Journal of Mathematical Biology

, Volume 63, Issue 5, pp 801–830 | Cite as

A decision-making Fokker–Planck model in computational neuroscience

  • José Antonio Carrillo
  • Stéphane Cordier
  • Simona Mancini
Article

Abstract

In computational neuroscience, decision-making may be explained analyzing models based on the evolution of the average firing rates of two interacting neuron populations, e.g., in bistable visual perception problems. These models typically lead to a multi-stable scenario for the concerned dynamical systems. Nevertheless, noise is an important feature of the model taking into account both the finite-size effects and the decision’s robustness. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker–Planck partial differential equation. In particular, in the Fokker–Planck setting, we analytically discuss the asymptotic behavior for large times towards a unique probability distribution, and we propose a numerical scheme capturing this convergence. These simulations are used to validate deterministic moment methods recently applied to the stochastic differential system. Further, proving the existence, positivity and uniqueness of the probability density solution for the stationary equation, as well as for the time evolving problem, we show that this stabilization does happen. Finally, we discuss the convergence of the solution for large times to the stationary state. Our approach leads to a more detailed analytical and numerical study of decision-making models applied in computational neuroscience.

Keywords

Computational neuroscience Fokker–Planck equation General relative entropy 

Mathematics Subject Classification (2000)

35Q84 35Q91 82C21 92D25 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • José Antonio Carrillo
    • 1
    • 2
  • Stéphane Cordier
    • 3
  • Simona Mancini
    • 3
  1. 1.ICREA (Institució Catalana de Recerca i Estudis Avançats)BellaterraSpain
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Fédération Denis Poisson (FR 2964), Department of Mathematics (MAPMO UMR 6628)University of OrléansOrléansFrance

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