as a neurotransmitter.
neuropharmacology: Study of how drugs alter neural function.
neurophysiology: Study of brain and other nervous tissue by observing
change in some measurable attribute such as voltage or the concentration
of a chemical.
neuropil: Areas of the nervous system with few cell bodies, dominated by
dendrites, axons, and synapses.
neurotransmitter: A chemical released across a chemical synapse to
communicate from one neuron to another.
nicotinic: Adjective referring to a set of acetylcholine receptors in the
central nervous system and at the neuromuscular junction.
NMDA receptor: An excitatory synaptic receptor that binds glutamate,
provides a long time-course excitatory postsynaptic potential. Activation
of NMDA underlies some forms of long-term potentiation (LTP). Abbrev.
for N-methyl-D-aspartate, an artificial agonist at the receptor.
NO: Nitric oxide. A gaseous neurotransmitter.
node of Ranvier: A gap in the myelin sheath where action potentials can
be generated.
noise: In signals and systems theory, random perturbations that interfere
with transmission of a signal.
nondominant hemisphere: The cerebral hemisphere that doesn’t control
language. The right hemisphere in most people, it is believed to play a role
in spatial tasks.
nonlinear: Having a response that is not directly proportional to the input.
normalization: In signal processing, reducing variation among signals of
differing amplitudes in order to handle them similarly.
NOT: A Boolean operator. Symbol: ∼. See Chap. 16, Section 16.3.
nucleus: In neuroanatomy, a collection of cell bodies in the central nervous
system. In cell biology, an organelle that holds the genome.
340 Glossary
nullcline: In dynamical systems, the set of points on the phase plane where
trajectories will either go straight up-down or straight left-right.
numerator: The top of a fraction, e.g., the 2 in 2
3 .
numerical integration: Approximating the solution of an integral
equation using numbers instead of infinitesimals.
object-oriented language: In computer science, a computer language
that defines types of objects to be manipulated in consistent ways. For
example, drawing programs use object-oriented commands such as rotate,
which will operate on many different graphical objects.
occipital cortex: The area of cortex at the back of the brain that is
involved in visual processing.
occipital lobe: Posterior area of cortex housing visual cortex.
octal: Base 8.
Ohm’s law: Voltage equals current times resistance (V = IR). Written in
terms of conductance: current equals conductance times voltage (I = gV ).
olfaction: Sense of smell.
oligodendroglia: A type of glial cell that provides axonal insulation
by wrapping myelin-filled processes around an axon. Also called
oligodendrocytes.
ommatidium: A single unit-eye of a compound eye.
one’s complement: In computer science, a method of doing subtraction,
see Chap. 4, Section 4.6.
operating system: The lowest-level software on a computer, used to integrate
control of disk drives, keyboard, and monitor with the CPU and
peripherals.
operator: In mathematics, a function. In computer science, a symbol that
takes one or more arguments and produces an output, e.g., + is an arithmetic
operator that takes two arguments; rotate is a common operator for
a graphical language that takes one argument.
optic chiasm: In neuroanatomy, the location where the optic nerves come
together and partially cross, forming the optic tracts.
optic nerve: Bundle of axons running from the retina to the optic chiasm.
The optic nerve is not really a nerve because the retina is actually part of
the central nervous system and not a peripheral receptor.
optic tract: Bundle of axons running from the optic chiasm to the lateral
geniculate nucleus.
OR: A Boolean operator. Symbol: ∨. See Chap. 16, Section 16.3.
orbit: In dynamical systems, a trajectory whereby one object (or state
variable) follows a closed path back to its original location, generally around
another object in the system.
ordinary differential equation: A differential equation with all derivatives
with respect to a single variable (e.g., time). Contrasted with partial
differential equation. Abbrev.: ODE.
organelle: In cell biology, cell organs including nucleus, Golgi apparatus,
mitochondria, etc.
Glossary 341
orthodromic: In electrophysiology, signal flow in the normal direction of
signaling. Contrasted with antidromic.
orthogonal: Lying or intersecting at right angles. In linear algebra, two
vectors whose dot product is zero.
oscillation: A signal or process that shows repeating activity.
outer product: A vector operation that produces a matrix out of a row
vector times a column vector. See Chap. 16, Section 16.4.
outward current: In electrophysiology, current that flows through a conductor
from cytoplasm to extracellular fluid. Outward conductance current
produces inward capacitative current, which will hyperpolarize.
outward rectifier: In electrophysiology, a channel that mediates outward
current with membrane depolarization.
overflow: In computer science, loss of bits that exceed word size after
performing an arithmetic operation.
overgeneralize: In learning theory, a tendency to apply learned patterns
too broadly and thereby make errors based on the use of a rule in a place
where there is an exception, e.g., using “haved” instead of “had.”
pair-wise: Combinations of n objects taken two at a time. Usually refers
to performing some operation on each possible couple of a list of objects;
e.g., pair-wise sums of 8,7,2,1 are 8+7, 8+2, 8+1, 7+2, 7+1, 2+1.
Papez circuit: The set of interconnected structures of the limbic system
believed (by Papez) to contribute to emotional responses. Recent data
suggest that this circuitry is more important for memory than for emotion.
parabola: A curve with a square of the independent variable: f(x) =
ax2 + b.
paradigm shift: Philosophy of science concept of Thomas Kuhn that
suggests that science does not progress by small incremental steps but
instead enjoys periodic revolutions where old theories are overthrown and
new theories are erected in their place.
Parallel Data Processor: Digital Equipment Corporation’s name for a
line of its computers. Abbrev.: PDP. Includes the PDP-8 and PDP-11.
parallel distributed processing: In artificial neural network theory, the
use of multiple simple units working at the same time (in parallel) with
information that is spread out (distributed) over the units. Abbrev.: PDP.
parallel fibers: Axon projection from granule cells to Purkinje cells in
cerebellum. See Fig. 15.3.
parallel processing: Calculating by using multiple data-flow paths
simultaneously. Opposed to serial processing.
parameter: In dynamical systems, a fixed value that defines the system.
Contrasted with state variables.
parameter variation: Altering the parameters that define a dynamical
systems. Typically done in order to make sure that the system will behave
similarly despite minor changes.
parameterization: Fitting data with functions and specific values.
342 Glossary
parasitic capacitance: Effect that reduces current that is being transmitted
using an oscillatory signal through capacitative coupling to
ground.
parietal lobe: Large middle region of cortex between occipital and frontal
lobes.
parity: In data transmission, descriptor as to whether previous set of bits
had even or odd number of ones. A parity bit is used to check that no bits
were reversed during transmission.
Parkinson disease: A degenerative brain disease involving loss of
dopamine-producing cells in the substantia nigra. Produces a movement
disorder causing tremor and reduced movement.
pars: Latin word for “part”; e.g., substantia nigra, pars compacta means
“the compact [meaning tightly packed cells] part of the black stuff.”
parse: In computer science, the task of interpreting lines of a program to
separate out arguments, operators, comments, subroutine names, etc. In
general, any effort to interpret a signal or language by breaking it up into
parts.
partial differential equation: A differential equation where different
derivatives are taken with respect to different variables (e.g., one derivative
with respect to time and another with respect to distance). Abbrev.: PDE.
particles: In Hodgkin-Huxley theory, the name given to the state variables
determining ion channel conductance in the Hodgkin-Huxley equations.
These were hypothesized to be little particles that blocked ion flow.
Pascal: A computer language.
passive channels: Ion channels that maintain same permeability (conductance)
at all times. Contrasted with active channels.
passive membrane: Membrane lacking voltage-sensitive conductances.
patch clamp: In electrophysiology, a technique for making direct measurements
of membrane by attaching an electrode tightly to the membrane
and pulling a small circle of membrane off of the cell on the electrode.
pattern completion: Ability of artificial neural network or natural
systems to fill in missing pieces of an incomplete input or memory.
PDP: Abbrev. for both parallel distributed processing (concept in neural
networks) and Parallel Data Processor (a model of computer)
peptide: A short sequence of amino acids.
perceptron: A unit of the single-layer linear neural networks developed
in the early 1960s.
perforant path: A pathway that projects from entorhinal cortex to
hippocampus.
perikaryon: Synonym for cell soma.
period: The duration of a single cycle of a repeating wave.
peripheral nervous system: The nervous system that lies outside of the
brain and spine. Abbrev.: PNS.
peripherals: In computer science, additional pieces stuck on to the
computer such as the monitor and keyboard.
Glossary 343
Perl: A computer utility language.
PET: See positron emission tomography.
phase: In signals and systems theory, the delay of an oscillatory signal
compared to a fixed signal of the same frequency. Measured in degrees or
radians.
phase plane: A two-dimensional graph illustrating the dynamics of a system
by mapping one state variable against another as they change in time.
This plane is a two-dimensional section through state space.
phase space: n-dimensional space where each axis represents one of the
n state variables of an n-dimensional dynamical system. The dynamics of
the system could be fully mapped as a curve in this space.
phase transition: In chemistry and physics, change from one state of
matter to another: e.g., liquid to gas. In neural network, analogous state
transition involving major changes in activity.
phasic: In signals and systems theory, a transient response to a signal.
phoneme: The minimum length sound unit that carries information in a
natural language. See grapheme.
phospholipid: Biological soap that is the main constituent of lipid bilayers.
They form with a charged phosphate (phosphorus and oxygen) head
and a long-chain hydrocarbon (the fatty part).
photoreceptor: An electronic device or biological cell that can detect light
or other electromagnetic radiation (photons).
phrenology: A discredited field of study that used skull shape to determine
mental abilities.
physiology: The study of functional interactions in biology. Contrasted
with anatomy.
pi: π: ratio of circumference to diameter of a circle.
piece-wise linear function: A function made of joined line segments.
pixel: Picture element. A single spot in a bit-mapped picture.
plasticity: Changeability. Used to refer to synapses or neural assemblies
that are altered by activity or environmental events.
point attractor: In dynamical system theory, a final steady state that is
a common target for some set of initial conditions.
point neuron: A model neuron without physical extent — represented as
a single isopotential compartment.
pointer: In computer science, a numerical address that indicates the
location of a data structure.
Poisson process: A probability model for events that occur rarely without
reference to past activity (memoryless). Can be used to describe neural
spiking as well as emission of radioactive particles and arrival of customers
at a service counter.
pons: The middle part of the brainstem.
pore: Another name for ion channel.
344 Glossary
port: To move or translate a program or algorithm from one type of machine
to another. For example, a new word processor will typically be ported
from one machine architecture or operating system to another.
positive feedback: A signal communicated back to a prior system
component that increases the output or activity of that component.
positive true: In computer science, the use of high voltage on the
transistor to represent binary 1 or true.
positron emission tomography: A physiological imaging technique that
localizes radioactive substances that release positrons. Abbrev.: PET.
posterior: Toward the rear. Depending on the position in the central
nervous system, this corresponds to either caudal or dorsal.
postsynaptic: The receiving side of a synapse, more generally used to
denote the follower side of any weight connection in a neural network.
postsynaptic potential: Change in voltage in the postsynaptic membrane
caused by synaptic activation.
potential: Electrical potential, voltage.
potential energy: Energy stored in a form from which it can later be
released, e.g., height in a gravitation field, a charge in an electric field.
presynaptic: The transmitting side of a synapse, more generally used to
denote the side of a weight connection that provides the signal.
primary sensory cortex: Area of cortex that receives sensory information
directly from thalamus or other noncortical area.
primary visual cortex: The primary sensory area for vision.
primates: Monkeys, apes, and people.
procedural memory: Human memory for how to perform a procedure.
This includes motor tasks and memories for rule-based activities such as
games.
program: In computer science, a sequence of steps written in a computer
language that is self-contained and produces some desired data or other
output.
program counter: In computer architecture, central processing unit
storage that used to provide a pointer to a word in memory to be executed.
projection cell: A neuron with axons that leave the area. Pyramidal cells
are the major projection cells of cortex.
projective field: Locations or movements that are produced by activity
of a particular motor neuron.
prokaryote: In biology, an organism that has a cells with no nucleus, such
as bacteria and some other single-celled organisms.
proprioception: The sensation of movement in muscles and joints.
prosopagnosia: Neurological condition in which faces of individual people
can’t be recognized
protein: Chain of amino acids that provides the building blocks of living
creatures. Proteins can be structural (e.g., hair) or may be enzymes
that make things (e.g., beer). Protein channels provide routes for ions and
chemicals to enter or leave a cell.
Glossary 345
proximal: Closer to the center, opposite of distal.
pseudo-code: In computer science, a style of writing out an algorithm in
the general form of a program without worrying about conforming to the
precise syntax required by a specific programming language.
psychophysics: Research techniques that use precisely defined stimuli to
quantify responses or behavior.
punch cards: An old technique of data storage that used holes in
cardboard cards as input to a computer.
Purkinje cell: One of the large projecting inhibitory cells of the
cerebellum.
putamen: An input area of the basal ganglia. A part of the striatum.
pyramidal cell: A characteristic neuron of the cortex that has a cell body
shaped like a pyramid.
pyramidal tract: Corticospinal tract.
Q10: A measure of the increase in rate of an active protein (e.g., an enzyme,
voltage-sensitive channel or pump) to temperature change. The rate
increase factor is equal to QΔtemperature/10
10 .
qualia: Term used by philosophers to connote the internal, subjective sense
of a sensation or experience.
quantum mechanics: Description of the motion of subatomic particles.
Distinct from the classical mechanics of dynamical systems.
queue: In computer science, a common data structure that stores and
retrieves multiple items in a first-in, first-out (FIFO) sequence
quod erat demonstrandum: In mathematics, a Latin phrase used to
denote the end of a proof. It means “that which was to have been proved.”
Abbrev.: QED.
radian: Unit of circular measure equal to ∼ 57.2958 degrees. There are
2 ・ π radians in a circle.
radix point: The period “.” used in number systems to indicate where the
whole part of a number ends and the fractional part begins. This is called
the decimal point in the base 10 system.
random-access memory: The main (primary) memory of a modern computer.
Abbrev.: RAM. Can be subdivided into sRAM (static RAM), which
cannot be altered by the computer and dRAM (dynamic RAM), which can
be altered. Software and data are stored in dRAM before execution.
range: In mathematics, the set of possible outputs of a function. For example,
the typical neural network squashing function has a range of 0 to
1. See domain.
rate coding: Frequency coding.
RC circuit: Resistor-capacitor circuit. Such circuits can be used to provide
different high-pass and low-pass signal filtering. The RC circuit is also a
model of the electrical properties of a passive cell membrane.
realistic neural network: Network that attempts to accurately portray
neurons and neural dynamics.
346 Glossary
receptive field: Locations or stimulation range that activates a particular
neuron. For touch this could be an area of skin; for vision this would a part
of the visual field or for hearing a range of frequencies.
receptor: In neuroscience, a molecule to which a chemical (ligand) can
bind.
reciprocal: Giving and taking. In neuroscience, describes to-and-fro connectivity
between areas or individual neurons: area A (or cell a) projects
to area B (or cell b), which then projects back to A (a).
rectify: In electronics, passing current in one direction only. The diode is
the main rectifying component.
recurrent collaterals: Axon collaterals that come back and synapse either
on the same neuron or on the same type of neuron.
reflex: A stereotyped closed stimulus-response loop.
register: In computer science, A specialized memory location residing on
the central processing unit.
reinforcement learning: An artificial neural network training technique
that utilizes a general reward signal rather than a specific target for
instruction.
relative refractory period: A period during which a second neuron
stimulation must be increased in order to generate another action potential.
relaxation: In physics, term referring to gradual reduction in energy of
a system. Used in artificial neural network theory to refer to a learning
process that reduces an energy function. See annealing.
relay cell: See thalamocortical cell.
relay nucleus: One of the sensory nuclei of the thalamus that transmits
information from the sensory periphery to sensory cortex.
repellor: In dynamical system theory, a point in state space that the
system moves away from. Also called an unstable equilibrium.
repolarization: In neurophysiology, reversal of depolarization.
resistor: A substance that impedes the flow of electricity linearly with
voltage (see Ohm’s law). Symbol R, units Ohms (Ω). Resistance is the
inverse of conductance: R = 1/g.
resting membrane potential: Voltage of neuron membrane in the
absence of stimulation. Usually somewhere near −70 mV. Abbrev.: RMP.
reticular activating system: Area of brainstem that maintains arousal.
Damage to this area causes coma.
reticular cell: Thin spindly cells found in various areas of the brain. The
reticular cells of thalamus are particularly well studied.
retina: The network of neurons at the back of the eye that detects light
and does early visual processing. The retina is part of the central nervous
system.
retinal slip: Lag of the eye during pursuit.
retinotopic: In neuroscience, a mapping onto an area of brain that preserves
an orderly representation of the retinal surface. Two points close
Glossary 347
to each other in retina are also represented by nearby cells in the brain
structure in question.
retrograde transmission: In neurophysiology, transmission of a chemical
signal backward across a synapse.
reversal potential: The electrical potential associated with an ion channel,
based on the type of ion or mix of ions passed by the channel. So
called because current through such a channel will reverse direction when
the membrane holding potential passes the reversal potential.
RISC: Reduced instruction set chip. In computer architecture, a central
processing unit design that executes only simple instructions, requiring
complex instructions to be coded in software. With growing CPU
complexity, the contrast with CISC is now historical.
RNA: Ribonucleic acid. The molecule that transmits information from the
gene (DNA) to produce proteins.
rod: A type of retinal photoreceptor that detects low levels of light.
ROM: Read-only memory. A form of computer memory that is written on
manufacture and can only be read thereafter.
Rosetta stone: A stone fragment containing three different scripts (Greek,
demotic, and hieroglyphics) that was the breakthrough document for
deciphering ancient Egyptian hieroglyphics.
rostral: The head end of the body. Opposite of caudal.
saccade: A rapid eye movement used to bring the eyes around to the object
of attention.
sagittal plane: A side-facing vertical plane for sectioning the body (see
Fig. 3.3).
salt: Ionically bonded crystalline compound produced by combining an
acid and a base. Table salt is NaCl.
saltatory conduction: The jumping of an action potential from one node
of Ranvier to the next in a myelinated axon.
sampling rate: In signals and systems theory, the frequency or density of
signal measurement. Interactions between frequency of sampling and the
frequency of the signal can lead to illusions of recurring patterns.
scalar: A single number. Contrasted with vector or matrix.
scale model: A reduced-size model that preserves relative dimensions.
scantron form: Form used for data entry that can be scanned by a computer.
