Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?

Chaos

Volumn 18, Issue 1, 2008, Pages

  Gottwald, Georg A ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41549145713     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2890430     Document Type: Article

References (68)

1. A. T. Winfree, When Time Breaks Down (Princeton University Press, Princeton, NJ, 1987).
2. J. M. Davidenko, A. M. Pertsov, R. Salomonsz, W. Baxter, and J. Jalife, " Stationary and drifting spiral waves of excitation in isolated cardiac muscle.," Nature (London) 335, 349 (1992).
3. F. Siegert and C. Weijer, " Analysis of optical density wave propagation and cell movement in the cellular slime mold dictyostelium discoideum.," Physica D 49, 224 (1991).
4. M. D. Berridge, P. Lipp, and M. J. Bootman, " The versatility and universality of calcium signalling.," Nat. Rev. Mol. Cell Biol. 1, 11 (2000).
5. A. T. Winfree, " Spiral waves of chemical activity.," Science 175, 634 (1972).
6. A. T. Winfree, " Stable particle-like solutions to the nonlinear wave equations of the three-dimensional excitable media.," SIAM Rev. 32, 1 (1990).
7. A. T. Winfree, " Electrical turbulence in three-dimensional heart muscle.," Science 266, 1003 (1994).
8. D. Margerit and D. Barkley, " Selection of twisted scroll waves in three-dimensional excitable media.," Phys. Rev. Lett. 86, 175 (2001).
9. D. Margerit and D. Barkley, " Cookbook asymptotics for spiral and scroll waves in excitable media.," Chaos 12, 636 (2002).
10. See review articles in the focus issue Chaos 8 (1) (1998).
11. M. Courtemanche, L. Glass, and J. P. Keener, " Instabilities of a propagating pulse in a ring of excitable media.," Phys. Rev. Lett. 70, 2182 (1993).
12. M. R. Guevara, G. Ward, A. Shrier, and L. Glass, " Electrical alternans and period-doubling bifurcations.," in IEEE Computers in Cardiology (IEEE Computer Society, Silver Spring, MD, 1984), pp. 167-170.
13. G. R. Mines, " On circulating excitations in heart muscles and their possible relation to tachycardia and fibrillation.," Trans R. Soc. Can. 4, 43 (1914).
14. L. H. Frame and M. B. Simson, " Oscillations of conduction, action potential duration, and refractoriness.," Circulation 78, 1277 (1988).
15. J. B. Nolasco and R. W. Dahlen, " A graphic method for the study of alternation in cardiac action potentials.," J. Appl. Physiol. 25, 191 (1968).
16. A. Karma, " Electrical alternans and spiral wave breakup in cardiac tissue.," Chaos 4, 461 (1994).
17. A. Karma, " Spiral breakup in model equations of action potential propagation in cardiac tissue.," Phys. Rev. Lett. 71, 1103 (1993).
18. F. H. Fenton, E. M. Cherry, H. M. Hastings, and S. J. Evans, " Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity.," Chaos 12, 852 (2002).
19. H. M. Hastings, F. H. Fenton, S. J. Evans, O. Hotomaroglu, J. Geetha, K. Gittelson, J. Nilson, and A. Garfinkel, " Alternans and the onset of ventricular fibrillation.," Phys. Rev. E 62, 4043 (2000).
20. H. Arce, A. López, and M. R. Guevara, " Triggered alternans in an ionic model of ischemic cardiac ventricular muscle.," Chaos 12, 807 (2002).
21. B. Echebarria and A. Karma, " Instability and spatiotemporal dynamics of alternans in paced cardiac dynamics.," Phys. Rev. Lett. 88, 208101 (2002).
22. B. Echebarria and A. Karma, " Spatiotemporal control of cardiac alternans.," Chaos 12, 923 (2002).
23. J. J. Fox, E. Bodenschatz, and R. F. Gilmour, " Period-doubling instability and memory in cardiac tissue.," Phys. Rev. Lett. 89, 138101 (2002).
24. H. Henry and W.-J. Rappel, " Dynamics of conduction blocks in a model of paced cardiac tissue.," Phys. Rev. E 71, 051911 (2005).
25. A. Vinet, " Memory and bistability in a one-dimensional loop of model cardiac cells.," J. Biol. Syst. 7, 451 (1999).
26. P. Comtois and A. Vinet, " Stability and bifurcation in an integral-delay model of cardiac reentry including spatial coupling in repolarization.," Phys. Rev. E 68, 051903 (2003).
27. P. Comtois and A. Vinet, " Multistability of reentrant rhythms in an ionic model of a two-dimensional annulus of cardiac tissue.," Phys. Rev. E 72, 051927 (2005).
28. G. A. Gottwald and L. Kramer, " A normal form for excitable media.," Chaos 16, 013122 (2006).
29. A. Karma, H. Levine, and X. Zou, " Theory of pulse instabilities in electrophysiological models of excitable tissues.," Physica D 73, 113 (1994).
30. M. Courtemanche, L. Glass, and J. P. Keener, " A delay equation representation of pulse circulation on a ring in excitable media.," SIAM J. Appl. Math. 56, 119 (1996).
31. R. FitzHugh, " Impulses and physiological states in theoretical models of nerve membranes.," Biophys. J. 0006-3495 1, 445 (1961); J. Nagumo, S. Arimoto, and S. Yoshizawa, " An active pulse transmission line simulating 1214-nerve axons.," Proc. IRE 50, 2061 (1962).
32. D. Barkley, " A model for fast computer simulation of waves in excitable media.," Physica D 49, 61 (1991).
33. M. Knees, L. S. Tuckerman, and D. Barkley, " Symmetry-breaking bifurcations in one-dimensional excitable media.," Phys. Rev. A 46, 5054 (1992).
