Numerical analysis of bump solutions for neural field equations with periodic microstructure

Abstract

We study numerically single bump solutions of a homogenized Amari equation with periodic microvariation. Two attempts are made to detect single bumps that depend on the microvariable. The first attempt which is based on a pinning function technique is applicable in the Heaviside limit of the firing rate function. In the second attempt, we develop a numerical scheme which combines the two-scale convergence theory and an iteration procedure for the corresponding heterogeneous Amari equation. The numerical simulations in both attempts indicate the nonexistence of single bump solutions that depend on the microvariable. Motivated by this result, we finally develop a fixed point iteration scheme for the construction of single bump solutions that are independent of the microvariable when the firing rate function is given by a sigmoidal firing rate function.