## J

and similarly for (5). The function gij is called the kernel of the distribution and represents the probability density function of the time delay. Since gij is a pdf it is normalized so that f0 gij (a) da = 1. Although distributions of delays are not commonly used in neural network models, they have been extensively used in models from population biology (Cushing, 1977; MacDonald, 1978). In this literature, the most commonly used distributions are the uniform distribution:

( 0 0 < a < Tmmin gij (*)=< l Tfmin < a < Tj + S, (7)

I 0 Timin < a and the gamma distribution:

j 0 0 < a <Tmmin gij (a) = S am (a_Tmin)m-lp-a{a-Tmin) Tmin< a , (8)

[ r(m) (a Tij ) e j Tij <a where a,m > 0 are parameters. r is the gamma function defined by r(0) = 1 and r(m +1) = mr(m). Both these distributions can be shown to approach a Dirac distribution in the appropriate limits (S ^ 0 for the uniform distribution and m ^ <x for the gamma distribution), which leads to a discrete delay in the coupling term. It is usual in the population biology literature

(Cushing, 1977; MacDonald, 1978) to take Tmfn = 0. In this case model with a gamma distribution can be shown to be equivalent to a system of m ordinary differential equations, which is amenable to the analysis techniques described elsewhere in this volume (Breakspear and Jirsa, 2006). However, as pointed out by Bernard et al. (2001), it makes more biological sense to take tmin > 0, since the probability of having zero delay is effectively zero in most applications. In this case, the model with a gamma distribution is equivalent to a system of m — 1 ordinary differential equations and one discrete delay differential equation.

In the next section I will review some tools for analyzing delay differential equations. To make the theory concrete, we will apply it to a particular example. Consider the following representation of the Fitzhugh-Nagumo model for a neuron (Fitzhugh, 1960; Nagumo et al., 1962)

Assume that the parameters are such that there is just one equilibrium point (v, w) of this equation, where v, W satisfy v3 — (a + 1)v2 + (a + - )d + I = 0 , (10)