From local to global spatiotemporal chaos in a cardiac tissue model

Zhilin Qu; James N. Weiss; Alan Garfinkel

doi:10.1103/PhysRevE.61.727

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From local to global spatiotemporal chaos in a cardiac tissue model

Zhilin Qu, James N. Weiss, and Alan Garfinkel

Departments of Medicine (Cardiology) and Physiological Science, University of California at Los Angeles, Los Angeles, California 90095

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Phys. Rev. E 61, 727 – Published 1 January, 2000

DOI: https://doi.org/10.1103/PhysRevE.61.727

Abstract

Two kinds of chaos can occur in cardiac tissue, chaotic meander of a single intact spiral wave and chaotic spiral wave breakup. We studied these behaviors in a model of two-dimensional cardiac tissue based on the Luo-Rudy I action potential model. In the chaotic meander regime, chaos is spatially localized to the core of the spiral wave. When persistent spiral wave breakup occurs, there is a transition from local to global spatiotemporal chaos.

References (39)

A. T. Winfree, Chaos 1, 303 (1991).
A. V. Panfilov and A. V. Holden, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 219 (1991).
H. Ito and L. Glass, Phys. Rev. Lett. 66, 671 (1991).
A. Karma, Chaos 4, 461 (1994).
I. R. Efimov, V. I. Krinsky, and J. Jalife, Chaos Solitons Fractals 5, 513 (1995).
H. Zhang and A. V. Holden, Chaos Solitons Fractals 5, 661 (1995).
M. Courtemanche, Chaos 6, 579 (1996).
A. F. M. Maree and A. V. Panfilov, Phys. Rev. Lett. 78, 1819 (1997).
V. N. Biktashev and A. V. Holden, Physica D 116, 342 (1998).
J. Beaumont et al., Biophys. J. 75, 1 (1998).
M. C. Strain and H. S. Greenside, Phys. Rev. Lett. 80, 2306 (1998).
Z. Qu, J. N. Weiss, and A. Garfinkel, Am. J. Physiol. 276, H269 (1999).
K. I. Agladze, V. I. Krinsky, and A. M. Pertsov, Nature (London) 308, 834 (1984).
M. Markus, G. Kloss, and I. Kusch, Nature (London) 371, 402 (1994).
G. Li et al., Phys. Rev. Lett. 77, 2105 (1996).
J. M. Davidenko et al., Nature (London) 355, 349 (1992).
A. M. Pertsov et al., Circ. Res. 72, 631 (1993).
R. A. Gray and J. Jalife, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 415 (1996).
A. Garfinkel et al., J. Clin. Invest. 99, 305 (1997).
Y. H. Kim et al., J. Clin. Invest. 100, 2486 (1997).
P.-S. Chen et al., Chaos 8, 127 (1998).
J. E. Poole and G. H. Bardy, in Cardiac Electrophysiology: From Cell to Bedside, edited by D. P. Zipes and J. Jalife (Saunders, Philadelphia, 1995), Vol. 2, p. 812.
D. Barkley, Phys. Rev. Lett. 68, 2090 (1992).
A. Karma, Phys. Rev. Lett. 65, 2824 (1990).
M. Bar and M. Eiswirth, Phys. Rev. E 48, R1635 (1993).
M. Courtemanche and A. T. Winfree, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 431 (1991).
A. Xu and M. R. Guevara, Chaos 8, 157 (1998).
F. Xie, Z. Qu, and A. Garfinkel, Phys. Rev. E 58, 6355 (1998).
C. H. Luo and Y. Rudy, Circ. Res. 68, 1501 (1991).
C. H. Luo and Y. Rudy, Circ. Res. 74, 1071 (1994).
S. Rush and H. Larsen, IEEE Trans. Biomed. Eng. 25, 389 (1978).
Z. Qu and A. Garfinkel, IEEE Trans. Biomed. Eng. 46, 1166 (1999).
J. W. Thomas, Numerical Partial Differential Equations (Springer-Verlag, New York, 1995).
P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos: Towards a Deterministic Approach to Turbulence (Wiley, New York, 1984).
We calculated λ by simultaneously integrating two copies of Eqs. (123) with the same initial condition, and gave a small perturbation to one of the integrations. We used the same definition of the maximum Lyapunov exponent as $λ = \underset{\vec{t} \infty}{limop} \frac{1}{t} \ln \frac{‖ w (t) ‖}{‖ w (0) ‖} \approx Σ {\vec{n} \infty}^{} \frac{λ_{n}}{n},$ where $λ_{n} = \frac{1}{Δ T} \ln \frac{‖ w (n Δ t) ‖}{‖ w ((n - 1) Δ T) ‖} .$ Here $w [(n - 1) Δ T]$ is the perturbation given at $(n - 1) Δ T$ and $w (n Δ T)$ is the difference of the integration at $n Δ T .$ At $n Δ T,$ we rescale $w (n Δ T)$ to be very small and reset the simulation. We used $Δ T = 20 ms .$
J. P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988).
Z. Qu and G. Hu, Phys. Rev. E 49, 1099 (1994).
D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971).
R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, New York, 1994).

