Emergence of pathological beta oscillation and its uncertainty quantification in a time-delayed feedback Parkinsonian model

It has been experimentally found that transmission delays play an important role in the emergence of excessively synchronized beta oscillations (13–30 Hz) related to the onset of Parkinson’s symptoms. In order to clarify the dynamical mechanism underlying beta oscillations, we generalize a conventional resonance model based on the subthalamic nucleus (STN)-external segment of globus pallidus (GPe)-cortex circuit to a feedback Parkinsonian model by incorporating the transmission delay within the basal ganglia (STN-GPe circuit) and the transmission delay from the STN to the cortex. By combining the theory of center manifolds and normal forms with numerical simulations, it is revealed how the pathological beta oscillations occur as the transmission delays increase beyond critical values. Furthermore, by regarding the transmission delays and the connection weights as random variables, variance-based sensitivity analysis is performed based on polynomial chaos expansion. It is identified that the transmission delays and connection strength in the long STN-cortex circuit are more significant than those in the STN-GPe circuit for beta oscillations. Our results also suggest that the severity of the Parkinsonian symptoms could be alleviated if a clinical therapy, such as deep brain stimulation, can enlarge the transmission delays in the STN-GPe and STN-cortex circuits simultaneously.