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Big homoclinic orbit bifurcation underlying postinhibitory rebound spike and a novel threshold curve of a neuron
1.  School of Mathematics and Science, Henan Institute of Science and Technology, Xinxiang 453003, China 
2.  School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China 
Postinhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarizationactive caution ($ I_h $) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without $ I_h $ current. The threshold curve for a spike evoked from a monostable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddlenode bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension2 bifurcation point related to the BHom orbit are acquired in the twoparameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.
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Y. Zhang, Y. Xu, Z. Yao and J. Ma, A feasible neuron for estimating the magnetic field effect, Nonlinear Dyn., 102 (2020), 18491867. Google Scholar 
[33] 
F. Zhang, W. Zhang, P. Meng and J. Z. Su, Bifurcation analysis of bursting solutions of two HindmarshRose neurons with joint electrical and synaptic coupling, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 637651. doi: 10.3934/dcdsb.2011.16.637. Google Scholar 
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show all references
References:
[1] 
G. A. Ascoli, S. Gasparini, V. Medinilla and M. Migliore, Local control of postinhibitory rebound spiking in CA1 pyramidal neuron dendrites, J. Neurosci., 30 (2010), 64346442. doi: 10.1523/JNEUROSCI.406609.2010. Google Scholar 
[2] 
A. Basu, C. Petre and P. E. Hasler, Dynamics and bifurcations in a silicon neuron, IEEE Trans. Biomed. Circuits Syst., 4 (2010), 320328. doi: 10.1109/TBCAS.2010.2051224. Google Scholar 
[3] 
S. Bertrand and J. Cazalets, Postinhibitory rebound during locomotorlike activity in neonatal rat motoneurons in vitro, J. Neurophysiol., 79 (1998), 342351. Google Scholar 
[4] 
P. Channell, G. Cymbalyuk and A. Shilnikov, Origin of bursting through homoclinic spike adding in a neuron model, Phys. Rev. Lett., 98 (2007), 134101. doi: 10.1103/PhysRevLett.98.134101. Google Scholar 
[5] 
L. Duan, Q. Cao, Z. Wang and J. Su, Dynamics of neurons in the preBötzinger complex under magnetic flow effect, Nonlinear Dyn., 94 (2018), 19611971. doi: 10.1007/s1107101844687. Google Scholar 
[6] 
L. Duan, Z. Yang, S. Liu and and D. Gong, Bursting and twoparameter bifurcation in the Chay neuronal model, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 445456. doi: 10.3934/dcdsb.2011.16.445. Google Scholar 
[7] 
B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems. A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195. Google Scholar 
[8] 
D. Fan, Y. Zheng, Z. Yang and Q. Wang, Improving control effects of absence seizures using singlepulse alternately resetting stimulation (SARS) of corticothalamic circuit, Appl. Math. MechEngl., 41 (2020), 12871302. Google Scholar 
[9] 
R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys., 17 (1955), 257278. doi: 10.1007/BF02477753. Google Scholar 
[10] 
L. Glass, Synchronization and rhythmic processes in physiology, Nature, 410 (2001), 277284. doi: 10.1038/35065745. Google Scholar 
[11] 
J.M. Goaillard, A. L. Taylor, S. R. Pulver and E. Marder, Slow and persistent postinhibitory rebound acts as an intrinsic shortterm memory mechanism, J. Neurosci., 30 (2010), 46874692. doi: 10.1523/JNEUROSCI.299809.2010. Google Scholar 
[12] 
L. Guan, B. Jia and H. Gu, A novel threshold across which negative stimulation evokes action potential near a saddlenode bifurcation in a neuronal model with $I_h$ current, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950198, 26 pp. doi: 10.1142/S0218127419501980. Google Scholar 
[13] 
J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHughNagumo equation: The singularlimit, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 851872. doi: 10.3934/dcdss.2009.2.851. Google Scholar 
[14] 
F. Han, B. Zhen, Y. Du, Y. Zheng and M. Wiercigroch, Global Hopf bifurcation analysis of a sixdimensional FitzHughNagumo neural network with delay by a synchronized scheme, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 457474. doi: 10.3934/dcdsb.2011.16.457. Google Scholar 
