Spectrum of a Linearized Amplitude Equation for Alternans in a Cardiac Fiber

Dai, Shu; Schaeffer, David G.

doi:10.1137/070711384

Cited by 6 publications

(11 citation statements)

References 8 publications

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“…In our previous paper [2], we found that for sufficiently large, as Λ −1 varies, the eigenvalues of (2.2) behave as illustrated in Figure 1. In the figure = 6 and only the real parts of all eigenvalues are graphed (Remark: In [2] we computed the eigenvalues asymptotically for large ). By the critical value of Λ −1 , say Λ (3) standing wave solution.…”

Section: Linear Stability: Bifurcation From a Simple Eigenvaluementioning

confidence: 62%

“…The trivial steady state solution a ≡ 0 of (1.3)-(1.4) loses its stability as σ increases. In the previous paper [2], we analyzed the eigenvalues of the linear operator that maps a function a(x) to and hence competition of multiple modes appear. The organization of this paper is the following: Section 2 reviews our previous result on the bifurcation from a simple eigenvalue.…”

Section: Introductionmentioning

confidence: 99%

“…A calculation in [2] shows that for sufficiently large , to the leading order, the dimensionless critical value equals…”

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confidence: 99%

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Bifurcations in a modulation equation for alternans in a cardiac fiber

Dai

Schaeffer

2010

ESAIM: M2AN

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Abstract. 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 travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose stability, as the pacing rate is increased, through either a Hopf or steady-state bifurcation. 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.Mathematics Subject Classification. 35B32, 92C30.

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Section: Linear Stability: Bifurcation From a Simple Eigenvaluementioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

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Bifurcations in a modulation equation for alternans in a cardiac fiber

Dai

Schaeffer

2010

ESAIM: M2AN

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“…In particular, because the dispersion effect may play an important role in long fibers as shown by Echebarria and Karma [20, 21] and by Dai and Schaeffer [22]. Nevertheless, the simple case sheds light on the effect of electrotonic coupling, the difference between APD and Ca-driven alternans, and the difference between negative and positive Ca in →V m couplings.…”

Section: Origin Of the Indeterminacymentioning

confidence: 99%

Indeterminacy of spatiotemporal cardiac alternans

Zhao

2008

Phys. Rev. E

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Cardiac alternans, a beat-to-beat alternation in action potential duration (at the cellular level) or in ECG morphology (at the whole heart level), is a marker of ventricular fibrillation, a fatal heart rhythm that kills hundreds of thousands of people in the US each year. Investigating cardiac alternans may lead to a better understanding of the mechanisms of cardiac arrhythmias and eventually better algorithms for the prediction and prevention of such dreadful diseases. In paced cardiac tissue, alternans develops under increasingly shorter pacing period. Existing experimental and theoretical studies adopt the assumption that alternans in homogeneous cardiac tissue is exclusively determined by the pacing period. In contrast, we find that, when calcium-driven alternans develops in cardiac fibers, it may take different spatiotemporal patterns depending on the pacing history. Because there coexist multiple alternans solutions for a given pacing period, the alternans pattern on a fiber becomes unpredictable. Using numerical simulation and theoretical analysis, we show that the coexistence of multiple alternans patterns is induced by the interaction between electrotonic coupling and an instability in calcium cycling.

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“…18, including its early bifurcations. 19 In Sec. III, choosing one set of ͑extreme͒ parameter values as an example, we show that solutions of the modulation equation are chaotic.…”

Section: Introductionmentioning

confidence: 99%

Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans

Dai

Schaeffer

2010

Chaos

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Instabilities in cardiac dynamics have been widely investigated in recent years. One facet of this work has studied chaotic behavior, especially possible correlations with fatal arrhythmias. Previously chaotic behavior was observed in various models, specifically in the breakup of spiral and scroll waves. In this paper we study cardiac dynamics and find spatiotemporal chaotic behavior through the Echebarria-Karma modulation equation for alternans in one dimension. Although extreme parameter values are required to produce chaos in this model, it seems significant mathematically that chaos may occur by a different mechanism from previous observations.

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Spectrum of a Linearized Amplitude Equation for Alternans in a Cardiac Fiber

Cited by 6 publications

References 8 publications

Bifurcations in a modulation equation for alternans in a cardiac fiber

Bifurcations in a modulation equation for alternans in a cardiac fiber

Indeterminacy of spatiotemporal cardiac alternans

Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans

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