## Time Delays in Neural Systems

Sue Ann Campbell

Department of Applied Mathematics, University of Waterloo, Waterloo ON N2L 3G1 Canada [email protected]

Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montreal Quebec H3C 3G7 Canada

### 1 Introduction

In this chapter I will give an overview of the role of time delays in understanding neural systems. The main focus will be on models of neural systems in terms of delay differential equations. Later in this section, I will discuss how such models arise. The goal of the chapter is two-fold: (1) to give the reader an introduction and guide to some methods available for understanding the dynamics of delay differential equations and (2) to review some of the literature documenting how including time delays in neural models can have a profound effect on the behaviour of those models.

### 1.1 Modelling Delay in Neural Systems

To begin, I will formulate a general model for a network of neurons and then determine how delays may occur in this model. Consider a network of n neurons modelled by the equations n xi (t) = Fi(xi(t))+53 fj (xi(t), xj (t)), i = 1,...,n. (1)

The variable xi represents all the variables describing the physical state of the cell body of the ith neuron in the network. For example, in the standard Hodgkin-Huxley model, it would represent the membrane voltage and gating variables: xi = (Vi,mi,ni,hi). The function Fi represents the intrinsic dynamics of the ith neuron and the function fij, often called the coupling function, represents the input to the ith neuron from the jth neuron. In neural models, the coupling is usually through the voltage, Vj, only, so fij = [fij, 0, 0,..., 0]T. I will primarily consider this case in the rest of the chapter.

If the jth neuron is connected to the ith via a chemical synapse, then the coupling function is given by