A decision-making Fokker-Planck model in computational neuroscience

José Antonio Carrillo (1) , Stéphane Cordier (2) , Simona Mancini (2)

1 UAB - Departament de Matemàtiques [Barcelona]
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

Minimal models for the explanation of decision-making in computational neuroscience are based on the analysis of the evolution for the average firing rates of two interacting neuron populations. While these models typically lead to multi-stable scenario for the basic derived dynamical systems, noise is an important feature of the model taking into account finite-size effects and robustness of the decisions. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker-Planck partial differential equation. In particular, we discuss the existence, positivity and uniqueness for the solution of the stationary equation, as well as for the time evolving problem. Moreover, we prove convergence of the solution to the the stationary state representing the probability distribution of finding the neuron families in each of the decision states characterized by their average firing rates. Finally, we propose a numerical scheme allowing for simulations performed on the Fokker-Planck equation which are in agreement with those obtained recently by a moment method applied to the stochastic differential system. Our approach leads to a more detailed analytical and numerical study of this decision-making model in computational neuroscience.

Domains: Analysis of PDEs [math.AP]

José Antonio Carrillo, Stéphane Cordier, Simona Mancini. A decision-making Fokker-Planck model in computational neuroscience. 2010. ⟨hal-00452994v1⟩