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

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