Quasi-patterns produced by a Mexican Hat coupling of quasi-cycles

A family of stochastic processes has quasi-cycle oscillations if its otherwise-damped oscillations are sustained by noise. Such a family forms the reaction part of a stochastic reaction-diffusion system when we insert a local Mexican Hat-type, difference of Gaussians, coupling on a one-dimensional and on a two-dimensional lattice. We find spatial patterns of oscillating quasicycles that resemble Turing patterns, called quasi-patterns. Specific properties of these patterns, such as local phase synchronization, can be predicted from the parameters of the reaction and of the Mexican Hat coupling. When the damping parameters of the reaction and diffusion parts are small and balanced, phase synchronization vanishes but amplitude patterns persist. These results extend our knowledge of the behaviour of coupled neural field equations and its dependence on stochastic fluctuations.