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Bursting and two-parameter bifurcation in the Chay neuronal model

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  • In this paper, we study and classify the firing patterns in the Chay neuronal model by the fast/slow decomposition and the two-parameter bifurcations analysis. We show that the Chay neuronal model can display complex bursting oscillations, including the "fold/fold" bursting, the "Hopf/Hopf" bursting and the "Hopf/homoclinic" bursting. Furthermore, dynamical properties of different firing activities of a neuron are closely related to the bifurcation structures of the fast subsystem. Our results indicate that the codimension-2 bifurcation points and the related codimension-1 bifurcation curves of the fast-subsystem can provide crucial information to predict the existence and types of bursting with changes of parameters.
    Mathematics Subject Classification: Primary: 34C15; Secondary: 92C20.


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  • [1]

    A. Kepecs, X. Wang and J. Lisman, Bursting neurons singale input slope, J. Neurosci., 22 (2002), 9053-9062.


    A. Kepecs and J. Lisman, Information encoding and computation with spikes and bursters, Network: Comput. Neural. Syst., 14 (2003), 103-9118.


    E. M. Izhikevichet.al. http://www.scholarpedia.org/article/Bursting.


    J. Rinzel, Bursting oscillations in an excitable membrane model, in "Ordinary and Partial Differential Equations, Lecture Notes in Mathematics" (eds. B.D. Sleeman, R.J. Jarvis), Springer, Berlin, 1151 (1985), 304-316.


    R. Bertram, M. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439.


    M. Rush and J. Rinzel, Analysis of Bursting in a thalamic neuron model, Biol. Cybern., 71 (1994), 281-291.doi: 10.1007/BF00239616.


    E. M. Izhikevich, Neural excitability, spiking and bursting, Int. J. of Bifurcation and Chaos, 10 (2000), 1171-1266.doi: 10.1142/S0218127400000840.


    X. J. Wang, Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle, Physica D, 62 (1993), 263-274.doi: 10.1016/0167-2789(93)90286-A.


    P. R. Shorten and D. J. Wall, A Hodgkin-Huxley model exhibiting bursting oscillations, Bull. Math. Biol., 62 (2000), 695-715.doi: 10.1006/bulm.2000.0172.


    M. Perc and M. Marhl, Different types of bursting calcium oscillations in non-excitable cells, Chaos, Solitons and Fractals, 18 (2003), 759-773.doi: 10.1016/S0960-0779(03)00027-4.


    L. Brusch, W. Lorenz, M. Or-Guil, M. Bär and U. Kummer, Fold-Hopf bursting in a model for calcium signal transduction, Z. Phys. Chem., 216 (2002), 487-97.doi: 10.1524/zpch.2002.216.4.487.


    V. N. Belykh, I. V. Belykh, M. Colding-Jørgensen and E. Mosekilde, Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, Eur. Phys. J. E, 3 (2000), 205-219.doi: 10.1007/s101890070012.


    Y. S. Fan and T. R. Chay, Generation of periodic and chaotic bursting in an excitable cell model, Biol. Cybern., 71 (1994), 417-431.doi: 10.1007/BF00198918.


    H. G. Gu, M. H. Yang, L. Li, Z. Q. Liu and W. Ren, Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios, Phys. Lett. A, 319 (2003), 89-96.doi: 10.1016/j.physleta.2003.09.077.


    Z. Q. Yang, Q. S. Lu, H. G. Gu and W. Ren, Gwn-induced bursting, spiking, and random subthreshold impulsing oscillation before Hopf bifurcations in the Chay model, Int. J. Bifurcation Chaos, 14 (2004), 4143-4159.doi: 10.1142/S0218127404011892.


    Z. Q. Yang and Q. S. Lu, Different types of bursting in Chay neuronal model, Sci. China Ser. G-Phys Mech. Astron., 6 (2008), 687-698.


    J. Guckenheimer and J. H. Tien, Bifurcation in the fast dynamics of neurons: implication for bursting, in "The Genesis of Rhythm in the Nervous System. Bursting" (eds. S. Coombes and P. C. Bressloff), World Scientific Publish, (2005), 89-122.doi: 10.1142/9789812703231_0004.


    X. Q. Wu and L. C. Wang, Hopf bifurcation of a class of two coupled relaxation oscillators of the van der pol type with delay, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 503-516.


    D. Liu, S. G. Ruan and D. M. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 151-168.doi: 10.3934/dcdsb.2009.12.151.


    L. X. Duan, Q. S. Lu and Q. Y. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomp, 72 (2008), 341-351.doi: 10.1016/j.neucom.2008.01.019.


    L. X. Duan, Q. S. Lu and D. Z. Cheng, Bursting of Morris-Lecar neuronal model with current-feedback control, Sci. China Ser. E-Tech. Sci., 52 (2009), 771-781.


    T. R. Chay, Chaos in the three-variable model of an excitable cell, Physica D, 16 (1985), 233-242.doi: 10.1016/0167-2789(85)90060-0.


    Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Springer-Verlag, New York, 2004.


    G. B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students," 1st edition, doi: 10.1137/1.9780898718195.

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