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Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

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Abstract

In this paper, Hopf bifurcation and Bogdanov–Takens bifurcation with codimension 2 in a coupled FitzHugh–Nagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov–Takens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov–Takens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of Bogdanov–Takens bifurcation indicates that the neuron signals can transit between resting and spiking.

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References

  1. Hodgkin, A.L.: The local electric changes associated with repetitive action in a non-modulated axon. J. Physiol. 107, 165–181 (1948)

    Google Scholar 

  2. Rinzel, J., Ermentrout, G.B.: Analysis of Neural Excitability and Oscillations, Methods in Neuronal Modeling. MIT, Cambridge (1989)

    Google Scholar 

  3. Izhikevich, E.M.: Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting. MIT, Cambridge (1967)

    Google Scholar 

  4. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 6, 1171–1266 (2000)

    MathSciNet  Google Scholar 

  5. Uçar, A.: On the chaotic behaviour of a prototype delayed dynamical system. Chaos Soliton Fractals 16, 187–194 (2003)

    Article  MATH  Google Scholar 

  6. Li, C., Liao, X., Yu, J.: Hopf bifurcation in a prototype delayed system. Chaos Soliton Fractals 19, 779–787 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane and its application to conduction and excitation innerve. J. Physiol. 117, 500–544 (1952)

    Google Scholar 

  8. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Article  Google Scholar 

  9. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Article  Google Scholar 

  10. Duarte, J., Silva, L., Ramos, J.S.: Types of bifurcations of FitzHugh–Nagumo maps. Nonlinear Dyn. 44, 231–242 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Burić, N., Todorović, D.: Dynamics of FitzHugh–Nagumo excitable systems with delayed coupling. Phys. Rev. E 67, 066222 (2003)

    Article  MathSciNet  Google Scholar 

  12. Burić, N., Todorović, D.: Bifurcations due to small time-lag in coupled excitable systems. Int. J. Bifurc. Chaos 15, 1775–1785 (2005)

    Article  MATH  Google Scholar 

  13. Burić, N., Ranković, D.: Bursting neurons with coupling delays. Phys. Lett. A 363, 282–289 (2007)

    Article  Google Scholar 

  14. Uçar, A., Lonngren, K.E., Bai, E.W.: Synchronization of the coupled FitzHugh–Nagumo systems. Chaos Soliton Fractals 20, 1085–1090 (2004)

    Article  MATH  Google Scholar 

  15. Fan, D., Hong, L.: Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays. Commun. Nonlinear Sci. Numer. Simul. 15, 1873–1886 (2010)

    Article  MathSciNet  Google Scholar 

  16. Zhen, B., Xu, J.: Simple zero singularity analysis in a coupled FitzHugh–Nagumo neural system with delay. Neurocomputing 73, 874–882 (2010)

    Article  Google Scholar 

  17. Zhen, B., Xu, J.: Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay. Commun. Nonlinear Sci. Numer. Simul. 15, 442–458 (2010)

    Article  MathSciNet  Google Scholar 

  18. Golubitsky, M., Stewart, I.N.: Hopf bifurcation in the presence of symmetry. Arch. Ration. Mech. Anal. 87, 107–165 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)

    Article  MATH  Google Scholar 

  20. Guo, S., Lamb, J.S.W.: Equivariant Hopf bifurcation for neutral functional differential equations. Proc. Am. Math. Soc. 136, 2031–2041 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995)

    Article  MATH  Google Scholar 

  22. Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity. J. Differ. Equ. 122, 201–224 (1995)

    Article  MATH  Google Scholar 

  23. Yuan, Y., Yu, P.: Computation of simplest normal forms of differential equations associated with a double-zero eigenvalues. Int. J. Bifurc. Chaos 12, 1307–1330 (2001)

    MathSciNet  Google Scholar 

  24. Wei, J., Jiang, W.: Stability and bifurcation analysis in van der Pol’s oscillator with delayed feedback. J. Sound Vib. 283, 801–819 (2005)

    Article  MathSciNet  Google Scholar 

  25. Jiang, W., Yuan, Y.: Bogdanov–Takens singularity in van der Pol’s oscillator with delayed feedback. Physica D 227, 149–161 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Guo, S.: Equivariant normal forms for neutral functional differential equations. Nonlinear Dyn. 61, 311–329 (2010)

    Article  MATH  Google Scholar 

  27. Li, X., Wei, J.: On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays. Chaos Soliton Fractals 26, 519–526 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hassard, B., Kazarinoff, N., Wan, Y.H.: Theory of application of Hopf bifurcation. In: London Math. Society Lecture Notes, vol. 41. Cambridge University, Cambridge (1981)

    Google Scholar 

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  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Yanqiu Li & Weihua Jiang

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  1. Yanqiu Li
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  2. Weihua Jiang
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Correspondence to Weihua Jiang.

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Li, Y., Jiang, W. Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay. Nonlinear Dyn 65, 161–173 (2011). https://doi.org/10.1007/s11071-010-9881-5

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  • DOI: https://doi.org/10.1007/s11071-010-9881-5

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