Spikes, Decisions, and Actions
Hugh R. Wilson, 1999

Introduction
Dynamics is the mathematics of the study of change. Its foundation is the differential equation, by which knowledge of the present state of a system can be used to predict a future state. The brain is a high;y complex and highly nonlinear system. This book will describe the techniques of nonlinear dynamics as required for the study of the brain. The level of abstraction treated in this book will vary from the general, in which each neuron is represented as a continuous variable describing it average firing rate, to the most detailed, incorporating diffusion of ionic potentials along dendrites with complex geometry. The book is intended to be useful to anyone with one or two years of calculus and who is familiar with the basics of vectors and matrices. This chapter includes a quick review of the exponential function, Taylor series and imaginary exponents. The book also assumes some knowledge of basic neurobiology.

First order linear differential equations
This chapter describes first-order differential equations, ie., having no higher than a first-order derivative, and which are linear and have constant coefficients. An equation of this type is applied to model a simple neuron and to describe both excitatory and inhibitory postsynaptic potentials.

Two-dimensional systems and state space

Higher dimensional linear systems

Approximation and simulation

Nonlinear neurodynamics and bifurcations

Computation by exitatory and inhibitory networks

Nonlinear oscillations

Action potentials and limit cycles

Neural adaptation and bursting

Neural chaos

Synapses and synchrony

Swimming and traveling waves

Lyapunov functions and memory

Diffusion and dendrites

Nonlinear dynamics and brain function

MatLab and the MatLab scripts


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