Title: Smoothed Noise and Mexican Hat Coupling Produce Pattern in a Stochastic Neural Field,
Abstract: 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 \emph{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.