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

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 KI_e Set 291

5.2.3. Self–Stabilization of the KI_e 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

T_a 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