Table of Contents
International Scholarly Research Notices
Volume 2014 (2014), Article ID 459675, 14 pages
http://dx.doi.org/10.1155/2014/459675
Research Article

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study

1Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan
2Graduate School of Advanced Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan

Received 11 July 2014; Revised 29 September 2014; Accepted 3 October 2014; Published 12 November 2014

Academic Editor: Weimin Han

Copyright © 2014 M. Osman Gani and Toshiyuki Ogawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. American Heart Association, 2001 Heart and Stroke Statistical Update, American Heart Association, Dallas, Tex, USA, 2000.
  2. http://www.thevirtualheart.org.
  3. A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of Physiology, vol. 117, no. 4, pp. 500–544, 1952. View at Google Scholar · View at Scopus
  4. D. Noble, “A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials,” The Journal of Physiology, vol. 160, pp. 317–352, 1962. View at Google Scholar · View at Scopus
  5. R. E. McAllister, D. Noble, and R. W. Tsien, “Reconstruction of the electrical activity of cardiac Purkinje fibres,” Journal of Physiology, vol. 251, no. 1, pp. 1–59, 1975. View at Google Scholar · View at Scopus
  6. G. W. Beeler and H. Reuter, “Reconstruction of the action potential of ventricular myocardial fibres,” Journal of Physiology, vol. 268, no. 1, pp. 177–210, 1977. View at Google Scholar · View at Scopus
  7. D. DiFrancesco and D. Noble, “A model of cardiac electrical activity incorporating ionic pumps and concentration changes,” Philosophical transactions of the Royal Society of London. Series B: Biological sciences, vol. 307, no. 1133, pp. 353–398, 1985. View at Publisher · View at Google Scholar · View at Scopus
  8. C.-H. Luo and Y. Rudy, “A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction,” Circulation Research, vol. 68, no. 6, pp. 1501–1526, 1991. View at Publisher · View at Google Scholar · View at Scopus
  9. R. L. Winslow, A. Kimball, T. Varghese, C. Adlakha, and D. Noble, “Generation and propagation of ectopic beats induced by Na-K pump inhibition in atrial network models,” Proceedings of the Royal Society B, vol. 254, pp. 55–61, 1993. View at Google Scholar
  10. G. H. Sharp and R. W. Joyner, “Simulated propagation of cardiac action potentials,” Biophysical Journal, vol. 31, no. 3, pp. 403–423, 1980. View at Publisher · View at Google Scholar · View at Scopus
  11. F. H. Fenton and A. Karma, “Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation,” Chaos, vol. 8, pp. 20–47, 1998. View at Google Scholar
  12. F. Fenton and A. Karma, “Fiber-rotation-induced vortex turbulence in thick myocardium,” Physical Review Letters, vol. 81, no. 2, pp. 481–484, 1998. View at Publisher · View at Google Scholar · View at Scopus
  13. F. H. Fenton, Theoretical investigation of spiral and scroll wave instabilities underlying cardiac_brillation [Ph.D. thesis], Northeastern University, Bostan, Mass, USA, 1999.
  14. R. FitzHugh, “Impulse and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, pp. 445–465, 1961. View at Google Scholar
  15. J. S. Nagumo, S. Arimoto, and S. Yoshizawa, “An active pulse transmission line simulating nerve axon,” Proceedings of the IRE, vol. 50, no. 10, pp. 2061–2071, 1962. View at Publisher · View at Google Scholar
  16. A. Karma, “Spiral breakup in model equations of action potential propagation in cardiac tissue,” Physical Review Letters, vol. 71, no. 7, pp. 1103–1106, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. R. R. Aliev and A. V. Panfilov, “A simple two-variable model of cardiac excitation,” Chaos, Solitons & Fractals, vol. 7, no. 3, pp. 293–301, 1996. View at Publisher · View at Google Scholar · View at Scopus
  18. S.-I. Kinoshita, M. Iwamoto, K. Tateishi, N. J. Suematsu, and D. Ueyama, “Mechanism of spiral formation in heterogeneous discretized excitable media,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 87, Article ID 062815, 6 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  19. K. Kuramoto and S. Koga, “Turbulized rotation chemical waves,” Progress of Theoretical Physics, vol. 66, pp. 1081–1085, 1981. View at Google Scholar
  20. A. T. Winfree, “Electrical instability in cardiac muscle: phase singularities and rotors,” Journal of Theoretical Biology, vol. 138, no. 3, pp. 353–405, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. A. V. Panfilov and A. V. Holden, “Self-generation of turbulent vortices in a two-dimensional model of cardiac tissue,” Physics Letters A, vol. 151, no. 1-2, pp. 23–26, 1990. View at Publisher · View at Google Scholar · View at Scopus
