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.
Computational neuroscience Fokker–Planck equation General relative entropy
Mathematics Subject Classification (2000)
35Q84 35Q91 82C21 92D25
This is a preview of subscription content, log in to check access.
Arnold A, Carlen E (2000) A generalized Bakry-Emery condition for non-symmetric diffusions. In: Fiedler B, Groger K, Sprekels J (eds) EQUADIFF 99—Proceedings of the international conference on differential equations, Berlin 1999. World Scientific, Singapore, pp 732–734Google Scholar
Arnold A, Markowich P, Toscani G, Unterreiter A (2001) On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Plack type equations. Commun PDE 26: 43–100MathSciNetzbMATHCrossRefGoogle Scholar
Arnold A, Carlen E, Ju Q (2008) Large-time behavior of non-symmetric Fokker-Planck type equations. Commun Stoch Anal 2(1): 153–175MathSciNetGoogle Scholar
Arnold A, Carrillo JA, Manzini C (2010) Refined long-time asymptotics for some polymeric fluid flow models. Commun Math Sci 8: 763–782MathSciNetzbMATHGoogle Scholar
Berglund N, Gentz B (2005) Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach. In: Probability and its applications. Springer, New YorkGoogle Scholar
Bogacz R, Brown E, Mohelis J, Holmes P, Choen JD (2006) The phyiscs of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks. Psychol Rev 113(4): 700–765CrossRefGoogle Scholar
Brody C, Romo R, Kepecs A (2003) Basic mechanisms for graded persistent activity: discrete attractors, continuous attractors, and dynamic representations. Curr Opin Neurobiol 13: 204–211CrossRefGoogle Scholar