Bifurcations in the Echebarria-Karma Modulation Equation for Cardiac Alternans in One Dimension

Abstract

While alternans in a single cardiac cell appears through a simpleperiod-doubling bifurcation, in extended tissue the exact natureof the bifurcation is unclear. In particular, the phase ofalternans can exhibit wave-like spatial dependence, eitherstationary or traveling, which is known as discordantalternans. We study these phenomena in simple cardiac modelsthrough a modulation equation proposed by Echebarria-Karma. Inthis dissertation, we perform bifurcation analysis for theirmodulation equation.Suppose we have a cardiac fiber of length l, which isstimulated periodically at its x=0 end. When the pacing period(basic cycle length) B is below some critical value Bc,alternans emerges along the cable. Let a(x,n) be the amplitudeof the alternans along the fiber corresponding to the n-thstimulus. Echebarria and Karma suppose that a(x,n) variesslowly in time and it can be regarded as a time-continuousfunction a(x,t). They derive a weakly nonlinear modulationequation for the evolution of a(x,t) under some approximation,which after nondimensionization is as follows: &partialt a = σ a + L a - g a 3,where the linear operator L a = &partialxxa - &partialx a -Λ-1 ∫ 0 x a(x',t)dx'.In the equation, σ is dimensionless and proportional toBc - B, i.e. σ indicates how rapid the pacing is,Λ-1 is related to the conduction velocity (CV) of thepropagation and the nonlinear term -ga3 limits growth after theonset of linear instability. No flux boundary conditions areimposed on both ends.The zero solution of their equation may lose stability, as thepacing rate is increased. To study the bifurcation, we calculatethe spectrum of operator L. We find that thebifurcation may be Hopf or steady-state. Which bifurcation occursfirst depends on parameters in the equation, and for one criticalcase both modes bifurcate together at a degenerate (codimension 2)bifurcation.For parameters close to the degenerate case, we investigate thecompetition between modes, both numerically and analytically. Wefind that at sufficiently rapid pacing (but assuming a 1:1response is maintained), steady patterns always emerge as the onlystable solution. However, in the parameter range where Hopfbifurcation occurs first, the evolution from periodic solution(just after the bifurcation) to the eventual standing wavesolution occurs through an interesting series of secondarybifurcations.We also find that for some extreme range of parameters, themodulation equation also includes chaotic solutions. Chaotic wavesin recent years has been regarded to be closely related withdreadful cardiac arrhythmia. Proceeding work illustrated somechaotic phenomena in two- or three-dimensional space, for instancespiral and scroll waves. We show the existence of chaotic waves inone dimension by the Echebarria-Karma modulation equation forcardiac alternans. This new discovery may provide a differentmechanism accounting for the instabilities in cardiac dynamics.