The perfect integrator driven by Poisson input and its approximation in the diffusion limit

TL;DR

This paper analyzes the perfect integrator driven by Poisson input, deriving its equilibrium and response characteristics, and compares these to diffusion approximation results, highlighting differences near the threshold affecting response behavior.

Contribution

It provides a detailed comparison between the exact Poisson-driven integrator and its diffusion approximation, revealing key differences in equilibrium density and response properties.

Findings

Probability density near threshold differs between models

Diffusion approximation alters response properties

Exact analysis improves understanding of integrator behavior

Abstract

In this note we consider the perfect integrator driven by Poisson process input. We derive its equilibrium and response properties and contrast them to the approximations obtained by applying the diffusion approximation. In particular, the probability density in the vicinity of the threshold differs, which leads to altered response properties of the system in equilibrium.