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The Handbook of
Brain Theory
and Neural Networks
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The Handbook of
Brain Theory
and Neural Networks
Second Edition
E
DITED BY
Michael A. Arbib
E
DITORIAL
A
DVISORY
B
OARD
Shun-ichi Amari • John Barnden • Andrew Barto • Ronald Calabrese
Avis Cohen • Joaquı´n Fuster • Stephen Grossberg • John Hertz
Marc Jeannerod • Mitsuo Kawato • Christof Koch • Wolfgang Maass
James McClelland • Kenneth Miller • Terrence Sejnowski
Noel Sharkey • DeLiang Wang
E
DITORIAL
A
SSISTANT
Prudence H. Arbib
A Bradford Book
THE MIT PRESS
Cambridge, Massachusetts

Biological Neurons and Synapses 47
Neural Plasticity 49
Neural Coding 52
Biological Networks 54
II.6. Dynamics and Learning in Artificial Networks 55
Dynamic Systems 55
Learning in Artificial Networks 58
Computability and Complexity 64
II.7. Sensory Systems 65
Vision 65
Other Sensory Systems 70
II.8. Motor Systems 71
Robotics and Control Theory 71
Motor Pattern Generators 73
Mammalian Motor Control 74
II.9. Applications, Implementations, and Analysis 77
Applications 77
Implementation and Analysis 78
Part III: Articles 81
The articles in Part III are arranged alphabetically by title.
To retrieve articles by author, turn to the contributors list,
which begins on page 1241.
Action Monitoring and Forward Control of
Movements 83
Activity-Dependent Regulation of Neuronal
Conductances 85
Adaptive Resonance Theory 87
Adaptive Spike Coding 90
Amplification, Attenuation, and Integration 94
Analog Neural Nets: Computational Power 97

Pattern Recognition 18
Associative Memory 19
Learning Rules 19
Hebbian Plasticity and Network
Self-Organization 19
Perceptrons 20
Network Complexity 20
Gradient Descent and Credit Assignment 21
Backpropagation 21
A Cautionary Note 22
Envoi 23
Part II: Road Maps: A Guided Tour of
Brain Theory and Neural Networks 25
How to Use Part II 27
II.1. The Meta-Map 27
II.2. Grounding Models of Neurons and Networks 29
Grounding Models of Neurons 29
Grounding Models of Networks 31
Associative Networks 117
Auditory Cortex 122
Auditory Periphery and Cochlear Nucleus 127
Auditory Scene Analysis 132
Axonal Modeling 135
Axonal Path Finding 140
Backpropagation: General Principles 144
Basal Ganglia 147
Bayesian Methods and Neural Networks 151
Bayesian Networks 157
Biologically Inspired Robotics 160
Biophysical Mechanisms in Neuronal Modeling 164

Series 276
Cooperative Phenomena 279
Cortical Hebbian Modules 285
Cortical Memory 290
Cortical Population Dynamics and Psychophysics 294
Covariance Structural Equation Modeling 300
Crustacean Stomatogastric System 304
Data Clustering and Learning 308
Databases for Neuroscience 312
Decision Support Systems and Expert Systems 316
Dendritic Learning 320
Dendritic Processing 324
Dendritic Spines 332
Development of Retinotectal Maps 335
Developmental Disorders 339
Diffusion Models of Neuron Activity 343
Digital VLSI for Neural Networks 349
Directional Selectivity 353
Dissociations Between Visual Processing Modes 358
Dopamine, Roles of 361
Dynamic Link Architecture 365
Dynamic Remapping 368
Dynamics and Bifurcation in Neural Nets 372
Dynamics of Association and Recall 377
Echolocation: Cochleotopic and Computational Maps
381
EEG and MEG Analysis 387
Electrolocation 391
Embodied Cognition 395
Emotional Circuits 398

Movements 507
vi Contents
Hebbian Learning and Neuronal Regulation 511
Hebbian Synaptic Plasticity 515
Helmholtz Machines and Sleep-Wake Learning 522
Hemispheric Interactions and Specialization 525
Hidden Markov Models 528
Hippocampal Rhythm Generation 533
Hippocampus: Spatial Models 539
Hybrid Connectionist/Symbolic Systems 543
Identification and Control 547
Imaging the Grammatical Brain 551
Imaging the Motor Brain 556
Imaging the Visual Brain 562
Imitation 566
Independent Component Analysis 569
Information Theory and Visual Plasticity 575
Integrate-and-Fire Neurons and Networks 577
Invertebrate Models of Learning: Aplysia and
Hermissenda 581
Ion Channels: Keys to Neuronal Specialization 585
Kalman Filtering: Neural Implications 590
Laminar Cortical Architecture in Visual Perception 594
Language Acquisition 600
Language Evolution and Change 604
Language Evolution: The Mirror System Hypothesis 606
Language Processing 612
Layered Computation in Neural Networks 616
Learning and Generalization: Theoretical Bounds 619
Learning and Statistical Inference 624

