ROPE: A Novel Method for Real-Time Phase Estimation of Complex Biological Rhythms

Abstract

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 paper, 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 to 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.

Figures (9)

• Figure 1: (a) Example of a pseudo-periodic signal $\mathbf{p}^\mathrm{c}$ and its discretization $\mathbf{p}$ (§ \ref{['sec:methods']}.\ref{['sec:pseudoperiodicity_baselines']}). (b) Two curves $\mathbf{\gamma}$ (blue) and $\mathbf{b}$ (red), and the shape reduction ($\mathbf{r}_\mathbf{b}(\mathbf{\gamma})$) (green) of $\mathbf{\gamma}$ onto $\mathbf{b}$ (§ \ref{['sec:methods']}.\ref{['sec:pseudoperiodicity_baselines']}). (c) Signals involved in the phase estimation problem, Problem \ref{['prob:estimate_phase']}.
• Figure 2: (a) Points associated to phase zero, on the baseline $\mathbf{b}$ and on the estimand signal $\mathbf{p}^\mathrm{c}$ (see § \ref{['sec:problem_formulation']}.\ref{['sec:multiple_signals']}). (b) Search domains used in \ref{['eq:minimization_search']} and \ref{['eq:smaller_search_domain']}. The first horizontal line reports the full search domain in \ref{['eq:minimization_search']}, the second and third lines depict the first case in \ref{['eq:smaller_search_domain']}, whereas the last line depicts the second case in \ref{['eq:smaller_search_domain']}.
• Figure 3: (a) Block scheme of the phase estimator for multidimensional periodic motion. (b) Flowchart of the phase estimation algorithm.
• Figure 4: Representation of key steps involved in the ROPE algorithm, as explained in Section \ref{['Sec:algorithm']}. (a) Process performed by the "loop estimator" block, in the estimation of the first loop. (b) Process performed by the "Phase computer " block. (c) Process performed by the "loop estimator" block, in the estimation of the second and later loops.
• Figure 5: (a) Representative loops of the validation signals used in this study: Rössler (chaotic dynamics), Spiral, Clockwise-Counterclockwise (C-CC), Macarena, Back & Forth (B&F), and Infinity. (b) A representative loop of the ECG signals.

Theorems & Definitions (4)

• definition 1: Pseudo-periodicity
• definition 2: Discrete pseudo-periodic signals
• definition 3: Shape reduction
• definition 4: Baseline