  In this article, we study optimal control problems of spiking neurons whose
dynamics are described by a phase model. We design minimum-power current
stimuli (controls) that lead to targeted spiking times of neurons, where the
cases with unbounded and bounded control amplitude are considered. We show that
theoretically the spiking period of a neuron, modeled by phase dynamics, can be
arbitrarily altered by a smooth control. However, if the control amplitude is
bounded, the range of possible spiking times is constrained and determined by
the bound, and feasible spiking times are optimally achieved by piecewise
continuous controls. We present analytic expressions of these minimum-power
stimuli for spiking neurons and illustrate the optimal solutions with numerical
simulations.
