Persistence of spatial patterns produced by neural field equations

We study an integro-differential equation defined on a spatially extended domain, proposed in the literature as a model of short term memory. Previous researchers have established the necessary conditions for the existence of two-bump stationary solutions of this equation when considering a Heaviside firing rate function and a coupling function of “Mexican hat” type. One such condition concerns the positiveness of the external stimulus which is quite difficult to establish under general assumptions. We address this problem and determine the necessary conditions for the existence of two-bump stationary solutions that assure this requirement is met. Furthermore, we prove the existence of symmetric two-bump stationary solutions for a class of coupling functions and establish their linear stability.