Delayed feedback and sensitivity analysis for oscillations in Parkinson's disease

Quanbao Ji; Hongjian Meng; Quanbao Ji; Hongjian Meng

doi:10.3934/era.2026050

Electronic Research Archive

2026, Volume 34, Issue 2: 1079-1094. doi: 10.3934/era.2026050

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Delayed feedback and sensitivity analysis for oscillations in Parkinson's disease

Quanbao Ji ^1,2,
Hongjian Meng ^{1
,
,}

1.
School of Mathematical Sciences, Guangxi Minzu University, Nanning 530006, China
2.
Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, Nanning 530006, China

Received: 09 December 2025 Revised: 24 January 2026 Accepted: 28 January 2026 Published: 30 January 2026

Pathological oscillations within the basal ganglia (BG) are a hallmark of Parkinson's disease (PD), a prevalent neurodegenerative disorder. This study aims to evaluate the effects of delayed feedback on oscillatory activity in a PD model and to analyze the roles of synaptic weights and transmission delays within the feedback circuit. Results indicate that the model induced by delayed feedback increase leads to the transition in firing activity, progressing from regular gamma activity to pathological beta oscillation states, and may even transition into complex states. This transition is further modulated by synaptic weights, which could not only change the model from a steady state to an oscillating state, but also cause a high-saturation steady state when sufficiently large. Furthermore, sensitivity analysis based on polynomial chaos expansions enables the quantification of the contributions of transmission delays and synaptic weights to the oscillation frequency variance of the model. The analysis shows that with increasing delayed feedback, the transmission delay parameter that contributes the most to model oscillation shifts from transmission delay within the BG circuit to that of the cerebral cortex circuit. Synaptic weights between the external globus pallidus and the subthalamic nucleus have a significant impact on oscillatory activity when delayed feedback is large. Our findings provide new directions for further clinical treatment of PD.
- Parkinson's disease,
- delayed feedback,
- oscillations,
- polynomial chaos expansion,
- sensitivity
Citation: Quanbao Ji, Hongjian Meng. Delayed feedback and sensitivity analysis for oscillations in Parkinson's disease[J]. Electronic Research Archive, 2026, 34(2): 1079-1094. doi: 10.3934/era.2026050

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Abstract

Pathological oscillations within the basal ganglia (BG) are a hallmark of Parkinson's disease (PD), a prevalent neurodegenerative disorder. This study aims to evaluate the effects of delayed feedback on oscillatory activity in a PD model and to analyze the roles of synaptic weights and transmission delays within the feedback circuit. Results indicate that the model induced by delayed feedback increase leads to the transition in firing activity, progressing from regular gamma activity to pathological beta oscillation states, and may even transition into complex states. This transition is further modulated by synaptic weights, which could not only change the model from a steady state to an oscillating state, but also cause a high-saturation steady state when sufficiently large. Furthermore, sensitivity analysis based on polynomial chaos expansions enables the quantification of the contributions of transmission delays and synaptic weights to the oscillation frequency variance of the model. The analysis shows that with increasing delayed feedback, the transmission delay parameter that contributes the most to model oscillation shifts from transmission delay within the BG circuit to that of the cerebral cortex circuit. Synaptic weights between the external globus pallidus and the subthalamic nucleus have a significant impact on oscillatory activity when delayed feedback is large. Our findings provide new directions for further clinical treatment of PD.

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