Synchronizing movements with the metronome: nonlinear error correction and unstable periodic orbits

Small perturbations in a finger-tapping task reveal inherent nonlinearities of the underlying error correction mechanism

Time processing in the few hundred milliseconds range is involved in the human skill of sensorimotor synchronization, like playing music in an ensemble or finger tapping to an external beat. In finger tapping, a mechanistic explanation in biologically plausible terms of how the brain achieves synchronization is still missing despite considerable research. In this work we show that nonlinear effects are important for the recovery of synchronization following a perturbation (a step change in stimulus period), even for perturbation magnitudes smaller than 10% of the period, which is well below the amount of perturbation needed to evoke other nonlinear effects like saturation. We build a nonlinear mathematical model for the error correction mechanism and test its predictions, and further propose a framework that allows us to unify the description of the three common types of perturbations. While previous authors have used two different model mechanisms for fitting different perturbation types, or have fitted different parameter value sets for different perturbation magnitudes, we propose the first unified description of the behavior following all perturbation types and magnitudes as the dynamical response of a compound model with fixed terms and a single set of parameter values.

Quantifying phase correction in sensorimotor synchronization: empirical comparison of three paradigms

Tapping in synchrony with a metronome requires phase error correction, a process often described by a single-parameter autoregressive model. The parameter (α) is a measure of sensorimotor coupling strength. This study compares α estimates obtained from three experimental paradigms: synchronization with (1) a perfectly regular metronome (RM), (2) a perturbed metronome containing phase shifts (PS), and (3) an "adaptively timed" metronome (AT). Musically trained participants performed in each paradigm at four tempi, with baseline interval durations ranging from 400 to 1300 ms. Two estimation methods were applied to each data set. Results showed that all α estimates increased with interval duration. However, the PS paradigm yielded much larger α values than did the AT paradigm, with those from the RM paradigm falling in between. Positional analysis of the PS data revealed that α increased immediately following a phase shift and then decreased sharply. Unexpectedly, all PS α estimates were uncorrelated with the RM and AT estimates, which were strongly correlated. These results suggest that abruptly perturbed sequences engage a different mechanism of phase correction than do regular or continuously modulated sequences.

Synchronization and chaotic dynamics of coupled mechanical metronomes

Synchronization scenarios of coupled mechanical metronomes are studied by means of numerical simulations showing the onset of synchronization for two, three, and 100 globally coupled metronomes in terms of Arnol'd tongues in parameter space and a Kuramoto transition as a function of coupling strength. Furthermore, we study the dynamics of metronomes where overturning is possible. In this case hyperchaotic dynamics associated with some diffusion process in configuration space is observed, indicating the potential complexity of metronome dynamics.

An internal clock generates repetitive predictive saccades

Previously we demonstrated the presence of a behavioral phase transition between reactive and predictive eye tracking of alternating targets. Prior studies of repetitive movements have proposed that an "internal clock" is the neural mechanism by which interval timing is achieved. In the present report we tested whether predictive oculomotor (saccade) tracking is based on an internal time reference (clock) by examining the effect of transient perturbations to the periodic pacing stimulus. These perturbations consisted of altering the timing of the stimulus (abruptly increasing or decreasing the inter-stimulus interval) or extinguishing the targets altogether. Although reactive tracking (at low pacing rates) was greatly affected by these timing perturbations, once predictive tracking was established subjects continued to time their eye movement responses at the pre-existing rate despite the perturbation. As expected from certain clock models, inter-stimulus intervals for predictive tracking followed Weber's law and the scalar property (timing variability increases in proportion to interval duration), but this was not true for reactive tracking. In addition, the perturbation results show that subjects can establish an internal representation of target pacing (the internal clock) in as little as two eye-movement intervals, which suggests that this mechanism is relevant for real-world situations. These findings are consistent with the presence of an internal clock for the generation of these predictive movements, and demonstrate that the neural mechanism responsible for this behavior is temporally accurate and flexible.

Sensorimotor synchronization: A review of the tapping literature

Sensorimotor synchronization (SMS), the rhythmic coordination of perception and action, occurs in many contexts, but most conspicuously in music performance and dance. In the laboratory, it is most often studied in the form of finger tapping to a sequence of auditory stimuli. This review summarizes theories and empirical findings obtained with the tapping task. Its eight sections deal with the role of intention, rate limits, the negative mean asynchrony, variability, models of error correction, perturbation studies, neural correlates of SMS, and SMS in musical contexts. The central theoretical issue is considered to be how best to characterize the perceptual information and the internal processes that enable people to achieve and maintain SMS. Recent research suggests that SMS is controlled jointly by two error correction processes (phase correction and period correction) that differ in their degrees of cognitive control and may be associated with different brain circuits. They exemplify the general distinction between subconscious mechanisms of action regulation and conscious processes involved in perceptual judgment and action planning.

Phase correction in sensorimotor synchronization: nonlinearities in voluntary and involuntary responses to perturbations

When finger taps are synchronized with an auditory sequence, both a global phase shift (PS) and a local event onset shift (EOS) in the sequence elicit a phase correction response (PCR) on the next tap. The PCR to an expected PS is intended and large, whereas that to an expected EOS is unintended and smaller. PCR magnitude increases linearly with perturbation magnitude up to about +/-15% of the sequence period (500 milliseconds). With larger perturbations, voluntary PCRs increase more slowly whereas involuntary PCRs reach an asymptote. These results, obtained previously in a blocked design [J. Exp. Psychol. Human Percept. Perform. (in press)], were replicated in a randomized design and in two additional task contexts that varied participants' intentions while neutralizing their expectations. Neither design nor expectations seemed to play a role. However, considerable individual differences were noted. The results confirm that phase correction is partially automatic and partially subject to voluntary control, and they provide empirical estimates of error correction functions that may be useful in formal modeling of sensorimotor synchronization behavior.