Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

One-dimensional Fokker-Planck reduced dynamics of decision making models in computational neuroscience

Pages: 523 – 540



José Antonio Carrillo (Department of Mathematics, Imperial College London, United Kingdom)

Stéphane Cordier (Université d’Orléans, Orléans, France)

Simona Mancini (Université d’Orléans, Orléans, France)


We study a Fokker-Planck equation modeling the firing rates of two interacting populations of neurons. This model arises in computational neuroscience when considering, for example, bistable visual perception problems and is based on a stochastic Wilson-Cowan system of differential equations. In a previous work [J.A. Carrillo, S. Cordier, and S. Mancini, J. Math. Biol., 63, 801–830, 2011], the slow-fast behavior of the solution of the two dimensional Fokker-Planck equation has been highlighted. Our aim is to demonstrate that the complexity of the model can be drastically reduced using this slow-fast structure. In fact, we can derive a one-dimensional Fokker- Planck equation that describes the evolution of the solution along the so-called slow manifold. This permits to have a direct efficient determination of the equilibrium state and its effective potential, and thus to investigate its dependencies with respect to various parameters of the model. It also allows to obtain information about the time escaping behavior. The results obtained for the reduced 1D equation are validated with those of the original 2D equation both for equilibrium and transient behavior.


computational neuroscience, slow-fast reduction, Fokker-Planck equation

2010 Mathematics Subject Classification

34E13, 35Q84, 35Q91, 82C22, 91E45

Published 7 December 2012