### Spikes, Decisions & Actions:

#### Dynamical Foundations of Neuroscience

by

### Hugh R. Wilson

#### Centre for Vision Research

York University of Toronto

##### Oxford University Press

### Excerpts from the Preface

There is one central reason why all neuroscientists and biopsychologists should be conversant with nonlinear dynamics. Nonlinear dynamics reveals and elucidates a range of phenomena that are simply inconceivable in the more mundane world of linear systems theory. Memory and forgetting, decision making, motor control, action potentials, and perhaps even free will and determinism can no longer be intelligently conceptualized without a basic understanding of nonlinear systems. At the deepest level, therefore, my decision to write this book was predicated on the belief that an understanding of brain function and behavior must be grounded in a conceptual framework informed by nonlinear dynamics.

Spikes, Decisions & Actions is intended for three audiences: advanced undergraduates in neuroscience or physiological psychology, graduate students, and professionals in these areas who wish to develop neural simulations relevant to their own interests.

The book is accompanied by a disk containing about 100 MatLabâ„˘ scripts that are intended to complement the text. These simulations will help to solidify and elucidate analytic results and concepts derived in the text, and they should also enhance the utility of the book for self-study. Beyond this, their inclusion brings the book closer to the realities of research in theoretical neuroscience, where a combination of analysis and simulation is to be found in much of the best work. MatLab was chosen because the scripts will run on the Macintosh, UNIX, or Windows platforms.

##### Please email any suggestions or typographical errors found to: hrwilson@yorku.ca

### Table of Contents

Introduction 1

First order linear differential equations 13

Two-dimensional systems and state space 28

Higher dimensional linear systems 48

Approximation & simulation 60

Nonlinear dynamics & bifurcations 72

Computation by excitatory & inhibitory networks 88

Nonlinear oscillations 116

Action potentials & limit cycles 136

Neural adaptation & bursting 156

Neural Chaos 173

Synapses & synchrony 186

Swimming & traveling waves 205

Lyapunov functions & memory 223

Diffusion & dendrites 251

Nonlinear dynamics and brain function 279

Appendix: MatLab & the MatLab scripts 287

References 291

### Spikes, Decisions & Actions

Corrections to the First Edition

p. 82, Eqn (6.16): right side of equation should read = 900 E12 (i.e., E1 should be squared).

p. 85, Eqn (6.19): upper right entry in matrix should be -5/(4t) rather than 0.

p. 87, next to last line: should read E2 = 1.

p. 91, Eqn (7.3): lower left entry in matrix should NOT include N. It should be: -S'k.

p. 91, paragraph below Eqn. (7.3): there should be a factor of 1/t multiplying all values of l.

p. 114: there should be a horizontal line marked F through the bottom left circle in the figure.

p. 147, Eqn (9.10): the last term in the dR/dt equation should be multiplied by the factor of 3.3, NOT 0.33.

p. 202: the program mentioned in the first line of problem #1 is EPSPinteractions.m. NOT EPEPâ€¦

p. 273, Eqn (15.48): the last term in the second equation should have (1 - p) in the denominator, not just p.

These are the only known error corrections at present.

If you find any additional corrections, please email me at: hrwilson@yorku.ca.