×

A decision-making Fokker-Planck model in computational neuroscience. (English) Zbl 1232.92033

Summary: 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.

MSC:

92C20 Neural biology
92-08 Computational methods for problems pertaining to biology
35Q84 Fokker-Planck equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann H (1976) Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev 18(4): 620–709 · Zbl 0345.47044 · doi:10.1137/1018114
[2] 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 732–734 · Zbl 0967.35057
[3] 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: 43–100 · Zbl 0982.35113 · doi:10.1081/PDE-100002246
[4] Arnold A, Carlen E, Ju Q (2008) Large-time behavior of non-symmetric Fokker-Planck type equations. Commun Stoch Anal 2(1): 153–175 · Zbl 1331.82042
[5] Arnold A, Carrillo JA, Manzini C (2010) Refined long-time asymptotics for some polymeric fluid flow models. Commun Math Sci 8: 763–782 · Zbl 1213.35094 · doi:10.4310/CMS.2010.v8.n3.a8
[6] Attneave F (1971) Multistability in perception. Sci Am 225: 63–71 · doi:10.1038/scientificamerican1271-62
[7] Berglund N, Gentz B (2005) Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach. In: Probability and its applications. Springer, New York · Zbl 1098.37049
[8] 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): 700–765 · doi:10.1037/0033-295X.113.4.700
[9] Brody C, Romo R, Kepecs A (2003) Basic mechanisms for graded persistent activity: discrete attractors, continuous attractors, and dynamic representations. Curr Opin Neurobiol 13: 204–211 · doi:10.1016/S0959-4388(03)00050-3
[10] Brown E, Holmes P (2001) Modelling a simple choice task: stochastic dynamics of mutually inihibitory neural groups. Stoch Dyn 1(2): 159–191 · Zbl 1060.92019 · doi:10.1142/S0219493701000102
[11] Deco G, Martí D (2007) Deterministic analysis of stochastic bifurcations in multi-stable neurodynamical systems. Biol Cybern 96(5): 487–496 · Zbl 1122.92008 · doi:10.1007/s00422-007-0144-6
[12] Deco G, Scarano L, Soto-Faraco S (2007) Weber’s law in decision making: integrating behavioral data in humans with a neurophysiological model. J Neuroscience 27(42): 11192–11200 · doi:10.1523/JNEUROSCI.1072-07.2007
[13] Evans LC (1998) Partial differential equations. AMS · Zbl 0898.35001
[14] Gardiner CW (1985) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer-Verlag, New York
[15] Gold JI, Shalden MN (2001) Neural computations that underline decisions about sensory stimuli. Trends Cogn Sci 5(1): 10–16 · doi:10.1016/S1364-6613(00)01567-9
[16] Gold JI, Shalden MN (2007) The neural basis of decision-making. Annu Rev Neurosci 30: 535–574 · doi:10.1146/annurev.neuro.29.051605.113038
[17] La Camera G, Rauch A, Luescher H, Senn W, Fusi S (2004) Minimal models of adapted neuronal response to in vivo-like input currents. Neural Comput 16(10): 2101–2124 · Zbl 1055.92011 · doi:10.1162/0899766041732468
[18] Michel P, Mischler S, Perthame B (2004) General entropy equations for structured population models and scattering. C R Acad Sci Paris Ser I 338(9): 697–702 · Zbl 1049.35070 · doi:10.1016/j.crma.2004.03.006
[19] Michel P, Mischler S, Perthame B (2005) General relative entropy inequality: an illustration on growth models. J Math Pures Appl 84(9): 1235–1260 · Zbl 1085.35042
[20] Moreno-Bote R, Rinzel J, Rubin N (2007) Noise-induced alternations in an attractor network model of perceptual bistability. J Neurophysiol 98: 1125–1139 · doi:10.1152/jn.00116.2007
[21] Renart A, Brunel N, Wang X (2003) Computational neuroscience: a comprehensive approach. Chapman and Hall, Boca Raton
[22] Rodriguez R, Tuckwell HC (1996) Statistical properties of stochastic nonlinear dynamical models of single neurons and neural networks. Phys Rev E 54: 5585–5590 · doi:10.1103/PhysRevE.54.5585
[23] Romo R, Salinas E (2003) Flutter discrimination: neural codes, perception, memory and decision making. Nat Rev Neurosci 4: 203–218 · doi:10.1038/nrn1058
[24] Usher M, Mc Clelland JL (2001) The time course of perceptual choice: the leaky, competing accumulator model. Phys Rev 108(3): 550–592
[25] Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12(1): 1–24 · doi:10.1016/S0006-3495(72)86068-5
[26] Wolka J (1987) Partial differential equations. Cambridge University Press, Cambridge
[27] Zhang J, Bogacz R, Holmes P (2009) A comparison of bounded diffusion models for choice in time controlled tasks. J Math Psychol 53: 231–241 · Zbl 1176.60075 · doi:10.1016/j.jmp.2009.03.001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.