Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Xianjun Wang; Huaguang Gu; Bo Lu; Xianjun Wang; Huaguang Gu; Bo Lu

doi:10.3934/era.2021023

Electronic Research Archive

2021, Volume 29, Issue 5: 2987-3015. doi: 10.3934/era.2021023

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Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Xianjun Wang ^{1
,},
Huaguang Gu ^{2
,
,},
Bo Lu ^{1
,}

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

Received: 01 December 2020 Revised: 01 February 2021 Published: 15 March 2021
Primary: 37G15, 92B20; Secondary: 37C29

Post-inhibitory 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 hyperpolarization-active 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 mono-stable 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 saddle-node 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 codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter 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.
- Bifurcation,
- big homoclinic orbit,
- threshold,
- post-inhibitory rebound spike
Citation: Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron[J]. Electronic Research Archive, 2021, 29(5): 2987-3015. doi: 10.3934/era.2021023

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Abstract

Post-inhibitory 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 hyperpolarization-active 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 mono-stable 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 saddle-node 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 codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter 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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