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

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.