MASS ACTION IN THE NERVOUS SYSTEM
Examination of the Neurophysiological Basis of Adaptive Behavior through the EEG
WALTER J. FREEMAN
Department of Physiology–Anatomy
University of California
Berkeley, California
ACADEMIC PRESS New York San Francisco London 1975
A Subsidiary of Harcourt Brace Jovanovich, Publishers
To my father
Copyright/Copyleft İ 2004 Walter J Freeman. Noncommercial reproduction in whole or in part is permitted with acknowledgment of the source.
ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
Freeman, Walter J.
Mass action in the nervous system.
Bibliography: p.
Includes indexes.
1. Neurophysiology–Mathematical models.
2. Adaptation (Physiology) –Mathematical models.
3. Electroencephalography.
I . Title.
DNLM: 1. Electroencephalography. 2. Neurophysiology.
WLJ5O.F855m1
QP356; F72 612; 822 74–27781
ISBN 0–12––267150–3
ORIGINALLY PRINTED IN THE UNITED STATES OF AMERICA
Contents
PREFACE XI
ACKNOWLEDGMENTS XI II
NOTATION XV
PREFACE II (Electronic Text Version) XXI
Chapter 1 Topological Properties
1.1. The Approach to Neural Masses 1
1.1.1. Direct and Indirect Observations 1
1.1.2. The Use of Models in a Hierarchy 3
1.1.3. Macroscopic Forms of Cooperative Neural Activity 5
1.2. Single Neurons 10
1.2.1. The Structures of Neurons 10
1.2.2. The Operations of Neurons 11
1.2.3. The State Variables of Neurons 13
1.2.4. Specification of the Active States and Operations 16
1.2.5. Input–Output Relations of Single Neurons 19
1.2.6. Multiple Stable States of Neurons 22
1.2.7. Basic Topologies of Networks of Neurons 24
1.3. Neural Masses 25
1.3.1. A Topological Hierarchy of Interactive Sets 25
1.3.2. The State Variables of KO and KI Sets 34
1.3.3. The Operations of Neural Sets 37
1.3.4. Feedback Gain as a Parameter for Interaction 39
1.3.5. Multiple Stable States and the Levels of Interaction 42
1.3.6. The Relation of Multiple Stabilities to Neural Signals 46
1.3.7. The Conditions for Realizability 47
1.3.8. The Use of Differential Equations 49
<Page vii>
Chapter 2 Time–Dependent Properties
2.1. Measurement of Neural Events 51
2.1.1. Representation of Events by Functions 51
2.1.2. Input–Output Functions 55
2.1.3. Linear Input–Output Functions 57
2.1.4. The Impulse and the Impulse Response 60
2.2. Linear Models for Neural Membrane 61
2.2.1. The Topology of the Membrane 61
2.2.2. Differential Equations 64
2.2.3. The Laplace Transform 67
2.2.4. Application of the Laplace Transform to the Membrane 70
2.3. Linear Models for Parts of Neurons 72
2.3.1. Convolution 72
2.3.2. The Convolution Theorem 76
2.3.3. Transfer Functions for Pulse Transmission 80
2.3.4. The Core Conductor Model 86
2.3.5. Synaptic Delay 91
2.4. Linear Models for Neurons 94
2.4.1. Formulation of the Topology 94
2.4.2. Input–Output Pairs and the Differential Equation 96
2.4.3. Interpretation of the Parameters 99
2.4.4. Linear Function for Wave to Pulse Conversion 101
2.5. Linear Models for Neural Masses 103
2.5.1. Use of Nonlinear Regression 103
2.5.2. The KO Neural Set 106
2.5.3. Oscillatory Responses from a KII Set 110
Chapter 3 Amplitude–Dependent Properties
3.1. Nonlinear Models for Neural Membranes 121
3.1.1. The Ionic Hypothesis 121
3.1.2. Metabolic Forces 125
3.1.3. The Concept of Equilibrium Potential 126
3.1.4. The Sodium Permeability Model 129
3.2. Nonlinear Models for Neurons and Parts of Neurons 134
3.2.1. Action Potentials in Axons 134
3.2.2. Threshold Uncertainty in Axons 138
3.2.3. Postsynaptic Potentials in Dendrites 140
3.2.4. Amplitude–Dependent Input–Output Relations 144
3.3. Nonlinear Models for Neural Masses 146
3.3.1. Background Activity in the Wave Mode 146
3.3.2. Background Activity in the Pulse Mode 150
3.3.3. Relations of Waves and Pulses 154
3.3.4. Wave to Pulse Conversion in the KI Set 159
3.3.5. Pulse to Wave Conversion in the KI Set 163
3.3.6. The Forward Gain of the KI Set 165
<Page ix>
Chapter 4 Space–Dependent Properties
4.1. Potential Fields of Single Neurons 172
4.1.1. Basis Functions for Measurement of Potential in Space 173
4.l.2. Basis Functions for Potential in Current Fields 177
4.1.3. Potential Functions for the Core Conductor 180
4.1.4. Potential Fields of Axons 185
4.1.5. Nodes and Branched Fibers 188
4.2. Potential Fields of Neural Masses 193
4.2.1. Measurement of Observed Fields 193
4.2.2. Basis Functions for Potential Fields of Neural Masses 196
4.2.3. Compound Potential Fields: Modular Analysis 202
4.3. Potential Fields in the Olfactory Bulb 211
4.3.1. Bulbar Geometry and Topology 212
4.3.2. Analysis of the Spatial Function of Potential 219
4.3.3. Time–Dependent Activity 228
4.4. Potential Fields in the Prepyriform Cortex 234
4.4.1. Cortical Geometry and Topology 234
4.4.2. Observed Fields of Cortical Potential 238
4.4.3. Relation of Potential Fields to Active States 245
4.5. Divergence and Convergence in Neural Masses 249
4.5.1. The Operation of Divergence 249
4.5.2. Evaluation of Spatial Distributions of Active States 253
4.5.3. Evaluation of Synaptic Divergence 260
4.5.4. Evaluation of Tractile Divergence 264
Chapter 5 Interaction: Single Feedback Loops with Fixed Gain
5.1. General Properties of Single Feedback Loops 270
5.1.1. Types of Neural Feedback 271
5.1.2. Derivation of the Lumped Piecewise Linear Approximation 273
5.1.3. Root Locus as a Function of Feedback Gain 278
5.1.4. Amplitude–Dependent Gain and Stability 284
5.2. Reduction from the KI Level 285
5.2.1. Topological Analysis of the Glomerular Layer 285
5.2.2. Differential Equations for the KIe Set 291
5.2.3. Self–Stabilization of the KIe Set 299
5.3. Reduction from the KII Level 305
5.3.1. Topological Analysis of the Olfactory Bulb 305
5.3.2. Differential Equations for the Open Loop Cases 309
5.3.3. Differential Equations for the Closed Loop Cases 314
5.4. Reduction from the KIII Level 321
5.4.1. Topological Analysis of the Prepyriform Cortex 321
5.4.2. Differential Equations for the Cortex 326
5.4.3. Transfer Function of the LOT Input Channel 330
5.4.4. Pulse–Wave Relations in Cortex and Bulb 334
5.4.5. Channels for Centrifugal Input 338
<Page x>
Chapter 6 Multiple Feedback Loops with Variable Gain
6.1. Equilibrium States: Characteristic Frequency 342
6.1.1. Definition of the Three Types of Feedback Gain 342
6.1.2. Solution of the Differential Equations 349
6.1.3. Experimental and Theoretical Root Loci 355
6.1.4. Bias Control of Characteristic Frequency 366
6.1.5. Root Loci Dependent on EEG Amplitudes 370
6.2. Limit Cycle States: Mechanisms of the EEG 378
6.2.1. Stability Properties of KII Sets 378
6.2.2. Limit Cycle States in the First Mode 381
6.2.3. Limit Cycle States in the Second Mode 386
6.2.4. Sources of Error and Limitation 390
6.2.5. Comparisons with Related Mathematical Models 396
Chapter 7 Signal Processing by Neural Mass Actions
7.1. Behavioral Correlates of Wave Activity in KII Sets 402
7.1.1. The Operational Basis for Correlation 402
7.1.2. Factor Analysis of AEPs 407
7.1.3. Patterns of Change in AEPS with Attention 414
7.1.4. A Proposed Cortical Mechanism of Attention 422
7.2. Transformations of Neural Signals by KII Sets 427
7.2.1. Neural Coding in the Olfactory Bulb 429
7.2.2. Bulbar Mechanisms for Phase Modulation 434
7.2.3. Attention and the Cortical Expectation Function 440
7.2.4. Possible Mechanisms of Cortical Output 446
7.3. Comments concerning Neocortical Mass Actions 448
7.3.1. Rhythmic Potentials and Rhythmic Stimulation 449
7.3.2. DC Polarization and Steady Potentials 452
7.3.3. Unit Activity Correlated with Sensory and Motor Events 455
References 462
AUTHOR INDEX 473
SUBJECT INDEX 477
<Page xi>
Preface (Original)
This book was written to answer the questions: What are the neural mechanisms, and what is the behavioral significance of the electroencephalogram (EEG)? The answers are partial, tentative, and predictably complex. Emphasis is given to observations made on the mammalian olfactory system for reasons stated below. Citations to the literature are restricted to reports exemplifying particular points. Extensive bibliographies can be found in several recent reviews of the olfactory system (LeGros Clark, 1957; Ottoson, 1963; Moulton & Tucker, 1964; Wenzel & Sieck, 1967; Shepherd, 1972). Some appropriate introductory textbooks in relevant fields of study are also suggested.
The book is organized as follows. Chapter 1 consists of a brief nonmathematical review of the concept of the neuron and the interrelations among neurons that lead to the formation of interactive masses. New terms are defined and the central argument is presented.
In Chapter 2 the linear properties of neurons and their parts are reviewed. This provides an opportunity to introduce the use of linear differential equations and the Laplace transform method for solution. Mathematical description is not a prerequisite for understanding single neurons and is usually deemphasized. Description and prediction of the properties of masses of neurons cannot, however, be undertaken without the use of mathematics, and the review provides both some experience in describing the lower level models and some equations to be used as elements in constructing models at a higher level.
In Chapter 3 the ionic hypothesis is reviewed, and the nonlinear input–output relations of neurons in masses are expressed in terms of amplitude–dependent coefficients in linear differential equations. Chapter 4 deals with the relations between the states of activity of neurons, both singly and in masses, and the electrical fields of potential which are the principle means for indirect observation of the activity. <Page xii> Chapter 5 describes the properties resulting from feedback within neural masses. Chapter 6 analyzes the effects of the nonlinearities in the input–output relations of neurons on the behavior of masses. Chapter 7 contains some inferences concerning the mechanisms of neural signal processing at the level of neural masses.
The book is intended as a model for an advanced text in neurophysiology, and some understanding is assumed of the elements of the fields of linear analysis (DiStefano et al., 1967), probability (Parzen , 1960), statistics Anderson, 1958), theory of potential (Rogers, 1954), neuroanatomy (Gardner, 1968), electrophysiology (Katz, 1966), neuropharmacology (Goodman & Gilman, 1970), and experimental psychology (Hebb, 1958). Introductory courses in neurobiology and calculus should suffice for understanding the basic approach, with the help of a textbook on linear systems analysis. Introductory materials have been included to provide a coherent argument from first principles, and to provide guidelines for extraction of essential background from standard textbooks in neurophysiology and linear analysis, but not as a substitute for the textbooks.
The greater part of the experimental detail in this book is drawn from the mammalian olfactory system. There are two reasons for this. The primary reason is that neural mass actions reflected in the EEG are mainly identified with the mechanisms of adaptive behavior in vertebrates. The neural machinery of the spinal cord, brainstem, and cerebellum has the property of modifiability, but only the forebrain is capable of elaborating adaptive, goal oriented, purposive, learned, teleological behavior. The neural masses in the forebrain are also the only brain structures that generate well–developed EEG waves in the range of 1 to 100 Hz. When the EEG is present and orderly, adaptive behavior is generally found. When the EEG is absent, or is disorganized as in deep sleep, epilepsy, or general anesthesia, there is no adaptive behavior. By inference, the EEG is like a Rosetta Stone for deciphering the neural coding of adaptive behavior. The olfactory system is the simplest part of the brain to elaborate both.
The more obvious reason for emphasizing the olfactory system is that a particular point of view is being presented which has evolved from the study of the properties of this system. The application of the theory and methods described here to other systems must be based on detailed reexamination of the anatomy, electrophysiology, and behavioral correlates of those systems and not on casual generalizations. The intention in giving examples is to illustrate what kinds of data are needed and how they are obtained, as much as to construct a general theory. Students of spinal, cerebellar, and brainstem machinery may find the means to break some intellectual log–jams with the methods and concepts described here, but the message is mainly directed to students of the cortex and basal ganglia.
<Page xi>
Acknowledgments
The work described here has been financially supported by grants from the National Institute of Mental Health, MH 06686, the Foundations' Fund for Research in Psychiatry, 59–204, and the Guggenheim Foundation. Many of the illustrations in this book were prepared with the help of Brian Burke, Charmane Thomson, The Scientific Photographic Laboratory, and the Computer Center on the Berkeley Campus. Computer programming was by Brian Burke. The manuscript was typed by Barbara Kitashima. Permission is acknowledged for reproduction of figures from Biophysical Journal, The Rockefeller Institute; Journal of Comparative Neurology, The Wistar Institute of Anatomy and Biology; Experimental Neurology, Academic Press, Inc.; The Conduction of the Nervous Impulse, Liverpool University Press; American Journal of Physiology, American Physiological Society; Brain Mechanisms, Progress in Brain Research, American Elsevier Publishing Co., Inc.; Studies from the Rockefeller Institute, Rockefeller Institute for Medical Research; Journal of Cellular and Comparative Physiology, Wistar Institute of Anatomy and Biology; Journal of Physiology, Cambridge University Press; Physiology of Nerve Cells, The Johns Hopkins Press; Transactions of Biomedical Engineering; Institute of Electronics and Electronic Engineers.
The author wishes to express appreciation to the students, former students, and colleagues on the Berkeley faculty, particularly Professor O. J. M. Smith for introducing us to systems analysis, Dr. Heinrich Bantli and Dr. Soo–Myung Ahn for advice and comment on the manuscript, and Professor T. Prigogine whose invitation to lecture as Titulaire de la Chaire Solvay 1974 at the Université Libre de Bruxelles provided an impetus for writing this book.
The first printing of this work was instigated by Bill Woodcock of Academic Press in 1972 and published in 1975. The last of a run of 2,200 copies was sold in 2000, and the book went out of print in 2000. The copyright was returned to me in 2002.
This electronic edition was prepared with the assistance and dedication of Mark Lenhart. It consists of 509 pages containing 7 Chapters, 8 footnotes, 185 figures, 239 symbols, and 664 equations. This was a monumental task of transliteration, correction of errors in the First Edition, and proof-reading, and his work has been greatly appreciated by myself and no doubt by all readers of this work.
<Page xiii>
Notation
A. Individual Neurons and Neural Sets
A1. Coordinate Variables
t real time 14, 52
T lag time (e.g., from stimulus) 55
Ta conduction (propagation) delay 83
s Laplace complex frequency 41, 68
∆T duration of an observation or time window 55
x, y, z Cartesian spatial coordinates 34
X vector denoting x, y, z 37
A2. Time–Dependent Functions and Operations
δ(t) Dirac delta function 60, 77
µ(t) step function 65, 77
o(t) time function for active state 17
f(t), v(t), p(t) time functions for observable events 52
, –1 Laplace transform and its inverse 69
F(s), V(s), P(s) linear operations in the frequency domain 68,272
v’(t) measured (digitized) time function in the wave mode 53
p’(t) measured (digitized) time function in the pulse mode 53
[v’(t)] = 
(T) wave mode ensemble
averages for fixed T 55
[p’(t)] = 
(T) pulse
mode ensemble averages for fixed T 55
, 
average of v’(t),
p’(t) over time t 207, 303
ε(t), ε(T, X) random
error, noise, or least mean square deviation, e.g., [
(T)–v(T)]=ε(T) 53
<Page xv>
A3. Equivalence Statements
= equals
is defined by
≈ is approximated by
is equivalent to or replaced by
B. Individual Neurons
B1. Subscripts Denoting Structure
a axonal 95
d dendritic 95
s soma 95
m membrane 64, 87
l longitudinal 65, 87
e external 64, 87
i internal 64, 87
B2. State Variables
o active state 14
i current 52
v potential difference 52
p pulse rate 52
j current density (vector) 177
j current source–sink density 179
q fixed charge equivalent to j 173
ξ fixed charge density 173
E electrical field intensity (vector) 173
B3. Amplitude–Dependent Functions
Gd(p, t) nonlinear time–varying function for pulse to wave conversion 17, 144
Gs(v, t) nonlinear time–varying function for wave to pulse conversion 17, 101
B4. Functional Parameters and Acronyms
τ passive membrane time coefficient (fixed number) 88
a passive membrane rate coefficient (fixed number) 99
Tb distributed delay time coefficient (fixed number) 92
b lumped cable delay rate coefficient (fixed number) 98
Tc, 2/c lumped synaptic delay time coefficient (fixed number) 93
<Page xvii>
λ membrane length coefficient (fixed number) 88
θ axonal conduction velocity 82
EPSP excitatory postsynaptic potential 20
IPSP inhibitory postsynaptic potential 20
vEPSP equilibrium potential for EPSP 141
vIPSP equilibrium potential for IPSP 141
B5. Structural Parameters (capital letter = total; lower case = specific)
cm, Cm transmembrane capacitance 63
rm, Rm transmembrane resistance 63
g, Gm transmembrane conductance 122
r1, R1 longitudinal resistance 87
re, Re external resistance 87
ri, ri internal resistance 87
ρ volume specific resistance 178
µy mobility of the yth ion species 123
Py membrane permeability to yth species 128
C. Neural Sets
C1. Topological Hierarchy Based on Interconnectivity
KOe noninteractive excitatory set 26
KOi noninteractive inhibitory set 26
KIe interactive excitatory set 29
KIi interactive inhibitory set 29
KII KIe set interactive with KIi set 31
KIII interactive KII sets 34
C2. State Variables
o, O active density in any form 34
op, Op pulse density (o, O in pulse mode) 35
ov, Ov wave amplitude (o, O in wave mode) 35
f, F lumped piecewise linear activity density 43
p, P pulse rate of neuron or of unit cluster 193
v, V potential difference in a neural mass 194
u, U o, O in wave and/or pulse mode 37
j current source–sink density at a point in space 179
q fixed charge at a point in space, equivalent to j 180
ψ, Ψ pulse density at a point in space, homologous to q 250
<Page xviii>
C3. Time–Dependent Functions and Operations
AEP average
evoked potential, a form of (T) 37
PSTH poststimulus
time histogram, a form of 
(T) 37
EEG electroencephalogram, a form of v’(t) 45, 49, See also A2.
C4. Amplitude–Dependent Functions and Operations
G(v) G[v (E(t)] nonlinear function for wave to pulse conversion 38, 161
G(p) G[p(T)] nonlinear function for pulse to wave conversion 38, 165
g(v) nonlinear gain function for wave to pulse conversion 39,168
g(p) nonlinear gain function for pulse to wave conversion 39,167
C5. Probability Functions in Pulse and Wave Modes
P(v) wave amplitude probability density 148
P(p∩v) joint pulse and wave amplitude probability density 155
P(p|v) conditional pulse probability on wave amplitude 155
P(p|T∩v) conditional pulse probability on wave time and amplitude 156
P(p|T∩v∩ω) predicted pulse probability conditional on wave time, amplitude and frequency 162
(p|T∩v∩ω) observed pulse probability
conditional on EEG time, amplitude and frequency 162
P(T), (T) predicted observed normalized
pulse probability wave 159
P(v), (v) predicted observed normalized
pulse probability sigmoid curve 159
C6. Space–dependent functions and operations
h(X, y) spatial component of a neural activity distribution 194
H(X)
H[h(X, y),
z] linear divergence
operation in the spatial domain 250
H–1(X) H–1[h(X, y),
z] linear convergence operation in the spatial domain 252
p(T, x, y, z), p(T, X) space–time distribution of action potentials 194
v(t, x, y, z), v(T, X) extracellular potential field 194
fX(T), pX(T) basis function in time for potential field 196
vT(X), PT(X) basis function in space for potential field 198
q(X) equivalent charge distribution for potential field 197
ψ(X) spatial distribution of activity in pulse mode 250
<Page xix>
x, y coordinates of a neural surface 34
x coordinate parallel to direction of propagated input over surface 219, 254
y coordinate orthogonal to direction of propagated input over surface 220, 254
z coordinate orthogonal to a neural surface 223, 254
x0, y0, z0 center of a response domain 223
x0, y0, z = 0 epicenter of a response domain 220
x=0, y=0, z=0 stimulus site 219
C7. Generalized Functions and Operations
o(t, x, y, u) activity density function 41, 249
ov(t, x, y, u) activity density function in the wave mode 41, 249
op(t, x, y, u) activity density function in the pulse mode 41, 250
O(s, X, U) operation in the domains of frequency, amplitude, and space; if separable, equivalent to G(U)H(X)F(s) 41, 275
Ω time and space operator denoting the form of neural signals transmitted by KII sets 434
o(Ω) wave packet carrying signal of a KII set and constituting a form of o(t, x, y, u) of a component KI set 434
φ(X) phase of p(T) as a function of X in P(p|T∩v∩ω) in a KI component of a KII set 432
~p(X) modulation amplitude of P(T) as a function of X in P(p|T∩v∩ω) in a KI component of a KII set 159, 433
C8. Parameters Relating to Interaction
e, + excitation 19, 26
i, – inhibition 19, 26
Kf, K', Kµ forward gains of input channels 293
ge, gi, go forward gains of transmitting to receiving subsets in KI sets 39, 166
Ke excitatory positive feedback gain 40, 277
Ki inhibitory positive feedback gain 41, 343
Kn negative feedback gain 42, 280
Ko reference feedback gain in steady state 44, 299
K2 lumped feedback gain–KII feedback modeled by a diffusion process 327
T2 lumped feedback delay–KII feedback modeled by a diffusion process 328
δe, δi operating bias coefficients of KIe, KIi, and KII sets 348
γe, γi rate constants for wave to pulse conversion 160
ζe, ζi rate constants for pulse to wave conversion 165
<Page xx>
vo, po mean wave amplitude and pulse density 160
e, i difference between v0 and vEPSP, v0 and vIPSP 164
r ratio of ve/vi or ζi/ζe 165
v*e, v*i effective operating amplitudes of KIe and KIi sets 170, 348
C9. Parameters Relating to Time and Space
a, b, c, d, ... open loop rate coefficients (fixed numbers) 108
α, ß, γ, ..., ω closed loop rate coefficients (gain–dependent variables) 61, 110
φj phase of the jth sinusoidal time function 61, 111
Aj, Bj,.., Pj, Vj amplitude coefficients of the jth time function 110
dj the jth pole of a transfer function 69, 108
zj the jth zero of a transfer function 69, 108
×, * open loop single pole, double pole 278
O open loop zero 278
∆ closed loop pole 278
□ closed loop zero 278
η distance (fixed number) 82
σx, σy standard deviations of a spatial Gaussian (normal) distribution h(X, y) 253,255
󵕵+l standard deviation of a Gaussian divergence (spatial) operation H(X) 260
σt standard deviation of a Gaussian dispersion (time) operation F(s) 83
C 10. Subscripts and Acronyms Referring to Structure
A Type A prepyriform neurons (excitatory), KIA set 47
B Type B prepyriform neurons (inhibitory), KIB set 47
C Type C prepyriform neurons (excitatory), KOC or KIC set 47
G bulbar granule cells (inhibitory), KIG set 32
M bulbar mitral–tufted cells (excitatory), KIM set 32
N anterior olfactory nucleus (unclassified) 47
P bulbar periglomerular neurons (excitatory), KIe set 33
R olfactory receptors (excitatory), KOR set 32
LOT lateral olfactory tract 215, 234
PON primary olfactory nerve 32, 212
Am(s) transfer function for PON, orthodromic to bulb 308
Ak(s) transfer function for LOT, antidromic to bulb 308
At(s) transfer function for LOT, orthodromic to cortex 325
<Page xxi>
Preface to the (electronic) Second Printing
The field of Neurodynamics has advanced steadily in the past quarter century but still can be regarded as a giant sleeping in infancy. I regard this work as a foundation on which to base further work. Its first premise is that brains are hierarchically organized, with multiple levels from quanta to neurons, and yet more levels from neurons to societies of brains. Its second premise is that within each level the optimal formulation of explanatory models is by use of linear systems analysis and ordinary differential equations, for which the domains of application can be expanded by piece-wise linearization with supplementary describing functions. Its third premise is that the function of brains can be approached by separation into domains with the independent variables of space, time, and amplitude, thereby avoiding, in first approximations, the complexity of nonlinear, time-varying, nonstationary equations. Its fourth premise is that reciprocal networks and recursive functions are essential components of molecular, neural, and systemic networks, and that the optimal groundwork is provided by linear feedback theory. Its fifth premise is that measurements of the functions of neurons and brains have little meaning, unless they are accompanied by measurements of the behaviors in which the subjects are engaged at the time of measurement. Concerning the functions of neurons and neural populations, the best approach is to experiment by trial-and-error and find domains in which superposition holds. Then it is possible to decompose and measure recorded events with linear basis functions, and to simulate the events with solutions of sets of ordinary linear differential equations. Owing to the Laplace Transform, the work of solution is largely done with look-up tables and the delights of high-school algebra, so as to avoid, or at least postpone, the complexities of arcane higher mathematics. Investigators who are exploring an unknown domain of dynamics can carve out an enclave and home base of certainty, based in combined mathematical and experimental proof, from which to foray into the future.
Advances have occurred in three main areas between the first and second printings. First, the forward gain function of neural populations embodied in what has come to be called the "sigmoid curve" is not at rest but is displaced to the excitatory side. The equation that fits the data is derived from the Hodgkin-Huxley equations. It opens the way to exploring an area of cortex like an axon as bistable, with a resting, receiving, diastolic state and an active, transmitting, systolic state. Second, the spatial patterns in the primary sensory areas of activity oscillating in what has come to be called the gamma range (so named by myself and Steve Bressler in 1980) lack invariance with respect to stimuli, and have been shown to depend on the state of the subjects, their history, and their expectancy of the future. This opens the way to studies of higher functions such as intention, the construction of meaning, and analysis of the genesis and functions of consciousness. Third, the development of the theory of chaos has enabled us to advance beyond the conception of limit cycle attractors embedded in noise, which dominates the present work, into an exploration of nonconvergent dynamics. The initial impact of chaos theory was from models based in three coupled nonlinear ordinary differential equations that displayed aperiodic solutions with infinite sensitivity to initial conditions. This proved to be a false start, because the conditions requisite for the models — stationarity, autonomy, and freedom from noise — do not hold for brains. The word "chaos" has lost its value as a prescriptive label and should be dropped in the dustbin of history, but the phenomenon of organized disorder constantly changing with fluctuations across the edge of stability is not to be discarded. It is directly addressed in this work by the root loci that cross the imaginary axis of the complex plane into the forbidden right half. That feature alone makes this work well worth the effort to keep it alive and moving across the electronic landscape.
<Page 1>
CHAPTER 1
Topological Properties
1.1. The Approach to Neural Masses
1.1.1. DIRECT AND INDIRECT OBSERVATIONS
The basis for our curiosity about the brain is the fact that we observe interesting and purposive behavior in ourselves and in animals with intact brains, and that this behavior is disorganized or absent when the brain is damaged or destroyed. We infer that the brain is a kind of biological machine that is susceptible to manipulation through specialized receptors on and in the body, that can store and modify past input, and that can organize complex sequences of muscular contraction and glandular secretion. For 2000 years we have been interested in the physicochemical properties of this machine, and new developments in our understanding have seldom lagged more than a generation behind the major advances in physical theory and technology.
Direct examination of the living brain is not very rewarding. When deprived of its protective coverings, the brain surface appears smoothly folded, pinkish grey, and faintly translucent with a web of pulsating red and purple blood vessels on its surface. It is yielding and moist to the touch. When removed from the cranium and cut, it has the texture and faint aroma of soft cheese. The cut surface is marbled pink and white. (The "grey matter" of the outer shell or cortex and the nuclei becomes grey only when it is hardened by formaldehyde.) Its features are so incongruous that artists notoriously have difficulty in drawing the brain, unless helped by a skilled anatomist. The point is instructive, because for no other organ is what <Page 2> we see so strongly dependent on what we expect or are told to see (Section 7.2.3).
Most of what we know and believe about the brain has come from indirect observations. These include stimulus–response relationships with or without brain lesions or the prior administration of drugs, recordings of amplified electrical potentials found or induced in the brain, and the structural residues seen under microscopes following exsanguination, chemical impregnation, and thin–sectioning of brain tissues. But the technologies of indirect observation are so complex, and the results are so dependent on variations of factors too numerous to explore systematically, that we cannot collect "facts" in the way a naturalist collects specimens for classification. Instead we maintain a dialectical process in which our expectations largely determine the techniques, and the results modify the expectations or are rejected.
This dialectic cannot be avoided, because fundamentally we have no conception of how the brain functions as a physicochemical machine. That is one of the main objectives of our search. Most new developments in understanding arise unexpectedly from the availability of new techniques for indirect observation and not from a basis in theory. They give limited insight into narrow aspects of brain function, falling far short of our ultimate aim. Often in our impatience to bridge this gap we apply to brain function certain broad concepts which accompany a new technology, but which hold for the brain only by analogy or metaphor. Familiar examples are the Cartesian pump, the Helmoltz telegraph system, and more recently the servomechanism, the digital computer, and the holograph. The use of analogies to guide experimentation is at best merely suggestive and at worst can be highly misleading.
Despite this and other pitfalls the circuitous and tentative dialectical process over the past 300 years has enabled us to accumulate a solid core of data on some elementary physicochemical properties of the brain and its parts. In order for us to interrelate and understand these data, we have evolved a series of symbolic (verbal and mathematical) representations or concepts, which are commonly called models. Some classic examples are the passive membrane, ionic permeability, the core conductor, the volume conductor, the synapse, the neuron, the neuron pool, and the neuron chain or reflex arc. These and related models constitute our best approximation to date of a systematic understanding of brain function on its own terms without recourse to metaphorical generalizations. They are characterized by flexibility and adaptiveness ; they may be presented in very simple form or made as elaborate as the needs of the user require. They are always abstractions; the only complete description of a system is the system itself. Because models are made to interrelate facts, they are used to predict <Page 3> facts. Their hallmark is immediate testability by means of comparison of prediction with experimental observation, and it is conformance to the predictions of a model that makes a "fact" significant and not trivial. They are the foundation for future developments in our understanding, and, as successful models, they provide us with some insights into what the epistemological characteristics of future models will be.
1.1.2. THE USE OF MODELS IN A HIERARCHY
Each of our models has four essential aspects. The first is a structure corresponding to an anatomical or material element in the brain, such as a membrane or a neuron, which has an inside, an outside, and some sort of boundary. Specification of structure includes also a list of connections and interconnections, which is the topology of the model. The second is an abbreviated list of input–output relations that summarizes the experimental observations on the element being modeled, including specification of the conditions and their limits within which the model is thought to be valid. The third is a set of statements describing the flow or transfer within or across the structure, the nature of what is being transferred, i.e., whether it is a form of energy, a substance, a signal, or a state of activity, and the magnitude of that entity in each part of the model. The fourth is a description of the relations of the model to its parts, which are models of a lower order, and to other models, which together comprise a model of a higher order. For example, the membrane model can be interpreted in terms of the properties of the bimolecular lipid layer, which is a model for a component of the membrane, or in terms of the model of a neuron of which it is a part.
The fourth aspect is by far the most difficult for comprehension and yet the most important for understanding the physicochemical properties of the brain. As a tissue, the brain consists of molecules, organelles, neurons, and masses of neurons. In parallel to this hierarchy of structure, we have a hierarchy of models at each of these levels as well as some in between. The power of these models for understanding lies in their use in predicting relations between data taken at different levels in the hierarchy. An example is the prediction by Hodgkin & Huxley (1952) of the amounts of sodium and potassium transferred across the axon membrane for each nerve impulse at the ionic level from measurements on the transmembrane potential and conductance at the cellular level. The difficulty lies in establishing correspondences between a part of a model at one level and the part conceived as a model in its own right. Strong models minimize the difficulty by achieving conceptual proximity, so that, for example, ionic conductance can plausibly be interpreted in terms of a diffusion channel <Page 4> through a membrane or a carrier molecule. Weak models are characterized by too great a jump between the parts and the whole, as when attempts are made to link ionic conductance changes to the process of memory as defined, for example, in relation to conditioned reflexes. It is not illogical to try this, but the interpretations are vague, ambiguous, and uncertain.
Yet in studies of the brain it is necessary to establish relations between physicochemical measurements and on–going behavior, because the brain is infinitely complex, and we wish to identify and emphasize those of its properties related to behavior, not merely those related to our choice of instruments. We establish these relations by means of hierarchical sets of models that form traditional views of the brain.
This book is based on the idea that two of the main traditions in neurophysiology are ripe for fusion. In each tradition it is held that the brain is composed of neurons, and that the behavior of animals depends both on the properties of neurons and on the ways in which they are functionally connected and interconnected. This is their topology. The older tradition emerged from 19th century studies of the reflex properties of the brain and spinal cord and was brought to fruition principally by Sherrington (1906). Owing to the limitations of his techniques consisting of surgical ablation, the inductorium, the focal electrode, and the muscle lever, the central nervous system in this tradition is conceived as a mosaic of centers. Each center is a pool of neurons acting in parallel with each other, and the action of each center is considered to be the sum of the actions of its neurons. Interconnections are between centers rather than within centers. There are centers for receiving each modality of input and for initiating each type of motor response. The main tasks of neurophysiologists for brain analysis are to locate the centers and chart the fixed and temporary connections between them. Fine structure of neural activity is not denied, but it is considered to be inaccessible to observation.
The introduction of the oscilloscope, the electronic amplifier, and the microelectrode has brought a new tradition in the second quarter of this century. The neural action potential has come to be viewed as the universal currency of the nervous system. Theorists such as McCulloch & Pitts (1943), Hebb (1949), and Walter (1953) have shown that networks of individual neurons might perform virtually all of the operations seen or inferred in animal behavior, and experimentalists such as Sperry (1951), Barlow (1953), Mountcastle (1957), Lettvin, Maturana, McCulloch & Pitts (1959), and others have demonstrated remarkable specificities in the topographical arrangements and behavioral correlates of single neurons. The traditional Sherringtonian central excitatory and inhibitory states are entirely too gross to account for these fine structures. The concept of the center is replaced in this tradition by the concept of the neuron as the bearer of behaviorally <Page 5> significant information. The telegraphic network of centers is replaced by the telegraphic network of individual neurons.
In this book it is proposed that both concepts are partially correct. The center is redefined as a domain of cooperative activity rather than as an anatomical pool of neurons. Activity sustained by a mass of neurons comprising a center is cooperative by virtue of synaptic interconnections among the neurons. Within the cooperative domain each neuron generates a pulse train, which has a different but related time sequence compared with those of its neighbors. The nature of the transmitted information can only be found by examining the pulse trains of many neurons recorded over the same period of time, because it is the interrelation of the output of all the neurons in the domain that defines the signals, and not the averaged activity across the domain or the activity of one or a few neurons.
The models for interactive neural masses are designated as neural sets. These models of masses differ as much from the model of the neuron as that does from the model of the membrane. The basis of the difference is the existence of locally dense synaptic interaction. When a large number of such highly nonlinear elements as neurons mutually influence each other diffusely over an extended range in time and space, a macroscopic entity emerges in the mass that requires a new model for description. If the flow of influence within the mass is sufficiently strong and over a sufficient number of channels, then sudden transitions occur in which completely new macroscopic properties emerge and define a new state or a new set of states for the mass. Such states are called macrostates and are characterized by widespread cooperative activity of neurons. Neither the macroscopic properties nor the cooperative activity can be explained or understood at the level of the individual neuron, any more than the properties of ice can be explained in terms of a molecule of water.
1.1.3. MACROSCOPIC FORMS OF COOPERATIVE NEURAL ACTIVITY
Models of cooperative activity have been advanced in numerous forms during the past 20 years by physiologists (Lilly & Cherry, 1954; Cragg & Temperley, 1954; Gerard, 1960) psychologists (Lashley, 1929; Köhler, 1940; Tolman, 1948; John, 1967; Pribram, 1971; Grossberg, 1974b), and theorists (von Neumann, 1958; Beurle, 1956; Griffith, 1963; Wilson & Cowan, 1972) among others. Until recently there have not been sufficient experimental data with which to test, elaborate, and adapt these concepts, nor an adequate language to describe them.
In relation to specific brain systems two developments have changed this situation. One is the availability of a new technology. It is based on the integrated circuit that permits construction of large numbers of reliable <Page 6> amplifiers, so that observations can be made simultaneously on many electrodes, and on the digital computer that permits the acquisition, storage, and processing of measurements in quantities that only 10 years ago were almost inconceivable. This technology will be as revolutionary in its impact as the Faradic stimulator and the cathode ray oscilloscope were in their times. We have only begun to learn how to use it.
The other development, which is more a promise than an actuality, comes from theoretical chemistry. When several chemical reactions in a distributed system are coupled by diffusion, so that the products of each reaction become the reactants of another, steady state inhomogeneities in the distribution of the reactants and products may occur. The inhomogeneities arise in a previously homogeneous system. At any instant the inhomogeneous concentrations conform to a geometric pattern in the space of the reaction system, so it is said to have a dynamic pattern. The pattern may or may not vary with time, and if it does, the variation may be periodic.
Four conditions are required for the formation of such dynamic patterns. There must be two or more nonlinear transformations of substance or energy in the system. The transformations must be coupled, so that the reactions occur topologically in one or more closed loops. There must be delays in either the transformations or in the diffusional flows or both. Finally, the system must have a continuing source of energy. The patterns are in fact dynamic structures, which feed on energy and dissipate it as heat. For this reason Prigogine (1969) used the term dissipative structures to distinguish them from equilibrium structures, such as crystals.
Because such reaction systems are open, the theory of equilibrium thermodynamics does not hold. Over the past 20 years the theory of irreversible thermodynamics has been further developed and applied by Turing (1952), Glansdorff & Prigogine (1971), Katchalsky, Rowland & Blumenthal (1974), and others, as a means for the mathematical description of diffusion–coupled chemical reactions leading to spatial forms.
Katchalsky (1971) and Katchalsky et al. (1974) were quick to leap beyond Liesegang rings and the Zhabotinsky reaction and to see an equivalence between diffusion–coupled reactions and the interactions within masses of neurons. Each neuron continually performs nonlinear transformations at synapses and trigger zones. Feedback connections exist between neurons at high densities, and neurons in masses are coupled with sufficient density to simulate the continuum of chemical reactions and thus to create neural macrostates. The flow of neural activity along axons and across synapses in masses is analogous to the diffusion of chemical reactants. Also self–sustaining levels of activity in excitatory neural masses with internal feedback provide the analog of varying levels of energy input to other masses.
