n:m Phase-Locking of Heterogeneous and Strongly Coupled Oscillators

Researchers have long struggled to model the complex phase-locking behavior of biological oscillators—from neurons firing in the brain to heart cells beating in unison—because real systems involve strong coupling and inherent heterogeneity between oscillators. Existing methods either assumed identical oscillators or were limited to weak coupling, failing to capture the nonlinear dynamics that govern phenomena like 4:1 or 3:2 locking patterns. In a new paper, Youngmin Park presents a scalar reduction technique that handles both nonlinear coupling strength and nonlinear heterogeneity simultaneously, reducing high-dimensional oscillator networks to a single scalar equation.

The method's power lies in its ability to predict exactly when n:m phase-locked states emerge and disappear as parameters vary. Park validated the approach on three distinct biological models: the nonradial isochron clock (relevant to circadian rhythms), a thalamic neural oscillator (important for sleep-wake cycles), and the classic Van der Pol oscillator (used in cardiac modeling). The analysis, spanning 58 pages with 28 figures, reveals that even tiny amounts of heterogeneity—effects that would be ignored by standard identical-oscillator assumptions—can dramatically shift locking patterns. This work opens the door to studying more biologically realistic, high-dimensional networks without sacrificing analytical tractability.