Two-bump solutions of Amari-type models of neuronal pattern formation

We study a partial integro-differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks. For a one-dimensional domain we develop criteria for the existence and stability of equal-width “2-bump” solutions under the assumption that the firing rate function is the Heaviside function. We apply these criteria to an example for which the connectivity is of lateral inhibition type (i.e. the coupling function has one positive zero) and find that families of 2-bump solutions exist, but none of the solutions are stable. Extensive numerical searches suggest that this is true for all coupling functions of this form. However, for a large class of coupling functions which have three positive zeros, we find the coexistence of both stable and unstable 2-bump solutions. We also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1-bump and 2-bump solutions. Our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity.