  The collective behavior of biological oscillators has been recognized as an
important problem for several decades, but its control has come into limelight
only recently. Much of the focus for control has been on desynchronization of
an oscillator population, motivated by the pathological neural synchrony
present in essential and parkinsonian tremor. Other applications, such as the
beating of the heart and insulin secretion, require synchronization, and
recently there has been interest in forming clusters within an oscillator
population as well. In this article, we use a formulation that allows us to
devise control frameworks to achieve all of these distinct collective behaviors
observed in biological oscillators. This is based on the Fourier decomposition
of the partial differential equation governing the evolution of the phase
distribution of a population of identical, uncoupled oscillators. Our first two
control algorithms are Lyapunov-based, which work by decreasing a positive
definite Lyapunov function towards zero. Our third control is an optimal
control algorithm, which minimizes the control energy consumption while
achieving the desired collective behavior of an oscillator population.
Motivated by pathological neural synchrony, we apply our control to
desynchronize an initially synchronized neural population. Given the proposed
importance of enhancing spike time dependent plasticity to stabilize neural
clusters and counteract pathological neural synchronization, we formulate the
phase difference distribution in terms of the phase distribution, and prove
some of its fundamental properties, and in turn apply our control to transform
the neural phase distribution to form clusters. Finally, motivated by
eliminating cardiac alternans, we apply our control to phase shift a
synchronous cardiac pacemaker cell population.
