Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?

TL;DR

This paper analyzes a normal form for traveling waves in excitable media, revealing that Hopf bifurcations are always subcritical, which challenges previous beliefs about the stability of cardiac alternans.

Contribution

It provides a detailed bifurcation analysis of a normal form for excitable media, showing the subcritical nature of Hopf bifurcations and their implications for cardiac alternans.

Findings

Hopf bifurcation is always subcritical

Revisits stability of cardiac alternans

Links bifurcation conditions to restitution theory

Abstract

We present a bifurcation analysis of a normal form for travelling waves in one-dimensional excitable media. The normal form which has been recently proposed on phenomenological grounds is given in form of a differential delay equation. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both, the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.