A decision-making Fokker–Planck model in computational neuroscience

Abstract

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.

On the a posteriori error analysis for linear Fokker–Planck models in convection-dominated diffusion problems

“…This paper derives and discusses a posteriori error estimates of the functional type for convection-diffusion problems, which are motivated by the decision-making Fokker-Planck model problem discussed by Carrillo et al [15] appearing in computational neuroscience. The mathematical model discussed in [15] is associated with stochastic dynamical systems, modelling the evolution of the decision-making (average firing rates) of two interacting neurone populations, see also [13] and [17]. In [15], the existence of a unique solution for stationary as well as evolutionary linear Fokker-Planck equations is discussed and proved under the assumption that the (regular enough) flux or drift is incoming in the bounded domain.…”

On the Exponential Convergence Rate for a Non-Gradient Fokker-Planck Equation in Computational Neuroscience

“…The exponential convergence of the solutions to evolution problems towards the steady states has been largely addressed for many years, see for example [1,5,7,9,10,11,12]. Techniques and proofs are usually based on the nature of each equation being the general entropy method developed in [12] for linear problems and used in computational neuroscience in [6,8] a powerful method for investigation of this question in the problem under consideration here.…”

“…We have assumed unit diffusion constant for simplicity without loss of generality. Under these hypothesis, existence, uniqueness and positivity of the solution u = u(t, x) of the evolution problem (1.1), and of the stable state (stationary solution) u ∞ (x) of the associated problem were proved in [8,Theorem 2], as well as the mass density conservation,…”

Asymptotic Behavior of the Solution of the Distribution Diffusion Equation for FENE Dumbbell Polymer Model

“…For some recent results on a related probability diffusion equation for rigid dumbbell polymers see Ciupercȃ and Palade [8] and on a Fokker-Planck model in computational neuroscience see Carillo, Cordier, Mancini [5].…”