[c1] A. T. Winfree, Chaos 1, 303 (1991).

[c2] A. V. Panfilov and A. V. Holden, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 219 (1991).

[c3] H. Ito and L. Glass, Phys. Rev. Lett. 66, 671 (1991).

[c4] A. Karma, Chaos 4, 461 (1994).

[c5] I. R. Efimov, V. I. Krinsky, and J. Jalife, Chaos Solitons Fractals 5, 513 (1995).

[c6] H. Zhang and A. V. Holden, Chaos Solitons Fractals 5, 661 (1995).

[c7] M. Courtemanche, Chaos 6, 579 (1996).

[c8] A. F. M. Maree and A. V. Panfilov, Phys. Rev. Lett. 78, 1819 (1997).

[c9] V. N. Biktashev and A. V. Holden, Physica D 116, 342 (1998).

[c10] J. Beaumont et al., Biophys. J. 75, 1 (1998).

[c11] M. C. Strain and H. S. Greenside, Phys. Rev. Lett. 80, 2306 (1998).

[c12] Z. Qu, J. N. Weiss, and A. Garfinkel, Am. J. Physiol. 276, H269 (1999).

[c13] K. I. Agladze, V. I. Krinsky, and A. M. Pertsov, Nature (London) 308, 834 (1984).

[c14] M. Markus, G. Kloss, and I. Kusch, Nature (London) 371, 402 (1994).

[c15] G. Li et al., Phys. Rev. Lett. 77, 2105 (1996).

[c16] J. M. Davidenko et al., Nature (London) 355, 349 (1992).

[c17] A. M. Pertsov et al., Circ. Res. 72, 631 (1993).

[c18] R. A. Gray and J. Jalife, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 415 (1996).

[c19] A. Garfinkel et al., J. Clin. Invest. 99, 305 (1997).

[c20] Y. H. Kim et al., J. Clin. Invest. 100, 2486 (1997).

[c21] P.-S. Chen et al., Chaos 8, 127 (1998).

[c22] J. E. Poole and G. H. Bardy, in Cardiac Electrophysiology: From Cell to Bedside, edited by D. P. Zipes and J. Jalife (Saunders, Philadelphia, 1995), Vol. 2, p. 812.

[c23] D. Barkley, Phys. Rev. Lett. 68, 2090 (1992).

[c24] A. Karma, Phys. Rev. Lett. 65, 2824 (1990).

[c25] M. Bar and M. Eiswirth, Phys. Rev. E 48, R1635 (1993).

[c26] M. Courtemanche and A. T. Winfree, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 431 (1991).

[c27] A. Xu and M. R. Guevara, Chaos 8, 157 (1998).

[c28] F. Xie, Z. Qu, and A. Garfinkel, Phys. Rev. E 58, 6355 (1998).

[c29] C. H. Luo and Y. Rudy, Circ. Res. 68, 1501 (1991).

[c30] C. H. Luo and Y. Rudy, Circ. Res. 74, 1071 (1994).

[c31] S. Rush and H. Larsen, IEEE Trans. Biomed. Eng. 25, 389 (1978).

[c32] Z. Qu and A. Garfinkel, IEEE Trans. Biomed. Eng. 46, 1166 (1999).

[c33] J. W. Thomas, Numerical Partial Differential Equations (Springer-Verlag, New York, 1995).

[c34] P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos: Towards a Deterministic Approach to Turbulence (Wiley, New York, 1984).

[c35] We calculated λ by simultaneously integrating two copies of Eqs. (123) with the same initial condition, and gave a small perturbation to one of the integrations. We used the same definition of the maximum Lyapunov exponent as $λ = \underset{\vec{t} \infty}{limop} \frac{1}{t} \ln \frac{‖ w (t) ‖}{‖ w (0) ‖} \approx Σ {\vec{n} \infty}^{} \frac{λ_{n}}{n},$ where $λ_{n} = \frac{1}{Δ T} \ln \frac{‖ w (n Δ t) ‖}{‖ w ((n - 1) Δ T) ‖} .$ Here $w [(n - 1) Δ T]$ is the perturbation given at $(n - 1) Δ T$ and $w (n Δ T)$ is the difference of the integration at $n Δ T .$ At $n Δ T,$ we rescale $w (n Δ T)$ to be very small and reset the simulation. We used $Δ T = 20 ms .$

[c36] J. P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988).

[c37] Z. Qu and G. Hu, Phys. Rev. E 49, 1099 (1994).

[c38] D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971).

[c39] R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, New York, 1994).

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From local to global spatiotemporal chaos in a cardiac tissue model

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References (39)

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