Katchalsky and co–workers reasoned that in any system containing a <Page 7> large number of nonlinear elements, which are diffusely coupled and therefore interactive in a continuum, the properties of the whole depend on the rates of energy inflow. If there is no inflow, the system stabilizes in a zero equilibrium state. If energy is fed in, the system is driven progressively away from equilibrium until a critical level is reached at which a sudden change in state occurs. A new macroscopic entity emerges that is characterized by a dynamic pattern or dissipative structure. With further increase in energy inflow, the system changes within this state and may then jump to yet another state, or through a series of new states, each with a dynamic pattern uniquely different from the others. Each new state may be a stable state and yet none is at zero equilibrium. †
† Katchalsky was enormously enthusiastic over the latent possibilities in further exploration of these concepts, both in mathematical and experimental analysis. His participation was cut short by his tragic death. Nevertheless he contributed some remarkable insights into the nature of neural activity, which stemmed from his deep knowledge of polymer chemistry. For purposes of analyzing different kinds of cooperativity a hierarchy of levels of interaction within neural masses has been suggested and in acknowledgment his name has been given to these levels.
The postulate that macroscopic forms of cooperative neural activity exist in the cortex, which transcend action potentials and synaptic potentials and which are analogous to diffusion–coupled chemical reaction rates, depends on several assumptions regarding the conditions of central neural activity.
(1) The global connections among neurons can be determined, in the sense of topographically organized tracts connecting areas of cortex and nuclei, but the precise local connections cannot be known, except for a small sample of representative neurons, and in any case need not be known with respect to most forms of animal behavior. That is, the precise trajectories of all pulse or wave transmissions in the cortex during a behavioral event are not knowable and need not be known, but only an average or instances representing the average (see Griffith, 1971, Chapter 8, for an equivalent statement regarding statistical mechanics) need be known (See also Goodwin, 1963).
(2) Macroscopic activity is continuously distributed in space in certain masses of the brain (Freeman, 1972f). We may then define a volume or surface element, which is sufficiently large that the level of activity of the element is an average over the ensemble of neurons in the element and which is sufficiently small with respect to the entire mass that the average is valid over the element, though the activity level over the mass is not homogeneous. This assumption is analogous to the assumption for an inhomogeneous chemical reaction system that a volume element exists around each point large enough to define a concentration for each ion <Page 8> species and small enough to represent the concentration across the element as uniform (Glansdoriff & Prigogine, 1971).
(3) The time and distance scales of macroscopic neural events are much longer than the scales for individual neural events. We can say that the relaxation times for action potentials and synaptic potentials range from .5 to 5 msec, and that the periods over which ensemble averages exist range upward from 100 msec. Similarly, chemical reaction rates in general are slower than the relaxation times of reactive ionic collisions. Also the dimensions of a macroscopic event conform to those of a nucleus or area of cortex and not to the dendritic arbors of individual neurons.
(4) Differential equations can be constructed to describe macroscopic states of an interactive mass of neurons without explicit reference to action potentials, thresholds, refractory periods, etc., because the proportion of neurons generating pulses at any moment in the mass is small, and that proportion tends to be uniformly distributed through the mass. The typical cortical neuron receives pulses continually but gives pulses at rates less than 10/sec, and its pulses for the most part appear to occur randomly in time. Less than 1% of its lifetime is spent in the state of pulse transmission despite continual pulse input. This condition is analogous to the predominance in a chemical system of elastic over reactive collisions and the tendency toward a locally Maxwellian momentum distribution function of the reactants (Prigogine & Nicolis, 1973).
(5) The activity level of each neuron, to the extent that it is determined by the activity levels of the neurons in its surround, must on the average be consistent with the ensemble average of the activity level in the surround, because it is a part of the surround. This assumption of self–consistency (Prigogine & Nicolis, personal communication) enables us to describe neural interactions involving mutual excitation or mutual inhibition among large numbers of neurons with feedback equations (Section 1.3.4).
Cooperative activity is not unique to masses of neurons and can be inferred or observed to exist in neural networks. These assumptions, however, lead to expectations of neural activity quite different from the discrete characteristics of activity in networks. In other words, that which emerges from the study of neural mass action is not merely an extension of current understanding; it is revolutionary in the sense defined by Kuhn (1970). In each field science grows not by continuous evolution, but by discontinuous stages. Each stage can be characterized as having a paradigm, which consists of a set of elements and assumptions, a collection of techniques, some characteristic observable events, classical experiments, and agreed–upon methods of description and proof. Most scientific efforts are directed toward the extension and elaboration of established paradigms. It is only when <Page 9> substantial experimental data are accumulated that are incompatible with or are unexplained by an accepted paradigm that a need for a new paradigm becomes clear. Once established, a new paradigm either replaces an older one or incorporates it into a broader view. The change is often marked by confusion and misunderstanding, because the techniques, data, and methods of proof may be so different that logical confrontation is not feasible.
To illustrate this point, a short list of the attributes that characterize the neural network paradigm is given in Table 1.1. This paradigm was introduced by McCulloch & Pitts (1943), was elaborated principally by Hebb (1949) and Eccles (1957), and has become the mainstay of contemporary studies of sensory and motor mechanisms. It has by no means completely replaced the Sherringtonian paradigm that preceded it, nor is it likely to be replaced by a paradigm for mass action. The list of attributes shows that paradigms need not be considered as true or false, but they are more or less appropriate in addressing certain questions, particularly those concerning the relation of brain to behavior. The way in which the questions arise or are cast depends on the techniques of observation, the part of the brain under study and the kind of behavior being pursued. In the best examples from the Sherringtonian paradigm the properties of the spinal cord, brainstem, and cerebellum in relation to postural and locomotor reflexes are studied with the techniques of stimulation and ablation. In the network paradigm the <Page 10> properties of single units are observed with microelectrodes in sensory and motor systems, in relation to sensory processing and the formation of stereotypic movements. There are, of course, as yet no classical experiments in the mass action paradigm, though a likely candidate is listed in Table 1.1 as the study of averaged evoked potentials (AEPs) and poststimulus time histograms (PSTHs) in relation to the formation of conditioned reflexes (John, 1967).
In the following chapters some evidence that macroscopic neural activity exists in the brain is presented and the methods by which the evidence is obtained is shown and evaluated. Descriptions are given for topological, time–, amplitude–, and space–dependent properties, starting with single neurons and parts of neurons and progressing to macrostates by use of the concept of neural sets. Formal descriptions of dynamics are restricted to lumped sets, because the data concerning spatial distributions of neural activity are still largely inadequate to guide the construction and conditions of solution of explanatory partial differential equations. In this and other respects, the more pressing needs and promising opportunities for further development are noted along the way.
1.2. Single Neurons
1.2.1. THE STRUCTURES OF NEURONS
The foundation of neurophysiology is the neuron doctrine. According to this concept the tissues of the central and peripheral nervous systems are composed of great numbers of tiny compartments or cells. Each cell consists of a collection of chemically active structures embedded in a watery medium called the cytoplasm, and it is bounded over its entire surface by a thin layer of fatty material called the membrane. Though it is very thin, the membrane is a relatively strong diffusion barrier and separates the intracellular compartment of each cell from the extracellular compartment common to all the cells.
There are two kinds of cells. First, the neuroglia in the brain and spinal cord and the Schwann cells of the peripheral nerves, which predominate numerically, are subsidiary cells serving to separate the neurons from widespread contact, to maintain the constancy of the chemical composition of the extracellular compartment, and possibly to effect slow or long–term changes in the functional properties of the neurons in the brain. Second, the neurons are the transactional cells, which receive input from peripheral and central receptors or other neurons, and which transmit over short and long distances within the brain or to peripheral effectors.
Each neuron has a spheroidal nucleus embedded in the cytoplasm, which <Page 11> serves to specify the location of its home base. The nucleus is essential for long–term survival and functioning, but it does not enter into the transactions discussed here. The expanded region of cytoplasm including the nucleus is the cell body or soma. From the soma extend one or more filaments or "processes" for varying distances, which usually branch repeatedly and taper as they do so. The entire set of branching trees is covered with the continuous membrane including the tips of the branches. Two types of cell filaments are distinguished on morphological grounds. The axon has more watery cytoplasm, a smoother surface, fewer branches, greater length, and commonly a laminar coat of fatty material called myelin. The dendrite or dendritic tree has denser cytoplasm, an irregular surface, and more branches, which are often studded with small protrusions called spines and gemmules. There is only one axon for each neuron (though a few types have none), but a neuron may have one or several dendritic trees (though some have none).
The filaments of a neuron are not randomly distributed in the brain. Though there is considerable variation in the size and shape of neurons, there is sufficient regularity to classify neurons into types. For each type there is a characteristic location, orientation (with respect to the brain), and length of axon, and a pattern of distribution of its branches. The branches may be distributed along a line, in a surface, or in a volume, and there is a measure for the width of distribution in each dimension. The dendrites also have reproducible characteristics in terms of the number of dendritic trees, orientation, length, degree of branching, etc. For each type of neuron, however, there is variation in the locations of terminal branches. The detailed geometry is known for only a very few neurons among the billions that comprise the brain. For most neurons we must work with averages of measured distributions. We assume as the null hypothesis that the local distributions around the means are random, in the sense that the precise locations of endings are neither predictable nor knowable and do not affect the performance of neurons. This is a working hypothesis (Section 1.1.3) that is continually subjected to challenge and restriction (Sholl, 1956).
1.2.2. THE OPERATIONS OF NEURONS
The typical neuron has a dendritic tree as its receptor pole and an axon conjoined to the dendrite at or near the soma as its effector pole. The dendritic membrane forms specialized membranous contacts with the axon tips of other neurons, which are the synapses. The membranes are closely apposed, but with few exceptions there is no cytoplasmic bridge between them. The dendrites receive input from as many as 105 axon tips from other neurons, combine the input, and deliver the resultant to the initial segment of the axon. The axon transmits the resultant to other parts of the <Page 12> nervous system, near or far, and distributes it in accordance with its pattern of branching.
The modes of operation of axons and dendrites are diametrically opposed. Dendrites receive pulse input and convert it to a continuously varying graded wave of ionic (synaptic) current (Bullock, 1959), which is manifested in the form of extracellular dendritic potentials or intracellular synaptic potentials. The pulse to wave conversion takes place at each synapse, whenever a pulse arrives on the axon terminal. The wave takes the form of a field of ionic current, which is distributed along the dendritic shaft. The output of the dendrites is the sum of waves resulting from all pulse inputs, which is delivered to the initial segment of the axon. Due to the electrical characteristics of the dendritic membrane and cytoplasm the sum is smoothed and attenuated (filtered), so that the irregularities resulting from trains of single pulses are minimized. The sum is transmitted (in the sense of translation in space and delay in time) to the initial segment of the axon. The amount of smoothing and attenuation of the part of the wave attributable to each pulse depends on the distance from the synapse to the initial segment. Due to the fact that synapses are distributed over the branches, the summation is carried out over space as well as over time. The convergence of the dendritic branches to the soma provides for the convergence of the wave activity in each neuron.
The axon responds to this wave input by generating a pulse train. Each pulse has a relatively fixed amplitude and duration. It is actively transmitted or propagated. It is said to be all–or–none because, except at the axon branching points, it tends to have constant velocity and amplitude. However, the peak amplitude of transmembrane potential for each pulse does vary depending on the resting amplitude of transmembrane potential just before the pulse and on other factors. The effectiveness of the pulse in synaptically generating a wave also depends on this and other factors, so the output of the axon is not really all–or–none, in terms of either the effect of single pulses or the frequency of pulses. Moreover, the output is distributed through the volume of the brain in accordance with the location of the axon branches and terminals. This constitutes divergence. Due to variation in lengths of axon branches or collaterals and to variation in conduction velocities in proportion to axon diameters, there is variation in the times of arrival of pulses at the terminals of branches of the same axon. This constitutes temporal dispersion.
Whereas dendrites provide for convergence, summing, and smoothing of their input, axons chop the resultant into pulses of relatively fixed size and varying interval, multiply it by means of branching and the state of its terminal membranes, and disperse it both temporally and spatially. The prime function of the dendrites is integration (the weighted summation of <Page 13> small quantities) in the wave mode, and that of the axon is transmission in the pulse mode. The model neuron representing the "typical" neuron is a localized unidirectional integrator and transmitter having several operations. It receives pulses and converts them to waves, sums and filters the waves over space and time, focuses the resultant, reconverts it to pulses, and transmits the pulses with delays, multiplication, and divergence (spatial dispersion).
1.2.3. THE STATE VARIABLES OF NEURONS
Each of these operations is dependent on the existence in the membrane of electromotive forces (emfs) that can move ions across the membrane in either or both directions. When an input activates a neuron, it causes the neuron from its metabolic energy sources to build emfs in some part of the membrane. That part of the membrane is said to be active, whereas the rest of the membrane is passive. When the emfs move a net electrical charge across an active part of the membrane, the same quantity of charge must move in the opposite direction across the passive membrane of another part of the same neuron (Section 4.1.2). This gives rise to a loop current in which current flows from the active region to the passive region inside the neuron and in the reverse direction in the extracellular space, or vice versa. The loop current is the only means by which one part of a neuron affects other parts of the same neuron in the operations of integration and transmission. (Metabolic and trophic phenomena are not at issue here.) For this reason the existence of a neural loop current is a necessary and sufficient condition for a neuron to be active. If there is no loop current, either the neuron is at rest, or it has uniform emf over the membrane and is neither transmitting nor integrating.
It is not possible to measure the strength of a loop current of a neuron unless the neuron is isolated in a closed chamber. However, the loop current is accompanied by changes in potential at points both in and around the neuron, with respect to a reference potential at a point some far distance from the neuron. Measurements of these potential differences are the main source of information about the properties of intact neurons. It is important to realize that all such measurements are indirect with respect to the state of a neuron. If it is active, there must be one or more loop currents, but these may or may not be detectable as potential differences with electrodes placed in and outside the neuron. The degree of activity of a neuron is dependent on the distribution and intensity of all its loop currents. However, the degree of activity is in no way the simple sum of the current loops, which are the basis for the activity.
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In order to describe, measure, and predict the activity of a neuron, we need to define precisely what is meant by its active state and not merely say that it is or is not generating synaptic currents and pulses. In the most general terms we must choose a set of measurements of potential differences (in the wave and pulse modes) and their rates of change, which are believed to manifest the activity of the neuron as a matter of inference and judgment based on extensive knowledge of the geometry and dynamic properties of the neuron. The loop currents vary with time, location, etc., and so do the numerical values derived from measurement.
By definition each ordered sequence of values constructed from the measurements of potential differences over time, distance, etc. constitutes a state variable. The set of all possible values of a state variable is called the state space of that variable. Any subset of values serves to define a domain in the state space of that variable. Then the state of the neuron is given by the minimal number of state variables necessary to describe the state uniquely. It should satisfy the following two conditions. First, the state at any time t1 determines uniquely the state of the neuron at any time t other than t1. Second, given the state at time t1 and the inputs at t1, which comprise the complete past history of the neuron, all of the state variables are uniquely determined. The set of all possible values of the minimal state variables constitutes the state space of the neuron.
The state of a neuron is more general than its active state. Other state variables may consist in the intracellular concentrations of various inorganic ions, ATP, protein, RNA, etc., their rates of turnover, size, length, location and rates of growth of fiber systems, and many others. We are not concerned with these, except indirectly to the extent that we can use them to explain in chemical and anatomical terms the loop currents already specified as essential to activity and the active state. Even with respect to loop currents, there are indefinitely many ways of constructing the minimal state variables in order to represent the same neuron uniquely with respect to its active state. We consider this in the next section. The important point here is that the active state of the neuron is given by an organized set of measurements, that describe its past input and its present levels of activity and their rates of change in such a way that the future activity of the neuron is uniquely predicted. The principle is exemplified in the measurement of action potentials by means of the Hodgkin–Huxley equations.
The problem of specifying the active state denoted o(t) can be greatly simplified when we adopt the postulate that if a neuron has only one output channel, then in so far as a neuron in a network is concerned, specification of only one state variable in terms of history comprising the initial conditions, present level, and rates of change suffices to determine its active state. If each input axon to the neuron is the output channel of another neuron, there is only one minimal variable for the active state of each input neuron. If the <Page 15> nonlinear transformations of pulse to wave and wave to pulse conversion are known, a set of measurements of either a pulse rate in the pulse mode p’(t) or a potential difference in the wave mode v’(t) provides the basis for evaluating the active state of each neuron.
If the neuron performs wave to pulse and pulse to wave conversions within its dendritic tree, or if it has more than one output channel, or if the nonlinear conversion functions are not known, then more than one state variable is minimally required to specify the active state of the neuron.
The reason for advancing the concept of the active state goes to the crux of the relation between a model and the thing it represents. In the real neuron, there is a dynamic on–going event of a certain magnitude, which is manifested by certain measurable physical quantities. The event and its manifestations are not identical. The event is represented in the model by a symbol, say o(t), and an observable state variable such as a pulse rate is represented by another symbol, say p(t). The experimental observations are measured and expressed as the state variable p’(t). The test of the model is the comparison of the prediction p(t) with the observation p’(t), but inference as to what is actually happening in the real neuron is from o(t) to o’(t), where o’(t) is the active state and o(t) is its representation:
The active state has often been considered to be a form of energy, because it is a scalar having relative magnitude, is subject to operations such as transmission and transformation, and is dependent on metabolic energy. It is not a form of energy, however, because it is not conserved (Freeman, 1972f). The simplest proof is that two action potentials propagating toward each other on the same axon undergo annihilation. Moreover, both excitation and inhibition, which result in raised and lowered active states, make positive demands on metabolic energy sources. The active state is therefore uniquely defined in the relation of the neuron to the neuron model, which specifies the active state as a form of neural output o(t), and it is not derived from or equivalent to analogous entities in any other physical system.
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1.2.4. SPECIFICATION OF THE ACTIVE STATES AND OPERATIONS
Let us recall that the neuron can be viewed either as a collection of parts or as a part in a collection of neurons (Section 1.1.2). When we talk about the "function" of the neuron as determined by its parts, we must assign an active state variable to each of its parts, but when we are concerned with the "function" of the neuron in a system of neurons, we assign one active state variable to each neuron and identify that variable with an observable in either the pulse mode or the wave mode. The observable in the wave mode is usually taken to be the level of transmembrane potential difference at the soma, because it is most accessible to measurement with an intracellular electrode at that site, and because that is the site of maximal dendritic convergence. It is also the usual site of anatomical origin of the axon and therefore the site of wave to pulse conversion. The observable in the pulse mode is usually taken to be the recording from the soma indicating a pulse or pulse train. The level or amplitude of activity in the pulse mode can be expressed as an instantaneous value (whether or not a pulse is present), an average over a time period of observation (pulse rate), or as the instantaneous pulse rate (the reciprocal of the time interval between two pulses). †
† Alternatively, in an ensemble of observations made with respect to some repeated external event, such as the onset of a stimulus or the firing of some other neuron, the active state can be estimated from the probability of pulse occurrence in each of an ordered sequence of short time intervals during a designated postevent time period. The pulse probability is the number of times a pulse occurs during a short time interval (equal to pulse duration) divided by the number of events (see Chapter 3).
