An analytical approach to initiation of propagating fronts

TL;DR

This paper develops an analytical criterion for wave initiation in a one-dimensional excitable fiber using the Nagumo equation, closely matching numerical results and enhancing understanding of propagation thresholds.

Contribution

It introduces an analytical approximation of the threshold surface for wave initiation in the Nagumo equation, improving theoretical understanding of excitable media.

Findings

Analytical criterion closely matches numerical simulations.

Approximation of the stable manifold effectively predicts initiation thresholds.

Provides a new theoretical tool for studying wave propagation in excitable systems.

Abstract

We consider the problem of initiation of a propagating wave in a one-dimensional excitable fibre. In the Zeldovich-Frank-Kamenetsky equation, a.k.a. Nagumo equation, the key role is played by the "critical nucleus'' solution whose stable manifold is the threshold surface separating initial conditions leading to initiation of propagation and to decay. Approximation of this manifold by its tangent linear space yields an analytical criterion of initiation which is in a good agreement with direct numerical simulations.