N-Body Oscillator Interactions of Higher-Order Coupling Functions

TL;DR

This paper presents a novel method to derive phase equations with N-body interactions for coupled oscillators, extending previous theories to arbitrary coupling types and strengths, enabling better analysis of oscillator network stability.

Contribution

The authors develop a general approach to identify phase equations with N-body interactions for arbitrary coupling, beyond weak coupling approximations, applicable to diverse oscillator models.

Findings

Accurately predicts stability loss in oscillator splay states.

Captures asymptotic limit-cycle dynamics in phase differences.

Outperforms weak and recent non-weak coupling theories.

Abstract

We introduce a method to identify phase equations that include $N$-body interactions for general coupled oscillators valid far beyond the weak coupling approximation. This strategy is an extension of the theory from [Park and Wilson, SIADS 20.3 (2021)] and yields coupling functions for $N\geq2$ oscillators for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions enable the study of oscillator networks in terms of phase-locked states, whose stability can be determined using straightforward linear stability arguments. We demonstrate the utility of our approach with two examples. First, we use $N=3$ diffusively coupled complex Ginzburg-Landau (CGL) model and show that the loss of stability in its splay state occurs through a Hopf bifurcation \yp{as a function of non-weak diffusive coupling. Our reduction also captures asymptotic limit-cycle dynamics in the phase differences}. Second, we use $N=3$ realistic conductance-based thalamic neuron models and show that our method correctly predicts a loss in stability of a splay state for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak and recent non-weak coupling theories can not.