A key pathology in the development of Parkinson’s disease is the occurrence of persistent beta oscillations, which are correlated with difficulty in movement initiation. We investigated the network model composed of the subthalamic nucleus (STN) and globus pallidus (GP) developed by A. Nevado Holgado et al. [(2010) Journal of Neuroscience, 30, 12340–12352], who identified the conditions under which this circuit could generate beta oscillations. Our work extended their analysis by deriving improved analytic stability conditions for realistic values of the synaptic transmission delay between STN and GP neurons. The improved conditions were significantly closer to the results of simulations for the range of synaptic transmission delays measured experimentally. Furthermore, our analysis explained how changes in cortical and striatal input to the STN–GP network influenced oscillations generated by the circuit. As we have identified when a system of mutually connected populations of excitatory and inhibitory neurons can generate oscillations, our results may also find applications in the study of neural oscillations produced by assemblies of excitatory and inhibitory neurons in other brain regions.