Desynchronization and inhibition of Kuramoto oscillators by scalar mean-field feedback

Abstract

Motivated by neuroscience applications, and in particular by the deep brain stimulation treatment for Parkinson’s disease, we have recently derived a simplified model of an interconnected neuronal population under the effect of its mean-field proportional feedback. In this paper, we rely on that model to propose conditions under which proportional mean-field feedback achieves either oscillation inhibition or desynchronization. More precisely, we show that for small natural frequencies, this scalar control signal induces an inhibition of the collective oscillation. For the closed-loop system, this situation corresponds to a fixed point which is shown to be almost globally asymptotically stable in the fictitious case of zero natural frequencies and all-to-all coupling and feedback. In the case of an odd number of oscillators, this property is shown to be robust to small natural frequencies and heterogencities in both the coupling and feedback topology. On the contrary, for large natural frequencies, we show that scalar proportional mean-field feedback is able to induce desynchronization. After having recalled a formal definition for desynchronization, we show how it can be induced in a network of originally synchronized oscillators.