Collective mode reductions for populations of coupled noisy oscillators

TL;DR

This paper compares three low-dimensional models for the collective dynamics of large populations of noisy coupled oscillators, finding that the two-cumulant approximation generally provides the most accurate results across different scenarios.

Contribution

It introduces a new two-cumulant truncation method that improves accuracy over existing models for noisy oscillator populations.

Findings

Two-cumulant approximation outperforms Ott-Antonsen and Gaussian ansatz.

Gaussian ansatz is more accurate only in high-synchrony states.

The proposed closure makes the two-cumulant approach a first-order correction to Ott-Antonsen.

Abstract

We analyze accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise, and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling, are considered as examples to test validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations.