Semi-analytical approach to criteria for ignition of excitation waves

TL;DR

This paper extends a semi-analytical method for predicting wave ignition thresholds in excitable systems, improving accuracy and applicability to complex models like cardiac excitation, with validation on five diverse test problems.

Contribution

The authors generalize and enhance an existing analytical approach to determine ignition criteria for a broader class of excitable systems, including multicomponent and non-self-adjoint cases.

Findings

Method successfully applied to five test problems.

Quadratic approximation improves accuracy in threshold prediction.

Approach applicable to realistic biological models like cardiac excitation.

Abstract

We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method (Idris and Biktashev, PRL, vol 101, 2008, 244101) for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich--Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognised that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically.