Traveling fronts for lattice neural field equations

Abstract

We show existence and uniqueness of traveling front solutions to a class of neural field equations set on a lattice with infinite range interactions in the regime where the kinetics of each individual neuron is of bistable type. The existence proof relies on a regularization of the traveling wave problem allowing us to use well-known existence results for traveling front solutions of continuous neural field equations. We then show that the traveling front solutions which have nonzero wave speed are unique (up to translation) by constructing appropriate sub and super solutions. The spectral properties of the traveling fronts are also investigated via a careful study of the linear operator around a traveling front in co-moving frame where we crucially use Fredholm properties of nonlocal differential operators previously obtained by the author in an earlier work. For the spectral analysis, we need to impose an extra exponential localization condition on the interactions.