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Nonlinear dynamics is identified to play very important roles in identifying the complex phenomenon, dynamical mechanism, and physiological functions of neural electronic activities. In the present paper, a novel viewpoint that the excitatory stimulus cannot enhance but reduce the number of the spikes within a burst, the novel viewpoint which is different from the traditional viewpoint, is proposed and is explained with the nonlinear dynamics. When the impulse current or the autaptic current with suitable strength is used in the suitable phase within the quiescent state of the bursting pattern of the Rulkov model, a novel firing pattern with reduced number of spikes within a burst is evoked. The earlier the application phase of the current within the quiescent state, the higher the threshold of the current strength to evoke the novel firing pattern is and the less the number of the spikes within a burst of the novel firing pattern. Moreover, such a novel phenomenon can be explained by the intrinsic nonlinear dynamics of the bursting combined with the characteristics of the current. The nonlinear behaviors of the fast subsystem of the Rulkov model are acquired by the fast and slow variable dissection method, respectively. For the fast subsystem, there exist a stable node with lower membrane potential, a stable limit cycle with higher membrane potential, a saddle serving as the border between the stable node and limit cycle, a saddle-node bifurcation, and a homoclinic orbit bifurcation. When external simulation is not received, the bursting pattern of the Rulkov model exhibits behavior alternating between the spikes corresponding to the limit cycle of the fast subsystem and quiescent state of the fast subsystem, which is located within the parameter region between the saddle-node bifurcation point and the homoclinic orbit bifurcation point of the fast subsystem. The spikes begin with the saddle-node bifurcation and end with the homoclinic orbit bifurcation. As the bifurcation parameter turns close to the homoclinic orbit bifurcation, the disturbation or stimulus that can induce the transition from the quiescent state to the spikes becomes strong. Therefore, as the application phase of the current within the quiescent state becomes earlier, the strength threshold of the current that can induce the transition from the quiescent state to the spikes becomes stronger, and the initial phase of the spikes becomes closer to the homoclinic orbit bifurcation, which leads the parameter region of the spikes to become shorter and then leads the number of spikes within a burst to turn less. It is the dynamical mechanism of the decrease of the spike number induced by the excitatory currents. The results enrich the nonlinear phenomenon and dynamical mechanism, present a novel viewpoint for the excitatory effect, and provide a new approach to modulating the neural bursting patterns.
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Keywords:
- neural bursting /
- bifurcation /
- excitatory autapse /
- time delay
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[35] Lisman J E 1997 Trends Neurosci. 20 38
[36] Izhikevich E M, Desai N S, Walcott E C, Hoppensteadt F C 2003 Trends Neurosci. 26 161
[37] Rulkov N F 2002 Phys. Rev. E 65 041922
[38] Rulkov N F 2001 Phys. Rev. Lett. 86 183
[39] Rinzel J 1987 Lecture Notes in Biomathematics (Berlin: Springer-Verlag) p267
