Bifurcations in a modulation equation for alternans in a cardiac fiber

Dai, Shu; Schaeffer, David G.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 6, pp. 1225-1238.

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.

Classification : 35B32, 92C30
Mots clés : bifurcation, cardiac alternans, modulation equation

[1] J. Carr, Applications of Centre Manifold Theory. Springer-Verlag, New York (1981). | Zbl

[2] S. Dai and D.G. Schaeffer, Spectrum of a linearized amplitude equation for alternans in a cardiac fiber. SIAM J. Appl. Math. 69 (2008) 704-719.

[3] B. Echebarria and A. Karma, Instability and spatiotemporal dynamics of alternans in paced cardiac tissue. Phys. Rev. Lett. 88 (2002) 208101.

[4] B. Echebarria and A. Karma, Amplitude-equation approach to spatiotemporal dynamics of cardiac alternans. Phys. Rev. E 76 (2007) 051911.

[5] A. Garfinkel, Y.-H. Kim, O. Voroshilovsky, Z. Qu, J.R. Kil, M.-H. Lee, H.S. Karagueuzian, J.N. Weiss and P.-S. Chen, Preventing ventricular fibrillation by flattening cardiac restitution. Proc. Natl. Acad. Sci. USA 97 (2000) 6061-6066.

[6] R.F. Gilmour Jr. and D.R. Chialvo, Electrical restitution, Critical mass, and the riddle of fibrillation. J. Cardiovasc. Electrophysiol. 10 (1999) 1087-1089.

[7] M. Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory. Springer-Verlag, New York (1985). | Zbl

[8] J. Guckenheimer, On a codimension two bifurcation, in Dynamical Systems and Turbulence, Warwick 1980, Lect. Notes in Mathematics 898, Springer (1981) 99-142. | Zbl

[9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dyanamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York (1983). | Zbl

[10] 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.

[11] 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. | Zbl

[12] W.F. Langford, Periodic and steady state interactions lead to tori. SIAM J. Appl. Math. 37 (1979) 22-48. | Zbl

[13] C.C. Mitchell and D.G. Schaeffer, A two-current model for the dynamics of the cardiac membrane. Bull. Math. Biol. 65 (2003) 767-793.

[14] D. Noble, A modification of the Hodgkin-Huxley equations applicable to Purkinje fiber actoin and pacemaker potential. J. Physiol. 160 (1962) 317-352.

[15] J.B. Nolasco and R.W. Dahlen, A graphic method for the study of alternation in cardiac action potentials. J. Appl. Physiol. 25 (1968) 191-196.

[16] A.V. Panfilov, Spiral breakup as a model of ventricular fibrillation. Chaos 8 (1998) 57-64. | Zbl