Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation

TL;DR

This paper develops a numerical method to compute adjoint eigenfunctions of spiral waves in reaction-diffusion systems, revealing their localization near the spiral tip and implications for controlling unstable spiral waves in atrial fibrillation models.

Contribution

Introduces a novel numerical approach for computing and analyzing adjoint eigenfunctions of spiral waves in reaction-diffusion models, with applications to atrial fibrillation.

Findings

All leading adjoint eigenfunctions are exponentially localized near the spiral tip.

Response functions show the strongest localization among marginal modes.

Localization of eigenfunctions explains spiral wave drift due to boundary interactions.

Abstract

This paper introduces a numerical method for computing the spectrum of adjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusion systems in arbitrary geometries. The method is illustrated by computing over a hundred eigenfunctions associated with an unstable time-periodic single-spiral solution of the Karma model on a square domain. We show that all leading adjoint eigenfunctions are exponentially localized in the vicinity of the spiral tip, although the marginal modes (response functions) demonstrate the strongest localization. We also discuss the implications of the localization for the dynamics and control of unstable spiral waves. In particular, the interaction with no-flux boundaries leads to a drift of spiral waves which can be understood with the help of the response functions.