Typically these have little circles that must be blackened with a
pencil to indicate a choice on an exam or questionnaire.
Schaeffer collaterals: Pathway in hippocampus from CA3 to CA1.
schizophrenia: A psychiatric disease characterized by disconnection from
reality, frequently with delusions and auditory hallucinations.
scientific notation: Standard for representation of large and small numbers
in science by using a mantissa and a base-10 exponent, e.g., 10,000
would be represented as 1 ・ 104. See Chap. 16, Section 16.2.
second derivative: The derivative of a derivative. For example, velocity
is the first derivative of position with respect to time. The first derivative
348 Glossary
of velocity with respect to time is acceleration. Acceleration is the second
derivative of position with respect to time.
second messenger: An intracellular chemical that transmits a message
from a postsynaptic release location to other locations in the cell.
segmentation error: In computer science, an error due to the attempt to
read or write to computer memory that is not currently accessible.
semantic memory: Human memory for general facts.
semicircular canal: An organ of inertial sensation in the ear. There are
three semicircular canals placed orthogonally in order to detect rotations
in any of the three dimensions of space.
sensitivity to initial conditions: In dynamical systems, an attribute of
chaotic systems. If a chaotic system is started at a slightly different point
(the initial condition), its subsequent behavior will be completely different.
sensorimotor: Denotes the coordinating of incoming sensory signals
with movement. This includes the modification of movement based on
kinesthetic feedback.
sensory channel: Psychophysics term for neural pathways communicating
a particular type of sensation.
serotonin: A neurotransmitter. Prozac and similar drugs work by blocking
serotonin reuptake, thereby increasing amount of transmitter at synapses.
set bit: Binary bit value corresponding to 1 or True.
shunting inhibition: Inhibition caused by increase in conductance
without hyperpolarization.
SI units: Abbrev. for the Syst`eme International d’Unit′es (International
System of Units). This is the standard system of scientific units.
siemens: The unit of conductance. Equal to inverse ohms. Abbrev.: S. Also
called mho.
sigmoid function: An equation giving an S-shaped curve (e.g., the logistic
function). Utilized as a squashing function in neural networks.
sign magnitude: In computer science, a method of representing negative
numbers by using a single bit as a code to indicate negative. See Chap. 4,
Section 4.6.
signal integration: In signals and systems theory, the adding up of signals
by a system.
signal transduction: Conversion of a physical signal in the environment
into a chemical or electrical form for neural processing. The first step in a
sensory system.
signals and systems: A field that studies generation, transmission, and
processing of time-dependent signals.
silent synapses: Anatomically defined synapses without physiological
effect.
silicon: A chemical element used in electronic circuits and devices. Used
as a short-hand to refer to computers and breast implants. Abbrev.: Si.
simulation: A numerical imitation of the behavior of a physical system on
a computer.
Glossary 349
single-unit recording: In electrophysiology, extracellular recording of
action potentials from one neuron. Contrasted with field potential or
multiunit recording.
sink: In electronics, a destination for current. Opposed to source. Ground
is the ultimate sink.
slow potential theory: Transduction of firing rate through temporal
summation of long-lasting postsynaptic potentials. See Chap. 11,
Section 11.7.
soap: A compound that has a fatty part that sticks to fat and a polar part
that sticks to water.
software: In computer science, programs. Contrasted with hardware.
solute: The dissolved material in a solution.
solution: In mathematics, an answer. In biology, a liquid with something
dissolved in it.
solvent: The liquid that a solute is dissolved in. Biologically, usually water.
soma: Large central area of a neuron from which extend the dendrites and
axon.
somatosensory: Relating to the sense of touch.
source: In electronics, a location from which current arises. Opposed to
sink.
space constant: The distance along a passive dendrite for a fixed voltage
to decline by 1
e ( 37%) Symbol: λ.
space-time trade-off: In computer science, the need to balance a program’s
requirement for computer memory (space) with the amount of time
the program will take to execute. In general, you can write a program that
will run faster if you devote more space to it.
spatial summation: Adding up of synaptic potentials arriving at different
places on a neuron.
spike: An action potential.
spike frequency: The frequency at which spikes occur. Usually between
5 and 100 Hz in most cell types.
spike initiation zone: A specialized area of soma or axon where a particularly
high concentrations of sodium channels allows initial generation
of an action potential.
spinal cord: The part of the central nervous system that lies in the spine.
spine: In neuroanatomy, a small thorn-like extension off a dendrite that is
often the location of an excitatory synapse. Also, the backbone of vertebrate
animals.
spontaneous activity: Resting firing of a neuron when not being
stimulated.
spurious attractors: In memory networks, additional attractors that
form during learning but do not correspond to any learned pattern.
squashing function: In neural networks, function that converts a broad
domain of input values into a limited range of output values. Used to limit
the range of state values for an artificial neural network unit.
350 Glossary
stack: In computer science, a common data structure that stores and
retrieves multiple items in a first-in, last-out (FILO) sequence.
state: The condition or value of a system or system component at a
particular time.
state space: Phase space representing progression of unit state vector in
a network.
state variable: In a dynamical system, an independent variable that
changes with time.
state vector: A vector giving the current value of all states of all units in
a neural network.
step function: A sharp thresholding function with discontinuity at 0.
Same as Heaviside function.
step size: In numerical integration, the Δt used to advance the calculation.
stimulus: Something that can activate a sensory response.
stochastic: In mathematics, a system whose behavior is governed by
random events. Contrasted with deterministic.
stomatogastric ganglion: A small (∼ 30 neuron) ganglion that runs parts
of the digestive and circulatory system in the lobster.
strange attractor: The type of attractor that is seen in chaotic dynamics.
A chaotic system never revisits the same spot in state space. A strange
attractor describes a region of state space that the trajectory will repeatedly
pass through.
stria of Gennari: White stripe in primary visual cortex due to the many
axons projecting from thalamus.
striate cortex: Another name for primary visual or occipital cortex, so
called because it has a white-matter stripe (stria of Gennari).
striatum: The caudate and putamen, the input area of the basal ganglia.
stroke: Death of brain tissue due to loss of blood flow. Also called
cerebrovascular accident (CVA).
subroutine: In computer science, a brief sequence of steps written in a
computer language that is utilized by other routines to perform a dataprocessing
task.
substantia nigra: An area of brainstem with connections to basal ganglia.
Loss of dopamine cells in this area is the cause of Parkinson disease. The
area is black due to accumulation of the melatonin formed from dopamine
breakdown.
subthreshold: In electrophysiology, not reaching the threshold for firing
an action potential.
sufficient and necessary: Science buzz phrase indicating that in a theory
showing that cause A is important for producing phenomenon B, A is
all you need (sufficient) to get B and that without A you won’t get B
(necessary). Synonymous with “if and only if.”
sulcus: The cerebral hemispheres are folded; a sulcus is one of the valleys.
summation: In mathematics, adding up, symbolized by Σ. In neuroscience,
the ability of neurons to combine synaptic inputs.
Glossary 351
summed-square error: A common error function used for matching a
single network output to a target: error = 1
2Σi(outi − ti)2. Using an error
vector _e = o_ut −_t, we can express this as 1
2 (_e ・ _e).
superior olive: A brainstem nucleus involved in hearing.
supervised learning: In artificial neural network learning theory, algorithms
to teach a neural network by presenting targets that are to be
learned.
suprachiasmatic nucleus: A nucleus of hypothalamus that contains cells
mediating the “master clock” of the circadian rhythm.
sylvian fissure: The large front-pointing indentation separating the
temporal from frontal and parietal lobes.
symbol processing: An artificial intelligence technique that manipulates
discrete descriptors of information in a particular knowledge domain.
symbol table: In computer science, a table that lists the symbols in a computer
program (e.g., names of variables, subroutines, and data structures)
alongside the physical address in memory where these will be stored.
synapse: Point at which information is transmitted from one neuron to
another. A connection between units in a neural network.
synaptic cleft: The region separating the presynaptic and postsynaptic
membranes at a chemical synapse.
synaptic transmission: The sending of information across the synapse,
usually by means of release and reception of a neurotransmitter.
synaptic triad: A synaptic complex in the thalamus involving co-localized
excitatory inputs from an afferent onto dendrites of both a thalamocortical
cell and a thalamic interneuron. A neighboring inhibitory dendrodendritic
synapse from the interneuron to the thalamocortical cell is the third part
of the triad.
synaptic vesicle: Organelles found in the presynaptic cell that store and
release neurotransmitter.
synchronous updating: In artificial neural network theory, the setting
of states for all units to their new values at the same time.
syncytium: A tissue of continuously connected cells that share a common
cytoplasm.
synesthesia: A benign neuropsychological condition that results in
crosstalk between sensory modalities, e.g., a synesthetic might experience
a particular smell when seeing a certain shape or experience a color when
hearing a particular melody.
syntax: In computer science, the detailed requirements for the order of
commands and arguments in a computer language.
systems neuroscience: Physiological investigation pertaining to perception
or behavior of an animal or person.
target: In artificial neural network theory, an explicitly provided value
used as a learning goal for a neural network.
352 Glossary
taxonomy: A set of descriptions for different manifestations of a central
phenomenon. In biology, the term is used for the descriptions of species.
teacher forcing: Clamping a dynamical neural network to sequences of
states in order to help it learn specific state progressions.
teleology: A point of view that assumes that natural objects are designed
for a purpose. Although most scientists believe that random evolution and
natural selection explain biological features, it is often easier to describe
an organ or organism as if it had been designed. For example, the lens is
placed where it can focus light on the retina.
temporal integration: In signals and systems theory, the adding up
of signals over time. In electrophysiology, the adding up of postsynaptic
potentials arriving over time at a synapse.
temporal lobe: A lobe of the brain lying below the sylvian fissure.
temporal summation: Adding together of postsynaptic potentials
arriving one after another at a synapse.
terminal arbor: A bunch of axon branches at the end of an axon.
terra incognita: Latin for “unknown country.”
tetrodotoxin: A poison from the puffer fish that blocks sodium channels.
Allegedly the poison used by the voodoos for creating zombies.
thalamic reticular nucleus: Collections of broad flat cells that wrap
around the principal thalamic nuclei. Also referenced by the Latin name:
nucleus reticularis thalami. Abbrevs.: TRN and nRT.
thalamocortical cell: The principal cells of thalamus that project to
cortex. Also called relay cells.
thalamus: Central gray matter nucleus that forms entry point for sensory
systems to the brain. It is also involved in sleep.
theta waves: Slow waves (4 to 7 Hz) in the electroencephalogram.
thought insertion: Placing thoughts into the head of a human or animal.
This is an occasional delusional complaint in schizophrenia.
three-body problem: In dynamical systems, the dynamics of three heavenly
bodies with mutual gravitational attraction (e.g., sun, earth, moon).
This problem cannot generally be solved analytically. Now that these systems
can be solved numerically, it has been shown that some parameter
choices can lead to chaotic dynamics.
threshold: In neuroscience, the voltage needed to generate an action potential.
In general, an activation level that causes a unit to go from a
quiescent to an excited state.
threshold logic unit: McCulloch-Pitts neuron. Abbrev.: TLU.
time constant: In a first-order differential equation, the time it takes for
a state variable to change by 1 − 1
e ∼ 63%.
tissue: In biology, a collection of connected cells in an organism.
tonic: A sustained response to a continuing stimulus, opposite of phasic.
torus: A geometrical shape that looks like a donut. From a point on the
surface of a torus you can follow a circular path all the way around in either
of two directions.
Glossary 353
tract: A bundle of axons that run together in the central nervous system.
Comparable to a nerve in peripheral nervous system.
tractable: Practicable, possible, doable.
trajectory: In mechanics, the movement of an object in a gravitational
field. More generally, the progress of a state variable in a dynamical system.
transcendental functions: Nonalgebraic functions, including trigonometric
functions such as sine and cosine.
transfer function: See squashing function.
transistor: An active electronic device that functions as a controllable
switch.
transistor-transistor logic: In computer science, the original technique
used to organize transistors to perform logical operations. This has since
been superseded by the organization of many transistors using VLSI.
