Pattern formation via stochastic neural field equations
Lawrence M. Ward · Open Collections · 2017
The formation of pattern in biological systems may be modeled by a set of reaction-diffusion equations. A form of diffusion-type coupling term biologically significant in neuroscience is a difference of Gaussian functions used as a space-convolution kernel. Here we study the simplest reaction-diffusion system with this type of coupling. Instead of the deterministic form of the model, we are interested in a \emph{stochastic} neural field equation, a space-time stochastic differential-integral equation. We explore, quantitatively, how the parameters of our model that measure the shape of the coupling kernel, coupling strength, and aspects of the spatially-smoothed space-time noise, control the pattern in the resulting evolving random field. We find that a spatial pattern that is damped in time in a deterministic system may be sustained and amplified by stochasticity, most strikingly at an optimal space-time noise level.