Parkinson's disease is one of the most common neurodegenerative diseases, yet despite its pervasiveness the underlying neural mechanisms of its onset and progression are still the subject of debate. One of the most important biomarkers of Parkinson 's disease is the occurrence of excessive oscillations in the beta frequency range 10 - 30Hz (dependent on species). These oscillations are found to be elevated in the basal ganglia, particularly within the subthalamic nucleus (STN) and globus pallidus external (GPe) in particular. Furthermore, the power of excessive beta oscillations is correlated with the symptoms of bradykinesia (slowness of movement) and muscle rigidity. A theoretical study by Holgado et al. (2010) has shown that the loop formed between the STN and GP can oscillate in the beta frequency range. In addition they identified the conditions under which this circuit could generate beta oscillations. This Thesis contributes to the field by first extending this 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, the analysis explained how changes in cortical and striatal input to the STN-GP network influenced oscillations generated by the circuit. Despite the success of the model at reproducing a significant amount of experimental data it did not explain a number of additional experimental results: 1) The coherence of synchronous oscillations between STN and Cortex, and 2) evidence that lesion or DBS outside the STN-GP nuclei could also affect the power of the synchronous oscillations. To address this data, the model was extended to include the dependence of cortical input to STN on the feedback provided by the STN itself. This feedback was via the output nuclei (globus pallidus internal) , thalamus and so called hyperdirect pathway. It was found from an analysis of this model that feedback can enhance the STN-GP's propensity to oscillate. In addition, this model of beta oscillation generation was found to reproduce experimental data by Tachibana et al. (2011), such as the effects of blocking connections on oscillations, in a way that is distinct and differentiable from other models of beta Dscillation generation.