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Tailoring relative phase estimation methods to movement shape. A practical approach to estimate the Arnold tongue with testing codes on human data and simulation of discrete and continuous signals.
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Tailoring relative phase estimation methods to movement shape. A practical approach to estimate the Arnold tongue with testing codes on human data and simulation of discrete and continuous signals.

Martin Le Guennec et Julien Lagarde
20/08/2026

Résumé

Relative phase Estimation Hilbert transform Stochastic simulation Arnold tongue Ghost attractor
The relative phase ϕis a key variable in the study of coordination dynamics across many disciplines. Several estimation methods exist and are used interchangeably in the study of sensorimotor synchronization, despite the limitations of certain methods when analyzing inharmonic (aka discrete) movements. In this study, we document differences in estimates of key variables used to describe coordination dynamics, the mean ϕand its standard deviation, calculated using a pointwise method and using the Hilbert transform with signal centering or with a correction of the non-linearity. We applied these methods to experimental data from a tapping task with an incremental increase in metronome frequency. The increase in frequency allowed us to observe differences between methods in inharmonic movements at low synchronization frequencies and in near-sinusoidal movements at higher frequencies. To complement this analysis, we conducted stochastic numerical simulations of non-linear oscillators and with ghost attractors to mimic signals from discrete movements. We compared the mean and SD ϕestimates to a gold standard based on the Arnold tongues framework. Our experimental and simulation results show that the mean and SD ϕestimates obtained from the Hilbert transform are biased for inharmonic signals, we recommend using the pointwise method in this context. With near-sinusoidal signals, both methods provide equivalent results. We propose a practical approach to estimate the quality of synchronization, sharing free testing codes, human data and simulation of discrete or continuous movements.

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