A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

作者：Caceres Maria J^*; Carrillo Jose A; Tao Louis

来源：Journal of Computational Physics, 2011, 230(4): 1084-1099.

DOI：10.1016/j.jcp.2010.10.027

• Neuronal network
• Kinetic equations
• Fokker-Planck equation
• Direct simulation Monte Carlo
• Deterministic simulations
• WENO methods
• Chang-Cooper method

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [15,14], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [15] to further simplify the kinetic theory.

• 出版日期2011-2-20
• 单位上海生物信息技术研究中心