Abstract

Collective synchronization in a large population of self-sustained units appears both in natural and engineered systems. Sometimes this effect is in demand, while in some cases, it is undesirable, which calls for control techniques. In this paper, we focus on pulsatile control, with the goal to either increase or decrease the level of synchrony. We quantify this level by the entropy of the phase distribution. Motivated by possible applications in neuroscience, we consider pulses of a realistic shape. Exploiting the noisy Kuramoto–Winfree model, we search for the optimal pulse profile and the optimal stimulation phase. For this purpose, we derive an expression for the change of the phase distribution entropy due to the stimulus. We relate this change to the properties of individual units characterized by generally different natural frequencies and phase response curves and the population’s state. We verify the general result by analyzing a two-frequency population model and demonstrating a good agreement of the theory and numerical simulations.

Synchronization naturally emerges in interacting oscillatory systems, but often, it is desirable to control its degree. A motivating example comes from neuroscience, where periodic pulse stimulation applied to the deep brain structures manipulates pathological rhythms in Parkinson’s disease and other pathologies and reduces the symptoms. Hypothetically, this stimulation suppresses synchrony in a large population of coupled neurons. This hypothesis triggered intensive research on control techniques that led to many feed-forward and closed-loop approaches to suppressing or enhancing synchrony. Potential applications in neuroscience imply additional requirements: the stimulation shall be pulsatile, and the pulses must be charge-balanced. It means that the total current provided by the stimulus shall be zero to avoid the charge accumulation in the live tissue. This paper uses a paradigmatic model of coherent collective activity, namely, the Kuramoto–Winfree model, to analyze the effect of a charge-balanced pulse shape. We link the properties of individual units—their phase response curves—and the current state of the oscillatory ensemble to a collective response of the system to pulse stimulation. In this way, we optimize the stimulus shape and find the proper phase of the collective mode for the onset of stimulation. We support our theoretical findings by numerical simulation and argue that the validity of our results goes beyond the exploited simplistic model.