A finite volume method for stochastic integrate-and-fire models

Abstract

The stochastic integrate and fire neuron is one of the most commonly used stochastic models in neuroscience. Although some cases are analytically tractable, a full analysis typically calls for numerical simulations. We present a fast and accurate finite volume method to approximate the solution of the associated Fokker-Planck equation. The discretization of the boundary conditions offers a particular challenge, as standard operator splitting approaches cannot be applied without modification. We demonstrate the method using stationary and time dependent inputs, and compare them with Monte Carlo simulations. Such simulations are relatively easy to implement, but can suffer from convergence difficulties and long run times. In comparison, our method offers improved accuracy, and decreases computation times by several orders of magnitude. The method can easily be extended to two and three dimensional Fokker-Planck equations.

Finite volume and asymptotic methods for stochastic neuron models with correlated inputs

“…The proof of this proposition is similar to the proof of the corresponding onedimensional result in Marpeau et al (2009), and we therefore omit it here.…”

“…The one dimensional Fokker-Planck equations (6) are discretized by using the onedimensional scheme from (Marpeau et al 2009) (see A.3). The main difference here is the presence of the age variables, s and r .…”

“…The advection equation in (16) is discretized by using the one-dimensional numerical fluxes in Marpeau et al (2009) in each direction. We set where the numerical fluxes are defined by…”

“…Let k be such that s n k + t n ≤ τ W , meaning that the population R n,k V will not exit its refractory state earlier than t n+1 . Equations (6) and (7) are discretized with the onedimensional numerical scheme in Marpeau et al (2009). We use the operator splitting technique…”

“…Previously, we proposed a fast and accurate finite volume method for modeling a general IF neuron driven by a stochastic input (Marpeau et al 2009). This method was significantly faster than Monte Carlo (MC) simulations and we proved several stability properties of the algorithm.…”