n:m Phase-Locking of Coupled Oscillators with Nonlinearities in Coupling Strength and Heterogeneity

We introduce a scalar reduction method for forced or coupled systems with nonlinearities in both heterogeneity and coupling strength. Heterogeneity is formulated as a relatively weak but nonlinear alteration of the vector field(s). The method can be used to determine the existence and stability of $n{:}m$ phase-locked states in a variety of forced or coupled biological oscillator models, including the nonradial isochron clock, a thalamic neural oscillator, and the Van der Pol oscillator. The proposed scalar reduction successfully captures the emergence and disappearance of phase-locked states as a function of nonlinear coupling strength and nonlinear heterogeneity. We find that even small amounts of heterogeneity can significantly alter phase-locked states in ways that cannot be captured by assuming identical oscillators. The proposed method enables a reduction and analysis of high-dimensional systems of coupled oscillators in more biologically realistic settings.