Abbrev.: TTL.
transmembrane protein: Protein that goes through the lipid bilayer to
face both extracellular and intracellular space. Such proteins can be pumps,
ion channels, receptors, etc.
transpose: In linear algebra, an operation that flips a matrix so as to make
the columns rows and rows columns.
tricarboxylic acid cycle: Krebs cycle.
trisynaptic pathway: Pathway in hippocampus from entorhinal cortex
to dentate gyrus to CA3 to CA1.
truth table: A list of outputs for a Boolean operator.
Turing test: Test of the ability of an artificial intelligence program to be
so damn intelligent that it could fool people into believing they’re talking
to a person.
two’s complement: In computer science, a method of doing subtraction,
see Chap. 4, Section 4.6.
two-point discrimination: The ability to distinguish two pins placed
close together on the skin.
ultrastructural: In anatomy, refers to structures that cannot be seen with
light microscopy but can be seen with electron microscopy.
unary operator: In computer science, an operator that takes one argument,
For example, “−” is a unary operator that turns a number into its
negative.
unit: A single element in a neural network.
unit analysis: See dimensional analysis.
Unix: An operating system popular for engineering and scientific applications.
unsupervised learning: In artificial neural network learning theory, algorithms
that allow a neural network to learn patterns by exposure to data
without the defining of specific tasks or targets.
update rule: In artificial neural network theory, an equation for assigning
the state of a unit based on its inputs. In numerical integration, the
assignments for state variables after a time step based on their prior values.
354 Glossary
upregulate: In biology, increasing the amount or activity of an enzyme,
receptor or other active protein. This is seen in the case of receptors that
are deprived of their ligand by drugs that block the receptor or prevent
transmitter release.
V1: Primary visual cortex.
vector: A one-dimensional array of numbers.
ventral: The belly side of the body. For the central nervous system this
would be anterior in the spinal cord and inferior (down) in the brain.
ventricles: Large cavities in the brain containing cerebrospinal fluid. Also
the large chambers of the heart.
vesicle: In cell biology, an organelle for chemical storage. See synaptic
vesicle.
vestibulo-ocular reflex: A reflex that stabilizes the eyes in the head when
the head moves to maintain vision. Abbrev.: VOR.
veto inhibition: Inhibition that completely turns off the neuron regardless
of amount of excitation.
virtual experiment: Process of exploring a simulation as if it were a
natural object. The complexity of many simulations makes it impossible to
understand them without running such experiments.
virus: In biology, cell parasites that contain genetic material (either DNA
and RNA) and proteins to get around but not to reproduce.
visual pursuit: Slow eye movements triggered by the passage of an object
through the visual field. Pursuit cannot be triggered consciously — there
must be something to look at.
VLSI: Very-large-scale integration. A technology that allows manufacturer
of silicon chips containing millions of transistors.
voltage: The work needed to move a unit positive charge from a reference
(typically ground) to a particular location. Symbol: V, units: volts. Also
called potential or electrical potential.
voltage clamp: Maintenance of a constant voltage in a neuron by injecting
or withdrawing current to compensate for membrane currents.
voltage-gated channel: Voltage-sensitive channel.
voltage-sensitive channel: An ion channel that opens and closes in
response to changes in membrane potential.
volume transmission: In neurophysiology, transmitters that spread
broadly and thereby signal to multiple neurons rather than point to point
at a synapse.
VOR: See vestibulo-ocular reflex.
voxel: Volume element. A single spot in an object being imaged with a
multidimensional imaging technique such as CT or MRI.
Weber-Fechner law: Psychophysical finding that perception of a stimulus
is typically proportionate to the logarithm of stimulus intensity in most
sensory systems: Perception = constant ・ log(stimulus).
weight: Numerical value representing strength of a connection between
units in a neural network.
Glossary 355
weight matrix: See connection matrix.
weight space: Phase space representing progression of weight matrix in a
network during learning.
weight vector: A row of a network weight matrix representing convergence
onto a single unit.
Wernicke’s area: An area of temporal lobe involved in processing of
language.
white matter: In neuroscience, brain areas that appear white due to
preponderance of myelinated axons.
white noise: In signals and systems theory, random perturbations that
occur equally at all frequencies.
whole-cell patch: In electrophysiology, a technique for making measurements
inside of a cell by attaching an electrode tightly to the outside of the
membrane and then blowing out a hole.
Widrow-Hoff rule: See delta rule.
winner-take-all: In neural networks, dynamics that permit the activity
of a single unit to emerge and dominate the network.
word: In computer science, a unit of memory. Differs in size in different
computer models.
wraparound: Organizing a neural network so that the units at one end
or edge connect with units at the other end or edge.
wulst: An area of avian brain that is involved in learning and memory.
XOR: A Boolean operator. See Chap. 16, Section 16.3.
Y2K problem: In computer science, Y2K means Year 2000, using Y for
year and K for kilo. Refers to the problem that arose due to the use of a
two-digit representation for calendar year in programs: “00” could be either
2000 or 1900.
zoology: The study of similarities and differences among animal species.
Index
ablate, 38, 39
absence epilepsy, 272
accumulator, 68
acetylcholine, 262
action potential, 111, 216
activation function, 94
active channels, 214
adaptation, 118
adaptive filter, 137
afferent, 116
afterhyperpolarization, 233
agonist, 249
algorithm definition, 18
all-or-none, 27
allosteric, 245
alpha waves, 113
AMPA receptor, 265
amplitude modulation, 206
analog, 14, 50
analog computer, 51
analytic, 23, 197
anatomical position, 30
anatomy, 35
anion, 219
apamin, 38
aphasia, 39
apical dendrite, 251
area MT, 116
art of modeling, 122, 139
artifact, 160, 181, 251
artificial intelligence, 63
artificial life, 64
Ascii, 55
assembler language, 65, 69
associative memory, 164, 167
associative property of multiplication,
164
asymptote, 200
asynchronous updating, 113, 174
ATP, 221
attention, 272
attractor, 101, 170
autoassociative memory, 167
axial resistance, 244
axoaxonic, 269
axon collateral, 271
back-propagating spikes, 254
back-propagation algorithm, 149, 162
basal ganglia, 275
BASIC, 65
basin of attraction, 170
belladonna, 38
beta waves, 113
358 Index
bias unit, 144
bilateral innervation, 107
binding problem, 38
bit-wise complement, 54
bitmap, 55
black box, 77
blind spot, 13
Boolean algebra, 48, 71
Boolean AND, 147
Boolean XOR, 149
bouton, 26
brain coordinate systems, 29
brain maps, 107
brain plasticity, 61
brain waves, 113
brainstem, 6
Brodmann area, 277
bus, 79
butterfly effect, 118
byte, 53
calcium, 36
calculus, 195, 295
capacitance, 193, 304
carrier frequency, 207
cation, 219
caudal, 29
cell assembly, 163, 259
cell impalement, 251
cell membrane, 190
central dogma, 37
central nervous system, 21, 29
cerebellum, 31, 273
chandelier cell, 277
chaos, 117
charge separation, 222
charge-couple device, 59
charging curve, 199, 201
charybdotoxin, 38
check sum, 52
chemical synapse, 215, 245
chunking, 111
circadian rhythm, 113
CISC, 79
clamp, 135, 144, 217
clear bit, 49
climbing fiber, 275
cocktail party effect, 175
command neuron, 108
compartment modeling, 239, 241
compilation, 65, 78
complex number, 53
compound eye, 122
computational neuroscience, 22
computer architecture, 65
computer programming, 64
computer science, 14
computer tomography, 35
computer virus, 76
computer word, 48, 53, 67
conditional statement, 72
conductance, 301
cone, 139
confabulation, 13
connectivity matrix, 96
contralateral innervation, 107
convergence, 96
coronal, 30
cortex, 21, 29, 31, 277
credit-assignment problem, 145
crosstalk, 176
cryptography, 52, 112
current clamp, 200
current injection, 202
cybernetics, 85
cytoplasm, 26, 242
Dale’s principle, 92
data management, 47
data structure, 46
database, 49
De Morgan’s theorem, 290
degenerative disease, 41
delta rule, 142, 145
delta waves, 113
dendrite, 17
dendrodendritic, 272
depersonalization, 39
depolarization blockade, 236
derealization, 39
description length, 60
desensitization, 245
deterministic system, 118
dielectric constant, 193
digital, 50
dimensional analysis, 286
dimensional reduction, 127
discrete mathematics, 187
Index 359
distributed code, 109
distributed coding, 152
divergence, 96
DNA, 38
domain, 131
dorsal, 29
dot product, 92, 93
downregulate, 246
driving force, 246
dual matrix hypothesis, 265
duty cycle, 89
dynamical system, 117
dyslexia, 60
edge effect, 128, 129
efferent, 115
electric eye, 78
electrical circuit, 192
electrical synapse, 215
electrocorticogram, 114
electrode, 251
electroencephalography, 37
electron microscopy, 36
electrophysiology, 36
electrotonically compact, 251
emergent properties, 41
emulator, 66
endogenous, 38
energy function, 147, 170
ensemble coding, 106
epiphenomenon, 21
equipotential, 241
error-minimization, 147
eukaryote, 190
Euler integration, 197, 297
exclusive OR, 149
existence proof, 162
explicit Euler integration, 197
extracellular, 37, 182
false unit, 144
feedback, 224
feedforward network, 102
field potential, 37
filter, 111
fixed point, 169
flux, 219
fMRI, 35, 36
FORTRAN, 65, 71
frequency coding, 114
frequency modulation, 206
gain, 237
gamma waves, 113
ganglion, 270
gap junction, 28, 186
Gaussian function, 131
gene, 38
generalization, 157
genetic algorithm, 147
genetic code, 45
glia, 26
globus pallidus, 276
glucose, 36
glutamate, 36
Goldman-Hodgkin-Katz equation,
221
Golgi stain, 36
graceful degradation, 109
gradient descent algorithm, 147
grandmother cells, 109
grapheme, 60
gray matter, 29, 31, 270
ground, 194
H.M., 270
habituation, 118
hack, 64, 181
hardware, 65
Heaviside function, 94
Hebb synapse, 14, 163, 254
heteroassociative memory, 167
hidden units, 102
hippocampus, 14, 270
histology, 129
history of action potential, 216
Hodgkin-Huxley equations, 228, 331
Hodgkin-Huxley model, 217
Hodgkin-Huxley particles, 226
Hodgkin-Huxley simulation, 231
hologram, 109
homologous evolution, 263
Hopfield network, 169
horseshoe crab, 122
humunculus, 107
hydrophilic, 190
hydrophobic, 190, 192
hypercube, 170
360 Index
I-beam, 182
I-f curve, 237
I/O, 236
IBM, 47
illusory conjunction, 38
implementation, 18
implicit Euler integration, 197
inertial guidance system, 77
inferior olive, 275
infinite loop, 71
infinitesimal, 196, 296
information theory, 112
initial condition, 98, 135, 198
inner product, 93
input impedance, 253
instantaneous frequency, 206
instruction register, 65, 68
instruction set, 64, 69
interpretation, 65, 79
interspike interval, 209
intracellular, 182
intracellular recording, 37
ion channel, 14, 38
ion channel memory, 233
isopotential, 241
Kirchhoff’s law, 195, 301
kludge, 75
knock-out, 38
Krebs cycle, 183
labeled line coding, 80
labeled-line coding, 122
lateral, 29
lateral geniculate nucleus, 33, 272
lateral inhibition, 123
learning rate, 146
length-constant, 251
levels of investigation, 18
levels of organization, 16
ligand, 38, 215
