Dynamics of the Langevin model subjected to colored noise: Functional-integral method

TL;DR

This paper investigates the dynamics of Langevin models with colored noise using the functional-integral method, comparing results with simulations and other approximations to understand their responses to various inputs.

Contribution

It introduces a combined functional-integral and equations of motion approach for analyzing Langevin models with colored noise, including complex cases with multiple noise sources.

Findings

FIM results agree well with direct simulations.

UCNA differs significantly in dynamical responses from FIM and simulations.

The study clarifies the limitations of approximate methods like UCNA for dynamic responses.

Abstract

We have discussed the dynamics of Langevin model subjected to colored noise, by using the functional-integral method (FIM) combined with equations of motion for mean and variance of the state variable. Two sets of colored noise have been investigated: (a) one additive and one multiplicative colored noise, and (b) one additive and two multiplicative colored noise. The case (b) is examined with the relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate-and-fire model including stochastic excitatory and inhibitory synapses and a noisy input. We have studied the stationary probability distribution and dynamical responses to time-dependent (pulse and sinusoidal) inputs of the linear Langevin model. Model calculations have shown that results of the FIM are in good agreement with those of direct simulations (DSs). A comparison is made among various approximate analytic solutions such as the universal colored noise approximation (UCNA). It has been pointed out that dynamical responses to pulse and sinusoidal inputs calculated by the UCNA are rather different from those of DS and the FIM, although they yield the same stationary distribution.