\bfitN -Body Oscillator Interactions of Higher-Order Coupling Functions\ast

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 and yields coupling functions for N \geq 2 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 a diffusely coupled complex Ginzburg-Landau (CGL) model with N = 3 and show that the loss of stability in its splay state occurs through a Hopf bifurcation viewing the nonweak diffusive coupling as the bifurcation parameter. 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 nonweak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak and recent nonweak coupling theories cannot.