Toward oscillations inhibition by mean-field feedback in Kuramoto oscillators

This note shows that the oscillations in a network of all-to-all coupled Kuramoto oscillators can be inhibited by a scalar output feedback. More precisely, by injecting an input proportional to the oscillators mean-field, a set of isolated equilibria is shown to be almost globally attractive when natural frequencies are zero. The normal hyperbolicity of all relevant equilibria let us conjecture that this property persists in the presence of natural frequencies that are sufficiently small with respect to the coupling gain. This work constitutes a first step in the direction of testing the possible neuronal inhibition arising in deep brain stimulation treatment for neurological diseases.