An Isochron-Free Framework for Phase Reduction and Coupling Inference
Phase description provides a compact and powerful framework for analyzing synchronization dynamics in weakly coupled limit-cycle oscillators. While its classical formulation relies on the asymptotic phase defined by isochrons, reconstructing isochrons from observed trajectories is often challenging for complex models and real-world systems. Here we develop an isochron-free framework based on a readily computable generalized phase, such as the polar angle computed from observed trajectories. We theoretically show that, under near-uniform rotation of the generalized phase and sufficiently stable amplitude dynamics, a one-period stroboscopic description yields a closed circle map. The interaction term of the resulting circle map coincides, to leading order, with the phase coupling function obtained from the conventional phase reduction. Based on this circle map, we propose a method to infer coupling from oscillatory time series. The method is validated using synthetic data from van der Pol oscillators. Our framework broadens the applicability of phase reduction and provides a theoretically grounded method for coupling inference from oscillatory data.