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

Abstract

<p>While alternans in a single cardiac cell appears through a simple</p><p>period-doubling bifurcation, in extended tissue the exact nature</p><p>of the bifurcation is unclear. In particular, the phase of</p><p>alternans can exhibit wave-like spatial dependence, either</p><p>stationary or traveling, which is known as <italic>discordant</italic></p><p>alternans. We study these phenomena in simple cardiac models</p><p>through a modulation equation proposed by Echebarria-Karma. In</p><p>this dissertation, we perform bifurcation analysis for their</p><p>modulation equation.</p><p>Suppose we have a cardiac fiber of length l, which is</p><p>stimulated periodically at its x=0 end. When the pacing period</p><p>(basic cycle length) B is below some critical value B_c,</p><p>alternans emerges along the cable. Let a(x,n) be the amplitude</p><p>of the alternans along the fiber corresponding to the n-th</p><p>stimulus. Echebarria and Karma suppose that a(x,n) varies</p><p>slowly in time and it can be regarded as a time-continuous</p><p>function a(x,t). They derive a weakly nonlinear modulation</p><p>equation for the evolution of a(x,t) under some approximation,</p><p>which after nondimensionization is as follows: </p><p> &partial_t a = σ a + <bold>L</bold> a - g a <super>3</super>,</p><p>where the linear operator</p><p> <bold>L</bold> a = &partial_xxa - &partial_x a -Λ<super>-1</super> ∫ <super>0</super> _x a(x',t)dx'.</p><p>In the equation, σ is dimensionless and proportional to</p><p>B_c - B, i.e. σ indicates how rapid the pacing is,</p><p>Λ<super>-1</super> is related to the conduction velocity (CV) of the</p><p>propagation and the nonlinear term -ga<super>3</super> limits growth after the</p><p>onset of linear instability. No flux boundary conditions are</p><p>imposed on both ends.</p><p>The zero solution of their equation may lose stability, as the</p><p>pacing rate is increased. To study the bifurcation, we calculate</p><p>the spectrum of operator <bold>L</bold>. We find that the</p><p>bifurcation may be Hopf or steady-state. Which bifurcation occurs</p><p>first depends on parameters in the equation, and for one critical</p><p>case both modes bifurcate together at a degenerate (codimension 2)</p><p>bifurcation.</p><p>For parameters close to the degenerate case, we investigate the</p><p>competition between modes, both numerically and analytically. We</p><p>find that at sufficiently rapid pacing (but assuming a 1:1</p><p>response is maintained), steady patterns always emerge as the only</p><p>stable solution. However, in the parameter range where Hopf</p><p>bifurcation occurs first, the evolution from periodic solution</p><p>(just after the bifurcation) to the eventual standing wave</p><p>solution occurs through an interesting series of secondary</p><p>bifurcations.</p><p>We also find that for some extreme range of parameters, the</p><p>modulation equation also includes chaotic solutions. Chaotic waves</p><p>in recent years has been regarded to be closely related with</p><p>dreadful cardiac arrhythmia. Proceeding work illustrated some</p><p>chaotic phenomena in two- or three-dimensional space, for instance</p><p>spiral and scroll waves. We show the existence of chaotic waves in</p><p>one dimension by the Echebarria-Karma modulation equation for</p><p>cardiac alternans. This new discovery may provide a different</p><p>mechanism accounting for the instabilities in cardiac dynamics.</p>