However, when we treat the neuron as part of a system of neurons, the most significant aspect of its active state is the level of its output, meaning the amplitude of its effect on other neurons. This presents a major difficulty, because the amplitude of effect is determinable only by measurement on the observables of target neurons, except of course for the effector neurons in the motor and endocrine systems. A substantial part of our efforts in neurophysiology is explicitly or implicitly devoted to determining the relations between observables and active states (see Chapter 4). For example, the pulse rate or pulse probability of a neuron is generally valid as an index of its state, because once a pulse is formed, it usually sweeps the entire axon tree (the divergence property). Instances occur, however, in which transmission over some or all branches is completely blocked or is either more or less effective than normal. In other instances pulses are not detectable or do not occur, and yet transmission from the neuron takes place. Extracellular dendritic potential or the level of intracellular potential may then replace measurement of pulses as the basis for estimating the active state of a neuron. The cytoplasm of neurons is negatively polarized <Page 17> with respect to the extracellular space. An increase of active state is associated with decreasing intracellular negativity or depolarization, and a decrease in active state is manifested by hyperpolarization. Once again, these relations do not always hold, so that the observation must be tied to the state of the neuron by inference. The observable state variables must not be confused with the active state variables.
The dendrites perform the serial operations of pulse to wave conversion, convergence, summing over space and time, and filtering the resultant. The axons have the serial operations of wave to pulse conversion, transmission, multiplication, and divergence. It is logically possible to define a state variable specifying an active state of a neuron before and after each of these operations. However, the realizable state variables are those conforming to measurable quantities, which are usually the pulse rate at the initial segment and the level of dendritic polarization at the soma. When the operations listed above cannot be distinguished by experimental observation, they are combined together into a single operation.
An operation is specified by a certain kind of relation between any two state variables. Suppose that we measure state variables v and p in successive pairs and find that over a range of values of v, for each occurrence of the same value of v, there is the same value for p. We then say that p is a function of v, and that the function p = G(v) represents uniquely the transformation of v to p by the operation G. This specification holds whether the state variables are from the same neuron or from two different neurons. Suppose that we make a surgical cut across a neuron or between two neurons and find that we cannot thereafter construct a function for v and p. Also suppose in the intact system that we can fix v experimentally at any value in a range and find that we can control p. Then we infer that p is caused by v. What is the entity that is transformed or transmitted? For each part of a neuron there maybe a different entity, such as loop current for dendrites, pulses for axons, transmitter substance for synapses, etc. If the operation includes a sequence of differing entities, then it is convenient to represent only the magnitude of the entity by a generic abstraction, which we call neural activity denoted o. (An alternative abstract term, signal, is reserved for other use in Chapter 7.)
We can now say that neural activity is transmitted from one part of a neuron to another or from one neuron to another, and that it is transformed by a neuron or a part of a neuron. The level of each activity is a component of the active state, and the function of any two active states specifies a transmission or a transformation, which are operations. Operations are represented by equations that contain the state variables and certain fixed terms called the parameters of the neural system. The parameters are either time, space, or gain coefficients, which depend on the relation between the <Page 18> observed rates of change in the system and the choice of units for measurement of time, distance, and amplitude. When the units of measurement are taken from a particular set of operations of a neural system, the analysis is dimensionless. For example, if we construct a function to describe the operation of transforming dendritic current to a pulse train, we need a gain coefficient to relate microamperes to pulses per second. But if we measure dendritic current as a function of input pulse rate to the dendrites and arbitrarily express its amplitude in units of pulses per second, the gain coefficient is dimensionless.
To summarize, we postulate that the active state of a neuron is given minimally by a single state variable, provided the functions are known for pulse to wave and wave to pulse conversions. Measurements are made optimally at the soma on pulse and wave activities, partly because of experimental feasibility and partly because of the convergence–divergence properties of the neuron. Operations within and between neurons are described by constructing functions from sets of measurements of pulses and waves. The main sequence consists in pulse to wave conversion between neurons from the pulse variable of one neuron to the wave variable of another neuron, which is described by a time–varying nonlinear function Gd(p, t), and in wave to pulse conversion within the neuron from the pulse and wave variables of the same neuron, which is described by a time–varying nonlinear function Gs(v, t).
In applications this general form may require elaboration. The classic example of a neuron performing these serial operations is the spinal motorneuron (Eccles, 1957), and many neurons in other parts of the central nervous system conform to this type. However, there are also numerous variants, such as a variety of central neurons with electrical coupling or electrotonic synapses (Pappas & Purpura, 1972), neurons in the retina that do not generate detectable action potentials (Byzov et al., 1970; Werblin & Dowling, 1969), somatosensory neurons without dendrites but with peripheral axon terminals that operate like dendrites (Loewenstein, 1971), etc. The description of appropriate state variables and operations for these variants requires only minor modifications of the general form.
On the other hand, it is not always desirable to reduce the minimal data variables to the membrane potential or pulse rate at the soma. In some neurons, e.g., the hippocampal pyramidal cell (Kandel & Spencer, 1961) and the cerebellar Purkinje cell (Llinás & Nicholson, 1971), there are trigger zones located in the dendrites, which imply that intermediate stages of wave to pulse and pulse to wave conversion occur in localized parts of dendritic trees. The possible existence of pulses in apical dendrites (Gusel'nikova, Gusel'nikov, Tsytolovskii, Engovatov & Voronkov, 1970) of mitral–tufted cells in the olfactory bulb suggests that the active states of <Page 19> these neurons may not be minimally represented by two state variables in the pulse and wave modes. Moreover, both the mitral–tufted cells and the periglomerular cells have output by axons in conventional axodendritic synapses, and they also have output directly from their dendrites to the dendrites of other cells by dendrodendritic synapses (Rall, Shepherd, Reese & Brightman, 1966). There are, in effect, at least two synaptic output channels for each type of neuron. Consideration of these interesting arrangements is limited in order to simplify the development. The analytic approach being described here is competent to handle these problems, but detailed applications in these directions have not yet been undertaken.
1.2.5. INPUT–OUTPUT RELATIONS OF SINGLE NEURONS
There are two fundamental classes of neurons in terms of the nature or "sign" of their effects on other neurons. Synaptic inputs that depolarize a neuron and increase its pulse rate or pulse probability are called excitatory denoted e or +, and the input neurons and synapses are also called excitatory. Synaptic inputs that hyperpolarize a neuron and decrease its pulse rate or pulse probability are called inhibitory denoted i or –, as are the input neurons and their synapses. Most neurons receive input from both excitatory and inhibitory neurons, but with at least one exception (Kandel, Frazier & Coggeshall, 1967) their output is either excitatory or inhibitory, but not both. Obviously the active state of a neuron is independent of the polarity of its output, and the output can be given a positive sign if it is excitatory and a negative sign if it is inhibitory.
The sign of action is a higher–order property of a neuron, because it refers to other neurons rather than itself, in terms of how it influences or is influenced by those neurons. Attempts thus far have been unsuccessful in determining the structural or chemical properties unequivocally specific to excitatory and inhibitory neurons, and it is still unclear whether the specialization of excitatory and inhibitory synapses lies in the presynaptic (input) or postsynaptic (receiving) membranes of synapses, or (most probably) both. The designation of the output of a neuron as excitatory or inhibitory should always include specification of the target neurons or effectors to which the action refers. The designation of its state as excited or inhibited refers only to the neuron, but from that state and a preceding input we can infer the sign of input.
The operations of the synapse and of the initial segment are now considered in more detail. If a pulse is delivered to a neuron at rest by way of the axons ending on its dendrites and soma, a brief wave of dendritic potential occurs, which is a postsynaptic potential (PSP). If the input is excitatory, <Page 20> the PSP is a depolarizing or excitatory PSP (EPSP), and if it is inhibitory, the PSP is a hyperpolarizing or inhibitory PSP (IPSP). If the size of an afferent volley (the number of axons having input pulses) is increased over a relatively narrow range, the amplitude of the EPSP or IPSP increases in proportion to the number. If two or more small inputs are given simultaneously, the responses are simply added, and the system is said to be linear. If two otherwise identical inputs are separated in time, and if the responses are identical except for time of onset, the system is time invariant. If the size of the afferent volley is increased over a relatively broad range, the amplitude of the EPSP or IPSP fails to increase in proportion to the input. The pulse to wave conversion process saturates and is nonlinear and may be time varying. It functions in a linear, time–invariant range only for small active states with bilateral saturation outside that range on either excessive excitation or excessive inhibition.
Irrespective of the range of amplitude of activity, time variance is an important property of pulse to wave conversion and is commonly found in many synapses. Two easily observed forms are called posttetanic potentiation and posttetanic depression. Following a brief period of high–frequency electrical stimulation of an axonal tract, which is called the conditioning stimulus, the PSP in response to single test shocks is found to undergo a brief decrease in amplitude (depression) relative to the pretetanic control PSP and then a multifold increase (potentiation) lasting several minutes. The period of high–frequency pulse activity of the presynaptic axon terminals causes prolonged electrochemical changes in the membranes, which at first decrease and then increase the effectiveness of each test impulse.
The same or similar mechanisms are involved in the phenomena called presynaptic inhibition and facilitation. Axon terminals are subject to modification of their electrochemical properties, either by changes in the extracellular ionic concentrations or by other axon terminals, which form axoaxonic synapses on them. If an axon terminal is partly depolarized in a sustained manner (not below its threshold), then, when a pulse does occur, the amplitude of the PSP is decreased. If the axon terminal is hyperpolarized, the amplitude of the PSP is increased above the control amplitude. The event causing the presynaptic modification has no direct effect on the postsynaptic neuron. Although this event is called "inhibition," the use of the term is inappropriate. Whereas the combination of synaptic excitation and synaptic inhibition is additive, the combination of presynaptic inhibition with either is multiplicative. A better term is presynaptic attenuation, corresponding to presynaptic facilitation.
The levels of presynaptic attenuation or facilitation may depend on the history of the synapse or on a combination of inputs on different tracts. For <Page 21> example, an afferent volley evoked on one spinal nerve may diminish or block the PSP induced in a motorneuron by stimulation of an adjacent spinal nerve. In either case the amplification of the axon as a transmission channel is time varying.
The wave to pulse conversion process at the initial segment or trigger zone depends on the steady state level of dendritic polarization. If a neuron is at rest, an IPSP whether small or large does not induce a pulse. If the magnitude of a brief EPSP is increased on successive trials from a small to a large amplitude, then at some amplitude known as the threshold a pulse occurs. If the EPSP is suprathreshold, the resulting pulse has the same amplitude, irrespective of the amplitude of the EPSP (all–or–none). Once the pulse is begun, there is a brief period corresponding approximately to the duration of the pulse (about 1 msec), during which no amount of additional depolarization can induce another pulse. This is the absolute refractory period. Subsequently, there is a period lasting several milliseconds in which an additional EPSP can induce a second pulse, only if the second EPSP is larger than the first. This is the relative refractory period. For some neurons, the required amplitude of the second EPSP may be smaller. This reflects a period of potentiation or supernormality.
Because of the refractory periods and the threshold property of the axon, wave to pulse conversion by the resting neuron is nonlinear. If, however, the neuron is given a steady excitatory input, which provides a depolarizing bias, the neuron generates a pulse train at some mean rate. If the depolarizing bias is increased, the mean pulse rate increases, and if it is decreased, the mean pulse rate decreases. Over a relatively narrow range, the bias level and mean pulse rate are proportional. For two or more inputs (excitatory or inhibitory or both) the bias levels sum, and the mean pulse rate is proportional to the sum. In this mode, wave to pulse conversion is linear. Over a relatively wide range of bias level, the proportionality fails. At the lower end, the pulse rate is limited by threshold, and at the upper end it is determined by the relative refractory period, or by supernormality. Over many operating ranges of neurons, saturation is bilateral.
Wave to pulse conversion also can display time variance. For example, suppose that a steady depolarizing current is passed across an axon membrane at rest at a subthreshold level. Shortly after it is initiated, the apparent threshold for triggering a pulse on the axon is decreased, but after many seconds, the threshold returns to the resting value, even though the current continues. This phenomenon is called accommodation. If a steady suprathreshold current is passed across the membrane, the resulting pulse train initially has a high frequency and subsequently may diminish to a lower frequency. This is called adaptation.
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1.2.6. MULTIPLE STABLE STATES OF NEURONS
The concepts of resting state and threshold lead to the concept of stability. If a neuron has no input, and if it settles to a low and unchanging active state (without pulses), it is at rest.
Suppose that we stimulate the neuron repeatedly with electrical pulses, and that on each trial the amplitude of its transmembrane potential jumps to a new value and decays exponentially to the resting amplitude. Then we infer that the resting state of the neuron is stable, and that there exists a domain of stability in its state space. If, when it is stimulated, the neuron generates a pulse, then it has left this domain. By repeatedly observing the output of the neuron we define a new domain in its state space. Because events in this domain are both bounded and predictable we infer that there is a second stable domain. Further exploration may reveal three or more stable domains. The level of the active state or set of states at which a neuron changes domains is a threshold. If for any conditions the neuron does not sustain reproducible or controllable output, so that we cannot construct a function, we say that it is unstable. For example, a damaged neuron that emits a stream of impulses and is then silent is unstable until proven otherwise.
The multiple stable domains exist in a hierarchy, and the transition of the active state of the neuron from one domain to the next occurs only when there is external input to the neuron, which raises its level of excitation. The input may be a sustained, unvarying excitatory synaptic input, or it may be a nonsynaptic chemical or metabolic condition, which is sustained and unvarying, as in a stretch receptor or a pacemaker neuron. Because of specialized leakage channels across its membrane, this neuron undergoes steady depolarization until it reaches threshold, fires, repolarizes, and repeats the process at regular intervals. It generates a pulse train in the absence of external input, characteristically with constant interpulse intervals. In either case, for analytic purposes we do not treat the sustained input or condition as an input. We treat it as a parameter of the equation that describes the neuron.
For example, suppose that we apply sustained, unvarying excitation to a neuron, either by high–frequency electrical stimulation of an excitatory afferent tract or by passing outward direct current across its membrane with an appropriate set of stimulating electrodes. Suppose that we additionally apply transient or pulse inputs by either or both means. 1f there is zero steady stimulus and no transient input, and if the membrane potential is at a fixed, unchanging level, we assume that the neuron is at rest. In the classical theory of membrane function, the membrane and the neuron are at equilibrium. If small transient inputs are given to the neuron and its membrane <Page 23> potential is changed and then returns to the rest level, the equilibrium is stable. The domain of input values over which the membrane potential returns monotonically to rest is the domain of resting or true equilibrium.
If a small steady stimulus is given, the membrane potential is displaced from equilibrium to a new value, but its rate of change is zero. If a transient input is given, and if the membrane potential returns monotonically to the preinput level and remains there without further change, the neuron is stable but not at rest. The domain of inputs for which this holds we will call the domain of stable zero equilibrium. It includes the rest point and a certain range of membrane potential.
If there is a sustained, unvarying stimulus of sufficient amplitude, the neuron enters a domain of instability in which pulses occur at unpredictable times. With further increase in the stimulus, the neuron generates pulses at either fixed or randomly varying intervals but at a sustained, unchanging mean rate. Now the invariant is the mean pulse rate and not the membrane potential. This steady state constitutes another form of equilibrium for the neuron. If the mean pulse rate is transiently increased or decreased by excitatory or inhibitory pulse or current inputs, and if it then returns to the preinput mean rate and stays there, then it is a stable nonzero equilibrium.
If the steady stimulus is increased still further, the neuron no longer fires at a steady mean rate, but in short bursts at high frequency with intervening periods of low–frequency firing or silence. If the duration and interval of bursts is constant, there is a new invariant, which is the burst rate. If the burst rate changes following transient inputs and then returns to the preinput burst rate and stays there, the active state is stable. If there is a range of burst rates, such that the burst rate monotonically increases or decreases with the level of stimulation, and if each burst rate is stable, then there is a domain which we will call the stable limit cycle domain.
For each domain, zero equilibrium, non–zero equilibrium and limit cycle, there is a stable invariant, respectively, membrane potential, pulse rate, and burst rate. Each domain has an upper and lower boundary or threshold. For analytic purposes the level of the stable active state in each domain, determined by the level of stimulation (treated as a parameter), is at an equilibrium point. The three domains exist in a hierarchy, yet higher–order domains may be defined and demonstrated experimentally.
Most experiments are done when the neuron is found or placed in one of its stable states. The definitions can be extended to include time variance within a stable state. For example, the phenomena of adaptation and accommodation (Section 1.2.5) are examples of time variance of the neuron, respectively, in the zero and nonzero equilibrium domains, not of instability. Furthermore, the input–output operation of the neuron may be in the linear <Page 24> range or in the nonlinear range in either of the three stable domains, depending on how the state variables are defined.
1.2.7. BASIC TOPOLOGIES OF NETWORKS OF NEURONS
There is no limit to the complexity of networks that can be constructed by connecting the outputs and inputs of model neurons. When properly interconnected and equipped with sensory and motor devices, as few as three to seven model neurons can generate remarkably versatile adaptive behavior simulating that of living animals, as shown by Walter (1953). The varieties of logical, behavioral, and electrophysiological operations that can be simulated with such networks have been described, respectively, by McCulloch & Pitts (1943), Hebb (1949), Caianiello (1967), Liebovic (1969), Brazier, Walter & Schneider (1973), and others.
The topological organization of a network is reduced to the set of lines representing discrete channels by which the activity of each neuron effects changes in the active states of other neurons. The lines connect nodes that represent the individual neurons. Although there is evidence that real neurons influence each other to some extent through diffuse chemical–electrical changes in the extracellular compartment or through the neuroglia, most such influences are considered here to be negligible. With few exceptions (see Section 5.2.1) the only admissible transmission channels are those corresponding to synaptic contacts between the filaments of neurons.
FIG. 1.1. Topological conventions representing patterns of connection among neurons. (a) Convergence. (b) Divergence. (c) Serial transmission. (d) Parallel transmission. (e) Autofeedback. (f) Excitatory positive feedback. (g) Inhibitory positive feedback. (h) Negative feedback.
There are six basic patterns as shown in Fig. 1.1. Convergence (Fig. 1.1a) <Page 25> takes place when the axons of two or more neurons end on the dendrites and soma of one neuron. Divergence (Fig. 1.1b) occurs when the axon of one neuron ends on two or more neurons. If the output of each of several neurons is transmitted to the next in succession, they are in series (Fig. 1.1e). If two or more neurons all receive input from another group of neurons, and transmit in common to yet another neuron, they are in parallel (Fig. 1.1d). If no neuron by any series or parallel channel transmits output to itself, the channels are all forward (Fig. 1.1a–d). If the output of a neuron is returned by any channel or set of channels to its input, then one or more feedback channels exist (Fig. 1.1e–h).
A channel may be excitatory (+) or inhibitory (–). A neuron with feedback onto itself is said to be autoexcitatory or autoinhibitory (Fig. 1.1e). Two neurons each with input to the other form a feedback loop. If both are either excitatory (Fig. 1.1f) or inhibitory (Fig. 1.1g), there is positive feedback. If either is excitatory and the other is inhibitory (Fig. 1.1h), there is negative feedback.
These simple topological conventions are essential for the study of both networks of neural sets and networks of neurons. The description of the properties of neurons is also essential for both. The dynamic properties of networks of neurons, however, are quite different from those of networks of neural sets and need not be studied in preparation for the study of neural masses.
1.3. Neural Masses
1.3.1. A TOPOLOGICAL HIERARCHY OF INTERACTIVE SETS
The brain as a tissue consists of a vast number of neurons. Their processes are complexly interwoven with each other and form innumerable synaptic connections between neurons. The observed behavior of each neuron depends as much on the properties of its connections as on the properties of its parts. It is not possible to describe all connections of each neuron in the brain, but it is possible on the basis of connections to represent the brain as consisting of sets of neurons, and to describe the principal connections within and between sets. The formal description of the main connections within a neural mass takes the form of a network of sets.