[40] Buschle L R, Kurz F T, Kampf T, Wagner W L, Dueer J, Stiller W, Konietzke P, Wünnemann F, Mall M A, Wielpütz M O, Schlemmer H P, Ziener C H 2017 Phys. Rev. E 95 022415
[41] Tsutome H, Yuichi H, Takao O, Masahiro T 2009 Phys. Rev. E 80 051921
[42] Guo D Q 2011 Cogn. Neurodyn. 5 293
[43] Chen F, Xia L, Li C G 2012 Chin. Phys. Lett. 29 070501
[44] Guo D Q, Wang Q Y, Perc M 2012 Phys. Rev. E 85 061905
[45] Wang Q Y, Murks A, Perc M, Lu Q S 2011 Chin. Phys. B 20 040504
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[1] Glass L 2001 Nature 410 277
[2] Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171
[3] Braun H A, Wissing H, Schäfer K, Hirsch M C 1994 Nature 367 270
[4] Braun H A, Schwabedal J, Dewald M, Finke C, Postnova S, Huber M T, Wollweber B, Schneider H, Hirsch M C, Voigt K, Feudel U, Moss F 2011 Chaos 21 047509
[5] Gu H G, Pan B B 2015 Nonlinear Dyn. 81 2107
[6] Jia B, Gu H G 2017 Int. J. Bifurcation Chaos 27 1750113
[7] Wang X J, Rinzel J 1992 Neural Comput. 4 84
[8] Wang X J, Rinzel J 1993 Neuroscience 53 899
[9] van V C, Abbott L F, Bard E G 1994 J. Comput. Neurosci. 1 313
[10] Cobb S R, Buhl E H, Halasy K, Paulsen O, Somogyi P 1995 Nature 378 75
[11] Bose A, Kunec S 2001 Neurocomputing 38 505
[12] Elson R C, Selverston A I, Abarbanel H D I, Rabinovich M 2002 J. Neurophysiol. 88 1166
[13] Belykh I, Shilnikov A 2008 Phys. Rev. Lett. 101 078102
[14] Gu H G, Zhao Z G 2015 PloS One 10 e0138593
[15] Jia B, Wu Y C, He D, Guo B H, Xue L 2018 Nonlinear Dyn. 93 1599
[16] Zhao Z G, Jia B, Gu H G 2016 Nonlinear Dyn. 86 1549
[17] Jia B 2018 Int. J. Bifurcation Chaos 28 1850030
[18] Tamas G, Buhl E H, Somogyi P 1997 J. Neurosci. 17 6352
[19] Bacci A, Huguenard J R, Prince D A 2003 J. Neurosci. 23 859
[20] Bacci A, Huguenard J R, Prince D A 2005 Trends Neurosci. 28 602
[21] Bacci A, Huguenard J R 2006 Neuron 49 119
[22] Saada R, Miller N, Hurwitz I, Susswein A J 2009 Curr. Biol. 19 479
[23] Deleuze C, Pazienti A, Bacci A 2014 Curr. Opin. Neurobiol. 26 64
[24] Straiker A, Dvorakova M, Zimmowitch A, Mackie K 2018 Mol. Pharmacol. 94 743
[25] Qin H X, Ma J, Wang C N, Wu Y 2014 PloS One 9 e100849
[26] Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China Phys. Mech. Astron. 57 1918
[27] Wang H T, Ma J, Chen Y L, Chen Y 2014 Commun. Nonlinear Sci. Numer. Simul. 19 3242
[28] Wang H T, Chen Y 2015 Chin. Phys. B 24 128709
[29] Guo D Q, Chen M M, Perc M, Wu S D, Xia C, Zhang Y S, Xu P, Xia Y, Yao D Z 2016 Europhys. Lett. 114 30001
[30] Guo D Q, Wu S D, Chen M M, Perc M, Zhang Y S, Ma J L, Cui Y, Xu P, Xia Y, Yao D Z 2016 Sci. Rep. 6 14
[31] Ma J, Xu Y, Wang C N, Jin W Y 2016 Physica A 461 586
[32] Yilmaz E, Baysal V, Ozer M, Perc M 2016 Physica A 444 538
[33] Gong Y, Wang B, Xie H 2016 Biosystems 150 132
[34] Yang X, Yu Y, Sun Z 2017 Chaos 27 083117
[35] Lisman J E 1997 Trends Neurosci. 20 38
[36] Izhikevich E M, Desai N S, Walcott E C, Hoppensteadt F C 2003 Trends Neurosci. 26 161
[37] Rulkov N F 2002 Phys. Rev. E 65 041922
[38] Rulkov N F 2001 Phys. Rev. Lett. 86 183
[39] Rinzel J 1987 Lecture Notes in Biomathematics (Berlin: Springer-Verlag) p267
[40] Buschle L R, Kurz F T, Kampf T, Wagner W L, Dueer J, Stiller W, Konietzke P, Wünnemann F, Mall M A, Wielpütz M O, Schlemmer H P, Ziener C H 2017 Phys. Rev. E 95 022415
[41] Tsutome H, Yuichi H, Takao O, Masahiro T 2009 Phys. Rev. E 80 051921
[42] Guo D Q 2011 Cogn. Neurodyn. 5 293
[43] Chen F, Xia L, Li C G 2012 Chin. Phys. Lett. 29 070501
[44] Guo D Q, Wang Q Y, Perc M 2012 Phys. Rev. E 85 061905
[45] Wang Q Y, Murks A, Perc M, Lu Q S 2011 Chin. Phys. B 20 040504
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