Abstract
AbstractMotivated by the development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of a proportional mean-field feedback. 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 show that, in general, no oscillating phase-locked solution can co-exist with any non-zero feedback gain. In the second part we propose a new characterization of phase-locking between Kuramoto oscillators. In particular we derive a fixed point equation for the Kuramoto system under mean-field feedback and we show how, generically, the “standard” (with zero feedback gain) Kuramoto fixed point equation is locally invertible in terms of the implicit function theorem.
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