A distinction must be made between anatomical and functional connections. An anatomical connection is a structural linkage, such as synapses or diffusion channels, between two sets of neurons. It implies the possibility of actions or interactions between the sets, but it cannot be used to infer that actions or interactions do occur, and it says little about the possible strength, spatial extent, or temporal course of actions or interactions.
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A functional connection is predicated on the existence of a function describing the relations between active states of two sets of neurons. It implies the necessity of anatomical connections, with two possibilities. Either the two sets receive anatomical forward connections from some third set, or they have anatomical interconnections between them. Correspondingly there are functional forward connections leading to coactive states and functional interconnections leading to interactive states.
Networks of multiple neural sets are based on functional connections and not anatomical connections. A set of neurons having fixed anatomical connections may admit to several network representations, depending on the functional state of each of its anatomical connections.
The dynamics of networks of sets can rapidly become so complex that care must be taken to restrict consideration to those networks that represent experimentally realizable neural masses. The restriction is enacted by designating a topological hierarchy for neural sets. Each set has as parts the sets at the lower levels and is a part at the higher levels of the hierarchy. The levels are designated KO, KI, KII, ... (for Katchalsky), so that for example, a neural mass having a topology at the KII level of complexity is designated a KII mass, and its representation is a KII set.
We begin with the lowest levels of the hierarchy. A KO set is defined as any set of neurons numbering from 103 to 108 having a common source of input and a common sign of output (+ or e, excitatory; – or i, inhibitory), and having no functional interconnections (Fig. 1.2a). The source of input may be a form of stimulus in relation to receptors, such as odor, white light, light of a particular wavelength, all cutaneous stimuli, a type of cutaneous stimulus such as heat or pressure, etc., or the source may be a set of primary sensory neurons previously defined, or it may be a bundle of axons, tract, or peripheral nerve at the convenience of the investigator. The set is not defined by a particular input but by a range of possible inputs from the common source. Once so defined, the set may be divided into subsets on the basis of location, grouping, or contiguity of its neurons, their cytoarchitecture or chemistry, their output paths and the locations of their target neurons, or their degree of accessibility to input at any moment in time or location in space. Such attributes define subsets but do not serve to define the set.
A common sign of output means that with respect to its effect on specified target neurons every neuron in the set is either excitatory (positive sign) or inhibitory (negative sign).
The first part of the definition is
flexible in respect to particulars, so that the specifications for the input in
any series of observations serves to define the set of neurons under
observation. For example, if only a part of a central tract is accessible to
stimulation, the number of neurons receiving <Page
27> input from that part of the tract constitutes the set.
If in later experiments a larger or different part of the tract is accessible,
the set is redefined. Similarly a pair of sets may be redefined into three sets
on the basis that a fraction of the neurons in each set receives input from a
double source. Neurons receiving input from tract A form set A, and those with
input from tract B form set B, whereas those with input from both tracts form set
A ∩ B (where ∩ means intersection
of the sets) as in Fig. 1.2b. Under this definition, however, A 
B, the union of sets A and B, does not
form a set.
FIG. 1.2. Topological conventions representing KO and KI sets. (a) KO set with one input channel and one output channel. (b) KO sets with partially overlapping input channels and one output channel. (e) KO set with one input channel and two output channels. (d) KI set with one input channel and one output channel.
The second part of the definition is rigid. If excitatory and inhibitory neurons both exist in a noninteractive mass, they must be treated as two sets, even if they have a common source of input. If they are not distinguished, the dynamics of the two types of neurons in higher–order networks are reduced to those of the type having the larger number, and the mixed nature of the mass is not recognizable. Special treatment is required for the case in which the neurons of a set A are all excitatory to set B, but all <Page 28> inhibitory to another set C, because the information concerning the dual sign of output of set A is preserved in the specifications of the signs of input to B (positive) and C (negative). In this case two output channels are assigned to the KO set (Fig. 1.2c).
A number of KO sets connected by forward channels in series or in parallel form a KO network. It is distinguished from networks of higher levels of complexity by the absence of functioning feedback channels between the KO sets.
Example A. The proprioceptive afferent axons from each muscle run through the dorsal root into the spinal cord, where they form synapses on the dendrites of spinal motorneurons. The axons from muscle spindles are excitatory to motorneurons innervating the same muscle and are called Group 1A fibres. They form a KOe set, which is designated KO1A (Fig. 1.3a), which excites the KOM1 set of motorneurons. The axons from the tendon organs are inhibitory to the same motorneurons and are called Group 1B fibres. They form a KOi set, or KO1B. This is an example of parallel forward excitation and inhibition of the KOM set (Sherrington, 1929). §
FIG. 1.3. Representations of the topology of the neural mass in the spinal cord by KO sets. (a) KOM set with two input channels. (b) Model according to Lloyd (1955). (c) Model according to Eccles (1957). (d) Introduction of inhibition by Renshaw cells (RN).
Example B. Group 1A axons are inhibitory to the set of motorneurons innervating antagonist muscles, KOM2. According to Lloyd (1955) the actions are direct, in which case Group 1A axons constitute the KOlA Set <Page 29> with two output channels in parallel (Fig. 1.3b). The accepted view (Eccles, 1957) is that the inhibitory actions are mediated by sets of inhibitory interneurons KOAN and KOBN that form KOi sets in parallel channels (Fig. 1.3c). §
Example C. When a motorneuron is excited by an orthodromic volley (in the normal direction of pulse conduction), its action potential is followed by a prolonged inhibitory event. That is lacking when the neuron is excited antidromically (in the direction opposite to normal pulse conduction) by electrical stimulation of its axon or by direct electrical stimulation of its soma. There is a set of neurons, called Renshaw cells, in the spinal cord located ventromedial to the motorneuron pool, which is excited by action potentials from motorneurons. The interneurons form a set designated KORN which is inferred to deliver inhibitory activity to motorneurons (Fig. 1.3d). The output of each motorneuron delivers by this channel inhibitory activity mainly to motorneurons other than itself. This is an example of parallel forward inhibition. It is called recurrent inhibition and is often cited as example of feedback. Doubtless under some experimental conditions the network of sets manifests feedback properties, but in the conditions in which it has been described, feedback is not manifested. §
A KI set is any set of neurons having a common source of input, a common sign of output, and dense interactions between neurons within the set (Fig. 1.2d). There are two types: The KIe set consists of neurons which are mutually excitatory, and the KIi set comprises mutually inhibitory neurons. Again, the restriction on sign of output is rigid, but the requirements on specification of input are flexible. A number of KO and KI sets connected only by forward channels comprises a KI network.
Example D. The compound eye of the horseshoe crab Limulus consists of an array of photoreceptor neurons comprising a KOe set designated KOS and an array of neurons that are densely interconnected by axons forming inhibitory synapses. These neurons form a KIi set designated KIN (Fig. 1.4a). The set of efferent axons carrying pulse trains from all parts of the KIN is shown as a pair of channels, on the basis that the positive feedback loop has two negative signs, and the output of the forward limb is opposite in sign to that of the feedback limb (Hartline & Ratliff, 1958; Knight, Toyoda & Dodge, 1970). §
Example E. Single–shock electrical stimulation of a dorsal root initiates a volley of action potentials on the primary sensory axons in the root forming a KOS set. The collaterals of many of the axons initiate prolonged <Page 30> activity of small neurons in the substantia gelatinosa in the dorsal sensory nucleus of the spinal cord, which is manifested by prolonged depolarization of the axons in the same KOS1 and adjacent KOR2 dorsal roots (Wall, 1962). The activity of the small neurons is presumably maintained by mutual excitation. If so, the neurons form a KIe set designated KISG in Fig. 1.4b. Volleys from single–shock dorsal root stimulation of KOR2 normally excite a set of second–order neurons KON2 and elicit output from them, which is measured in the wave or pulse mode. During the maintained activity of the KISG set, there is reduction in the amplitude of output from the KON2 set, which is caused by presynaptic inhibition (attenuation) of the KOR2 axonal input by the KISG neurons. The attenuating effect by the KISG excitatory set is designated by –x. The locus of the action is shown by the square A, which is a topological element in the network and not a neural set. §
FIG. 1.4. Representation of the topology of (a) Limulus eye, (b) spinal cord, and (c) and (d) cerebellum by KO and KI sets.
Example F. The efferent neurons in the cortex of the cerebellum are the Purkinje cells, which are inhibitory and are interconnected by inhibitory axon collaterals. They form a KIi set (Fig. 1.4c). The KIP set receives <Page 31> excitatory axons called climbing fibers from neurons in the inferior olivary nucleus. Although there is evidence for interaction of the neurons in the nucleus, in respect to electrical stimulation, the climbing fibers can be treated as the KOCF set. Another cerebellar input channel consists of mossy fibers, the KOMF set, which are excitatory to the cerebellar granule cells forming the KOGS set. The granule cells emit axons called parallel fibers, which are excitatory both to Purkinje cells and to two types of interneurons. For simplicity these are represented as a single set KO1N, which sends inhibitory axons to the Purkinje cells. This is an example of a network of sets having two converging inputs and output from a KIi set (Eccles, Ito & Szentagothai, 1967; Banlti, 1974a, b). §
Example G. Another important set of neurons in the cerebellum consists of the Golgi cells, which can be represented as the KOGC Set (Fig. 1.4d). Axons from these neurons end in synaptic complexes called glomeruli (g in Fig. 1.4d), which consist of the axon terminals of mossy fibers, the dendrites of granule cells, and the axon terminals of the Golgi cells. The effect of Golgi cell input is to attenuate transmission through the mossy fiber–granule cell synapse, which is designated as –x. Because the input to Golgi cells is delivered through these synapses, the topology contains a feedback loop, which involves both excitatory and inhibitory neurons. However, the synaptic action is multiplicative rather than additive, so that special techniques are required to describe its dynamics. These are discussed in Section 5.2.2. §
These examples are merely sketches that convey some of the main features of operation of the several mechanisms as they are described in the literature. They are not definitive models. For example, there is evidence that Renshaw cells interact with each other (Example C) and form a KIi set, that cerebellar inhibitory neurons (Examples F and G) likewise form a KIi set, and that motorneurons may form a KIe set. These channels deserve careful experimental evaluation by the methods to be described.
The KII set is formed by the existence of dense functional interconnections between two KI sets. If both are excitatory, it is a KIIee set, and if both are inhibitory it is designated KIIii. If one component is KIe and the other is KIi, it is designated KIei, or KIie, or simply the KII set, irrespective of which set receives a specified input (Fig. 1.5a). In the KII set, each excitatory neuron interacts with inhibitory neurons as well as with other excitatory neurons, and each inhibitory neuron interacts with both excitatory and inhibitory neurons. Only for a special case, which will be discussed, can the functional topology be reduced to actions of excitatory neurons onto inhibitory neurons, and vice versa. This is the reduced KII set (Fig. 1.5b).
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FIG. 1.5. Topological conventions representing (a) the KII set, (b) the reduced KII Set, (c) the neural masses of the olfactory bulb and cortex at the KII level, and (d) the neural masses of the olfactory bulb and cortex at the KIII level.
Example H. The olfactory receptors in the nasal mucosa form a noninteractive set KOS, and their axons are sent to the olfactory bulb in the primary olfactory nerve (PON). The terminals form excitatory synapses on the apical dendrites of mitral and tufted cells, which are excitatory to each other and form a KIe set KIM. The basal dendrites of the mitral–tufted cells form reciprocal dendrodendritic synapses with granule cells in the bulb, which have no axons, but which interact with each other, possibly through stellate cells, by mutual inhibition and form a KIi set KIG. The mitral–tufted cells excite granule cells and are inhibited by them. The interaction gives rise to the KII set KIIMG. The KIIMG set responds to electrical stimulation of the PON by generating a damped sine wave output, which arises from repeated cycles of excitation and inhibition (Fig. 1.5c) (Freeman, 1972c). §
The oscillation occurs only in waking or lightly anesthetized animals. Under deep anesthesia the evoked response may be reduced to a single cycle, which consists of initial excitation of each KI set followed by <Page 33> inhibition of each KI set. This so–called biphasic evoked potential is commonly observed on stimulation of many parts of the paleocortex and the neocortex. It is strong evidence that in most if not all types of cortices the excitatory neurons excite inhibitory neurons and are in turn inhibited by them. With few exceptions there is little or no experimental evidence for or against the occurrence of mutual excitation and mutual inhibition. In the absence of such evidence the excitatory neurons are treated as if they formed a KOe set, and the inhibitory neurons are treated as if they formed a KOi set. The interaction then forms a reduced KII set (Fig. 1.5b). This is useful as a first approximation for analyzing the KII set and in some conditions is a very good approximation, but no example has yet been demonstrated experimentally.
The biphasic evoked potential is the manifestation of recurrent inhibition, because the excitatory input is delivered to the KOe set and, by its axon collaterals, to the KOi set, which inhibits the KOe set. This is not an example of negative feedback, because the recurrent inhibition initiated by each neuron in the KOe set is returned mainly to neurons other than itself (see also Example C and Fig. 1.3d). Only if there is more than one cycle of the oscillation can feedback be said to occur (Section 5.1.1).
Example I. The PON input in Example H is also delivered to a set of mutually excitatory † neurons, the periglomerular KI set, which delivers their output to the KIIMG set (Fig. 1.5c). The response of the KI set to a PON electrical stimulus is prolonged excitation, which causes prolonged secondary excitation of the KIM and KIG sets. The output of the KIIMG set consists of an oscillatory component resulting from its internal interaction and a nonoscillatory component resulting from the KI input. §
† For an alternate view of periglomerular neurons as inhibitory, see Pinching & Powell (1971d), Reese & Shepherd (1972), Shepherd (1972), and Freeman (1974e).
Example J. The prepyriform or primary olfactory cortex also contains a KII set that receives input from the bulbar KII set (Fig. 1.5d). The properties are analyzed in Chapters 5 and 6. This example is given in order to show how the graphic notation is used to describe complex topologies, including another example of multiplicative attenuation in the olfactory bulb. (See Figs. 1.4b and 1.4d.) §
The designation of KIIee and KIIii sets has been included for the sake of logical completeness, but as yet there are no physiologically verified examples. Networks of KII sets and lower–level sets occur at the KII level of complexity provided all of the sets have only forward connections as, <Page 34> for example, in Fig. 1.5d (without the dashed lines). If there are one or more feedback channels (dashed lines), the network and the mass exist at the KIII level. KIII sets and higher–order sets have not been adequately studied and are not yet defined.
Three aspects of this hierarchical design deserve emphasis. The definitive attribute of the neural mass is its set of functional connections, which are the channels through which neural activity is transmitted. The assignment of a level of complexity depends on the number and the scope of feedback connections. The minimal elements for the analysis of neural masses are the KO or noninteractive set and the KI or interactive set.
1.3.2. THE STATE VARIABLES OF KO AND KI SETS
The neurons of a set are distributed in three dimensions, with afferent and efferent tracts approaching and leaving the same or different sides. For topological modeling a set is conceived as a surface distribution of model neurons. Afferent axons approach one face and efferent axons leave the other face, and both tracts are orthogonal to the surface. Divergence and convergence of afferent and efferent axons and the internal interconnections of axons and dendrites take place in distributions parallel to the surface.
The active state for a neural set must be assigned a value for each point of the whole surface. This cannot be done by measuring or assigning a value to each of the 103–108 neurons comprising a set, so an alternative approach must be used. We begin by assuming that the input activity to a KO set is continuously distributed in both time and space (Section 1.1.3).
The basis for this assumption is that most realizable inputs to a set consist of pulse trains on more than one afferent axon in the input tract. Characteristically the axons branch and the terminal branches diverge over the surface of the set, intersperse with branches of other axons, and end on multiple neurons in the set. Each neuron in the set shares with its neighbors some common input from afferent axons. To the extent that its active state is dependent on its input, its state will resemble that of its neighbors. The closer the spatial proximity of two neurons, the more similar is their state of activity. This implies that the active state of the set in the vicinity of a point is the average of the active states of the neurons of the same set in that vicinity. The value of the average may change as the location of the point is changed, but if the change in location is made vanishingly small, then it may be assumed that the change in the average will also be vanishingly small. The level of activity may then be said to change in a continuous manner across the surface.
At each time t1 and point in the surface (x1, y1) there is an activity density denoted o(t, x1, y1), which is the local active state of the set. It is manifested <Page 35> by the time sequence in mean local level of dendritic polarization or pulse rate divided by the surface area over which the mean is taken. Any set of measurements on the manifestation of activity density at a set of points in the KO set serves to describe a neural activity distribution in either the pulse mode p(t, x, y, z) or the wave mode v(t, x, y, z), which is a discrete sample from a continuously varying distribution. If the sample is adequate, then a continuous function may be derived by appropriate transformations to describe the active state of the whole surface. This time and space function is called an activity density function o(t, x, y). It is the best attainable description of the active state of the set and may be expressed either in the pulse mode op(t, x, y) or the wave mode ov(t, x, y).
If the active state is assumed to be uniform across the KO set, it has a single value for the set. It is then identical to the Sherringtonian (1929) central excitatory state (ces) or central inhibitory state (cis). Its value is the same as the mean of the states of all the neurons in the set. Any change in value may be due to a relatively large increase in depolarization or pulse rate of a few cells or to a relatively small change in many cells. In either case, the set can be modeled as if it were a single "average" neuron for the purposes of constructing a topological flow diagram. Examples are the "motorneuron pool" and the "respiratory centers," which are known to be sets, but which in networks can be represented by single model neurons.
The KIe set, which is the interactive set of excitatory neurons, requires no restrictions on the continuity of its input activity. The existence of excitatory interconnections implies that if any neuron receives input and becomes active, its output is delivered to many neurons in its vicinity. Those neurons in turn deliver input to the initiating neuron in varying degree and with varying delay. This interaction implies that the active states of neurons in the vicinity of a point tend to the same value, and that the active state of an interactive set can be described in terms of a continuum. Therefore we describe the active state of a KIe mass by means of a continuous activity density function o(t, x, y), which is evaluated by inference from measurements of the manifestations of neural activity density at selected points.
Continuity need not hold for the KIi set if it receives input from a KO set. If, however, it receives input from a KIe set, or if the activity distribution from a KO set is continuously distributed, then its activity density function is also continuous.
In the event that continuity holds, the state variables of a KI set, which are functions of time and distance, are evaluated from multiple sequential measurements on averages across neurons of dendritic potentials and axonal pulse rates in the vicinity of each of a number of recording sites in the set.
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Interaction implies that variation in the active states of neurons in the set with time occur in a coordinated manner, though not necessarily in synchrony. The sequences of values for the active states of any two neurons in a KI set, which constitute their state variables, are correlated or covariant to a degree depending on the level of interaction in their vicinity of the KI set and not on their private interaction. While interaction implies covariance, the converse is not valid, because the activity of two neurons may be correlated, if they have some degree of common input and are not members of a KI set.
Because interaction exists in a KI set, the active state of neither the set nor any of its subsets can be represented as that of an "average" neuron, as in the instance of the KO set. By virtue of the interaction a new entity arises that has properties peculiar to itself. These properties are to some extent manifested in the behavior of single neurons in the set, but they cannot properly be considered as properties of the single neurons. In the first place the state variables are predictable and measurable only for averages across numerous measurements on the activity of many single neurons. The averages are taken either across ensembles of neurons at any time, or across ensembles of observations on one neuron at a set of times, or both. In the second place the cooperative states are incommensurable with single neural events. They last longer, are more widespread, and have slower rates of change than events from single neurons and cannot exist in the time and space dimensions of activity in single neurons. In the third place the KI set has stable domains of operation in its state space, which are qualitatively different from those of single neurons.
