Dissertation titled Neuronal Clustering and Phase Relationships in Basal Ganglia Circuits. 

Pathological beta oscillations in Parkinson's disease arise from complex basal ganglia circuit interactions. This dissertation develops the time-to-spike (TTS) function framework to investigate clustering dynamics and phase relationships in these networks.

We first examined how slow adaptation currents drive cluster formation in excitatory-inhibitory networks. Using a novel adaptation distribution map with cutting, contraction, and gluing operations, we demonstrated that the steepness of the time-to-spike function determines transitions between synchronized and clustered states. This analysis revealed bifurcation mechanisms underlying oscillatory dynamics in subthalamic nucleus circuits.

Extending the TTS framework to phase relationships, we discovered that different integrate-and-fire neuron models exhibit distinct phase preferences under periodic inhibition. Quadratic integrate-and-fire neurons establish stable in-phase relationships, while leaky integrate-and-fire neurons prefer anti-phase coupling. These single-neuron properties shape population-level dynamics.

Through rate models and spiking simulations, we demonstrated that phase relationship critically influences beta oscillation amplitude. In-phase subthalamopallidal relationships enhance beta power, while anti-phase coupling suppresses it. This provides a mechanistic explanation for contradictory findings in recent Parkinson's disease studies.

Our results suggest that seemingly opposing conclusions about subthalamic nucleus function in beta generation stem from neuronal model choices rather than fundamental circuit disagreements. Different integrate-and-fire implementations produce opposite phase relationships, determining whether the subthalamopallidal loop enhances or suppresses oscillations.
This work establishes that single neuron dynamics can fundamentally alter network-level pathological activity, emphasizing the importance of model selection in computational neuroscience. The TTS framework provides a powerful tool for bridging single-neuron dynamics to population-level behaviors, with broad potential applications for studying oscillatory phenomena in excitatory-inhibitory neural circuits.

Advisors: Dr. Jonathan Rubin (primary), Dr. Bard Ermentrout