Hopf bifurcation control of the ML neuron model with Hc bifurcation type

Qinghua Zhu; Meng Li; Fang Han; Qinghua Zhu; Meng Li; Fang Han

doi:10.3934/era.2022032

Electronic Research Archive

2022, Volume 30, Issue 2: 615-632. doi: 10.3934/era.2022032

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Hopf bifurcation control of the ML neuron model with Hc bifurcation type

College of Information Science and Technology, Donghua University, Shanghai 201620, China

Received: 20 September 2021 Revised: 28 October 2021 Accepted: 02 November 2021 Published: 15 February 2022

It is shown that many neurological diseases are caused by the changes of firing patterns induced by bifurcations. Therefore, the bifurcation control may provide a potential therapeutic method of these neurodegenerative diseases. In this paper, we investigate the Hopf bifurcation control of the Morris-Lecar (ML) model with Homoclinic (Hc) bifurcation type by introducing a dynamic state-feedback control. The results indicate that the linear term can change the ML model from Hc bifurcation type to SNIC bifurcation type without changing the firing patterns. The cooperation of linear and cubic term can transform the ML model from the Hc bifurcation type to the Hopf bifurcation type, resulting in the transformation of firing patterns from type I to type II. Besides, we utilize the Poincare Birkhoff (PB) normal form method to derive the analytical expression of the bifurcation stability index for the controlled ML model with Hc bifurcation type, and the results show that the cubic term can regulate the criticality of the Hopf bifurcation. Numerical simulation results are consistent with the theoretical analysis.
- ML model,
- hopf bifurcation control,
- dynamic state-feedback control,
- bifurcation stability index,
- neurological dynamical diseases
Citation: Qinghua Zhu, Meng Li, Fang Han. Hopf bifurcation control of the ML neuron model with Hc bifurcation type[J]. Electronic Research Archive, 2022, 30(2): 615-632. doi: 10.3934/era.2022032

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Abstract

It is shown that many neurological diseases are caused by the changes of firing patterns induced by bifurcations. Therefore, the bifurcation control may provide a potential therapeutic method of these neurodegenerative diseases. In this paper, we investigate the Hopf bifurcation control of the Morris-Lecar (ML) model with Homoclinic (Hc) bifurcation type by introducing a dynamic state-feedback control. The results indicate that the linear term can change the ML model from Hc bifurcation type to SNIC bifurcation type without changing the firing patterns. The cooperation of linear and cubic term can transform the ML model from the Hc bifurcation type to the Hopf bifurcation type, resulting in the transformation of firing patterns from type I to type II. Besides, we utilize the Poincare Birkhoff (PB) normal form method to derive the analytical expression of the bifurcation stability index for the controlled ML model with Hc bifurcation type, and the results show that the cubic term can regulate the criticality of the Hopf bifurcation. Numerical simulation results are consistent with the theoretical analysis.

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