High-Order Accuracy Computation of Coupling Functions for Strongly Coupled Oscillators

TL;DR

This paper introduces a comprehensive framework for deriving high-order accurate phase reduction equations for strongly coupled oscillators, extending beyond weak coupling assumptions and applicable to various coupling types.

Contribution

It extends existing phase reduction theory to higher-order accuracy for arbitrary coupling, enabling analysis of nonlinear oscillators in regimes beyond weak coupling.

Findings

Accurately predicts bifurcations in complex Ginzburg-Landau models

Correctly predicts phase differences in conductance-based neuron models

Outperforms weak coupling theories in strong coupling regimes

Abstract

We develop a general framework for identifying phase reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from [Wilson and Ermentrout, Phys. Rev. Lett 123, 164101 (2019)] and yields coupling functions that are valid to higher-order accuracy in the coupling strength for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions can be used to understand the behavior of potentially high-dimensional, nonlinear oscillators in terms of their phase differences. The proposed formulation accurately replicates nonlinear bifurcations that emerge as the coupling strength increases and is valid in regimes well beyond those that can be considered using classic weak coupling assumptions. We demonstrate the performance of our approach through two examples. First, we use diffusively coupled complex Ginzburg-Landau (CGL) model and demonstrate that our theory accurately predicts bifurcations far beyond the range of existing coupling theory. Second, we use a realistic conductance-based model of a thalamic neuron and show that our theory correctly predicts asymptotic phase differences for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak coupling theories can not.