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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. [email protected] 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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References

J. Carr, Applications of Centre Manifold Theory. Springer-Verlag, New York (1981).
Dai, S. and Schaeffer, D.G., Spectrum of a linearized amplitude equation for alternans in a cardiac fiber. SIAM J. Appl. Math. 69 (2008) 704–719. CrossRef
Echebarria, B. and Karma, A., Instability and spatiotemporal dynamics of alternans in paced cardiac tissue. Phys. Rev. Lett. 88 (2002) 208101. CrossRef
Echebarria, B. and Karma, A., Amplitude-equation approach to spatiotemporal dynamics of cardiac alternans. Phys. Rev. E 76 (2007) 051911. CrossRef
Garfinkel, A., Kim, Y.-H., Voroshilovsky, O., Qu, Z., Kil, J.R., Lee, M.-H., Karagueuzian, H.S., Weiss, J.N. and Chen, P.-S., Preventing ventricular fibrillation by flattening cardiac restitution. Proc. Natl. Acad. Sci. USA 97 (2000) 6061–6066. CrossRef
Gilmour Jr, R.F.. and D.R. Chialvo, Electrical restitution, Critical mass, and the riddle of fibrillation. J. Cardiovasc. Electrophysiol. 10 (1999) 1087–1089. CrossRef
M. Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory. Springer-Verlag, New York (1985).
J. Guckenheimer, On a codimension two bifurcation, in Dynamical Systems and Turbulence, Warwick 1980, Lect. Notes in Mathematics 898, Springer (1981) 99–142.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dyanamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York (1983).
M.R. Guevara, G. Ward, A. Shrier and L. Glass, Electrical alternans and period doubling bifurcations, in Proceedings of the 11th Computers in Cardiology Conference, IEEE Computer Society, Los Angeles, USA (1984) 167–170.
P. Holmes, Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation, in Nonlinear Dynamics, R.H.G. Helleman Ed., New York Academy of Sciences, New York (1980) 473–488.
Langford, W.F., Periodic and steady state interactions lead to tori. SIAM J. Appl. Math. 37 (1979) 22–48. CrossRef
Mitchell, C.C. and Schaeffer, D.G., A two-current model for the dynamics of the cardiac membrane. Bull. Math. Biol. 65 (2003) 767–793. CrossRef
Noble, D., A modification of the Hodgkin-Huxley equations applicable to Purkinje fiber actoin and pacemaker potential. J. Physiol. 160 (1962) 317–352. CrossRef
Nolasco, J.B. and Dahlen, R.W., A graphic method for the study of alternation in cardiac action potentials. J. Appl. Physiol. 25 (1968) 191–196.
Panfilov, A.V., Spiral breakup as a model of ventricular fibrillation. Chaos 8 (1998) 57–64. CrossRef