On a Kinetic Fitzhugh–Nagumo Model of Neuronal Network

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• 作者：S. Mischler ; C. Quiñinao ; J. Touboul
• 刊名：Communications in Mathematical Physics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：342
• 期：3
• 页码：1001-1042
• 全文大小：4,942 KB
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• 作者单位：S. Mischler (1)
  C. Quiñinao (2) (3)
  J. Touboul (4)
  
  1. Université Paris-Dauphine & IUF CEREMADE, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France
  2. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, CNRS UMR, 7598, 4 place de Jussieu, 75005, Paris, France
  3. Mathematical Neuroscience Team, CIRB - Collège de France, 11 place Marcelin-Berthelot, 75005, Paris, France
  4. Mathematical Neuroscience Team & INRIA Paris-Rocquencourt, Mycenae Team, CIRB - Collège de France, 11 place Marcelin-Berthelot, 75005, Paris, France
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

We investigate existence and uniqueness of solutions of a McKean–Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh–Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we demonstrate uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we show exponential nonlinear stability in the small connectivity regime.