Bifurcation analysis of a Parkinson's disease model with two time delays

In this paper, a cortex-basal ganglia model about Parkinson's disease with two time delays is studied, and the critical conditions for Hopf bifurcation are derived. The results show that time delays can change the state of basal ganglia. The basal ganglia is stable when the delays are small. However, when the time delay is greater than the corresponding bifurcation critical point, different types of oscillations occur in the basal ganglia. The larger the time delays, the more active the neuronal clusters in the basal ganglia. Furthermore, the bidirectional Hopf bifurcation is found by studying the connection weights between different neural nuclei. Finally, the influence of connection weight and time delay which are related to the internal segment of the globus pallidus on its oscillation is discussed. Research shows that reducing the connection weight and the corresponding time delay in excitatory neuronal clusters, or increasing the connection weight and decreasing the corresponding time delay in inhibitory neuronal clusters, can improve the oscillation of Parkinson's disease.