Smoothed Noise and Mexican Hat Coupling Produce Pattern in a Stochastic Neural Field

The formation of pattern in biological systems may be modeled by a set of reaction-diffusion equations. A diffusion-type coupling operator biologically significant in neuroscience is a difference of Gaussian functions (Mexican Hat operator) used as a spatial-convolution kernel. We are interested in the difference among behaviors of stochastic neural field equations, namely space-time stochastic differential-integral equations, and similar deterministic ones. 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 spatio-temporal noise level. In addition, we find that spatially-smoothed noise alone causes pattern formation even without spatial coupling.