Oscillatory spiking neurons have been shown to play an important role in dynamical diseases of the nervous system, particularly Parkinson's disease This dissertation seeks to understand the population-level response dynamics of such neurons, and proposes several event-based feedback control strategies to control their spike timing and degree of synchronization. These control schemes are a significant departure from the existing open-loop electrical deep brain stimulation protocols.

In the first part of this dissertation, we consider population-level dynamics of oscillatory neurons. We study the ensemble response to independent spike trains with interspike intervals drawn from a Poisson distribution, which has been shown to be a reasonable representation of background spiking activity in the brain. The concept of partial phase synchronization, a quantitative tool with which to characterize the state of a population, is presented and shown to provide useful information that is distinct from measurements of spiking synchrony. Then, we investigate parametric resonance to sinusoidal stimulus, and how nonlinearity and coupling lead to a wide variety of stable and unstable solutions, which are categorized based on symmetry considerations and solution types.

In the second part of the dissertation, we consider several event-based feedback control schemes for spike timing control. The first approach uses biologically-inspired impulsive and quasi-impulsive stimulus protocols to drive a neural oscillator to spike in phase with a reference oscillator. We show how these control schemes can be used to desynchronize populations of neurons that, in absence of control, are driven to spike in synchrony by a pacemaker. The second approach uses a minimum-time-to-reach optimal control scheme, based on the Hamilton-Jacobi-Bellman framework, to drive a neural oscillator's state to a point at which its asymptotic phase is extremely sensitive to background noise, which in practice provides effective phase randomization. An extension of this control scheme is presented which prevents pathological synchronous spiking in a network of all-to-all coupled neurons. Finally, we propose a set of algorithms for stabilizing the interspike time interval between a pair of oscillatory neurons that is tailored to experimental implementation.