Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Neural field equations are used to describe the spatio-temporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain. In order to determine finite-size effects—deviations from the mean-field limit due to the finite size of the populations in the network—we analyze the fluctuations of this Markov chain and set up an approximating system of diffusion processes. We show that a well-posed stochastic neural field equation with a noise term accounting for finite-size effects on traveling wave solutions is obtained as the strong continuum limit.