ROPE: a novel method for real-time phase estimation of complex biological rhythms

Accurate phase estimation-the process of assigning phase values between 0 and 2π to repetitive or periodic signals-is a cornerstone in the analysis of oscillatory signals across diverse fields, from neuroscience to robotics, where it is fundamental, e.g. to understanding coordination in neural networks, cardiorespiratory coupling and human-robot interaction. However, existing methods are often limited to offline processing and/or constrained to one-dimensional signals. In this article, we introduce ROPE, which, to the best of our knowledge, is the first phase-estimation algorithm capable of (i) handling signals of arbitrary dimension and (ii) operating in real time, with minimal error. ROPE identifies repetitions within the signal to segment it into (pseudo-)periods and assigns phase values by performing efficient, tractable searches over previous signal segments. We extensively validate the algorithm on a variety of signal types, including trajectories from chaotic dynamical systems, human motion-capture data and electrocardiographic recordings. Our results demonstrate that ROPE is robust against noise and signal drift and achieves significantly superior performance compared with state-of-the-art phase-estimation methods. This advancement enables real-time analysis of complex biological rhythms, opening new pathways, for example, for early diagnosis of pathological rhythm disruptions and developing rhythm-based therapeutic interventions in neurological and cardiovascular disorders.