Bifurcation and uncertainty quantification of abnormal β oscillations in Parkinson’s model with GPe heterogeneity

Experimental studies have found that the basal ganglia neurons (mainly the subthalamic nucleus (STN) and the globus pallidus pars externa (GPe)) and the cortical neurons produce synchronous abnormal β oscillations with frequencies ranging from 13~30 Hz in Parkinson’s patients. Moreover, it was found that the GPe exists function heterogeneity in the disease, and it can be classified into two types: type-A and type-Ⅰ. In order to investigate the effect of synaptic transmission delay on the occurrence of abnormal β oscillations, a mean field model which includes STN, A-type GPe, Ⅰ-type GPe, as well as excitatory and inhibitory cortical neuron populations was established. The regulatory effect of synaptic transmission delay on abnormal β oscillations was studied with Hopf bifurcation theory. Considering the difficulty in accurately measuring synaptic weights in this model, we further conducted uncertainty quantification on the parameters with polynomial chaos expansion and Sobol’ sensitivity analysis. It was revealed that the excitatory connection weight from cortex to the STN and the inhibitory connection weight from type-A GPe to the STN are the main factors regulating the oscillation frequency in the model. Meanwhile, within the synaptic weights that indirectly regulate STN activity, changes in cortical connection weights also have a significant impact on the oscillation frequency of the model.