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 firstorder differential equations, ie.,
having no higher than a firstorder 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.
Twodimensional 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