The state variables of KII and higher–order sets and networks are given only by the state variables of the KO and KI component sets. For this reason, the KO and KI sets are essential parts in KII and higher–order sets. The definition of the state space for a single neuron requires that a minimal number of state variables be known, such that every state variable can be uniquely predicted from knowledge of the history contained in the initial conditions and the values at any time. For a neuron the active state can be given by a single state variable. The same definition holds for the neural set with the following difference. The activity of a set is an activity density function, and the state of the set is given by a state variable, which is a time function for the entire surface of the set. One activity density function is sufficient for each KO and KI set, irrespective of the organization of the KO and KI sets into KII and higher–order sets. The KOµ and KIµ activity density functions oµ(t, x, y) are the minimal state variables that define the states of KII and higher–order sets. The oµ(t, x, y) for the KO and KI sets comprising a neural mass such as the olfactory bulb constitute together the macrostate of the mass.
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1.3.3. THE OPERATIONS OF NEURAL SETS
Neural activity in masses occurs in the wave mode ov referring to synaptic current or potential v and in the pulse mode referring to action potentials p. Either or both forms are denoted by u. As in the case of single neurons the operations of masses include conversions of activity between the two modes. Integration occurs predominantly in the wave mode, and transmission takes place mainly in the pulse mode. Integration comprises weighted summation over both space and time, including filtering in both dimensions; transmission in either mode comprises translation, delay, dispersion, convergence, and divergence. Each operation in a mass is specified by measurement of a pair of input and output state variables and by construction of a function describing their relation to each other.
The most easily controlled form of input to a neural mass is the volley of action potentials induced in a tract or nerve by an electrical stimulus or impulse, which is given once on each of a sequence of trials. The measured output in the wave mode is an average of the extracellular potential recorded at an appropriate point in the mass on a sequence of trials, each serial trial beginning with the time of onset of the stimulus. This output is known as an averaged evoked potential (AEP). It reflects an average of the dendritic currents from many neurons in accordance with their cellular geometry and an average across the ensemble of trials. The measured output in the pulse mode is a histogram in which time after the stimulus is divided into intervals minimally equal to the duration of the action potential. On sequential trials the occurrence of a pulse in a given interval on any trial is noted by adding a unit to that interval. Over a stated number of trials, a poststimulus time histogram (PSTH) emerges. If the recording electrode in the set is optimally positioned with respect to one neuron, so that its pulse can be clearly distinguished, the output is a unit PST histogram, which is an average over an ensemble of trials. If the pulses are counted from more than one neuron, the result is a multiple unit cluster PSTH, which is an average over ensembles of trials and the neurons in the vicinity of the recording site.
The AEPs and PSTHs from an appropriate set of points spaced in a mass describe the neural activity distribution, from which activity density functions for the KI sets in the mass can be determined. The functions are generated by equations, which are the solutions to differential equations proposed to describe the dynamics of the interactive sets. They are tested and evaluated by fitting curves to the AEPs and PSTHs. The variables in the equations fitted to the AEPs and PSTHs, which represent potential or pulse density as functions of time and distance, v(T, X) or p(T, X), respectively, represent state variables of the neural sets. In accordance with the <Page 38> definition of state variables in Section 1.2.3, the value of v(T, X) or p(T, X) at any time T uniquely determines the value of v(T1, X) or p(T1, X) at another time T1 at that place.
In general there are two minimal state variables required to specify the active state of each KI set, one in the wave mode v(T, X) and one in the pulse mode p(T, X). The function G, which is given by the relation between any two state variables, defines an operation between two KI sets or within a KI set. For example, suppose we have two sets KI1 and KI2, which generate the state variables v1(T, X), p1(T, X), V2(T, X), and p2(T, X). We can construct four functions. Two of these are functions representing operations between the sets v1 = G12(p2) and V2 = G21(p1). The other two functions represent operations within sets, p1 = G11(v1) and p2 = G22(v2). If the function G11 or G22 has been determined, then the minimal number of state variables required to specify the active state of set KI1 or set KI2 is one, in either the wave or pulse mode. Typically the input p1(T, X) on the axons of set is known, because it is the response to an electrical stimulus. The observed output of the target set KI2 is v2(T, X) or p2(T, X) or both, which gives the means for determining G21 and G22. That is, each fixed electrical stimulus and AEP or PSTH constitutes an input–output pair, from sets of which the operations Gd(p), Gs(v) or more generally G(u) within and between KI sets can be determined.
When a large number of input–output pairs have been collected over a sufficient range of variation of input, a state space is defined for the state variables. The extent and complexity of the state spaces of even the simplest neural masses are staggering. We cannot hope to describe all the operations the masses are capable of, nor do we wish to. We restrict consideration to certain limited domains in several ways. The choice of inputs is limited to those that give reasonable opportunities for measurement and control of input. The permissible domain of input amplitude is limited, so that the range of output is similar in terms of amplitude, spatial extent, rates of change, etc. to the output observed from the same masses when the same or a similar animal is engaged in normal behavioral activity, presumably involving activity of the mass. The state of the animal under observation is kept as close to that of appropriately defined behavior as the recording conditions allow. Unstable or nonreproducible responses are not considered as an explicit basis for modeling.
Following these limitations, the resulting domain of the state space is subdivided by separating the main independent variables. If responses are observed in spatially uniform activity distributions, the relations between state variables are reduced to operations only in the time domain. If they are measured at many sites and are constant over some time period ∆t, the state variables yield operations in the space domain, Most of our discussion will be concerned with time–dependent operations.
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1.3.4. FEEDBACK GAIN AS A PARAMETER FOR INTERACTION
Even the foregoing restrictions are insufficient to limit the field of study of a set, because there are too many unspecified degrees of freedom in the choices of the nature and location of the input, the precise definition of the set, the location of recording sites, the procedures for averaging, and many aspects of the condition of the animal. We must have precise predictions on what to look for, so that these circumstances of indirect observation can be properly controlled. Our best recourse is to the theory of neural masses. As it is thus far developed, the theory can be used to generate elementary curves or functions. These elementary forms we call basis functions, and they tell us what AEPs and PSTHs ought to look like if our theory of neural masses is valid. We adjust our experimental conditions until the expected response forms are observed and then fit the predicted curves to the response forms.
In order to apply the theory, we must convert our topological model to a computational model, in the following way. We note in Section 1.2.5 that when operations of the neuron are described in the form of equations, certain fixed terms emerge as the parameters. The time and space coefficients have values reflecting the relation between the natural rates of change of the neuron and the units of measurement of time and distance. The gain coefficients reflect the relative amplitudes of output and input state variables and are given by the ratio of output to input amplitude. If both active states are measured with the same units, the gain is dimensionless. The gain for the single neuron is forward gain. For any desired interval of time if the neuron transmits as many pulses as it receives, its gain is 1. If it transmits more than it receives, its gain exceeds 1, and if less its gain is less than 1.
The KO set and the KI set have time, space, and forward gain coefficients g as parameters in equations describing their operations. The KI set has one additional parameter, which is called its feedback gain K and which describes the level of interaction within the set. The feedback gain at each point in the set is defined by the ratio of the activity density at the point for any two successive time intervals of appropriate separation and duration, provided there is no external input. If the activity density is constant but not zero, feedback gain is 1. If the activity is decreasing, gain must be less than 1, and if the activity is increasing, gain must be greater than 1. Moreover, the feedback gain is always dimensionless, because it is a measure of increment or decrement in a closed loop.
Feedback gain is a pure collective property that cannot be reduced to or measured as the level of transmission between any two neurons. It is locally specified as a dimensionless quantity for every subset having sufficient density of functional interconnections to sustain an average over the subset. It is a continuous variable over the surface of a set with a value at each <Page 40> point on the surface equal to the mean over the vicinity of the point. Specification of the level of interaction for a set is by a surface function K(X). If the level of interaction is sufficiently uniform over the set, then it has a single value for the set.
For the purpose of evaluating feedback gain, a KI set can be divided into two KO subsets on the following basis. In the presence of ongoing activity of unspecified origin, in which each neuron generates pulses at random at some mean rate, at any moment some neurons are giving pulses and most are not. Those neurons transmitting pulses are unable to receive pulses at that time because they are refractory. The other neurons are receiving pulses but are not transmitting pulses. The transmitting and receiving subsets are mutually exclusive and together include all members of the set. The neurons in the two subsets are homogeneously distributed over the surface. Each neuron in the set switches at random between the two subsets, and across the set the switching is continuous in time and uniform in space. By this definition an external input must go to the receiving subset. If the input is effective, a transmitting subset emerges from within the receiving subset and transmits to the continuously reconstituted receiving subset.
FIG. 1.6. Representation of (b) KI and (c)KII sets by networks of (a)KO sets.
By this partitioning (see also Section 1.1.3) the KI set can be reduced to a single feedback loop between two KO sets (Fig. 1.6b) in a topological flow diagram. The level of interaction is the feedback gain of the loop. Because the neurons all have the same sign of output, the feedback gain of a KI set is always positive K≥ 0, whether the set is excitatory with Ke ≥ 0 <Page 41> or inhibitory with Ki ≥ 0. For uniformly distributed activity in the steady state its value must be unity. Otherwise, for the set having a spatially nonuniform active state, which is continuously varying with time and distance, the feedback gain is a space–time–dependent variable. For present purposes, the representation of the spatially uniform interactive set as a single feedback loop without spatial dimensions with fixed feedback gain (not necessarily unity) suffices as a topological model. It is called the lumped circuit model.
The topological representation of the
KI set does not include channels for autoexcitation or autoinhibition. These
are excluded on the premise that the proportion of its total input that each
neuron receives directly from its own output in the interactive set is
vanishingly small. In this respect the KI model differs from other proposed
interactive neural models (e.g., Wilson & Cowan, 1972; Grossberg, 1973a;
also see Section 6.2.5). Further, the exclusion of autoexcitatory and
autoinhibitory channels makes it possible to include the refractory periods
with other determinants of interaction strength and to avoid having to specify
channels in which they play special roles. Then, both kinds of nonlinear
conversions G(u) can
be conceived to occur instantaneously at points and are independent of time and
distance. The activity density function of the µth
set oµ(t, x, y, u) is dependent on time, distance, and
amplitude in either or both modes. Operations on the activity density function,
which take place in the Cartesian dimensions of the neural set X = x,
y, z, are denoted by
capital letters as in Oµ(s, X, U),
where s is a complex frequency
(see Section 2.2.3) replacing time t or T. Because the amplitude–dependent
nonlinearity is independent of time and distance, we can separate Oµ (s, X,
U) into an equivalent sequence of a linear space–time
operation FH(s, X) and a nonlinear operation G(U) as in Oµ(s, X, U) Gµ(U) FHµ(s, X), where G(U) incorporates the feedback gains.
From this analysis we can predict that the observed output of a KI set (in the form of AEPs or PSTHs) should conform to the output of two identical KO sets with positive feedback between them. This can be tested by measuring the response of a KO set to impulse input to evaluate its time constants and calculating the expected impulse response (a set of basis functions) for a KI set by means of a feedback equation. The set of basis functions then consists of a sum of exponential terms. The impulse response of a KO set closely resembles the impulse response of a single neuron in its fast rise and less rapid decay (Fig. 1.6a). The impulse response of a KI set has a still faster rise and very slow monotonic decay (Fig. 1.6b). This conforms to the prediction from the set of basis functions. The predicted curve is fitted to the AEP or PSTH. The value for feedback gain can be calculated from the required decay rate of the fitted curve.
The impulse responses of both forward and feedback elements of the KIe set are both upward for excitatory input or downward for inhibitory <Page 42> input. On focal excitation of the distributed KIe set the activity density must either be increased in or around the region of input, or be at background level at sufficient distances. For the KIi set the output of the feedback element is inverted in sign from the output of the forward channel. On focal excitation the KIi set can display surround (locally distributed) inhibition and possibly secondary surround excitation with increasing distance from the center of the focus. That is, for brief focal excitation the impulse response of the KIe set decays monotonically from the input with both time and distance, whereas the impulse response of the KIi set decays monotonically with time but may oscillate with distance. Therefore, complex spatial frequencies can be expected from KIi sets but not from KIe sets.
There is a wide range of experimental conditions in which the KII set may also be represented by interacting KOe and KOi sets (Fig. 1.6c). In addition to positive feedback there is negative feedback denoted by Kn. The sign of action by convention is incorporated into the topological representation of the feedback loop so that Kn is a real number ≥ 0. The predicted impulse response or set of basis functions contains a sum of damped sine waves, which conforms to observations on KII sets. In this case, three gain coefficients can be evaluated: positive excitatory feedback Ke, positive inhibitory feedback Ki, and negative feedback Kn, representing the three interactions of excitatory and inhibitory neurons.
A beginning analysis should give two very general results. First, we should have a realizable point in the state space of a KI or KII set. Around this point, we can expect to develop domains of increasing scope, in which the activities of these sets can be predicted and understood. Second, we should obtain values for the essential parameters of interactive sets. As we find other means for evaluating the same parameters, we can hope to establish both their ranges and their significance in relation to larger aspects of brain function.
1.3.5. MULTIPLE STABLE STATES AND THE LEVELS OF INTERACTION
We have seen that in theory a KI set or a KII set can be described by means of a network of KO sets with feedback between them. The solutions to the appropriate equations describing the dynamics of feedback for impulse input specify a set of basis functions, respectively, a sum of real exponentials for a KI set, and a sum of damped sine waves for a KII set. We shape the experimental conditions so that an AEP or PSTH conforms to the set of basis functions. We fit the observed wave form with a set of basis functions and evaluate the coefficients in the basis functions. The coefficients then serve to evaluate the feedback gains, which are the numerical estimates of the interaction levels.
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Next we systematically vary one of the experimental conditions, such as input amplitude or depth of anesthesia. We observe changes in the AEP or PSTH. We fit the changed wave forms with the same basis functions to obtain sets of rate coefficients, and from the sets we calculate sets of feedback gain coefficients. With each experimental variable we deduce that a systematic change occurs in the value of one or more feedback gain coefficients.
We infer that the levels of interaction in KI and KII sets are dependent on input amplitude, on depth of anesthesia, and on many other experimental variables. The changes in levels of interaction are continuous in the sense that each change can be made as small as desired by reducing the change in experimental variable to a small step. Three facts emerge. First, there is virtually no range of output of KI and KII sets over which superposition holds and which can be said to be linear. Second, over small domains of change in input about many amplitudes of input, the range of output can be treated as if it were linear with a fixed value for the feedback gain coefficients. Third, we can combine a set of linear domains in the form of a continuous successive linear approximation by describing the feedback gain coefficients as functions of the relevant experimental variable, such as input amplitude, etc.
These facts imply that we can predict
the shapes of sets of AEPs and PSTHs by using three equations to describe the
dynamics of KI and KII sets: a linear equation in time, a linear equation in
space, and a nonlinear equation in amplitude. We can separate the linear
time– and space–dependent properties from the nonlinear
amplitude–dependent properties. We use linear analysis to treat the
dynamics in the time and space domains by means of linear operators F(s)
and H(X), where FH(s, X) H(X)F(s). This frees us to examine the nonlinear
properties G(p) and G(v) in greatly simplified form. When the
feedback gains are written as functions of some controlled experimental
variable, we can make direct inference from an experimental manipulation to its
effect on the levels of interaction in KI and KII sets. For example, we can
measure the set of AEPs from a KI or KII set for a set of pulse stimuli over a
domain of input amplitudes, and from the measurements we can calculate how the
change in input amplitude affects the levels of interaction in the KII sets.
Then Oµ(s, X, U) Gµ(U) Hµ(X)Fµ(S)
in the serial order of operation Fµ(s), Hµ(X), and
Gµ(U) for each set and mode.
The value of this approach stems from the definition of the KI and KII sets as interactive. If interaction is so basic as to be definitive, we must have direct access to the defining characteristic in the form of a measure as our first major step toward understanding.
There is good reason to expect the levels of interaction in KI and KII sets to vary. As noted in Section 1.3.1 the existence of anatomical interconnections does not imply interactions but only their possibility. If the <Page 44> neurons in an anatomically interconnected set are in a resting state, which is so far below their thresholds that no available input can cause them to transmit pulses to each other, they form a KO set. A set of neurons that normally functions at the KI or KII level can be reduced to the KO level by a strong general anesthetic. In this state the set has no interaction and therefore its feedback gain is zero. It is said to be in the open loop state. This is a useful state because the rate constants of the neurons in the set can be measured in the open loop state.
The open loop state provides the starting point for description of the multiple stable states of KI and KII sets, which are analogous to the stable states of the neuron (Section 1.2.6). For each KI or KII set there is at least one sustained, unvarying input, or a combination of such inputs, which can be treated as a parameter of the system. The parameter may be a sustained synaptic excitatory input, or an electrochemical property of the membranes of the constituent neurons, or some combination of these. It is a principal determinant in KI or KII sets of the steady state magnitude of interaction. When that magnitude serves as a reference magnitude with respect to gain changes induced by transient input, it is designated K0.
If there is no sustaining input, the value of K0 is zero. If K0 is zero and there is no transient input, the KI or KII set is at the resting equilibrium, and the active states are all zero. If for small transient inputs the active states of sets are displaced and then return to their zero state and stay there, the resting equilibrium is stable. If there is a nonzero value for K0 that is not too great, one or more of the active states in the wave mode (but not the pulse mode) may be nonzero, but the rates of changes of all the active states are zero. That is, some or all neurons in the sets in the steady state may have steady membrane potentials other than resting, but none has pulse trains. The range of values for the parameter and the transient input conforming to this condition defines the stable zero equilibrium domain of the KI and KII sets. It includes the open loop state but is not identical with the open loop state, in which feedback gain is zero at all times and neither transient nor steady transmission can occur in any loop channel in the sets. The zero equilibrium range of outputs includes responses to transient inputs, which may consist of short trains of pulses and waves, but which terminate in zero pulse rates and unvarying wave amplitudes.
If, when K0 is increased sufficiently, the KI and KII active states are increased and are manifested by pulse trains and wave amplitudes at steady, nonzero mean rates, the sets are at a new equilibrium. If the active states are changed by transient inputs and then return to the preinput levels and stay there, the equilibrium is stable. This is the stable nonzero equilibrium state of KI and KII sets. Within this domain, the constant values of the <Page 45> mean pulse rates and the wave amplitudes depend on K0 and increase with increasing K0.
For the KII set and the KI sets within KII sets, there is another state, which is induced by yet further increase in K0. This state is characterized by limit cycle oscillation of the active states of subsets in the domains of the component KI sets in both the wave and pulse modes. The value for the frequency of the oscillation may or may not depend on the value of K0 over a limit cycle domain. It is in the limit cycle state that certain types of electroencephalographic potentials (EEG waves) arise. It the oscillatory active state has a certain frequency and amplitude in the absence of input, and if activity manifested in the EEG is changed by transient input and then returns to the same frequency and amplitude, the limit cycle is stable. For example, in certain unusual conditions, such as following a period of hyperthermia with brain temperature of about 42°C, the KII set in the prepyriform cortex generates sustained high–amplitude sinusoidal EEG waves at a frequency of about 28 Hz. This EEG activity manifests a limit cycle. The activity is only briefly altered by strong electrical stimulation of the afferent path to the cortex. It is totally suppressed by a brief period of asphyxia, but it recurs at the same frequency and amplitude on recovery from asphyxia. Therefore, the limit cycle is stable.
The existence of nonzero equilibrium and limit cycle stable states depends on the presence of sufficient sustaining excitation represented by the parameter K0. If K0 is decreased sufficiently following the induction of a limit cycle state, the KII set returns to the nonzero equilibrium state or further to the zero or resting equilibrium states. Whether the thresholds for K0 and the active states are the same for entering and leaving each of these states is not known from experimental data. The possibility that the entering and leaving thresholds may differ significantly and give rise to hysteresis effects has been shown by theorists (Section 6.2.5). Such effects may occur in both KI and KII sets.
