Predicting critical ignition in slow-fast excitable models

TL;DR

This paper analyzes the asymptotic behavior of eigenfunctions around unstable travelling waves in slow-fast excitable models, improving predictions of critical ignition curves in FitzHugh-Nagumo systems.

Contribution

It introduces two new heuristics that enhance the accuracy of strength-extent curve predictions by addressing eigenfunction asymptotics in slow-fast excitable systems.

Findings

Asymptotics of eigenspaces differ in full systems versus reduced models.

New heuristics outperform previous methods in predictive accuracy.

Critical nucleus convergence observed in certain models.

Abstract

Linearization around unstable travelling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable travelling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type, and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable travelling wave solution converges to a stationary "critical nucleus" of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.