Neocortex: Chemical and Electrical Synapses 725
Neural Automata and Analog Computational Complexity
729
Neuroanatomy in a Computational Perspective 733
Neuroethology, Computational 737
Neuroinformatics 741
Neurolinguistics 745
Neurological and Psychiatric Disorders 751
Neuromanifolds and Information Geometry 754
Neuromodulation in Invertebrate Nervous Systems 757
Neuromodulation in Mammalian Nervous Systems 761
Neuromorphic VLSI Circuits and Systems 765
NEURON Simulation Environment 769
Neuropsychological Impairments 773
Neurosimulation: Tools and Resources 776
NMDA Receptors: Synaptic, Cellular, and Network
Models 781
NSL Neural Simulation Language 784
Object Recognition 788
Object Recognition, Neurophysiology 792
Object Structure, Visual Processing 797
Ocular Dominance and Orientation Columns 801
Olfactory Bulb 806
Olfactory Cortex 810
Optimal Sensory Encoding 815
Optimality Theory in Linguistics 819
Optimization, Neural 822
Optimization Principles in Motor Control 827
Orientation Selectivity 831
Oscillatory and Bursting Properties of Neurons 835

Rate Coding and Signal Processing 941
Reaching Movements: Implications for Computational
Models 945
Reactive Robotic Systems 949
Reading 951
Recurrent Networks: Learning Algorithms 955
Recurrent Networks: Neurophysiological Modeling 960
Reinforcement Learning 963
Reinforcement Learning in Motor Control 968
Respiratory Rhythm Generation 972
Retina 975
Robot Arm Control 979
Robot Learning 983
Robot Navigation 987
Rodent Head Direction System 990
Schema Theory 993
Scratch Reflex 999
Self-Organization and the Brain 1002
Self-Organizing Feature Maps 1005
Semantic Networks 1010
Sensor Fusion 1014
Sensorimotor Interactions and Central Pattern
Generators 1016
Sensorimotor Learning 1020
Sensory Coding and Information Transmission 1023
Sequence Learning 1027
Short-Term Memory 1030
Silicon Neurons 1034
Simulated Annealing and Boltzmann Machines 1039
Single-Cell Models 1044

Temporal Dynamics of Biological Synapses 1156
Temporal Integration in Recurrent Microcircuits 1159
Temporal Pattern Processing 1163
Temporal Sequences: Learning and Global Analysis 1167
Tensor Voting and Visual Segmentation 1171
Thalamus 1176
Universal Approximators 1180
Unsupervised Learning with Global Objective
Functions 1183
Vapnik-Chervonenkis Dimension of Neural Networks
1188
Vestibulo-Ocular Reflex 1192
Visual Attention 1196
Visual Cortex: Anatomical Structure and Models of
Function 1202
Visual Course Control in Flies 1205
Visual Scene Perception, Neurophysiology 1210
Visual Scene Segmentation 1215
Visuomotor Coordination in Frog and Toad 1219
Visuomotor Coordination in Salamander 1225
Winner-Take-All Networks 1228
Ying-Yang Learning 1231
Editorial Advisory Board 1239
Contributors 1241
Subject Index 1255
viii Contents
Preface to the Second Edition
Like the first edition, which it replaces, this volume is inspired by two great questions:
“How does the brain work?” and “How can we build intelligent machines?” As in the first
edition, the heart of the book is a set of close to 300 articles in Part III which cover the

