A Fokker-Planck formalism for diffusion with finite increments and absorbing boundaries

TL;DR

This paper develops a Fokker-Planck formalism for systems driven by finite increments and absorbing boundaries, revealing new boundary conditions and response behaviors in threshold models like neurons.

Contribution

It introduces a boundary condition for stationary densities with finite-grained Poisson noise, altering the understanding of threshold unit responses.

Findings

Response is instantaneous, not low-pass.

Response is highly non-linear and asymmetric.

Mechanism applies to pulse-coupled threshold systems.

Abstract

Gaussian white noise is frequently used to model fluctuations in physical systems. In Fokker-Planck theory, this leads to a vanishing probability density near the absorbing boundary of threshold models. Here we derive the boundary condition for the stationary density of a first-order stochastic differential equation for additive finite-grained Poisson noise and show that the response properties of threshold units are qualitatively altered. Applied to the integrate-and-fire neuron model, the response turns out to be instantaneous rather than exhibiting low-pass characteristics, highly non-linear, and asymmetric for excitation and inhibition. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled systems of threshold units.