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ISSN 2542-1905 (Online)

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Slepnev A. V., Shepelev I. A., Vadivasova T. E. 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: 10.18500/0869-6632-2014-22-2-50-61

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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
Vadivasova Tatjana Evgenevna, 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;16(3):205–265.
  2. Rinzel J, Keller JD. Traveling wave solutions of a nerve conduction equation. J. Biophys. 1973;13(12):1313--1337. DOI: 10.1016/S0006-3495(73)86065-5
  3. Winfree AT. The geometry of biological time. New York: Springer; 1980. 543 p.
  4. Romanovsky YUM, Stepanova NV, Chernavsky DS. Mathematical modeling in biophysics. Moscow-Izhevsk: ICR; 2003. 402 p. (In Russian).
  5. Keener J, Sneyd J. Mathematical physiology. New York: Springer; 1998. 766 p.
  6. Winfree A. Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media. Chaos. 1991;1(3):303–334. DOI: 10.1063/1.165844
  7. Loskutov AU, Mikhailov AS. Fundamentals of complex systems theory. Moscow-Izhevsk: RCD, ICR; 2007. 620 p.
  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 AM, Ermakova EA, Panfilov AV. Rotating spiral waves in a modified FitzHugh–Nagumo model. Phys. D. 1984;14(1):117–124.
  10. Zaritski RM, Pertsov FM. Stable Spiral structures and their interaction in two-dimensional excitable media. Physical Review E. 2002;66(6):066120(1–6). DOI: 10.1103/PhysRevE.66.066120
  11. Jones KRT. Stability of the traveling wave solution of the FitzHugh–Nagumo system. Trans. Amer. Math. Soc. 1984;286:431–469. DOI: 10.1090/S0002-9947-1984-0760971-6
  12. Neu JC, Preissig RS. and Krassowska W. Initiation of propagation in a one-dimensional excitable medium. Phys. D. 1997;102(3-4):285–299. DOI: 10.1016/S0167-2789(96)00203-5
  13. Nagai Y, Gonzalez H, Shrier A, Glass L. Paroxysmal starting and stopping ofcirculating waves in excitable media. Phys. Rev. Letters. 2000;84(18):4248–4251. DOI: 10.1103/PhysRevLett.84.4248
  14. Cytrynbaum E, Keener JP. Stability conditions for the traveling pulse: Modifying the restitution hypothesis. Chaos. 2002;12(3):788–799. DOI: 10.1063/1.1503941
  15. Alford JG, Auchmuty G. Rotating wave solutions of the FitzHugh–Nagumo equations. J. Math. Biol. 2006;53(5):797–819. DOI: 10.1007/s00285-006-0022-1
  16. Buric N, Todorovic D. Dynamics of FitzHugh–Nagumo excitable systems with delayed coupling. Phys. Rev. E. 2003;67(6):066222. DOI: 10.1103/PhysRevE.67.066222
  17. Scholl E, Hiller G, Hovel P, Dahlem MA. Time-delayed feedback in neurosystems. Phil. Trans. R. Soc. A. 2009;367(1891):1079–1096. DOI: 10.1098/rsta.2008.0258
  18. Rosenblum MG, Pikovsky A, Kurths J. Synchronization-a universal concept in nonlinear sciences. Cambridge, UK: Cambridge University Press; 2001. 432 p.
  19. Han SK, Yim TG, Postnov DE, Sosnovtseva OV. Interacting coherence resonance oscillators. Phys. Rev. Lett. 1999;83(9):1771–1774. DOI:10.1103/PhysRevLett.83.1771
  20. Neiman A, Schimansky-Geier L, Cornell-Bell A, Moss F. Noise-enhanced phase synchronization in excitable media. Phys. Rev. Lett. 1999;83(23):4896–4899. DOI:10.1103/PHYSREVLETT.83.4896
  21. Hu B, Zhou Ch. Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance. Phys. Rev. E. 2000;61(2):R1001(1–4). DOI: 10.1103/physreve.61.r1001
  22. Nomura T, Glass L. Entrainment and termination of reentrant wave propagation in a periodically stimulated ring of excitable media. Phys. Rev. E. 1996;53(6):6353–6360. DOI: 10.1103/physreve.53.6353
  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;139:83–89. DOI:10.1143/PTPS.139.83
  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;65(2):021908(1–10). DOI:10.1103/PhysRevE.65.021908
  25. FitzHugh RA. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1961;1(6):445–466. DOI: 10.1016/s0006-3495(61)86902-6
  26. Nagumo JS, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the Institute of Radio Engineers. 1962;50(10):2061–2070. DOI: 10.1109/JRPROC.1962.288235
  27. Slepnev AV, Vadivasova TE. Two kinds of auto-oscillations in active medium with periodical border conditions. Nelin. Dinam. 2012;8(3):497–505.
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