  22. A. V. Panfilov and A. V. Holden, “Spatiotemporal irregularity in a two-dimensional model of cardiac tissue,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 1, no. 1, pp. 219–225, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  23. M. Courtemanche and A. T. Winfree, “Re-entrant rotating waves in a Beeler-Reuter based model of two-dimensional cardiac electrical activity,” International Journal of Bifurcation and Chaos, vol. 1, no. 2, pp. 431–444, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  24. M. Gerhardt, H. Schuster, and J. J. Tyson, “A cellular automaton model of excitable media including curvature and dispersion,” Science, vol. 247, no. 4950, pp. 1563–1566, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H. Ito and L. Glass, “Spiral breakup in a new model of discrete excitable media,” Physical Review Letters, vol. 66, no. 5, pp. 671–674, 1991. View at Publisher · View at Google Scholar · View at Scopus
  26. 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, vol. 12, no. 3, pp. 852–892, 2002. View at Publisher · View at Google Scholar · View at Scopus
  27. M. Courtemanche, “Complex spiral wave dynamics in a spatially distributed ionic model of cardiac electrical activity,” Chaos, vol. 6, no. 4, pp. 579–600, 1996. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Karma, “Electrical alternans and spiral wave breakup in cardiac tissue,” Chaos, vol. 4, no. 3, pp. 461–472, 1994. View at Publisher · View at Google Scholar · View at Scopus
  29. M. Bär and M. Eiswirth, “Turbulence due to spiral breakup in a continuous excitable medium,” Physical Review E, vol. 48, no. 3, pp. R1635–R1637, 1993. View at Publisher · View at Google Scholar · View at Scopus
  30. A. F. M. Marée and A. V. Panfilov, “Spiral breakup in excitable tissue due to lateral instability,” Physical Review Letters, vol. 78, no. 9, pp. 1819–1822, 1997. View at Publisher · View at Google Scholar · View at Scopus
  31. A. Giaquinta, S. Boccaletti, and F. T. Arecchi, “Superexcitability induced spiral breakup in excitable systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 6, no. 9, pp. 1753–1759, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. A. Panfilov and P. Hogeweg, “Spiral breakup in a modified FitzHugh-Nagumo model,” Physics Letters A, vol. 176, no. 5, pp. 295–299, 1993. View at Publisher · View at Google Scholar · View at Scopus
  33. A. V. Panfilov, “Spiral breakup as a model of ventricular fibrillation,” Chaos, vol. 8, pp. 57–64, 1998. View at Publisher · View at Google Scholar
  34. C. W. Zemlin and A. V. Panfilov, “Spiral waves in excitable media with negative restitution,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 63, Article ID 041912, 4 pages, 2001. View at Publisher · View at Google Scholar
  35. A. V. Panfilov and C. W. Zemlin, “Wave propagation in an excitable medium with a negatively sloped restitution curve,” Chaos, vol. 12, no. 3, pp. 800–806, 2002. View at Publisher · View at Google Scholar · View at Scopus
  36. J. Jalife, R. A. Gray, G. E. Morley, and J. M. Davidenko, “Self-organization and the dynamical nature of ventricular fibrillation,” Chaos, vol. 8, no. 1, pp. 79–93, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  37. M. O. Gani and T. Ogawa, “Instability of periodic traveling wave solutions in a generalized fitzhugh-nagumo model for excitable media,” In preparation.
  38. E. Meron, “Pattern formation in excitable media,” Physics Reports: A Review Section of Physics Letters, vol. 218, no. 1, pp. 1–66, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. J. Pinnell, S. Turner, and S. Howell, “Cardiac muscle physiology,” Continuing Education in Anaesthesia, Critical Care and Pain, vol. 7, no. 3, pp. 85–88, 2007. View at Publisher · View at Google Scholar · View at Scopus
  40. M. P. Nash and A. V. Panfilov, “Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias,” Progress in Biophysics & Molecular Biology, vol. 85, no. 2-3, pp. 501–522, 2004. View at Publisher · View at Google Scholar · View at Scopus
  41. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press, Cambridge, UK, 2005. View at MathSciNet
  42. D. W. Peaceman and H. H. Rachford Jr., “The numerical solution of parabolic and elliptic differential equations,” Journal of the Society for Industrial & Applied Mathematics, vol. 3, no. 1, pp. 28–41, 1955. View at Google Scholar
  43. J. J. Douglas, “On the numerical integration of uxx+uyy=ut by implicit methods,” Journal of the Society for Industrial and Applied Mathematics, vol. 3, pp. 42–65, 1955. View at Google Scholar · View at MathSciNet
  44. A. Hagberg and E. Meron, “Propagation failure in excitable media,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 57, no. 1, pp. 299–303, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. M. Markus, G. Kloss, and I. Kusch, “Disordered waves in a homogeneous, motionless excitable medium,” Nature, vol. 371, no. 6496, pp. 402–404, 1994. View at Publisher · View at Google Scholar · View at Scopus
  46. J. N. Weiss, A. Garfinkel, H. S. Karagueuzian, Z. Qu, and P.-S. Chen, “Chaos and the transition to ventricular fibrillation: a new approach to antiarrhythmic drug evaluation,” Circulation, vol. 99, no. 21, pp. 2819–2826, 1999. View at Publisher · View at Google Scholar · View at Scopus
  47. M. Bär and L. Brusch, “Breakup of spiral waves caused by radial dynamics: eckhaus and finite wavenumber instabilities,” New Journal of Physics, vol. 6, pp. 1–22, 2004. View at Google Scholar · View at Scopus