Bifurcations in the Echebarria -Karma modulation equation for cardiac alternans in one dimension
While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or traveling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. In this dissertation, we perform bifurcation analysis for their modulation equation.
Suppose we have a cardiac fiber of length ℓ, which is stimulated periodically at its x=0 end. When the pacing period (basic cycle length) B is below some critical value B c, alternans emerges along the cable. Let a( x,n) be the amplitude of the alternans along the fiber corresponding to the n-th stimulus. Echebarria and Karma suppose that a(x,n) varies slowly in time and it can be regarded as a time-continuous function a(x,t). They derive a weakly nonlinear modulation equation for the evolution of a(x,t) under some approximation, which after nondimensionization is as follows: [special characters omitted]where the linear operator [special characters omitted]In the equation, σ is dimensionless and proportional to Bc – B, i.e. σ indicates how rapid the pacing is, Λ-1 is related to the conduction velocity (CV) of the propagation and the nonlinear term –ga 3 limits growth after the onset of linear instability. No flux boundary conditions are imposed on both ends.
The zero solution of their equation may lose stability, as the pacing rate is increased. To study the bifurcation, we calculate the spectrum of operator L. We find that the bifurcation may be Hopf or steady-state. Which bifurcation occurs first depends on parameters in the equation, and for one critical case both modes bifurcate together at a degenerate (codimension 2) bifurcation.
For parameters close to the degenerate case, we investigate the competition between modes, both numerically and analytically. We find that at sufficiently rapid pacing (but assuming a 1:1 response is maintained), steady patterns always emerge as the only stable solution. However, in the parameter range where Hopf bifurcation occurs first, the evolution from periodic solution (just after the bifurcation) to the eventual standing wave solution occurs through an interesting series of secondary bifurcations.
We also find that for some extreme range of parameters, the modulation equation also includes chaotic solutions. Chaotic waves in recent years has been regarded to be closely related with dreadful cardiac arrhythmia. Proceeding work illustrated some chaotic phenomena in two- or three-dimensional space, for instance spiral and scroll waves. We show the existence of chaotic waves in one dimension by the Echebarria-Karma modulation equation for cardiac alternans. This new discovery may provide a different mechanism accounting for the instabilities in cardiac dynamics.