Biological Neurons and Synapses
Neural Plasticity
Neural Coding
Biological Networks
Dynamics and Learning in Artificial Networks
Dynamic Systems
Learning in Artificial Networks
Computability and Complexity
Sensory Systems
Vision
Other Sensory Systems
Motor Systems
Robotics and Control Theory
Motor Pattern Generators
Mammalian Motor Control
Applications, Implementations, and Analysis
Applications
Implementation and Analysis
The authors of the articles in Part III come from a broad spectrum of disciplines—such
as biomedical engineering, cognitive science, computer science, electrical engineering,
linguistics, mathematics, physics, neurology, neuroscience, and psychology—and have
worked hard to make their articles accessible to readers across the spectrum. The utility
of each article is enhanced by cross-references to other articles within the body of the
article, and lists at the end of the article referring the reader to road maps, background
material, and related reading.
To get some idea of how radically the new edition differs from the old, note that the
new edition has 285 articles in Part III, as against the 266 articles of the first edition. Of
the articles that appeared in the first edition, only 9 are reprinted unchanged. Some 135
have been updated (or even completely rewritten) by their original authors, and more than
30 have been written anew by new authors. In addition, there are over 100 articles on new

charts the immense progress made in recent years in answering many related, but far more
specific, questions.
The term neural networks has been used for a century or more to describe the networks
of biological neurons that constitute the nervous systems of animals, whether invertebrates
or vertebrates. Since the 1940s, and especially since the 1980s, the term has been used for
a technology of parallel computation in which the computing elements are “artificial neu-
rons” loosely modeled on simple properties of biological neurons, usually with some adap-
tive capability to change the strengths of connections between the neurons.
Brain theory is centered on “computational neuroscience,” the use of computational
techniques to model biological neural networks, but also includes attempts to understand
the brain and its function through a variety of theoretical constructs and computer analo-
gies. In fact, as the following pages reveal, much of brain theory is not about neural
networks per se, but focuses on structural and functional “networks” whose units are in
scales both coarser and finer than that of the neuron. Computer scientists, engineers, and
physicists have analyzed and applied artificial neural networks inspired by the adaptive,
parallel computing style of the brain, but this Handbook will also sample non-neural ap-
proaches to the design and analysis of “intelligent” machines. In between the biologists
and the technologists are the connectionists. They use artificial neural networks in psy-
chology and linguistics and make related contributions to artificial intelligence, using neu-
ron-like unites which interact “in the style of the brain” at a more abstract level than that
of individual biological neurons.
Many texts have described limited aspects of one subfield or another of brain theory
and neural networks, but no truly comprehensive overview is available. The aim of this
Handbook is to fill that gap, presenting the entire range of the following topics: detailed
models of single neurons; analysis of a wide variety of neurobiological systems; “connec-
tionist” studies; mathematical analyses of abstract neural networks; and technological ap-
plications of adaptive, artificial neural networks and related methodologies. The excite-
ment, and the frustration, of these topics is that they span such a broad range of disciplines,
including mathematics, statistical physics and chemistry, neurology and neurobiology, and
computer science and electrical engineering, as well as cognitive psychology, artificial

of themselves, give some sense of the sweep of the Handbook:
Connectionism: Psychology, Linguistics, and Artificial Intelligence
Dynamics, Self-Organization, and Cooperativity
Learning in Artificial Neural Networks
Applications and Implementations
Biological Neurons and Networks
Sensory Systems
Plasticity in Development and Learning
Motor Control
A more detailed view of the structure of the book is provided in the introductory section
“How to Use this Book.” The aim is to ensure that readers will not only turn to the book
to get good brief reviews of topics in their own specialty, but also will find many invitations
to browse widely—finding parallels amongst different subfields, or simply enjoying the
discovery of interesting topics far from familiar territory.
Acknowledgments
My foremost acknowledgment is to Prue Arbib, who served as Editorial Assistant during
the long and arduous process of eliciting and assembling the many, many contributions to
Part III; we both thank Paulina Tagle for her help with our work. The initial plan for the
book was drawn up in 1991, and it benefited from the advice of a number of friends,
especially George Adelman, who shared his experience as Editor of the Encyclopedia of
Neuroscience. Refinement of the plan and the choice of publishers occupied the first few
months of 1992, and I thank Fiona Stevens of The MIT Press for her support of the project
from that time onward.
As can be imagined, the plan for a book like this has developed through a time-consum-
ing process of constraint satisfaction. The first steps were to draw up a list of about 20
topic areas (similar to, but not identical with, the 23 areas surveyed in Part II), to populate
these areas with a preliminary list of over 100 articles and possible authors, and to recruit
the first members of the Editorial Advisory Board to help expand the list of articles and
focus on the search for authors. A very satisfying number of authors invited in the first
round accepted my invitation, and many of these added their voices to the Editorial Ad-

