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

Desynchronization; Kuramoto oscillators; Mean-field feedback; Oscillations inhibition; Collective oscillations; Deep brain stimulation; Feedback topology; Fixed points; Formal definition; Globally asymptotically stable; Mean-field; Neuronal populations; Parkinson's disease; Proportional feedback; Scalar control; Simplified models; Synchronized oscillator; Control and Systems Engineering; Signal Processing; Control and Optimization; Applied Mathematics

Abstract :

[en] 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. © 2012 Springer-Verlag London Limited.

Disciplines :

Engineering, computing & technology: Multidisciplinary, general & others

Author, co-author :

Franci, Alessio ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Brain-Inspired Computing ; L2S, University Paris Sud 11, Supélec, Gif-sur-Yvette 91192, France

Chaillet, Antoine; L2S, University Paris Sud 11, Supélec, Gif-sur-Yvette 91192, France

Panteley, Elena; CNRS, L2S, Gif-sur-Yvette 91192, France

Lamnabhi-Lagarrigue, Françoise; CNRS, L2S, EECI, Gif-sur-Yvette 91192, France

Language :

English

Title :

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

Publication date :

April 2012

Journal title :

Mathematics of Control, Signals and Systems

ISSN :

0932-4194

eISSN :

1435-568X

Publisher :

Springer Science and Business Media LLC

Volume :

Issue :

1-2

Pages :

169 - 217

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://link.springer.com/content/pdf/10.1007/s00498-011-0072-9.pdf

Funding text :

The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n257462 HYCON2 Network of excellence, and by the French CNRS through the PEPS project TREMBATIC.

Available on ORBi :

since 26 May 2023

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