Phase-based spatial ordinal patterns for characterizing oscillatory dynamics

Abstract

The emergence of organized spatiotemporal patterns is ubiquitous in oscillatory systems, from neural populations to engineered networks. Identifying these patterns and tracking how they evolve over time remains challenging, particularly when systems exhibit transient dynamics. Here, we introduce a framework based on spatial ordinal patterns to characterize the spatiotemporal dynamics of oscillatory systems. Our approach acts directly on the phase rather than the amplitude, with additional patterns introduced to account for near-equal phases. This symbolic representation encodes local spatial ordering relations, capturing both phase gradients and synchronized clusters within a single framework. From this construction, we define a spatial permutation entropy that quantifies the diversity of spatiotemporal patterns at each point in time, enabling the detection of transient dynamics and regime transitions as they occur. We show that this approach distinguishes phase-locked states with identical levels of global synchronization but distinct spatial organization and also characterizes partially synchronized states. We demonstrate the method on synthetic oscillator networks across multiple spatiotemporal regimes, and on resting-state EEG recordings from human volunteers, where it distinguishes different conditions within individual volunteers.