Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network

The advance of Parkinson's disease is associated with the existence of abnormal oscillations within the basal ganglia with frequencies in the beta band (13-30 Hz). While the origin of these oscillations remains unknown, there is some evidence suggesting that oscillations observed in the basal ganglia arise due to interactions of two nuclei: the subthalamic nucleus (STN) and the globus pallidus pars externa (GPe). To investigate this hypothesis, we develop a computational model of the STN-GPe network based upon anatomical and electrophysiological studies. Significantly, our study shows that for certain parameter regimes, the model intrinsically oscillates in the beta range. Through an analytical study of the model, we identify a simple set of necessary conditions on model parameters that guarantees the existence of beta oscillations. These conditions for generation of oscillations are described by a set of simple inequalities and can be summarized as follows: (1) The excitatory connections from STN to GPe and the inhibitory connections from GPe to STN need to be sufficiently strong. (2) The time required by neurons to react to their inputs needs to be short relative to synaptic transmission delays. (3) The excitatory input from the cortex to STN needs to be high relative to the inhibition from striatum to GPe. We confirmed the validity of these conditions via numerical simulation. These conditions describe changes in parameters that are consistent with those expected as a result of the development of Parkinson's disease, and predict manipulations that could inhibit the pathological oscillations.