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Multiple time delay induced Hopf bifurcation of a cortex - basal ganglia model for Parkinson’s Disease

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Abstract

Exploring the origin of beta - band oscillation in the cortex - basal ganglia model plays an important role in understanding the mechanism of Parkinson’s disease. In this paper, we investigate the effect of three synaptic transmission time delays in the subthalamic nucleus(STN) - the globus pallidus external segment(GPe) loop, the excitatory neurons in the cortex(EXN) - the inhibitory neurons in the cortex(INN) loop and EXN - STN loop on critical conditions of occurrence of beta - band oscillation through Hopf bifurcation theory including linear stability analysis, center manifold theorem and normal form analysis. Our results reveal that suitable transmission time delay can induce beta - band oscillation through Hopf bifurcation, and the critical condition under which Hopf bifurcation occurs is more sensitive to the transmission time delay in EXN - STN loop \(T_3\), where if \(T_3 > 0.00185\), beta - band oscillation always occurs for any transmission time delay in STN - GPe, EXN - INN loops. Our theoretical analyses provide some ideas for the future research of Parkinson’s disease.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 12062017 and 12262025), Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grants 2021ZD01), and Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region of China (No. NMGIRT2208). The authors acknowledge the reviewers for their valuable reviews and kind suggestions.

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  1. Qiaohu Zhang and Yuanhong Bi have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Qiaohu Zhang & Quansheng Liu

  2. School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot, 010070, China

    Yuanhong Bi

  3. Inner Mongolia Key Laboratory for Economic Data Analysis and Mining, Hohhot, 010070, China

    Yuanhong Bi

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  1. Qiaohu Zhang
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  2. Quansheng Liu
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Correspondence to Quansheng Liu.

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Zhang, Q., Liu, Q. & Bi, Y. Multiple time delay induced Hopf bifurcation of a cortex - basal ganglia model for Parkinson’s Disease. Cogn Neurodyn (2024). https://doi.org/10.1007/s11571-024-10071-7

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