light microscopy, 35
limit cycle, 102, 171
limulus equation, 135
linearity, 127
lipid bilayer, 190
LISP, 79
lissencephaly, 32
local code, 108
local minimum, 156
long-term depression, 164
long-term potentiation, 14, 249, 264
longitudinal resistance, 244
lumen, 132
luminance, 124
Mach bands, 136
machine language, 65
macroscopic, 36
mantissa, 282
mapping, 167
Markov model, 239
Marr, 63
matrix, 92
matrix multiplication, 166
medial, 29
memory management, 80
metabolism, 36, 183
microelectrode, 37
micropipette, 37
microtubule, 14
mirror attractor, 170
model design, 126
models and metaphors, 11, 192
molecular biology, 37
monotonic, 236
mossy fibers, 271
motor, 115
MRI, 35
multiplex, 181
multiplexing, 185
multiunit recording, 37
myelin, 217
natural language, 79
negative feedback, 34
negative number, 52
neglect, 40
neocortex, 32, 277
Nernst potential, 221
network architecture, 102
network clamping, 144
network layers, 144
network target, 143
neural code, 45
neural state coding, 110
neuroanatomy, 26
neuron, 14, 26
Index 361
neuron doctrine, 26, 254
neuron state, 184
neuron weight, 185
neuropharmacology, 37
neurophysiology, 36
neuropil, 270, 277
neuroscience, 14
neurotransmitter, 17, 183
nitric oxide, 192
NMDA receptor, 265
nonlinear, 90
normalization, 253
normalize, 169
nucleus, 29, 31
numerical accuracy, 201
numerical integration, 197, 296
numerical stability, 201
occipital cortex, 272
octal, 65
Ohm’s law, 195, 301
olfaction, 177
ommatidium, 123
one’s complement, 54
organelle, 26, 184
orthogonal, 168
outer product, 165
overflow, 54
overgeneralize, 152
paradigm shift, 182
parallel distributed processing, 152
parallel fiber, 274
parallel processing, 14, 185
parallel-conductance model, 217
parameter dependence, 198
parameterization, 130
parameterize, 130
parasitic capacitance, 305
Parkinson disease, 41
particle, 223
particles, 226
passive membrane, 205, 214, 250
PDP-8, 66
perforant path, 271
PET, 35, 36
phase, 207
phoneme, 60
phospholipids, 190
photoreceptor, 106, 122
phrenology, 32
physiology, 35
piece-wise linear function, 131
pixel, 55
place notation, 287
point attractor, 101, 102, 147, 169
point neuron, 184, 202
pointer, 67, 74
positive true, 50, 89
problem definition, 18
program, 58
program counter, 68
prokaryote, 190
protein, 38
pseudo-code, 73
psychophysics, 38
punch card, 48
Purkinje cell, 34
Q10, 229
qualia, 20, 137
radix point, 287
RAM, 67
rate coding, 21, 114, 234
receptive field, 106
recurrent network, 102
reflex, 6
register, 67
reinforcement learning, 141
relaxation, 174
relay cell, 272
relay nucleus, 33, 272
repolarization, 232
resistance, 301
resting membrane potential, 37, 214
reticularis cell, 272
retinal slip, 152
retinotopic, 106
retrograde transmission, 186
RISC, 79
RNA, 38
rostral, 29
running average, 113
saccade, 152
sagittal, 30
sampling, 124
362 Index
scalar, 92
scaling, 126
scantron form, 49
Schaeffer collateral, 271
schizophrenia, 40, 116
scientific notation, 282
second messenger, 21, 251
segmentation error, 71
semicircular canal, 152
sensitivity to initial conditions, 118
sensory, 116
serial processing, 185
set bit, 49
shunting inhibition, 246
SI units, 284
sigmoid, 94
sign magnitude, 53
signal integration, 206
signal transduction, 122
signals and systems, 198
silent synapse, 266
silicon, 18
simulation, 22
single-unit recording, 37
sink, 194
slow potential theory, 205
soap bubble, 192
software, 65, 78
space-time trade-off, 49
sparse matrix, 102
spatial summation, 202
spike, 21, 111
spine, 17, 26, 186
spontaneous firing, 114
spurious attractor, 170
squashing function, 94
state, 90, 110
state space, 101
state variable, 229
state vector, 93
steady state, 100
step function, 94
stomatogastric ganglion, 42
strange attractor, 118, 176
stria of Gennari, 277
stroke, 36
substantia nigra, 276
subthreshold, 118
subtraction, 53
summed-square error, 147, 155
supervised learning, 142
suprachiasmatic nucleus, 114
symbol table, 71
synapse, 14, 17
synaptic cleft, 26
synaptic triad, 186, 272
synchronous updating, 113, 174
synesthesia, 139
target, 141
temporal coding, 111
temporal integration, 112, 113, 201
tetrodotoxin, 38
thalamic reticular nucleus, 272
thalamocortical cell, 272
thalamus, 272
theta waves, 113
thought-insertion, 116
three-body problem, 230
threshold, 90
time constant, 201
tissue, 36
transcendental function, 291
transistor, 14, 89
transistor-transistor logic, 187
transmembrane protein, 192
transpose, 291
tricarboxylic acid cycle, 183
trisynaptic pathway, 271
truth table, 290
two’s complement, 53
two-point discrimination, 107
ultrastructural, 17
unit analysis, 286
unsupervised learning, 141
update equation, 94, 103, 197
update rule, 91
upregulate, 246
V1, 277
vector, 92
vector language, 59
vector plot, 57
ventral, 29
vestibulo-ocular reflex, 152
veto inhibition, 90
virtual experiment, 159
Index 363
visual pursuit, 152
voltage clamp, 217
voltage divider, 303
volume transmission, 187
VOR, 152
weather forecasting, 3, 47
Weber-Fechner law, 127
weight, 90
weight matrix, 96
weight space, 147
weight vector, 93
white matter, 29, 31, 270
whole-cell patch, 251
winner-take-all, 140
wraparound, 128, 129
Y2K problem, 64 | l******8
发帖数: 1691 | 2 Pretty good basics to fresh PHDs. The title is misleading.
【在 f**d 的大作中提到】
: 这是一本计算神经科学的优秀著作，全文拷贝这里（图和公式缺），有兴趣的同学可以
: 阅读
: 如需要，我可以分享PDF文件（－－仅供个人学习，无商业用途）
: From Computer to Brain
: William W. Lytton
: From Computer to Brain
: Foundations of Computational Neuroscience
: Springer
: William W. Lytton, M.D.
: Associate Professor, State University of New York, Downstato, Brooklyn, NY
| d*w
发帖数: 74 | | f**d
发帖数: 768 | 4 email address to my mitbbs email please
【在 d*w 的大作中提到】
: 可以分享下PDF吗
: 这看起来太累人了
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