Radiophysics and Quantum Electronics

, Volume 49, Issue 11, pp 910–921 | Cite as

Chaotic synchronization in ensembles of coupled neurons modeled by the FitzHugh-Rinzel system

  • V. N. Belykh
  • E. V. Pankratova


We consider various networks of diffusively coupled identical neurons modeled by a system of FitzHugh-Rinzel coupled differential equations. The mathematical model of a solitary nerve cell is analyzed theoretically and numerically. Regions corresponding to the qualitatively different behavior of a neuron are separated in the parameter space of the system. We present synchronization conditions in a network in which the central element modeling the pacemaking neuron is linked to the group of uncoupled neurons (star configuration). Within the framework of the connection graph stability method [1], which allows us to determine the character of variation in the threshold values of the coupling strengths sufficient for establishing the complete-synchronization regime, we study the influence of the network structure on the synchronization thresholds in different ensembles. In particular, we consider a configuration in the form of diffusively coupled stars structures.


Lyapunov Function Coupling Strength Network Element Coupling Matrix Complete Synchronization 
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  1. 1.
    V. N. Belykh, I. V. Belykh, and M. Hasler, Physica D, 195, 159 (2004).MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Fundamental Concept in Nonlinear Sciences, Cambrdge Univ. Press, New York (2001).Google Scholar
  3. 3.
    S. Boccaletti, J. Kurths, G. Osipov, et al., Phys. Rep., 366, 1 (2002).MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    C. Sparrow, The Lorenz Equations, Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York (1982).MATHGoogle Scholar
  5. 5.
    O. E. Roessler, Phys. Lett. A, 57, 397 (1976).CrossRefADSGoogle Scholar
  6. 6.
    P. R. Shorten and D. J. Wall, Bull. Math. Biol., 62, No. 4, 695 (2000).CrossRefGoogle Scholar
  7. 7.
    I. Belykh, E. Lange, and M. Hasler, Phys. Rev. Lett., 94, 188101 (2005).Google Scholar
  8. 8.
    V. N. Belykh, I. V. Belykh, M. Colding-Jorgensen, and E. Mosekilde, Eur. Phys. J. E, 3, 205 (2000).CrossRefGoogle Scholar
  9. 9.
    E. Izhikevich, Int. J. Bifurcation and Chaos., 10, 1171 (2000).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Rinzel, in: Lecture Notes in Biomath. Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, Vol. 71 Springer-Verlag, Berlin, 267 (1987).Google Scholar
  11. 11.
    E. Izhikevich, SIAM Review, 43, 315 (2001).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    C. A. Del Negro, C.-F. Hsiao, S. H. Chandler and A. Garfinkel, Biophys. J., 75, 174 (1998).CrossRefGoogle Scholar
  13. 13.
    F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, New York (1973).MATHGoogle Scholar
  14. 14.
    N. N. Bautin, The Behavior of Dynamical Systems near the Boundaries of the Domain of Stability [in Russian], Nauka, Moscow (1984).Google Scholar
  15. 15.
    V. N. Belykh and Yu. S. Chertkov, in: Boundary-Value Problems. Collection of Papers [in Russian], Perm’ Polytech. Inst., Perm’ (1980), p. 120.Google Scholar
  16. 16.
    I. Belykh, M. Hasler, M. Lauret, and H. Nijmeijer, Int. J. Bifurcation and Chaos., 15, No. 11, 3423 (2005).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    V. N. Belykh, I. V. Belykh, and M. Hasler, Physica D, 195, 188 (2004).MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. N. Belykh
    • 1
  • E. V. Pankratova
    • 1
  1. 1.Volga State Academy of Water TransportNizhny NovgorodRussia

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