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How to Use This Book
More than 90% of this book is taken up by Part III, which, in 285 separately authored
articles, covers a vast range of topics in brain theory and neural networks, from language
to motor control, and from the neurochemistry to the statistical mechanics of memory.
Each article has been made as self-contained as possible, but the very breadth of topics
means that few readers will be expert in a majority of them. To help the reader new to
certain areas of the Handbook, I have prepared Part I: Background and Part II: Road Maps.
The next few pages describe these aids to comprehension, as well as offering more infor-
mation on the structure of articles in Part III.
Part I: Background: The Elements of Brain Theory and Neural Networks
Part I provides background material for readers new to computational neuroscience or
theoretical approaches to neural networks considered as dynamic, adaptive systems. Sec-
tion I.1, “Introducing the Neuron,” conveys the basic properties of neurons and introduces
several basic neural models. Section I.2, “Levels and Styles of Analysis,” explains the
interdisciplinary nexus in which the present study of brain theory and neural networks is
located, with historical roots in cybernetics and with current work going back and forth
between brain theory, artificial intelligence, and cognitive psychology. We also review the
different levels of analysis involved, with schemas providing the functional units inter-
mediate between an overall task and neural networks. Finally, Section I.3, “Dynamics and
Adaptation in Neural Networks,” provides a tutorial on the concepts essential for under-
standing neural networks as dynamic, adaptive systems. We close by stressing that the full
understanding of the brain and the improved design of intelligent machines will require
not only improvements in the learning methods presented in Section I.3, but also fuller
understanding of architectures based on networks of networks, with initial structures well
constrained for the task at hand.
Part II: Road Maps: A Guided Tour of Brain Theory and Neural Networks
The reader who wants to survey a major theme of brain theory and neural networks, rather
than seeking articles in Part III one at a time, will find in Part II a set of 22 road maps
that, among them, place every article in Part III in a thematic perspective. Section II.1

Applications, Implementations, and Analysis
Applications
Implementation and Analysis
This ordering of the themes has no special significance. It is simply one way to approach
the richness of the Handbook, making it easy for you to identify one or two key road maps
of special interest. By the same token, the order of articles in each of the 22 road maps
that follow the Meta-Map is one among many such orderings. Each road map starts with
an alphabetical listing of the articles most relevant to the current theme. The road map
itself will provide suggestions for interesting traversals of articles, but this need not imply
that an article provides necessary background for the articles it precedes.
Part III: Articles
Part III comprises 285 articles. These articles are arranged in alphabetical order, both to
make it easier to find a specific topic (although a Subject Index is provided as well, and
the alphabetical list of Contributors on page 1241 lists all the articles to which each author
has contributed) and because a given article may be relevant to more than one of the
themes of Part II, a fact that would be hidden were the article to be relegated to a specific
section devoted to a single theme. Most of these articles assume some prior familiarity
with neural networks, whether biological or artificial, and so the reader new to neural
networks is encouraged to master the material in Part I before tackling Part III.
Most articles in Part III have the following structure: The introduction provides a non-
technical overview of the material covered in the whole article, while the final section
provides a discussion of key points, open questions, and linkages with other areas of brain
theory and neural networks. The intervening sections may be more or less technical, de-
pending on the nature of the topic, but the first and last sections should give most readers
a basic appreciation of the topic, irrespective of such technicalities. The bibliography for
each article contains about 15 references. People who find their favorite papers omitted
from the list should blame my editorial decision, not the author’s judgment. The style I
chose for the Handbook was not to provide exhaustive coverage of research papers for the
expert. Rather, references are there primarily to help readers who look for an introduction
to the literature on the given topic, including background material, relevant review articles,

Michael A. Arbib
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I.1. Introducing the Neuron 3
How to Use Part I
Part I provides background material, summarizing a set of concepts
established for the formal study of neurons and neural networks by
1986. As such, it is designed to hold few, if any, surprises for
readers with a fair background in computational neuroscience or
theoretical approaches to neural networks considered as dynamic,
adaptive systems. Rather, Part I is designed for the many readers—
be they neuroscience experimentalists, psychologists, philosophers,
or technologists—who are sufficiently new to brain theory and
neural networks that they can benefit from a compact overview of
basic concepts prior to reading the road maps of Part II and the
articles in Part III. Of course, much of what is covered in Part I is
also covered at some length in the articles in Part III, and cross-
references will steer the reader to these articles for alternative ex-
positions and reviews of current research. In this exposition, as
throughout the Handbook, we will move back and forth between
computational neuroscience, where the emphasis is on modeling
biological neurons, and neural computing, where the emphasis
shifts back and forth between biological models and artificial neural
networks based loosely on abstractions from biology, but driven
more by technological utility than by biological considerations.
Section I.1, “Introducing the Neuron,” conveys the basic prop-
erties of neurons, receptors, and effectors, and then introduces sev-
eral simple neural models, including the discrete-time McCulloch-
Pitts model and the continuous-time leaky integrator model.
References to Part III alert the reader to more detailed properties
of neurons which are essential for the neuroscientist and provide

