Optimal Control of Weakly Forced Nonlinear Oscillators

Optimal control of nonlinear oscillatory systems poses numerous theoretical and computational challenges. Motivated by applications in neuroscience, we develop tools and methods to synthesize optimal controls for nonlinear oscillators described by reduced order dynamical systems. Control of neural oscillations by external stimuli has a broad range of applications, ranging from oscillatory neurocomputers to deep brain stimulation for Parkinson\u27s disease. In this dissertation, we investigate fundamental limits on how neuron spiking behavior can be altered by the use of an external stimulus: control). Pontryagin\u27s maximum principle is employed to derive optimal controls that lead to desired spiking times of a neuron oscillator, which include minimum-power and time-optimal controls. In particular, we consider practical constraints in such optimal control designs including a bound on the control amplitude and the charge-balance constraint. The latter is important in neural stimulations used to avoid from the undesirable effects caused by accumulation of electric charge due to external stimuli. Furthermore, we extend the results in controlling a single neuron and consider a neuron ensemble. We, specifically, derive and synthesize time-optimal controls that elicit simultaneous spikes for two neuron oscillators. Robust computational methods based on homotopy perturbation techniques and pseudospectral approximations are developed and implemented to construct optimal controls for spiking and synchronizing a neuron ensemble, for which analytical solutions are intractable. We finally validate the optimal control strategies derived using the models of phase reduction by applying them to the corresponding original full state-space models. This validation is largely missing in the literature. Moreover, the derived optimal controls have been experimentally applied to control the synchronization of electrochemical oscillators. The methodology developed in this dissertation work is not limited to the control of neural oscillators and can be applied to a broad class of nonlinear oscillatory systems that have smooth dynamics

Control and Synchronization of Neuron Ensembles

Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we analytically derive optimal controls for spiking single- and two-neuron systems, and analyze the applicability of the latter to an ensemble system. Finally, we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations. The methodology developed here is universal to the control of general nonlinear phase oscillators.Comment: 29 pages, 6 figure

Charge-Balanced Minimum-Power Controls for Spiking Neuron Oscillators

In this paper, we study the optimal control of phase models for spiking neuron oscillators. We focus on the design of minimum-power current stimuli that elicit spikes in neurons at desired times. We furthermore take the charge-balanced constraint into account because in practice undesirable side effects may occur due to the accumulation of electric charge resulting from external stimuli. Charge-balanced minimum-power controls are derived for a general phase model using the maximum principle, where the cases with unbounded and bounded control amplitude are examined. The latter is of practical importance since phase models are more accurate for weak forcing. The developed optimal control strategies are then applied to both mathematically ideal and experimentally observed phase models to demonstrate their applicability, including the phase model for the widely studied Hodgkin-Huxley equations.Comment: 24 pages, 12 figure