Asymptotics of conduction velocity restitution in models of electrical excitation in the heart

TL;DR

This paper develops and compares asymptotic methods to analyze conduction velocity restitution in cardiac models, providing insights into arrhythmia prediction and addressing complex eigenvalue problems in excitable media.

Contribution

It introduces a non-Tikhonov asymptotic approach for calculating conduction velocity restitution in ionic cardiac models, extending previous methods to handle stiff nonlinear eigenvalue problems.

Findings

Asymptotic methods effectively approximate restitution curves in cardiac models.

Comparison between generic and ionic models reveals new mathematical features.

Application to Beeler-Reuter model demonstrates practical utility.

Abstract

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest nontrivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a nonlinear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh-Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified "caricature" of the Noble (1962) model and the Beeler and Reuter (1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler-Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method.