Propagation failure in heterogeneous FitzHugh--Nagumo systems via coupled upper and lower solutions

Abstract

Traveling waves are fundamental objects in reaction--diffusion systems; however, spatial heterogeneities can distort or inhibit their propagation. Determining whether a given heterogeneity results in propagation failure remains challenging, particularly in systems with coupled components. In this paper, we present explicit constructions of stationary coupled upper and lower solutions tailored to specific heterogeneities for a one-dimensional FitzHugh--Nagumo system. By leveraging the mixed quasimonotone structure of the system, these barriers yield sufficient conditions for the failure of propagation for families of trapped initial conditions. The stationary constructions establish directional signal blocking in FitzHugh--Nagumo neuron models with localized depressed dynamics or abrupt geometric variations, and the resulting stationary barriers persist for sufficiently small recovery diffusion. Additionally, we introduce explicit almost stationary barriers, which provide conditional bounds on the expansion of componentwise threshold regions in homogeneous media.