Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift–Hohenberg equation

This work studies the spatio-temporal dynamics of a generic integral–differential equation subject to additive random fluctuations. It introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for Fokker–Planck equations, and studies analytically the systems’ stability near Turing bifurcations. In addition two types of fluctuation are studied, namely fluctuations uncorrelated in space and time, and global fluctuations, which are constant in space but uncorrelated in time. We show that the global fluctuations shift the Turing bifurcation threshold. This shift is proportional to the fluctuation variance. Applications to a neural field equation and the Swift–Hohenberg equation reveal the shift of the bifurcation to larger control parameters, which represents a stabilization of the system. All analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral–differential equation. To gain some insight into experimental manifestations, the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively.