Asymptotic effects of boundary perturbations in excitable systems

TL;DR

This paper analyzes how boundary perturbations influence the long-term behavior of solutions in excitable systems, specifically the Fitzhugh Nagumo model, revealing conditions under which boundary effects diminish or persist over time.

Contribution

It provides a detailed asymptotic analysis of boundary effects in excitable systems, establishing conditions for their vanishing or persistence as time progresses.

Findings

Boundary effects vanish when initial data effects diminish over time.

Boundary disturbances that converge lead to bounded solutions.

Effects of boundary derivatives depend on their integrability properties.

Abstract

A Neumann problem in the strip for the Fitzhugh Nagumo system is consid- ered. The transformation in a non linear integral equation permits to deduce a priori estimates for the solution. A complete asymptotic analysis shows that for large t the effects of the initial data vanish while the effects of bound- ary disturbances depend on the properties of the data. When they are convergent for large t, the solution is everywhere bounded; when theirs first derivatives belong to L one too, the effects are vanishing.