Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback

Motivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phase-locked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). This general result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections.