Numerical simulations of one- and two-dimensional stochastic neural field equations with delay.

Abstract

Neural Field Equations (NFE) are intended to model the synaptic interactions between neurons in a continuous neural network, called a neural field. This kind of integro-differential equations proved to be a useful tool to describe the spatiotemporal neuronal activity from a macroscopic point of view, allowing the study of a wide variety of neurobiological phenomena, such as the sensory stimuli processing. The present article aims to study the effects of additive noise in one- and two-dimensional neural fields, while taking into account finite axonal velocity and an external stimulus. A Galerkin-type method is presented, which applies Fast Fourier Transforms to optimise the computational effort required to solve these equations. The explicit Euler-Maruyama scheme is implemented to obtain the stochastic numerical solution. An open-source numerical solver written in Julia was developed to simulate the neural fields in study.