# Bifurcations in a modulation equation for alternans in a cardiac fiber

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 6, page 1225-1238
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topDai, Shu, and Schaeffer, David G.. "Bifurcations in a modulation equation for alternans in a cardiac fiber." ESAIM: Mathematical Modelling and Numerical Analysis 44.6 (2010): 1225-1238. <http://eudml.org/doc/250814>.

@article{Dai2010,

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.
},

author = {Dai, Shu, Schaeffer, David G.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Bifurcation; cardiac alternans; modulation equation; bifurcation; one space dimension; Hopf or steady-state bifurcation; series of secondary bifurcations},

language = {eng},

month = {10},

number = {6},

pages = {1225-1238},

publisher = {EDP Sciences},

title = {Bifurcations in a modulation equation for alternans in a cardiac fiber},

url = {http://eudml.org/doc/250814},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Dai, Shu

AU - Schaeffer, David G.

TI - Bifurcations in a modulation equation for alternans in a cardiac fiber

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/10//

PB - EDP Sciences

VL - 44

IS - 6

SP - 1225

EP - 1238

AB -
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.

LA - eng

KW - Bifurcation; cardiac alternans; modulation equation; bifurcation; one space dimension; Hopf or steady-state bifurcation; series of secondary bifurcations

UR - http://eudml.org/doc/250814

ER -

## References

top- J. Carr, Applications of Centre Manifold Theory. Springer-Verlag, New York (1981). Zbl0464.58001
- 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. Zbl1218.34021
- B. Echebarria and A. Karma, Instability and spatiotemporal dynamics of alternans in paced cardiac tissue. Phys. Rev. Lett.88 (2002) 208101.
- B. Echebarria and A. Karma, Amplitude-equation approach to spatiotemporal dynamics of cardiac alternans. Phys. Rev. E76 (2007) 051911.
- 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. USA97 (2000) 6061–6066.
- R.F. Gilmour Jr. and D.R. Chialvo, Electrical restitution, Critical mass, and the riddle of fibrillation. J. Cardiovasc. Electrophysiol.10 (1999) 1087–1089.
- M. Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory. Springer-Verlag, New York (1985). Zbl0607.35004
- J. Guckenheimer, On a codimension two bifurcation, in Dynamical Systems and Turbulence, Warwick 1980,Lect. Notes in Mathematics898, Springer (1981) 99–142.
- J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dyanamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York (1983). Zbl0515.34001
- 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. Zbl0506.34034
- W.F. Langford, Periodic and steady state interactions lead to tori. SIAM J. Appl. Math.37 (1979) 22–48. Zbl0417.34030
- 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. Zbl1334.92097
- D. Noble, A modification of the Hodgkin-Huxley equations applicable to Purkinje fiber actoin and pacemaker potential. J. Physiol.160 (1962) 317–352.
- 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.
- A.V. Panfilov, Spiral breakup as a model of ventricular fibrillation. Chaos8 (1998) 57–64. Zbl1069.92509

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.