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Emergence of Oscillatory Behaviors for Excitable Systems with Noise and Mean-Field Interaction: A Slow-Fast Dynamics Approach

  • Eric Luçon
  • Christophe PoquetEmail author
Article
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Abstract

We consider the long-time dynamics of a general class of nonlinear Fokker–Planck equations, describing the large population behavior of mean-field interacting units. The main motivation of this work concerns the case where the individual dynamics is excitable, i.e. when each isolated dynamics rests in a stable state, whereas a sufficiently strong perturbation induces a large excursion in the phase space. We address the question of the emergence of oscillatory behaviors induced by noise and interaction in such systems. We tackle this problem by considering this model as a slow-fast system (the mean value of the process giving the slow dynamics) in the regime of small individual dynamics and by proving the existence of a positively stable invariant manifold, whose slow dynamics is at first order the dynamics of a single individual averaged with a Gaussian kernel. We consider applications of this result to Stuart–Landau and FitzHugh–Nagumo oscillators, and to the Cucker–Smale alignment model.

Notes

Acknowledgements

We would like to thank Giambattista Giacomin and Charles-Edouard Bréhier for very fruitful discussions. Funding was provided by Agence Nationale de la Recherche (Grant No. ANR-17-CE40-0030). We would like to thank the referees for their careful reading and useful comments that helped to improve consequently the presentation of the paper.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.MAP5 (CNRS UMR 8145)Université de ParisParisFrance
  2. 2.Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208Institut Camille JordanVilleurbanneFrance

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