and efficaciously shaped by experience. Moreover, the full under-
standing of the brain and the improved design of intelligent ma-
chines will require not only improvements in these learning meth-
ods and their initialization, but also a fuller understanding of
architectures based on networks of networks. Cross-references to
articles in Part III will set the reader on the path to this fuller
understanding. Because Part I focuses on the basic concepts estab-
lished for the formal study of neurons and neural networks by 1986,
it differs hardly at all from Part I of the first edition of the Hand-
book. By contrast, Part II, which provides the road maps that guide
readers through the radically updated Part III, has been completely
rewritten for the present edition to reflect the latest research results.
I.1. Introducing the Neuron
We introduce the neuron. The dangerous word in the preceding
sentence is the. In biology, there are radically different types of
neurons in the human brain, and endless variations in neuron types
of other species. In brain theory, the complexities of real neurons
are abstracted in many ways to aid in understanding different as-
pects of neural network development, learning, or function. In neu-
ral computing (technology based on networks of “neuron-like”
units), the artificial neurons are designed as variations on the ab-
stractions of brain theory and are implemented in software, or VLSI
or other media. There is no such thing as a “typical” neuron, yet
this section will nonetheless present examples and models which
provide a starting point, an essential set of key concepts, for the
appreciation of the many variations on the theme of neurons and
neural networks presented in Part III.
An analogy to the problem we face here might be to define ve-
hicle for a handbook of transportation. A vehicle could be a car, a
train, a plane, a rowboat, or a forklift truck. It might or might not

mammals in general and humans in particular), this network is
called the central nervous system (CNS), and the brain constitutes
the most headward part of this system, linked to the receptors and
effectors of the body via the spinal cord. Invertebrate nervous sys-
tems (neural networks) provide astounding variations on the ver-
tebrate theme, thanks to eons of divergent evolution. Thus, while
the human brain may be the source of rich analogies for technol-
ogists in search of “artificial intelligence,” both invertebrates and
vertebrates provide endless ideas for technologists designing neural
networks for sensory processing, robot control, and a host of other
applications. (A few of the relevant examples may be found in the
Part II road maps, Vision, Robotics and Control Theory, Motor
Pattern Generators, and Neuroethology and Evolution.)
The brain provides far more than a simple stimulus-response
chain from receptors to effectors (although there are such reflex
paths). Rather, the vast network of neurons is interconnected in
loops and tangled skeins so that signals entering the net from the
receptors interact there with the billions of signals already travers-
ing the system, not only to yield the signals that control the effec-
tors but also to modify the very properties of the network itself, so
that future behavior will reflect prior experience.
The Diversity of Receptors
Rod and cone receptors in the eyes respond to light, hair cells in
the ears respond to pressure, and other cells in the tongue and the
mouth respond to subtle traces of chemicals. In addition to touch
receptors, there are receptors in the skin that are responsive to
movement or to temperature, or that signal painful stimuli. These
external senses may be divided into two classes: (1) the proximity
senses, such as touch and taste, which sense objects in contact with
the body surface, and (2) the distance senses, such as vision and

effectors, we must have a closer look at “the” neuron. As already
emphasized, there is no such thing as a typical neuron. However,
we will summarize properties shared by many neurons. The “basic
neuron” shown in Figure 1 is abstracted from a motor neuron of
mammalian spinal cord. From the soma (cell body) protrudes a
number of ramifying branches called dendrites; the soma and den-
drites constitute the input surface of the neuron. There also extrudes
from the cell body, at a point called the axon hillock (abutting the
initial segment), a long fiber called the axon, whose branches form
the axonal arborization. The tips of the branches of the axon, called
nerve terminals or boutons, impinge on other neurons or on effec-
tors. The locus of interaction between a bouton and the cell on
which it impinges is called a synapse, and we say that the cell with
the bouton synapses upon the cell with which the connection is
made. In fact, axonal branches of some neurons can have many
varicosities, corresponding to synapses, along their length, not just
at the end of the branch.
We can imagine the flow of information as shown by the arrows
in Figure 1. Although “conduction” can go in either direction on
the axon, most synapses tend to “communicate” activity to the den-
drites or soma of the cell they synapse upon, whence activity passes
to the axon hillock and then down the axon to the terminal arbo-
Figure 1. A “basic neuron” abstracted from a
motor neuron of mammalian spinal cord. The
dendrites and soma (cell body) constitute the ma-
jor part of the input surface of the neuron. The
axon is the “output line.” The tips of the branches
of the axon form synapses upon other neurons or
upon effectors (although synapses may occur
along the branches of an axon as well as at the