34. G. A. Gottwald and L. Kramer, " On propagation failure in 1 and 2 dimensional excitable media.," Chaos 14, 855 (2004).
35. N. Krasovskii, Stability of Motion (Stanford University Press, Stanford, CA, 1963).
36. J. K. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York, 1977).
37. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer-Verlag, New York, 1993).
38. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther, Delay Equations, Applied Mathematics and Sciences (Springer-Verlag, New York, 1995), Vol. 110.
39. W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, and J. Groslambert, " Delay-induced instabilities in nonlinear feedback systems.," Phys. Rev. E 49, 203 (1994).
40. B. F. Redmond, V. G. LeBlanc, and A. Longtin, " Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry.," Physica D 166, 131 (2002).
41. J. Sieber and B. Krauskopf, " Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity.," Nonlinearity 17, 85 (2004).
42. J. Carr, Application of the Center Manifold Theory, Applied Mathematics and Sciences (Springer-Verlag, Berlin, 1981), Vol. 35.
43. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
44. K. Engelborghs, T. Luzyanina, and G. Samaey, " DDE-BIFTOOL v. 2.00 user manual: A Matlab package for bifurcation analysis of delay differential equations.," Technical Report TW-330, Department of Computer Science, K. U. Leuven, Leuven, Belgium (2001).
45. G. Giacomelli and A. Politi, " Relationship between delayed and spatially extended dynamical systems.," Phys. Rev. Lett. 76, 2686 (1996).
46. G. Giacomelli and A. Politi, " Multiple scale analysis of delayed dynamical systems.," Physica D 117, 26 (1998).
47. M. Schanz and A. Pelster, " Analytical and numerical investigations of the phase-locked loop with time delay.," Phys. Rev. E 67, 056205 (2003).
48. M. Nizette, " Front dynamics in a delayed-feedback system with external forcing.," Physica D 183, 220 (2003).
49. A. Amann, E. Schöll, and W. Just, " Some basic remarks on eigenmode expansions of time-delay dynamics.," Physica A 373, 191 (2006).
50. M. Wolfrum and S. Yanchuk, " Eckhaus instability in systems with large delay.," Phys. Rev. Lett. 96, 220201 (2006).
51. M. G. Zimmermann, S. O. Firle, M. A. Natiello, M. Hildebrand, M. Eiswirth, M. Bär, A. K. Bangia, and I. G. Kevrekidis, " Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model.," Physica D 110, 92 (1997).
52. J. Krishnan, I. G. Kevrekidis, M. Or-Guil, M. G. Zimmermann, and M. Bär, " Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction-diffusion model.," Comput. Methods Appl. Mech. Eng. 170, 253 (1999).
53. M. Or-Guil, J. Krishnan, I. G. Kevrekidis, and M. Bär, " Pulse bifurcations and instabilities in an excitable medium: Computations in finite ring domains.," Phys. Rev. E 64, 046212 (2001).
54. I. S. Aranson and L. Kramer, " The world of the complex Ginzburg-Landau equation.," Rev. Mod. Phys. 74, 99 (2002).
55. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, New York, 2000).
56. G. W. Beeler and H. Reuter, " Reconstruction of the action potential of ventricular myocardial fibers.," J. Physiol. (London) 268, 177 (1977).
57. W. Quan and Y. Rudy, " Unidirectional block and reentry of cardiac excitation: A model study.," Circ. Res. 66, 367 (1990).
58. A. Vinet, " Quasiperiodic circus movement in a loop model of cardiac tissue: Multistability and low dimensional equivalence.," Ann. Biomed. Eng. 28, 704 (2000).
59. D. Noble, " A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker potentials.," J. Physiol. (London) 160, 317 (1962).
60. M. R. Guevara, L. Glass, and A. Shrier, " Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells.," Science 214, 1350 (1981).
61. T. J. Lewis and M. R. Guevara, " Chaotic dynamics in an ionic model of the propagated cardiac action potential.," J. Theor. Biol. 146, 407 (1990).
62. F. H. Fenton, S. J. Evans, and H. M. Hastings, " Memory in an excitable medium: A mechanism for spiral wave breakup in the low-excitability limit.," Phys. Rev. Lett. 83, 3964 (1999).
63. G. M. Hall, S. Bahar, and D. J. Gauthier, " Prevalence of rate-dependent behaviors in cardiac muscle.," Phys. Rev. Lett. 82, 2995 (1999).
64. I. Banville and R. A. Gray, " Effect of action potential duration and conduction velocity restitution and their spatial dispersion on alternans and the stability of arrhythmias.," J. Cardiovasc. Electrophysiol. 13, 1141 (2002).
65. E. Cytrynbaum and J. P. Keener, " Stability conditions for the traveling pulse: Modifying the restitution hypothesis.," Chaos 12, 788 (2002).
66. E. M. Cherry and F. H. Fenton, " Suppression of alternans and conduction blocks despite steep APD restitution: Electronic, memory, and conduction velocity restitution effects.," Am. J. Physiol. Heart Circ. Physiol. 286, H2322 (2004).
67. S. Bauer, G. Röder, and M. Bär, " Alternans and the influence of ionic channel modifications: Cardiac three-dimensional and one-dimensional numerical bifurcation analysis.," Chaos 17, 015104 (2007).
68. B. Echebarria and A. Karma, " Amplitude equation approach to spatial dynamics of cardiac alternans.," Phys. Rev. E 76, 051911 (2007).