Hopf bifurcation mechanism analysis in an improved cortex-basal ganglia network with distributed delays: An application to Parkinson’s disease

Abstract

In this paper, we introduce general distributed delays (in pathways between different neurons) and self-feedback distributed delays (in self-feedback pathways) into an improved cortex-basal ganglia (BG) network to study possible Hopf bifurcation mechanisms for Parkinson’s oscillations. Hopf bifurcation critical conditions for discrete delays are obtained, which compare well to numerical simulations. General distributed delays inhibit oscillations in BG, but self-feedback distributed delays promote oscillations. The different effects of strong and weak kernels on Hopf bifurcations mainly depend on self-feedback distributed delays, strong kernels are more conducive to the generation of oscillations when self-feedback distributed delays exceed a certain value. The excitatory feedback and inhibitory feedback from cortex to BG have different effects on oscillation. We define four different states (absolutely stable, conditional stable, conditional oscillation and absolutely oscillation) in this model, which can explain different mechanisms of oscillation origin. An increase in distributed delays can induce the transition between supercritical (SPH) and subcritical (SBH) Hopf bifurcations, which in turn may cause the transition of oscillations in different frequency bands, such as beta oscillations and alpha oscillations. Near the SBH, frequencies increase with an increase in discrete delays; and the trend is opposite at the SPH. In general, the amplitude of oscillation increases with the increase of firing activation level. Discrete delays can improve the firing activation level of the BG and suppress oscillations in its adjacent circuits. With the increase of distributed delays, oscillatory regions evolve regularly and the roles of general distributed delays and self-feedback distributed delays are contrary. Different types of delay have different effects on oscillation frequency. Discrete delay and self-feedback distributed delay have similar effects on oscillation frequency, but are different from general distributed delay. Whether the frequency will change with an increase in delay depends on coupling weights.