in much the same passive way that heat is conducted down a rod
of metal: a normal change in potential difference across the cell
membrane can propagate in a passive way so that the change occurs
later, and becomes smaller, the farther away we move from the site
of the original change. This passive propagation is governed by the
cable equation
2
ץV ץ V
ס
2
ץt ץx
If the starting voltage at a point on the axon is V
0
, and no further
conditions are imposed, the potential will decay exponentially, hav-
ing value V
(x)
ס V
0
e
מx
at distance x from the starting point, where
the length unit, the length constant, is the distance in which the
potential changes by a factor of 1/e. This length unit will differ
from axon to axon. For “short” cells (such as the rods, cones, and
bipolar cells of the retina), passive propagation suffices to signal a
potential change from one end to the other; but if the axon is long,
this mechanism is completely inadequate, since changes at one end
will decay almost completely before reaching the other end. For-
tunately, most nerve cells have the further property that if the

SCILLATORY
AND
B
URSTING
P
ROPERTIES OF
N
EURONS
), and explicating trig-
gered long-distance propagation. Hodgkin and Huxley used curve
fitting from experimental data to determine the terms for conduc-
tance change in their model. Subsequently, much research has
probed the structure of complex molecules that form channels
which selectively allow the passage of specific ions through the
membrane (see I
ON
C
HANNELS
:K
EYS TO
N
EURONAL
S
PECIALI
-
ZATION
). This research has demonstrated how channel properties
not only account for the terms in the Hodgkin-Huxley equation,
but also underlie more complex dynamics which may allow even
small patches of neural membrane to act like complex computing

either excitatory, tending to move the potential difference across
the postsynaptic membrane in the direction of the threshold (de-
polarizing the membrane), or inhibitory, tending to move the po-
larity away from the threshold (hyperpolarizing the membrane).
There are some exceptional cell appositions that are so large or
have such tight coupling (the so-called gap junctions) that the im-
pulse affects the postsynaptic membrane without chemical media-
tion (see N
EOCORTEX
:C
HEMICAL AND
E
LECTRICAL
S
YNAPSES
).
Most neural modeling to date focuses on the excitatory and in-
hibitory interactions that occur on a fast time scale (a millisecond,
more or less), and most biological (as distinct from technological)
models assume that all synapses from a neuron have the same
“sign.” However, neurons may also secrete transmitters that mod-
ulate the function of a circuit on some quite extended time scale.
Modeling that takes account of this neuromodulation (see S
YN
-
APTIC
I
NTERACTIONS
and N
EUROMODULATION IN

from right to left (D). (From Arbib,
M. A., 1989, The Metaphorical Brain 2:
Neural Networks and Beyond, New York:
Wiley-Interscience, p. 60. Reproduced
with permission. Copyright ᭧ 1989 by
John Wiley & Sons, Inc.)
D
ENDRITIC
P
ROCESSING
). A highly simplified example (Figure 2)
shows how the properties of nervous tissue just presented would
indeed allow a simple neuron, by its very dendritic geometry, to
compute some useful function (cf. Rall, 1964, p. 90). Consider a
neuron with four dendrites, each receiving a single synapse from a
visual receptor, so arranged that synapses A, B, C, and D (from
left to right) are at increasing distances from the axon hillock. (This
is not meant to be a model of a neuron in the retina of an actual
organism; rather, it is designed to make vivid the potential richness
of single neuron computations.) We assume that each receptor re-
acts to the passage of a spot of light above its surface by yielding
a generator potential which yields, in the postsynaptic membrane,
the same time course of depolarization. This time course is prop-
agated passively, and the farther it is propagated, the later and the
lower is its peak. If four inputs reached A, B, C, and D simulta-
neously, their effect may be less than the threshold required to
trigger a spike there. However, if an input reaches D before one
reaches C, and so on, in such a way that the peaks of the four
resultant time courses at the axon hillock coincide, the total effect
could well exceed threshold. This, then, is a cell that, although very