Abstract
A piece-wise linear planar neuron model, namely, two-dimensional McKean model with periodic drive is investigated in this paper. Periodical bursting phenomenon can be observed in the numerical simulations. By assuming the formal solutions associated with different intervals of this non-autonomous system and introducing the generalized Jacobian matrix at the non-smooth boundaries, the bifurcation mechanism for the bursting solution induced by the slowly varying periodic drive is presented. It is shown that, the discontinuous Hopf bifurcation occurring at the non-smooth boundaries, i.e., the bifurcation taking place at the thresholds of the stimulation, leads the alternation between the rest state and spiking state. That is, different oscillation modes of this non-autonomous system convert periodically due to the non-smoothness of the vector field and the slow variation of the periodic drive as well.
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References
Chay TR, Fan YS, Lee YS (1995) Bursting, spiking, chaos, fractals, and university in biological rhythms. Int J Bifurc Chaos 5:595–635
Coombes S (2008) Neuronal networks with gap junctions: a study of piece-wise linear planar neuron model, SIAM. J Appl Dyn Syst 7:1101–1129
Dong J, Zhang GJ, Xie Y, Yao H, Wang J (2014) Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn Neurodyn 8:167–175
Du Y, Lu QS, Wang RB (2010) Using interspike intervals to quantify noise effects on spike trains in temperature encoding neurons. Cogn Neurodyn 4:199–206
Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1182:445–466
Harris-Warrick RM, Flamm RE (1987) Multiple mechanisms of bursting in a conditional bursting neuron. J Neurosci 7:2113–2128
Hodgkin AL, Huxley AF (1952a) A quantitative description of membrane current and its application to conduction and excitation in nerve tissue. J Physiol (Lond) 116:449–472
Hodgkin AL, Huxley AF (1952) Propagation of electrical signals along giant nerve fibres. Proc R Soc Ser B Biol Sci 140:177–183
Izhikevich EM (2000) Neural excitability, spiking and bursting. Int J Bifurc Chaos 10:1171–1266
Izhikevich EM (2003) Simple model of spiking neurons. IEEE Trans Neural Netw 14:1569–1572
Jaume L, Manuel O, Enrique P (2013) On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry. Nonlinear Anal Real World Appl 14:2002–2012
Johnson SW, Seutin V, North RA (1992) Burst firing in dopamine neurons induced by N-methyl-d-aspartate: role of electrogenic sodium pump. Science 258:665–667
Leine RI (2006) Bifurcations of equilibria in non-smooth continuous systems. Phys D 223:121–137
Leine RI, van Campen DH (2006) Bifurcation phenomena in non-smooth dynamical systems. Eur J Mech A/Solids 25:595–616
Marszalek W, Trzaska Z (2010) Mixed-mode oscillations in a modified Chua’s circuit. Circuits Syst Signal Process 29:1075–1087
McKean HP (1970) Nagumo’s equation. Adv Math 4:209–223
Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070
Qin ZY, Lu QS (2009) Map analysis for non-smooth bifurcations. J Vib Shock 28:79–81
Rinzel J (1985) Bursting oscillation in an excitable membrane model. In: Sleeman BD, Jarvis RJ (eds) Ordinary and partial differential equations. Springer, Berlin, pp 304–316
Rinzel J, Lee YS (1987) Dissection of a model for neuronal parabolic bursting. J Math Biol 25:653–675
Sherman A, Rinzel J (1992) Rhythmogenic effects of weak electrotonic coupling in neuronal model. Proc Natl Acad Sci USA 89:2471–2474
Simpsona DJW, Meiss JD (2012) Aspects of bifurcation theory for piecewise-smooth, continuous systems. Phys D 241:1861–1868
Smolen P, Terman D, Rinzel J (1993) Properties of a bursting model with two slow inhibitory variables. SIAM J Appl Math 53:861–892
Tiesinga PHE (2002) Precision and reliability of periodically and quasiperiodically driven integrate-and fire neurons. Phys Rev E 65:041913
Tonnelier A (2003) The McKean’s caricature of the Fitzhugh–Nagumo model I. The space-clamped system. SIAM J Appl Math 63:459–484
Xu X, Wang RB (2014) Neurodynamics of up and down transitions in a single neuron. Cogn Neurodyn 8:509–515
Yamashita Y, Torikai H (2014) Theoretical analysis for efficient design of a piecewise constant spiking neuron model. IEEE Trans Circuits Syst II Express Br 61:54–58
Yang ZQ, Lu QS (2008) Different types of bursting in Chay neuronal model. Sci China Ser G-Phys Mech Astron 51:687–698
Zhang F, Lubbe A, Lu QS, Su JZ (2014) On bursting solutions near chaotic regimes in a neuron model. Discrete Contin Dyn Syst Ser S 7:1363–1383
Acknowledgments
Supported by the National Natural Science Foundation of China (Grant Nos. 11302086, 11374130, 11474134, and 11302136), Natural Science Foundation of Jiangsu province (Grant No. BK20141296), Post-doctoral Science Fund of China (Grant No. 2014M561574), Post-doctoral Science Fund of Jiangsu province (Grant No. 1302094B).
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Ji, Y., Zhang, X., Liang, M. et al. Dynamical analysis of periodic bursting in piece-wise linear planar neuron model. Cogn Neurodyn 9, 573–579 (2015). https://doi.org/10.1007/s11571-015-9347-z
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DOI: https://doi.org/10.1007/s11571-015-9347-z