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

Abstract

Abstract. To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13,12], 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 [13] to further simplify the kinetic theory.

Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models

“…Our analytical and numerical results contribute to support the NNLIF system as an appropriate model to describe well known neurophysiological phenomena, as for example synchronization/asynchronization of the network, since the blow-up in finite time might depict a synchronization of a part of the network, while the presence of a unique asymptotically stable stationary solution represents an asynchronization of the network. In addition, the abundance in the number of steady states, in terms of the connectivity parameter values, that can be observed for this simplified model, probably will help us to characterize situations of multi-stability for more complete NNLIF models and also other models including conductance variables as in [6]. In [7] it was shown that if a refractory period is included in the model, there are situations of multi-stability, with two stable and one unstable steady state.…”

“…Other PDE models describing the spiking neurons, which are related to the Fokker-Planck system considered in the present work, are based on Fokker-Planck equations including conductance variables, [26,6,25] (and references therein) and on the time elapsed models [22,23,24]. In [15] the authors study the connection between the Fokker-Planck and the elapsed models.…”

Analysis of Nonlinear Noisy Integrate&Fire Neuron Models: blow-up and steady states

“…This method was also used in [20] for a computational neuroscience model with variable voltage and conductance. In order to use this method, the first step is to rewrite the Fokker-Planck equation (1.4) in terms of the Maxwellian $Mfalse(vfalse)=e^{-\frac{{(v-bN)}^2}{2afalse(Nfalse)}}$ as follows, …”

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

“…The formulation presented here is equivalent and more suitable for mathematical treatment. Other more complicated microscopic models including the conductance and leading to kinetic-like Fokker-Planck equations have been studied recently, see [4] and the references therein. Finally, the nonlinear Fokker-Planck equation can be rewritten as…”