Phase distribution control of a population of oscillators

Highlights: • Developed a framework to control a population of oscillators. • Used Lyapunov-based and optimal control for the desired collective behavior. • Control can synchronize, desynchronize, cluster, and phase shift the population. • Pseudo-spectral method and modified Newton iteration for numerical implementation. -- Abstract: 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. For the systems considered in this paper, the control algorithms can be applied to achieve any desired traveling-wave phase distribution, as long as the combination of the initial phase distribution and phase response curve is non-degenerate. To demonstrate the effectiveness of our control for each of these applications, we show that a population of 100 phase oscillators with the applied control mimics the desired phase distribution.