[15] 
A. L. Hodgkin, The local electric changes associated with repetitive action in a nonmedullated axon, J. Physiol., 107 (1948), 165181. doi: 10.1113/jphysiol.1948.sp004260. Google Scholar 
[16] 
E. M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 11711266. doi: 10.1142/S0218127400000840. Google Scholar 
[17] 
E. M. Izhikevich, Which model to use for cortical spiking neurons?, IEEE T. Neural Network, 15 (2004), 10631070. Google Scholar 
[18]  E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT press, Cambridge, 2007. Google Scholar 
[19] 
P. Jiang, X. Yang and Z. Sun, Dynamics analysis of the hippocampal neuronal model subjected to cholinergic action related with alzheimer's disease, Cogn. Neurodyn., 14 (2020), 483500. doi: 10.1007/s11571020095866. Google Scholar 
[20] 
T. Malashchenko, A. Shilnikov and G. Cymbalyuk, Bistability of bursting and silence regimes in a model of a leech heart interneuron, Phys. Rev. E, 84 (2011), 041910. doi: 10.1103/PhysRevE.84.041910. Google Scholar 
[21] 
Y. Mao, Dynamic transitions of the FitzhughNagumo equations on a finite domain, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 39353947. doi: 10.3934/dcdsb.2018118. Google Scholar 
[22] 
J. Mitry, M. McCarthy, N. Kopell and M. Wechselberger, Excitable neurons, firing threshold manifolds and canards, J. Math. Neurosci., 3 (2013), Art. 12, 32 pp. doi: 10.1186/21908567312. Google Scholar 
[23] 
J. Rinzel and G. Ermentrout, Analysis of neural excitability and oscillations, in Methods in Neuronal Modeling (eds. C. Koch and I. Segev), MIT Press, Cambridge, (1998), Chap. 7,251–291. Google Scholar 
[24] 
M. Rush and J. Rinzel, The potassium Acurrent, low firing rates and rebound excitation in HodgkinHuxley models, Bull. Math. Biol., 57 (1995), 899929. Google Scholar 
[25] 
Z. Song and J. Xu, Codimensiontwo bursting analysis in the delayed neural system with external stimulations, Nonlinear Dyn., 67 (2012), 309328. doi: 10.1007/s1107101199794. Google Scholar 
[26] 
V. A. Straub and P. Benjamin, Extrinsic modulation and motor pattern generation in a feeding network: A cellular study, J. Neurosci., 21 (2001), 17671778. doi: 10.1523/JNEUROSCI.210501767.2001. Google Scholar 
[27] 
X. Sun and G. Li, Synchronization transitions induced by partial time delay in an excitatoryinhibitory coupled neuronal network, Nonlinear Dyn., 89 (2017), 25092520. doi: 10.1007/s1107101736004. Google Scholar 
[28] 
A. Tonnelier, Threshold curve for the excitability of bidimensional spiking neurons, Phys. Rev. E, 90(2014), 022701. doi: 10.1103/PhysRevE.90.022701. Google Scholar 
[29] 
F. Q. Wu and H. Gu, Bifurcations of negative responses to positive feedback current mediated by memristor in neuron model with bursting patterns, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2030009, 22 pp. doi: 10.1142/S0218127420300098. Google Scholar 
[30] 
F. Wu, H. Gu and Y. Li, Inhibitory electromagnetic induction current induced enhancement instead of reduction of neural bursting activities, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104924, 15 pp. doi: 10.1016/j.cnsns.2019.104924. Google Scholar 
[31] 
X. Zhang, H. Gu and L. Guan, Stochastic dynamics of conduction failure of action potential along nerve fiber with hopf bifurcation, Sci. China Technol. Sci., 62 (2019), 15021511. doi: 10.1007/s1143101895154. Google Scholar 
[32] 
Y. Zhang, Y. Xu, Z. Yao and J. Ma, A feasible neuron for estimating the magnetic field effect, Nonlinear Dyn., 102 (2020), 18491867. Google Scholar 
[33] 
F. Zhang, W. Zhang, P. Meng and J. Z. Su, Bifurcation analysis of bursting solutions of two HindmarshRose neurons with joint electrical and synaptic coupling, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 637651. doi: 10.3934/dcdsb.2011.16.637. Google Scholar 
[34] 
Z. Zhao and H. Gu, Transitions between classes of neuronal excitability and bifurcations induced by autapse, Sci. Rep., 7 (2017), 6760. doi: 10.1038/s41598017070519. Google Scholar 
[35] 
Z. Zhao, L. Li and H. Gu, Dynamical mechanism of hyperpolarizationactivated nonspecific cation current induced resonance and spiketiming precision in a neuronal model, Front. Cell. Neurosci., 12 (2018), 62. doi: 10.3389/fncel.2018.00062. Google Scholar 
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