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Slepnev A. V., Shepelev I. A. External synchronization of traveling waves in an active medium in self-sustained and excitable regime. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 2, pp. 50-61. DOI:


External synchronization of traveling waves in an active medium in self-sustained and excitable regime

Slepnev Andrej Vjacheslavovich, Saratov State University
Shepelev Igor Aleksandrovich, Saratov State University

The model of a one-dimensional active medium, which cell represents FitzHugh–Nagumo oscillator, is studied with periodical boundary conditions. Such medium can be either self-oscillatory or excitable one in dependence of the parameters values. Periodical boundary conditions provide the existence of traveling wave regimes both in excitable anself-oscillatory case without any deterministic or stochastic impacts. The local periodic force influence on the medium is under study. In addition to the uniform medium study the single FitzHugh–Nagumo oscillator with complementary time-delayed feedback is considered. The comparison of synchronization effects in excitable and self-oscillatory regimes of the active medium and its analogue is carried out.


1. Wiener N., Rosenblueth A. The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle // Arch. Inst. Cardiol. Mexico. 1946. Vol. 16. P. 205. 2. Rinzel J., Keller J.D. Traveling wave solutions of a nerve conduction equation // J. Biophys. 1973. Vol. 13. P. 1313. 3. Winfree A.T. The geometry of biological time. New York: Springer, 1980. 4. Романовский Ю.М., Степанова Н.В., Чернавский Д.С. Математическое моделирование в биофизике. М.-Ижевск: Институт компьютерных исследований, 2003. 5. Keener J., Sneyd J. Mathematical physiology. New York: Springer, 1998. 6. Winfree A. Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media // Chaos. 1991. Vol. 1, No 3. P. 303. 7. Лоскутов А.Ю., Михайлов А.С. Основы теории сложных систем. М.-Ижевск: НИЦ «РХД», Институт компьютерных исследований, 2007. 8. Bub G., Shrier A., Glass L. Spiral Waves Break Hearts: New research stresses the importance of communication between cardiac cells. Inside Science News Services, 2005 9. Pertsov A.M., Ermakova E.A., Panfilov A.V. Rotating spiral waves in a modified FitzHugh–Nagumo model // Phys. D. 1984. Vol. 14. P. 117. 10. Zaritski R.M., Pertsov F.M. Stable Spiral structures and their interaction in two-dimensional excitable media // Physical Review E. 2002. Vol. 66, No 6. P. 066120(1–6). 11. Jones K.R.T. Stability of the traveling wave solution of the FitzHugh–Nagumo system // Trans. Amer. Math. Soc. 1984. Vol. 286. P. 431. 12. Neu J.C., Preissig R.S. and Krassowska W. Initiation of propagation in a one-dimensional excitable medium // Phys. D. 1997. Vol. 102. P. 285. 13. Nagai Y., Gonzalez H., Shrier A., Glass L.  ? Paroxysmal starting and stopping ofcirculating waves in excitable media // Phys. Rev. Letters. 2000. Vol. 84, No 18. P. 4248. 14. Cytrynbaum E. and Keener J.P. Stability conditions for the traveling pulse: Modifying the restitution hypothesis // Chaos. 2002. Vol. 12 P. 788. 15. Alford J.G., Auchmuty G. Rotating wave solutions of the FitzHugh–Nagumo equations // J. Math. Biol. 2006. Vol. 53, No 5. P. 797. 16. Buric N., Todorovic D. Dynamics of FitzHugh–Nagumo excitable systems with delayed coupling // Phys. Rev. E. 2003. Vol. 67. P. 066222. 17. Scholl E., Hiller G., H  ? ovel P., Dahlem M.A.  ? Time-delayed feedback in neurosystems// Phil. Trans. R. Soc. A. 2009. Vol. 367. P. 1079. 18. Rosenblum M. G., Pikovsky A., Kurths J. Synchronization-a universal concept in nonlinear sciences. Cambridge, UK: Cambridge University Press, 2001. 19. Han S.K., Yim T.G., Postnov D.E., Sosnovtseva O.V. Interacting coherence resonance oscillators // Phys. Rev. Lett. 1999. Vol. 83, No 9. P. 1771. 20. Neiman A., Schimansky-Geier L., Cornell-Bell A., Moss F. Noise-enhanced phase synchronization in excitable media // Phys. Rev. Lett. 1999. Vol. 83, No 23. P. 4896. 21. Hu B., Zhou Ch. Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance // Phys. Rev. E. 2000. Vol. 61, No 2. P. R1001(1–4). 22. Nomura T., Glass L. Entrainment and termination of reentrant wave propagation in a periodically stimulated ring of excitable media // Phys. Rev. E. 1996. Vol. 53, No 6. P. 6353. 23. Gonzalez H., Nagai Y., Bub G., Glass L.  ? Resetting and annihilating reentrant waves in a ring of cardiac tissue: Theory and experiment // Progress of Theor. Phys. Supplement. 2000. Vol. 139. P. 83. 24. Glass L., Nagai Y., Hall K., Talajic M., Natte S. Predicting the entrainment of reentrant cardiac waves using phase resetting curves // Phys. Rev. E. 2002. Vol. 65. P. 021908(1–10). 25. FitzHugh R.A. Impulses and physiological states in theoretical models of nerve membrane // Biophys. J. 1961. Vol. 1. P. 445. 26. Nagumo J.S., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon // Proceedings of the Institute of Radio Engineers. 1962. Vol. 50. P. 2061. 27. Слепнев А.В., Вадивасова Т.Е. Два вида автоколебаний в активной среде с периодическими граничными условиями // Нелинейная динамика. 2012. Т. 8, No 3. С. 497.

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