Exploring dynamic behavior and bifurcations in a Filippov neuronal system with a double-tangency singularity

Yi Yang; Rongfeng Li; Xiangguang Dai; Haiqing Li; Changcheng Xiang; Yi Yang; Rongfeng Li; Xiangguang Dai; Haiqing Li; Changcheng Xiang

doi:10.3934/math.2024924

AIMS Mathematics

2024, Volume 9, Issue 7: 18984-19014. doi: 10.3934/math.2024924

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Research article

Exploring dynamic behavior and bifurcations in a Filippov neuronal system with a double-tangency singularity

1.
College of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing, 404100, China
2.
Key Laboratory of Intelligent Information Processing and Control of Chongqing Municipal Institutions of Higher education, Chongqing Three Gorges University, Chongqing, 404100, China
3.
School of Intelligent Technology, Chongqing Preschool Education College, Chongqing, 404047, China
4.
School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China

Received: 26 February 2024 Revised: 28 May 2024 Accepted: 29 May 2024 Published: 06 June 2024

We investigated the phenomenon of pseudo-Hopf bifurcation in a Filippov Hindmarsh-Rose neuronal system with threshold switching, and the existence of crossing limit cycles was proved by constructing the half-return mapping. Through the threshold control, the firing state of the system could be switched, allowing transitions from a non-periodic state to a periodic state, as well as the evolution from spiking to bursting. Furthermore, through threshold switching, the system exhibited the coexistence of multiple attractors, the system could be in multiple stable states, or have multiple stable sets that could attract system trajectories. This meant that neuronal system could exhibits diverse dynamical behaviors than being limited to a single stable state. The phenomenon of period-doubling bifurcation also indicated that the system will eventually enter a chaotic state. By extending the analysis to nonlinear neuronal systems, this study contributes to a deeper understanding of complex dynamics and provides valuable insights for designing state switching in the application of neural dynamics.
- Hindmarsh-Rose,
- pseudo-Hopf bifurcation,
- crossing limit cycle,
- Filippov neuronal system,
- double-tangency singularity
Citation: Yi Yang, Rongfeng Li, Xiangguang Dai, Haiqing Li, Changcheng Xiang. Exploring dynamic behavior and bifurcations in a Filippov neuronal system with a double-tangency singularity[J]. AIMS Mathematics, 2024, 9(7): 18984-19014. doi: 10.3934/math.2024924

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Abstract

We investigated the phenomenon of pseudo-Hopf bifurcation in a Filippov Hindmarsh-Rose neuronal system with threshold switching, and the existence of crossing limit cycles was proved by constructing the half-return mapping. Through the threshold control, the firing state of the system could be switched, allowing transitions from a non-periodic state to a periodic state, as well as the evolution from spiking to bursting. Furthermore, through threshold switching, the system exhibited the coexistence of multiple attractors, the system could be in multiple stable states, or have multiple stable sets that could attract system trajectories. This meant that neuronal system could exhibits diverse dynamical behaviors than being limited to a single stable state. The phenomenon of period-doubling bifurcation also indicated that the system will eventually enter a chaotic state. By extending the analysis to nonlinear neuronal systems, this study contributes to a deeper understanding of complex dynamics and provides valuable insights for designing state switching in the application of neural dynamics.

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