Dynamics of a subthalamic nucleus-globus palidus network with three delays

This paper analyses a model of the subthalamic nucleus (STN)-globus palidus (GP) network with three different transmission delays. A time-shift transformation reduces the model to a system with two time delays, for which the existence of a unique steady state is established. Conditions for stability of the steady state are derived in terms of system parameters and time delays. Numerical stability analysis is performed using traceDDE to investigate different dynamical regimes in the STN-GP model, and to obtain critical stability boundaries separating stable (healthy) and oscillatory (Parkinsonian-like) neural firing. Direct numerical simulations of the fully nonlinear system are performed to confirm analytical findings, and to illustrate different dynamical behaviours of the system.