To summarize, multiple stable states can exist in the activity of neural sets, which are characterized by the stable invariance of one or more state variables. In the zero equilibrium state the pulse density function and the rate of change with time of wave amplitude are everywhere zero. In the nonzero equilibrium state the rates of change with time of pulse densities and wave amplitudes are zero. In the stable limit cycle state the pulse densities and wave amplitudes vary at a fixed frequency. These stable states are the set of conditions in which the KI and KII sets are placed or are assumed to be prior to measurement of their observable state variables. The experimental evidence and the neural mechanisms for these stables state are described in Chapter 6.
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1.3.6. THE RELATION OF MULTIPLE STABILITIES TO NEURAL SIGNALS
The importance of sustained excitatory input as a determinant of the properties of neural sets is well known. For example, the interesting linear interactive properties of the Limulus eye depend on the presence of diffuse illumination of the photoreceptors that provides background excitation to the KIi set and places it in a stable nonzero equilibrium state.
It is not generally appreciated that these stable states are ubiquitous in the mammalian central nervous system, lasting for time periods ranging from 0.1 sec to many seconds, as shown by the near–universal presence of background activity in the form of seemingly random pulse trains. If we use the concept of multiple stable states to interpret our electrophysiological and behavioral data, we open a new approach to understanding the neural dynamics reflected in these data.
For example, suppose that the neurons in a KIe set at rest in a zero equilibrium stable state are brought sufficiently close to threshold by a brief excitatory input so that each neuron giving a pulse to its neighbors excites them and receives more than one pulse in return. The feedback gain is greater than unity, and the active state of the mass must increase. The increase continues until the pulse rate of each neuron is high enough to be limited by the relative refractory period of each neuron. Under certain conditions corresponding to a normal physiological state of the animal, the input can be terminated, and a new steady state develops. In this state each neuron in the set fires at nearly random intervals of time and independently of its neighbors but at a steady mean rate. If the set is further perturbed by input, the active state of the set changes momentarily but returns to the prestimulus steady state. It is therefore a self–sustained, nonzero equilibrium stable state. It may be terminated by a brief inhibitory pulse.
This mechanism involving one KIe set with output to a motorneuron pool can account for an action in which a limb or an eye is moved to a certain position, held there for a certain time, and then returned to its starting point. The actual position may depend on the value for K0 in the KIe set, which may be determined by the membrane properties of the neurons in the KIe set or by sustained input from a KO set or another KI set.
In the absence of sustained excitatory input the KII set stabilizes in the stable zero equilibrium domain. The introduction of an excitatory bias from a KO, KIe, or another KII set can induce stable nonzero equilibrium and limit cycle states in the KII set. Examples are given in Chapters 5 and 6.
In order to explore these possibilities to maximal advantage, however, we must pause to recognize the complexity of the undertaking and to adopt some guidelines for experimental analysis. The most important restrictions are: (1) We must have clearly in hand the topological and electrophysiological <Page 47> data on a neural mass before proposing a K–set analysis, in order to specify the number of sets involved, the constraints on the operations of each set, and the K–level of analysis; and (2) we must know the behavioral correlates of the activities of the neural mass so as to determine the domains of normal function.
As stated in Section 1.3.1 the combination of KII and lower–order sets having feedback between them gives a system at the KIII level. This approaches in complexity the level of the classical sensory, motor, or corticonuclear systems. An example is shown in Fig. 1.5d of the primary olfactory system, consisting of the receptors (the KOR set), the olfactory bulb (the KI and KIIMG Sets), the olfactory cortex (the KIIAB and KOC sets), and the anterior olfactory cortex (N, an unidentified set). These sets together with their forward connections and feedback interconnections (as well as others not shown) comprise the mass of the primary olfactory system.
Whereas KO, KI, and KII are the levels of electrophysiological analysis, KIII is the lowest level of behavioral analysis. At this level a space–time pattern of neural activity, which is represented by an activity density function, can be said to be a neural signal, in the sense that it constitutes a unique effect and correlate of an external stimulus. For example, an odor (which, as distinct from an odorous substance, is an effective stimulus) must be represented by a space–time pattern of activity in each KO and KI set in the olfactory system, including the receptors, the bulb, and the cortex. Each pattern is a signal at a particular stage in a transmission sequence through the olfactory system. When these patterns have been described and measured in order to specify activity density functions, the operations within and between the sets can be determined, including transmission or translation and delay, transformation, storage, comparison, classification, and elective readout from stage to stage.
A concluding hypothesis of this monograph is that some neural signals may exist by virtue of the interactions of neurons in masses, and they may then be identified with specific active states (macrostates) of neural sets at the KO and KI levels. The complex space–time dynamic patterns of the active states, which are the neural equivalents to dissipative structures, are the result of stable limit cycle states in KII sets. Such states arise primarily because of continuing input from a KIe set into a KII set, which drives the KII set from a nonzero equilibrium range into a limit cycle state, in which the signal is held for as long as transforming operations require.
1.3.7. THE CONDITIONS FOR REALIZABILITY
In order for the function of a KO, KI, or KII set to be observable, four conditions must hold. First, it must be possible to isolate the set, so that the pattern of transmission within it conforms to the proposed topology.
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This can be done by control of input or by use of a variety of surgical or pharmacological techniques. When it is not possible to do this, it may be possible to dissect a complex pattern of responses into components, one of which represents the operation of the set in the designated mode. If neither approach is feasible, there is no alternative except to construct a higher–order model.
This condition may be very difficult to satisfy by surgical isolation of a part of the brain such as cortical undercutting, because the sets in such a part may depend on an external bias for maintenance of certain internal connections in a functional state. A KII set that is normally capable of oscillation may be reduced by deafferentation to the functional level of its KO components, and the analysis may reveal little more than the properties of its component neurons. In general it is necessary to work mainly with intact preparations, to propose a set of alternative models based on local or more distant topologies, and to weed out alternatives as opportunities arise.
The second condition is that the component neurons must generate detectable forms of activity, either in the form of dendritic potentials or pulse trains or preferably both. Many and perhaps most neurons in the brain are too small to permit intracellular recording. Many do not generate extracellularly detectable pulses and may not generate pulses at all. Others generate no extracellularly detectable dendritic potentials. In some instances there may be no measurable extracellular potential for a set, and its existence and actions must be inferred from the behavior of other sets with which it is connected anatomically. Analysis of an unknown mass into sets requires formulation of several simple alternative topologies and dynamics on the basis of anatomical examination, which can predict the existence of classes of responses of the mass and its neurons for specified input. The failure of predicted classes of response to exist and the existence of supernumerary classes give clues as to how the simple models are to be modified or elaborated. This is the dialectical process between theory and experiment.
Third, the experimental techniques must permit multiple simultaneous measurements at different points in the mass. (Uncommonly, the function of the mass may be shown to be sufficiently constant in time that sequential sampling at different points is permissible.) This condition follows from the definition of a set as a number of neurons, which must have a certain distribution in space. The existence of cooperative activity cannot be inferred without demonstration of correlated activity at many points in the set.
Fourth, the measurements of active states of single neurons must be averaged over time and space to give estimates of the active states of sets of neurons. The duration of observation or time window ∆T for averaging and the size of the domain of spatial averaging cannot exceed the temporal <Page 49> size and spatial limits (lower as well as upper) of the extent of a cooperative event. This condition severely limits the usefulness of the pulse train or intracellular dendritic potential of a single neuron. It places the greater value on the extracellularly recorded dendritic potentials and groups of action potentials p called unit clusters, which result from the weighted sum of extracellular currents near a recording site. Such events represent spatial averages that in the appropriate experimental situation can be interpreted as or used to infer the time–varying means for the active states ov and op of a set or a subset near the site.
For example, one of the manifestations of a limit cycle in a KII mass is the occurrence of extracellular fields of potential that vary in amplitude in a rhythmic manner–the EEG waves. They can occur as background activity without reference to a specific external stimulus. The background pulses of single neurons in such a mass may appear to occur at random, but if their times of occurrence are averaged in an appropriate way with respect to the ensemble mean taken from the EEG, it is seen that the probability of pulse discharge may oscillate in time at the frequency of the EEG. This oscillation is called the pulse probability wave of the neuron. Although the pulses are generated by a single neuron, the pulse probability wave is a collective property of the KII set within which the neuron is embedded.
1.3.8. THE USE OF DIFFERENTIAL EQUATIONS
An overview has now been given in general language for analysis of neural masses along with a proposed set of models to describe them. Each of these models is described in detail in subsequent chapters. As noted above each has four aspects.
First, the anatomical structure is reviewed and expressed in the form of a geometry and a topological flow diagram. The geometry is important with respect to observation and measurement, and the topology is essential for the dynamics.
Second, the input–output relations are summarized and expressed by means of generic wave forms in pairs. These input–output pairs serve to classify the system under study in relation to certain general properties, such as linearity, time invariance, and level of complexity. They serve to specify the kind of descriptive equation needed for each set and its degree of complexity.
Third, a set of differential equations is constructed to represent the dynamics of the mass. The class of each equation is determined by the input–output relations, and its internal structure and parameters are chosen in accordance with the topology. The solution of the equations for the initial conditions prescribed by the input is compared with the output of <Page 50> each input–output pair. The differential equations are modified until the solution conforms to the output, and they are then said to represent the dynamics of the set.
Fourth, the parameters of the equation are interpreted in terms of the parts at the lower levels in the hierarchy. The differential equations are then simplified, so that they can serve as a part in relation to models of sets at higher levels in the hierarchy.
The mathematical approach is required for two reasons. It is the only means by which quantitative comparisons can be made between the observed and predicted responses of selected sets, so that proposed dynamics can be accepted or rejected. Also it is the easiest way to combine the models constructed at one level of the hierarchy to formulate a model for testing at the next level. Because the mathematics is a means and not an end, the forms of the equations and the means of solution have been kept as simple as the experimental observations would allow. The peculiar properties of neural masses even in multiple stable states have lent them to an unanticipated degree to the use of linear differential equations with amplitude–dependent coefficients. Therefore, the analysis of the dynamics of masses is not limited by the intractibility of nonlinear equations but by the need for better definitions and measurements of observable events. Much of the material in this monograph is devoted to the problems of how to observe and how to measure events in neural masses, so that proposed models can be adequately tested.
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CHAPTER 2
Time–Dependent Properties
The activity of neurons is manifested by time–varying neural fields of potential. The procedure for analysis of a temporal sequence of magnitudes for a source for an electrical field consists of (1) repeated measurement of the field potential, (2) postulation of a model expressed in the form of a differential equation, (3) solution of the equation for specified initial conditions and input to obtain the time response of the system, and (4) comparison of the observed and computed field potentials. For the simplest class of neural events the models can be based on ordinary and partial differential equations with constant coefficients. In this chapter the experimental conditions are described for which such equations are applicable, and an operational method for solution is demonstrated with neural phenomena.
2.1. Measurement of Neural Events
2.1.1. REPRESENTATION OF EVENTS BY FUNCTIONS
Electrophysiology is based on observations of continuously varying currents and potential differences in and around neurons. Such observations are usually milliseconds to hours or more. Because neural systems under study have causal dynamic processes, each pair is inferred to have an antecedent or input member and a subsequent or output member. If the observer has adequate or complete control of the input member, it is called <Page 52> a stimulus, and the output member is a response. This may be called a stimulus response pair. If control is inadequate or not imposed, the pair should be called an input–output pair.
Each member varies with time. It is measured by assigning to it a real number at each moment in time and a real number to that moment. The set of real numbers assigned to time forms a domain, and the set assigned to the variable forms a range. The relation between the pairs of real numbers constitutes a function. A function is defined as the relation between a set of ordered pairs of real numbers in which no two numbers in the range correspond to the same number in the domain. In the case of neural events no variable has two values at the same time, but the same value of the variable can occur at two or more times. The input functions and output functions constituting input–output pairs are represented by symbols, such as v for potential, i for current, p for pulse, and f for any of these and are called functions of time, such as v(t), i(t), p(t), and f(t).
The process of measurement by which the sets of real numbers are derived has two requirements. The first is a set of basis functions. A basis function is an elementary function of time that is defined over all time. Some examples of common basis functions that we will use are the exponential
the sine wave
and the rectangular pulse
with ∆t = t2 – t1 . The typical output functions of K–sets are complicated functions, which we like to express in a simpler way. One possibility is by using sums of simple functions such as exponentials and sine waves. A family of basis functions that lets us do this is a basis. A set of basis functions is a specified number of basis functions that is added together to fit or reproduce a function being measured.
The second requirement is for units of measurement, such as seconds, volts, amperes, pulses per second, etc., and their submultiples. When a unit of time has been selected, such as a millisecond, an appropriate real number for α, ω, and ∆t in Eqs. (1)–(3) is then assigned to each basis function in the set. When the unit of the variable has been selected, such as <Page 53> millivolts, a real number is assigned to v0 in each of the basis functions. The real numbers must be chosen so that the sum of basis functions optimally represents the observed function.
Example A. An event is observed in the nervous system that consists of a stimulus current i(t) and response potential difference v(t). The period of observation from t = 0 to t = T msec is divided into N time intervals(commonly N = 100), so that ∆t= T/N msec. Each basis function is a rectangular unit pulse pn(t) ∆t, having duration ∆t, location in time tn and amplitude of 1 µA or 1 µV.
Each basis function is multiplied by a real number i(t) or v(t), so that the amplitude of the rectangular pulse optimally approximates the amplitude of the input or output member at tn. The set of basis functions is added over time
where pn(t) ∆t is a unit vector and i(tn) and v(tn) are real numbers. §
This procedure is called digitizing. We use primes to designate events measured in this way. The process is complete when the set of real numbers suffices to reconstruct an event to any specified degree of precision.
Example B. Suppose that an input–output pair is observed to undergo periodic oscillation in time. We can measure each function by digitizing to obtain i’(t) and v’(t). We can then fit these functions with another basis function.
The process of measurement of the two functions consists of finding values for i0, v0, ω, and φ, such that
where εi(t) and εv(t) are minimized deviations (errors) between i(t) and i’(t) or v(t) and v’(t). The usual criterion is least mean squares deviation. This aspect of measurement is a problem in curve–fitting, and is treated in many statistical texts. For some basis functions, such as straight lines and single exponentials, we can use linear regression. For most sets of basis functions, <Page 54> such as sums of exponentials and sine waves, we use nonlinear regression (Section 2.5.1). §
Each basis function has one or more parameters other than its amplitude. These may be fixed as is, for example, the duration of the rectangular digitizing pulse. When we vary the parameters, as in the case of ω and φ in Eqs. (6), we have an adaptive basis function. A basis function can be considered to establish a coordinate space for measurement, similar to a unit of length, volume, pressure, etc. Some set of basis functions is necessary and sufficient to represent any observable neural event.
The nature of the representing function depends on the choice of basis functions. The outstanding virtue of the rectangular basis function or digitizing interval is that it permits measurement to any desired degree of precision without the restriction as to the nature of the wave form being measured, and without inference regarding the underlying dynamics. All of the measurements from electrophysiological observations described in this study, which are the data, have been made by digitizing. They are represented graphically by points.
Other basis functions are derived from two sources. Theoretical basis functions result from solving differential equations representing the dynamics of neural systems. Experimental basis functions are derived by inspection of observed events in time. All of these basis functions are displayed graphically as smoothed curves.
Example C. Two microelectrodes are inserted into a neuron, and two electrodes are placed in the tissue outside it. A brief current pulse is passed between one inside and one outside electrode, and the potential difference is recorded between the other inside and outside electrodes. If the duration of the current pulse is shorter than the digitizing interval, the stimulus member of the input–output pair can be represented as
where i’(0) is the amplitude of the applied transmembrane current pulse in microamperes. The response member is represented by
where vm(t) is a continuously varying transmembrane potential difference, and v’m(t) is the sequence of digitized values of amplitude of transmembrane potential difference in millivolts.
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By inspection the sequence v’m(t) often takes the form of an exponential decay. A new basis function is chosen, such that
where v0 is amplitude in millivolts at t = 0 and τ is a time constant in units of milliseconds. By proper choice of v0 and τ, vm(t) may be fitted to v’m(t) to an acceptable degree of approximation. §
Unless otherwise stated, the values of stimulus response pairs are always zero for times less than zero. In order to simplify the notation, the functions of input and output are represented as
and
Zero values for t < 0 are implied if not stated. Equation (12) may not represent vm(t) as precisely as Eq. (9), but it reduces the parameters of the output to two, as opposed to the number of digital values, and it suggests something about the nature of membrane dynamics, which Eq. (9) does not.
2.1.2. INPUT–OUTPUT FUNCTIONS
Multiple observations of input–output pairs are made in an experiment or a set of experiments. A set of ordered input–output pairs constitutes a function. Again, no two different output members can have the same input member. That is, the same input repeated two or more times cannot result in two or more different outputs. If it does, then either there is some other uncontrolled input, such that the input members are not completely described, or the neural system is changing with time, so that the set of input output pairs does not constitute a function.
If the variation of the input or the system is not of interest, then we average across repeated trials. We can do this by digitizing the input or output functions on each trial w so that, for example,
where ∆T = T/N and ε denotes the ensemble average. We average the values of i’w(T) or v’w(T) over W trials for each n to estimate the <Page 56> ensemble average (Fig. 2.1):
FIG. 2.1. Derivation of an ensemble average.
If on repeated trials the
measurements either of input or output are randomly distributed, then the mean 
w(T) is the most reasonable estimate for v'w(T) and îw(T) is likewise for i'w(T). The ordered pairs then constitute
a function F such that
If the variation is of interest, or if the measurements cannot be averaged, then the input–output pairs must be ordered into more than one set, so that other functions are defined.
Example A. Suppose that in Example C (Section 2.1.1) the measured output v0 is found to very randomly on successive trials but without variation in or observable variation in i'f(t). We may ignore the variation by taking ensemble averages [•] as in Eq. (14) or we may infer that there is unexplained variation in our input and look for it by describing <Page 57> another set of functions
where iε(t) and vε(t) are random variables. We may infer that
or more generally
We may also infer time variation in the system, which is formulated in a way depending on the results, one of which might be
where k(T) may be a randomly varying parameter. §
If a system gives the same responses on successive trials of the same input, or the same mean responses on successive sets of trials with the same input, then it is time invariant. If it is rapidly time varying, then an explicit time–dependent function of a variable or parameter is required, as above. If the time variance is slow with respect to the duration of each trial, the input–output pairs are ordered into a sequence of sets or functions, each function being valid only for a limited time, and only by approximation. This condition holds for most physiological research.
2.1.3. LINEAR INPUT OUTPUT FUNCTIONS
The order in a set of input–output pairs is imposed by ranking them with regard to one or more of the variables describing the input members, such as stimulus or amplitude. The entire sample of input–output pairs is never complete, because new pairs can always be added. The set can be divided into subsets by imposing other constraints. The input members of a subset constitute a domain of input, and the corresponding output members define a range of output.
The most important subset for our purposes is established by imposing the constraints of proportionality and additivity. Proportionality or homogeneity means that the input and output increase by the same factor:
Additivity means that when two inputs are given at the same time, the output is equal to the sum of the outputs of the two inputs given separately:
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These two conditions are equivalent to the single condition of super–imposibility:
This condition establishes a domain of input and a range of output that are said to be linear. This subset of input–output pairs defines a linear function, and the neural dynamics are said to be linear in this range. The linear domain of input for membranes and for neurons is a small part of the total input domain, comprising inputs with low amplitudes and relatively short durations. However, a large part of neural activity takes place within the linear range, and, in fact, the range is broader than it at first appears.
