Nonlinear Dynamics

, Volume 78, Issue 3, pp 2085–2099 | Cite as

Stability analysis and rich oscillation patterns in discrete-time FitzHugh–Nagumo excitable systems with delayed coupling

  • Xiujuan Wang
  • Mingshu Peng
  • Ranran Cheng
  • Jinchen Yu
Original Paper


In this paper, we give a detailed study of the stable region in discrete-time FitzHugh–Nagumo delayed excitable Systems, which can be divided into two parts: one is independent of delay and the other is dependent on delay. Two different new states are to be observed, which are new steady states (equilibria-the excitable FitzHugh–Nagumo) or limit cycles/higher periodic orbits (the FitzHugh–Nagumo oscillators) as the origin loses its stability, and usually, one is synchronized and the other asynchronized. We also find out that there exist critical curves through which there occur fold bifurcations, flip bifurcations, Neimark–Sacker bifurcations and even higher-codimensional bifurcations etc. It is also shown that delay can play an important role in rich dynamics, such as the occurrence of chaos or not, by means of Lyapunov exponents, Lyapunov dimensions, and the sensitivity to the initial conditions. Multistability phenomena are also discussed including the coexistence of synchronized and asynchronized oscillators, or synchronized/asynchronized oscillators and multiple stable nontrivial equilibria etc.


Neural network models Delay Stability Bifurcations Synchronization/asynchronization Chaos 


  1. 1.
    Atay, F.M., Jost, J., Wende, A.: Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 92, 144101 (2004)CrossRefGoogle Scholar
  2. 2.
    Atay, F.M., Karabacak, Özkan: Stability of Coupled Map Networks with Delays. SIAM J Appl. Dyn. Syst 5, 508–527 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baldi, P., Atiya, A.: How delays affect neural dynamics and learning. IEEE T. Neural Networ. 5, 612–621 (1994)CrossRefGoogle Scholar
  4. 4.
    Buri\(\grave{c}\), N., Todorovi\(\grave{c}\), D.: Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. Phys. Rev. E 67, 066222 (2003)Google Scholar
  5. 5.
    FitzHugh, R.: Impulses and physillogical states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)CrossRefGoogle Scholar
  6. 6.
    Guo, S., Tang, X., Huang, L.: Bifurcation analysis in a discrete-time single-directional network with delays. Neurocomputing 71(7–9), 1422–1435 (2008)CrossRefGoogle Scholar
  7. 7.
    Guo, S., Tang, X., Huang, L.: Stability and bifurcation in a discrete system of two neurons with delays. Nonlinear Anal RWA 9, 1323–1335 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hadeler, K.P.: Effective computation of periodic orbits and bifurcation diagrams in delay equations. Numer. Math. 34, 457–67 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane and its application to conduction and excitation in nerve. J. Physiol 117, 500–544 (1952)Google Scholar
  10. 10.
    Hopfield, J.J.: Neural networks and physical systems with emergent collective computational properties. Proc. Nat. Acad. Sci. 79, 2554–2558 (1982)Google Scholar
  11. 11.
    Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two state neurons. Proc. Nat. Acad. Sci. 81, 3088–3092 (1984) Google Scholar
  12. 12.
    Hopfield, J.J., Tank, D.W.: Computing with neural circuits: a model. Science 233, 625–633 (1986)CrossRefGoogle Scholar
  13. 13.
    Iooss, G.: Bifurcation of Maps and Applications. North-Holland, Amsterdam (1979)Google Scholar
  14. 14.
    Ioos, G., Los, J.E.: Quasi-genericity of bifurcations to high dimensional Invariant tori for maps. Commun. Math. Phys. 119, 453–500 (1988)CrossRefGoogle Scholar
  15. 15.
    Kakiuchi, N., Tchizawa, K.: On an explicit duck solution and delay in the FitzHugh-Nagumo equation. J. Differ. Equations. 141, 327–339 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kaslik, E., Balint, S.: Chaotic Dynamics of a Delayed Discrete-Time Hopfield Network of Two Nonidentical Neurons with no Self-Connections. J Nonlinear Sci. 18, 415–432 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kim, S., Park, S.H., Ryu, C.S.: Multistability in coupled oscillator systems with time delay. Phys. Rev. Lett. 79, 2911–2914 (1997)CrossRefGoogle Scholar
  18. 18.
    Kuruklis, S.A.: The asymptotic stability of \(x_{n+1}-ax_n+bx_{n-k}=0\). J. Math. Anal. Appl. 188, 719–731 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer-Verlag, New York (2004)CrossRefzbMATHGoogle Scholar
  20. 20.
    Murray, J.D.: Mathematical Biology. Springer, New York (1990)zbMATHGoogle Scholar
  21. 21.
    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)CrossRefGoogle Scholar
  22. 22.
    Peng, M.: Bifurcation and chaotic behavior in the Euler method for a Uçar prototype delay model prototype delay model. Chaos Soliton Fract 22, 483–493 (2004)CrossRefzbMATHGoogle Scholar
  23. 23.
    Peng, M., Yuan, R.: Higher-codimension bifurcations caused by delay. Nonlinear Dyn 58, 453–467 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Peng, M., Yang, X.: New stability criteria and bifurcation analysis for nonlinear discrete-time coupled loops with multiple delays. Chaos 20, 13125 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nikolai, F.: Rulkov, Regularization of Synchronized Chaotic Bursts. Phys. Rev. Lett. 86, 183–186 (2001)CrossRefGoogle Scholar
  26. 26.
    Zhen, B., Xu, J.: Simple zero singularity analysis in a coupled FitzHugh-Nagumo neural system with delay, Neurocomputing, 73 874–882 (2010)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Xiujuan Wang
    • 1
    • 2
  • Mingshu Peng
    • 1
  • Ranran Cheng
    • 1
  • Jinchen Yu
    • 1
    • 3
  1. 1.Department of MathematicsBeijing Jiao Tong UniversityBeijing People’s Republic China
  2. 2.Weifang UniversityWeifang People’s Republic China
  3. 3.Shandong Jiaotong UniversityJinan People’s Republic China

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