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Bifurcations in a modulation equation for alternansin a cardiac fiber

Published online by Cambridge University Press:  15 April 2010

Shu Dai  and
David G. Schaeffer
Shu Dai
Affiliation:
Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, USA. sdai@mbi.osu.edu Department of Mathematics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA.
David G. Schaeffer
Affiliation:
Department of Mathematics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA.
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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 travelling, which is known as discordantalternans. We study these phenomena in simple cardiac modelsthrough a modulation equation proposed by Echebarria-Karma. Asshown in our previous paper, the zero solution of their equationmay lose stability, as the pacing rate is increased, througheither a Hopf or steady-state bifurcation. Which bifurcationoccurs first depends on parameters in the equation, and for onecritical case both modes bifurcate together at a degenerate(codimension 2) bifurcation. For parameters close to thedegenerate case, we investigate the competition between modes,both numerically and analytically. We find that at sufficientlyrapid pacing (but assuming a 1:1 response is maintained), steadypatterns always emerge as the only stable solution. However, inthe parameter range where Hopf bifurcation occurs first, theevolution from periodic solution (just after the bifurcation) tothe eventual standing wave solution occurs through an interestingseries of secondary bifurcations.

Keywords

Bifurcationcardiac alternansmodulation equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 6 , November 2010 , pp. 1225 - 1238
Copyright
© EDP Sciences, SMAI, 2010

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