Existence of pulses in excitable media with nonlocal coupling

TL;DR

This paper proves the existence of fast traveling pulse solutions in excitable media with non-local coupling using a PDE-oriented approach, expanding previous results beyond local diffusive and finite-range non-local cases.

Contribution

It introduces a PDE-based method to establish pulse existence in non-local media, replacing geometric singular perturbation techniques.

Findings

Established existence of pulses in non-local media

Developed PDE-based analytical framework

Extended previous results to broader non-local coupling scenarios

Abstract

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.