Motivated by issues related to treating certain neurological diseases such as Parkinson's disease by a method called electrical deep brain stimulation, we consider applying different control methods to both mathematical models of neurons and in vitro neurons. Patients suffering from Parkinson's disease experience involuntary tremors that typically affect the distal portion of their upper limbs. It has been hypothesized that these tremors are associated with simultaneous spiking of a cluster of neurons in the thalamus and basal ganglia regions of the brain. In a healthy situation, the periodic ring of neurons is not synchronized, but they can engage in a pathological synchrony and all re at the same time which results in release of strong action potentials that trigger the downstream muscles with periodic shocks, manifested as tremors.

This dissertation investigates the control of different neuronal systems using methods of optimal control. The neuronal systems considered range from simple one-dimensional phase models to multi-dimensional conductance-based models, both on a single neuron level and on a population level. The optimal control methods considered produce event-based, continuous-time, typically bounded input stimuli that can optimally achieve the desired control objective. The optimality criteria considered are minimum energy and minimum time. The control objectives of interest are the interspike interval for single neurons and desynchrony for populations of neurons.

There are three parts to this dissertation. In the first part, (Chapters 2 and 3), event-based time optimal and energy optimal control is presented to achieve desired interspike intervals for phase models of single neurons. In the second part, (Chapters 4 and 5), the problem of desynchronizing a network of pathologically synchronized coupled neurons is considered. In the third part, (Chapter 6), the theoretical method of Chapter 3 is adopted and the applicability of this method is shown in practice by testing the controller on in vitro pyramidal neurons in the CA1 region of rat hippocampus.