Example A. A pair of electrodes 3 mm apart is placed on one end of an axon suspended in mineral oil, and a second pair 3 mm apart is placed 20 mm down the same axon. A single shock consisting of a brief current pulse with a fixed amplitude is delivered to the axon at one end, and the external potential difference is measured further down the nerve. The input amplitude is varied on successive trials. Below a certain input level, there is no output; above that threshold there is an action potential, the amplitude of which is independent of the input. This so–called all–or–none property is an example of nonproportionality of output to input. Such a system cannot be linear. §
Example B. Paired shocks are given, the first or conditioning shock being supra threshold, the second or test shock being adjusted to threshold. With decreasing interval between the two shocks, the amplitude of the second must be increased in order to obtain a second response (relative refractory period), and for an interval corresponding approximately to the duration of the first response (the action potential) there is no second response for any intensity of input (absolute refractory period). This is an example of nonadditivity of output for the sum of two inputs displaced in time and is evidence for nonlinearity in the system. It is also evidence for time variance. §
Example C. Let the conditioning stimulus current be considerably below the amplitude for threshold (thr) ithr and have amplitude i1. At times long after the first shock, the required threshold amplitude for the second shock i2 to elicit a response is that threshold value ithr needed in the absence of the conditioning shock. For shorter intervals, the experimental evidence (Katz, 1939) is illustrated in the relation
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where T is a constant that depends on the nature of the axon and the external medium. Now let i1 be either negative (cathodal) or positive (anodal) but not be given values near ithr . In a certain small domain, the required amplitude of the test shock i2 is linearly dependent on the amplitude of the conditioning shock i1 by Eq. (23). This is an example of proportionality. §
Example D. Let two subthreshold conditioning shocks, i1 and i2, not necessarily equal in sign or magnitude, be given at an interval T. The threshold voltage ithr and test input i3 required to trigger the impulse are experimentally determined as before. They conform to the relation
The effect on the nerve produced by the second conditioning shock adds to that produced by the first. Both effects decay independently of each other, and the time course depends only on the initial amplitude and time of onset of each. This is an example of additivity. §
In these examples the nerve axon responds linearly to subthreshold inputs, and its main nonlinear characteristic (the threshold) can be used as a tool for measuring its function in this range.
Example E. Let a number n of suprathreshold stimuli in a train be given the same intensity i(tn) at times tn, separated by an interval greater than the relative refractory period. A set of n action potentials p(tn) results which is identical to the sum of the responses of n single stimuli given at times tn. This is an example of additivity. Now let us redefine the input variable, calling it the average number of stimuli per second in each second î(Tn), and the output variable, calling it p(T) the average number of action potentials or pulses per second. Output is then proportional to input, and it is additive for two or more inputs, provided that no interval between two or more stimuli is less than the refractory period or greater than the averaging period, and provided that the amplitude of each stimulus train is fixed above threshold. This defines a set of linear domains and linear functions for the axon, there being one domain for each suprathreshold amplitude of input. §
These examples show that the terms "linear" and "nonlinear" are applied to the domain of input, the range of output, and to the function in that range, but not to the neuron. A membrane or an axon is a system having both linear and nonlinear functions, and each function is nothing more than the description of the operation for transforming or mapping a subset of inputs into a subset of outputs with correspondence between pairs.
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2.1.4. THE IMPULSE AND THE IMPULSE RESPONSE
The single shock has long been recognized in physiology as a powerful driving energy applied for a time interval that is very brief in comparison to the duration of the response. The shock is measured by the product of its magnitude and duration if it is a rectangular function. More generally, it is the integrated area under the input curve.
The mathematical impulse or Dirac delta function may be defined as a rectangular pulse of unit area. Let An(t) = n for 0 ≤ t ≤ 1/n (Fig. 2.2), and A(t) = 0 elsewhere. As the duration 1/n approaches zero, the unit impulse is
Under certain conditions it is permissible to represent the single–shock input to a neural system by the delta function and call the output the impulse response. Experimentally, these conditions may be established by applying a rectangular pulse of appropriate magnitude and duration to an axon, nerve, or afferent tract. The duration is then decreased and the amplitude is increased in such fashion that their product is kept constant. For relatively long durations the system may show varying, mixed, or long latency components in the response, such that the form of the response changes as the pulse duration is decreased. Below some value for the duration, the output function is independent of the pulse duration. Below that value, the shock can be treated as an impulse and can be represented by the delta function δ(t) multiplied by a constant representing intensity. †
† This procedure should not be confused with that for determining the strength–duration curve. A rectangular current pulse I is applied to a nerve for a time interval T just long enough to elicit an action potential, i.e., threshold is determined. For low pulses, long durations are required. There is some level of current Ithr below which excitation will not take place for any duration (rheobase or threshold current). The values for I and T when plotted in rectangular coordinates conform to a hyperbola having the equation
where Qthr is the total charge displaced across the nerve membrane by the exciting current at threshold. The chronaxie is the duration required for a threshold current having twice the amplitude of the rheobase. It is an empirical measurement, and a pulse at chronaxie may or may not be regarded as equivalent to the impulse, depending on whether the response being observed is independent of changes in duration about the chronaxie.
These concepts are particularly useful in neurophysiology for two reasons. First, the axon responds to a variety of input wave forms by generating one or more action potentials. If each action potential causes an event in other neurons that does not depend on the duration of the action potential, it can be treated as an impulse, so that irrespective of the nature of the input to the axon, the input to some further stage may be treated in terms <Page 61> of impulse functions. Second, the impulse response f(t) of a linear system consists typically of the sum of exponential terms (including the step and impulse) and damped sinusoids
where A and B are the initial amplitudes in volts, the αn are rate constants in reciprocal seconds, ωn are the frequencies of sinusoidal oscillation radians per second (equal to 2π times the frequency in cycles per second) –φn are phase of onset expressed in radians or degrees of the sinusoid, and ßn are the rate constants in reciprocal seconds of the envelopes of the sinusoids. Both αn and ßn are real numbers.
FIG. 2.2. Derivation of the delta function.
The term cos (cot) may be written
where j = (–1)1/2. Using Euler's theorem ejy = cos y + j sin y this may be written as
Therefore the general form of the impulse response of a linear system may be written as
where αn may be either real or complex, and the complex terms occur always in conjugate pairs.
2.2. Linear Models for Neural Membrane
2.2.1. THE TOPOLOGY OF THE MEMBRANE
The anatomical structure of each neuron generating or affecting a field of potential consists of a closed surface (the membrane) across which ionic currents flow in closed loops. The internal and external compartments and <Page 62> the currents have three dimensions, and there is great variability in their geometries. In two dimensions the structure and the flow can be represented by an arbitrary closed boundary and by closed lines of current (Fig. 2.3). For topological purposes, the structure and lines are reduced to linear configurations. Two patterns of current are distinguished. Transmembrane current im flows across the membrane and longitudinal current flows parallel to the membrane. By convention, outward flow and flow to the right are positive (Fig. 2.4).
FIG. 2.3. Topological representation of loop current for a neuron of arbitrary shape.
FIG. 2.4. Reduction of loop current to transmembrane, extracellular, and intracellular geometrical channels.
From Kirchhoffs current law, the sum of currents entering and leaving every node at all times is zero:
From Kirchoff’s voltage law around any closed path in a topological representation the sum of potential differences at all times is zero:
The properties of the system are derived by describing the relation between current and voltage for each node and path in the topological diagram.
How do we know what properties to assign to the current paths across the membrane and in the two compartments? Our best information comes <Page 63> from input–output pairs. We know that when the input member is a subthreshold step or a pulse, the output member can be measured with a set of basis functions containing at least one exponential term with a real rate coefficient α. Further, we know that the exponential function is the solution to a first–order differential equation. We infer that the dynamics of passive membrane can be described by such an equation.
From elementary circuit theory we know that current paths exist in which the current i(t) is proportional to the rate of change of potential difference dv(t)/dt by a fixed parameter called capacitance C:
We know that other current paths exist in which current is proportional to potential difference by a fixed parameter called resistance R:
Here C and R are defined by Eqs. (33) and (34) where R is measured in units of ohms (volts per ampere) and capacitance is measured in farads (ampere × second per volt) or in microfarads (10–6 F).
Salt solutions have pure resistance over all rates of change of potential that concern us here. Because the inner and outer compartments usually behave like salt solutions, and because the membrane has an appropriate molecular structure, we infer that an element behaving like a capacitance is in the membrane. Electrodes can be represented by an equivalent circuit composed of resistors and capacitors, but attempts to approximate this circuit by a single resistor R and a single capacitor, C in parallel yield a circuit element for which R and C increase with decreasing frequency or rate of change of potential (Cole & Curtis, 1939). Thus, a parallel combination of fixed R and C is merely an approximation for the elements of an electrode. Similarly, the equivalent resistance and capacitance of the membrane depend on the rate of change. At relatively low or high rates of change, they are treated as fixed parameters. They are functions describing the relations between current and voltage in limited domains, and should not be identified literally with physical circuit elements.
The topology of the membrane is elaborated to introduce the specific functions that are assigned to each current path. As shown in Fig. 2.5, Rm, designates transmembrane resistance and Cm the transmembrane capacitance of each part of the membrane, Re is extracellular resistance, and Ri is intracellular resistance.
An element across which the voltage is proportional to the rate of change of current is an inductance measured in henries:
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FIG. 2.5. Representation of channels for loop current by dynamic elements defined by functions.
Although this has been introduced several times into physiology in the past half century, its use has been discarded in favor of nonlinear circuit elements that is, variable voltage–dependent membrane conductances (Hodgkin & Rushton, 1946). The present discussion is restricted to the linear range of neuron function and does not require use of an equivalent inductance.
2.2.2. DIFFERENTIAL EQUATIONS
A set of examples is given to illustrate several aspects of analysis of the model for neural membrane.
Example A. Electrodes are placed in an axon as in Example C, Section 2.1.1, and are used to pass a unit current impulse (t) across the membrane to a distant external electrode. The output function to be predicted is transmembrane potential vm(t) and its dependence on the input current. A simplified representation is shown in Fig. 2.6. From the definition of the impulse, we assert that the input pulse lasts for a negligibly small time interval during which the capacitance is charged to a certain potential vo. After the pulse ends, current flows only as indicated by the arrows in Fig. 2.6. By Kirchoff’s laws, in Section 2.2.1,
FIG. 2.6. Reduction of representation of loop current to lumped circuit model for membrane.
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Therefore
where a is a rate constant in seconds and a = 1/RmCm. The solution is
where v0 is the voltage across the capacitor at time t = 0 by the impulse δ(t). The charge q delivered by a unit current impulse is one coulomb:
Thus the response for a unit current impulse is
The same model holds to a first approximation for the response of a synaptic membrane to an afferent volley in which the current source is a region of activated membrane and the RC network is the passive membrane. This is discussed in Section 2.3.4.
FIG. 2.7. Elaboration of Fig. 2.6 by introduction of longitudinal resistance.
Example B. The same electrode is used to pass a current step µ(t) across the membrane. The external resistance Re is included in the model (Fig. 2.7). Again by Kirchhoffs laws
The input is
Therefore
where the rate constant is given by a = 1/RmCm in units of reciprocal seconds. The solution is
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The ratio of output to input is
in volts per unit current step.
This model is used to measure the parameters of membrane with intracellular stimulation and recording. As t –> ∞, vc(t) –> V∞, and Rm = V∞/I in ohms. When t = 1/a in milliseconds, vc(t) = V∞ (1 – 1/e). Then Cm = 1/aRm in microfarads.
Example C. The same circuit is used as in Example B, but the input is a step voltage vi(t) = V0µ(t). From Kirchoff’s laws,
Therefore
Let
Equation (47) then becomes
The solution has the same form as for Eq. (42),
or as the ratio of output to input,
The impulse response for a voltage pulse can be found by differentiating Eq. (50) with respect to time and is
The rate constant b1 is not that for passive membrane a, because when the output in response to a voltage input v0 depends on both the transmembrane and external resistances, the external path short–circuits the membrane and causes the potential to change at a more rapid rate. This <Page 67> model conforms to the conditions obtaining with stimulating electrodes commonly used in which the output voltage and not the output current is regulated (characteristic of transformers, batteries, and other sources with low internal impedance), or to extracellular stimulation in vivo when the extraneuronal fluid short–circuits the stimulating current.
Example D. Let two widely spaced electrodes be placed on a nerve in air and be connected through a switch to a battery (Fig. 2.8). It is proposed that the transmembrane voltage v2 at the cathode can be predicted as a function of v1(t), i.e., the ratio v2(t)/v1(t). From Kirchoff’s laws,
Because of the rectification characteristic of nerve membrane, R1 ≠ R2 and the classical methods exemplified here are very cumbersome. The solution is deferred to Section 2.2.4.
FIG. 2.8. Elaboration of Fig. 2.7 by introducing a difference in the membrane at site of current inflow i1 and outflow i2.
These examples illustrate the general property that in the linear range of function, neuron systems are described by linear differential equations in the input and output. The solutions represent predictions of what the output will be for a specified input. For realistic problems representing more complex topologies, the solutions are too difficult to obtain by the conventional methods described. An operational method will now be introduced to provide solutions to such problems more easily.
2.2.3. THE LAPLACE TRANSFORM
The study of a neuronal system consists of analysis of its output for specified input conditions and synthesis of a model expressed by differential <Page 68> equations. In diagrammatic form there is an input u(t) and an output v(t),
both functions of time, which are related to yield a statement about the intervening transforming process F.
In Section 2.1.4 it is stated that for an impulse input, the output of a linear system has the form
which is the sum of the exponential and sinusoidal components. This holds for a variety of inputs including impulses, steps, exponentials, etc., so that for all commonly used inputs
In Section 2.2.2 it was stated that the characteristics of a linear system can be described using ordinary differential equations in the input and output functions. The generic form is
where the aj and bj, 0 ≤ j ≤ n, 0 ≤ j ≤ m, are constants. The problem for analysis is to evaluate Aj, bj, α and ßj from observation of the neural system. The problem for synthesis is to set up Eq. (56) in the effort to bring together all sources of information about the system in a form suitable for comparison of predicted output with measured output.
In the event that all initial conditions are zero as with impulse input, we can think intuitively (for our present purposes) of a variable s as an operator,
expressing the operation of taking the derivative with respect to time. This operator s has the dimensions of reciprocal seconds. Substitution of s for d/dt in Eq. (56) yields a polynomial in s:
where the input and output are expressed as functions of s rather than <Page 69> of time. This can be rewritten by collecting terms:
The ratio for output to input is then
The substitution of s for d/dt when the initial conditions are all zero constitutes the Laplace transformation of Eq. (56) to Eq. (58), which is expressed in generic form as
The inverse transform is by reverse substitution:
The transfer function of a linear system can now be expressed as a function of s and is defined as the ratio of the input function to the output function,
so that
The differential equation is now in the form of the ratio of two polynomials. The next step toward solution is to factor these polynomials into the form
where cj and dj are real or complex, and where complex numbers always occur in conjugate pairs (see Section 2.1.4). When m or n is equal to 2, the quadratic formula is used. For higher–order polynomials, other methods are required (e.g., DiStefano et al., 1967, pp. 64–80).
Because the roots in Eq. (64) take the form of complex numbers, the operator s may be given values in the form of complex numbers. When s approaches –cj for 0 ≤ j ≤ n, then F(s) approaches zero. Characteristically, the output of a system does not contain frequencies for inputs whose frequencies are such that s approaches – cj for some j where 0 < j ≤ n. On the other hand, when s approaches some –dj, 0 ≤ j ≤ m, then F(s) approaches infinity. This represents a characteristic or natural frequency or rate of change for the system. The values for s = –cj are known as the zeros and the values for s = –dj are known as the poles of the system.
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The poles and zeros together completely describe the characteristics of a linear system. The values for the poles and zeros are readily susceptible to graphic display in the coordinates of α and ω, which is known as the s plane, where α is the real part and ωis the imaginary part of s = –α ħ jω.
The ratio of factored polynomials provides the solution to the differential equation. The inverse transform, Eqs. (62), yields the output function for the impulse input. This consists of the sum of exponential terms having rate constants specified by the poles
The amplitude coefficients Bm are found by the method of residues (DiStefano et al., 1967). Let the value of s approach the value of one of the coefficients in the denominator
Therefore
The procedure is repeated m times to find the m values for Bm. Conjugate pairs of complex exponentials are transformed to sinusoids by use of Euler's theorem (see Section 2.1.4).
2.2.4. APPLICATION OF THE LAPLACE TRANSFORM TO THE MEMBRANE
The operational method is now demonstrated using the examples in Section 22.2.
Example A. This is the same as Example A in Section 2.2.2, Fig. 2.6. The equation relating input to output is
The Laplace transform of the impulse function δ(t) is equal to unity (see Section 2.3.2). Therefore the Laplace transformation of Eq. (68) is
Algebraic reformulation to solve for the ratio of output Vm(S) to input, which is F(s) = I, gives
The inverse transform is
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Example B. The system (Fig. 2.7) in Example B, Section 2.2.2, has the input function
a constant times the unit step function. The Laplace transform for µ(t) is 1/s (see Section 2.3.2). Therefore, from (43)
By rearrangement
The solution is
or
Example C. Equation (47) in Section 2.2.2 can be rewritten as
This transforms to
The solution is found as for Eq. (73), and it is identical to Eq. (51) in Section 2.2.2. §
Example D. Equations (53) in Section 2.2.2 are written as time functions and combined (see Fig. 2.8):
The Laplace transform gives
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By substituting Eq. (80) into Eq. (81) and regrouping,
where
The solution by inverse transform is
This reduces to
For the case where R1 = R2 and C1 = C2,
Further examples of lumped circuit analysis using conventional and operational methods can be found in any of several texts on linear analysis. The examples selected here represent common problems in the analysis of neural membrane dynamics, particularly in regard to whether the input is a current or voltage, when the output is measured as a potential difference across the membrane or across the external resistance of the medium.
2.3. Linear Models for Parts of Neurons
An operational method has now been demonstrated for solving simple differential equations from the network model of passive membrane. Before the method is applied to more complex systems, of which the membrane is a part, the operation must be defined, and some of its basic properties must be clarified.
2.3.1. CONVOLUTION
A property of a system in a linear, time–invariant range is that when the impulse response is known, the output can be predicted for any other input. This is done by treating the input as if it were an ordered sequence of impulses and adding the sequence of impulse responses.
In Section 2.1.1 we note that a neural event i(t) can be measured by using a set of N digitizing basis functions (Fig. 2.9) pn(t), n = 1, . . . , N.
Each rectangular pulse has unit amplitude, time of occurrence tn, and <Page 73> duration ∆t. The value of an input function i’(t) representing an input event, is equal to the sum of the set of basis functions.
FIG. 2.9. Representation of continuous activity in the wave mode ∆(t) by a sequence of pulses i’(t) as the basis for constructing a wave function.
where p(t) = i for t = tn, and p(t) = 0 elsewhere.
Here we can treat the input as if it consisted of an ordered sequence of delta functions or impulses with amplitude u(tn)∆t. If each input impulse occurs at time T measured with respect to a reference time t the amplitude of the input function at time T is u'(Tn)∆T. The function u'(t) is given by
where δ(t–Tn) replaces pn(t). Each impulse initiates an impulse response at t–Tn, which has the value f(t–Tn), because the system is time invariant. For any time t the predicted output v(t) is the sum of all the impulse responses weighted by u(Tn)∆T, because the system is linear.
This is illustrated in Fig. 2.10.
FIG. 2.10. Representation of a wave function by the sum of a set of impulse responses in convolution,
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If the input function and impulse response are both measured by digitizing at n intervals, u'(Tn) and f'(tm–Tn), then the output v'(tm) is predicted by a set of discrete sums
