Fredholm theory for the mean first-passage time of integrate-and-fire oscillators with colored noise input

TL;DR

This paper introduces a Fredholm theory-based method to analyze how different noise timescales affect the mean first-passage time in nonlinear oscillators, providing a unified framework for slow and fast noise regimes.

Contribution

It develops an exact integral equation for the mean event rate of leaky integrate-and-fire oscillators with correlated noise, extending analysis to arbitrary noise timescales.

Findings

Derived an exact integral equation for mean event rate

Unified framework for slow and fast noise scaling behaviors

Showed non-reciprocal scaling in different noise limits

Abstract

We develop a method to investigate the effect of noise timescales on the first-passage time of nonlinear oscillators. Using Fredholm theory, we derive an exact integral equation for the mean event rate of a leaky-integrate-and-fire oscillator that receives constant input and temporally correlated noise. Furthermore, we show that Fredholm theory provides a unified framework to determine system scaling behavior for small and large noise timescales. In this framework, the leading order and higher-order asymptotic corrections for slow and fast noise are naturally emerging. We show the scaling behavior in the both limits are not reciprocal. We discuss further how this approach can be extended to study the first-passage time in a general class of nonlinear oscillators driven by colored noise at arbitrary timescales.