Time intervals production in tapping and oscillatory motion

Simultaneous event-based and emergent timing: synchronization, continuation, and phase correction

It has been claimed that rhythmic tapping and circle drawing represent fundamentally different timing processes (event-based and emergent, respectively) and also that circle drawing is difficult to synchronize with a metronome and exhibits little phase correction. In the present study, musically trained participants tapped with their left hands, drew circles with their right (dominant) hands, and also performed both tasks simultaneously. In Experiment 1, they synchronized with a metronome and then continued on their own, whereas in Experiment 2, they synchronized with a metronome containing phase perturbations. Circle drawing generally exhibited reliable synchronization, although with greater variability than tapping, and also showed a clear phase-correction response that evolved gradually during the cycle immediately following a perturbation. When carried out simultaneously in synchrony, with or without a metronome, the two tasks affected each other in some ways but retained their distinctive timing characteristics. This shows that event-based and emergent timing can coexist in a dual-task situation. Furthermore, the authors argue that the two timing modes usually coexist in each individual task, although one mode is often dominant.

The effect of entrainment on the timing of periodic eye movements

We performed an experiment in which eight healthy individuals made periodic eye movements at five pacing interval conditions (500 ms, 750 ms, 1000 ms, 1250 ms, and 1500 ms). Three methods of entrainment were used in the synchronization phase: saccade, continuous pursuit and discontinuous pursuit. The stimulus train was extinguished and in the continuation phase, subjects made saccadic eye movements at the entrained movement frequencies between two static targets. Using the Wing-Kristofferson model, clock and motor variance were extracted from the time series of continuation trials for all three entrainment conditions. Our results revealed a main effect of time interval on total variance clock variance (as predicted by Weber's law) and on motor variance. We also report that the pursuit entrainment conditions resulted in and mean duration and variance to the saccade entrainment. These results suggest that the neural networks recruited to support a periodic motor timing task depend on the method used to establish the temporal reference.

Long-range correlation in synchronization and syncopation tapping: a linear phase correction model

We propose in this paper a model for accounting for the increase in long-range correlations observed in asynchrony series in syncopation tapping, as compared with synchronization tapping. Our model is an extension of the linear phase correction model for synchronization tapping. We suppose that the timekeeper represents a fractal source in the system, and that a process of estimation of the half-period of the metronome, obeying a random-walk dynamics, combines with the linear phase correction process. Comparing experimental and simulated series, we show that our model allows accounting for the experimentally observed pattern of serial dependence. This model complete previous modeling solutions proposed for self-paced and synchronization tapping, for a unifying framework of event-based timing.

Estimating long-range dependence in time series: an evaluation of estimators implemented in R

Recent studies have shown that many physiological and behavioral processes can be characterized by long-range correlations. The Hurst exponent H of fractal analysis and the fractional-differencing parameter d of the ARFIMA methodology are useful for capturing serial correlations. In this study, we report on different estimators of H and d implemented in R, a popular and freely available software package. By means of Monte Carlo simulations, we analyzed the performance of (1) the Geweke-Porter-Hudak estimator, (2) the approximate maximum likelihood algorithm, (3) the smoothed periodogram approach, (4) the Whittle estimator, (5) rescaled range analysis, (6) a modified periodogram, (7) Higuchi's method, and (8) detrended fluctuation analysis. The findings-confined to ARFIMA (0, d, 0) models and fractional Gaussian noise-identify the best estimators for persistent and antipersistent series. Two examples combining these results with the step-by-step procedure proposed by Delignières et al. (2006) demonstrate how this evaluation can be used as a guideline in a typical research situation.

Two different processes for sensorimotor synchronization in continuous and discontinuous rhythmic movements

To account for sensorimotor synchronization, the information processing and the dynamical systems perspectives have developed different classes of models. While the former has focused on cycle-to-cycle correction of the timing errors, the latter deals with a continuous, state-dependent within-cycle coupling between the oscillating limb and the metronome. The purpose of the present study was to investigate the extent to which the two modeling frameworks partially capture the same behavior or, instead, account for different aspects of synchronization. A comparative two-level analysis (time intervals and movement trajectories) of synchronized tapping and synchronized oscillation data revealed distinct patterns of results with regard to (1) the relationship between the (a)symmetry of movement cycles and the achievement of timing goals, and (2) the sequential or within-cycle organization of synchronization processes. Our results support the idea that movement trajectories contribute to the achievement of synchronized movement timing in two different ways as a function of the (dis)continuous nature of movement. We suggest that the two modeling frameworks indeed account for different synchronization processes involved in the process of keeping time with the beat.

Theories and models for 1/f(beta) noise in human movement science

Human motor behavior is often characterized by long-range, slowly decaying serial correlations or 1/f(beta) noise. Despite its prevalence, the role of the 1/f(beta) phenomenon in human movement research has been rather modest and unclear. The goal of this paper is to outline a research agenda in which the study of 1/f(beta) noise can contribute to scientific progress. In the first section of this article we discuss two popular perspectives on 1/f(beta) noise: the nomothetic perspective that seeks general explanations, and the mechanistic perspective that seeks domain-specific models. We believe that if 1/f(beta) noise is to have an impact on the field of movement science, researchers should develop and test domain-specific mechanistic models of human motor behavior. In the second section we illustrate our claim by showing how a mechanistic model of 1/f(beta) noise can be successfully integrated with currently established models for rhythmic self-paced, synchronized, and bimanual tapping. This model synthesis results in a unified account of the observed long-range serial correlations across a range of different tasks.

Tapping with intentional drift

When tapping a desired frequency, subjects tend to drift away from this target frequency. This compromises the estimate of the correlation between inter-tap intervals (ITIs) as predicted by the two-level model of Wing and Kristofferson which consists of an internal timer ('clock') and motor delays. Whereas previous studies on the timing of rhythmic tapping attempted to eliminate drift, we compared the production of three constant frequencies (1.5, 2.0, and 2.5 Hz) to the production of tapping sequences with a linearly decreasing inter-tap interval (ITI) (corresponding to an increase in tapping frequency from 1.5 to 2.5 Hz). For all conditions a synchronization-continuation paradigm was used. Tapping forces and electromyograms of the index-finger flexor and extensor were recorded and ITIs were derived yielding interval variability and model parameters, i.e., clock and motor variances. Electromyographic recordings served to study the influence of tapping frequency on the peripheral part of the tap event. The condition with an increasing frequency was more difficult to perform, as evidenced by an increase in deviation from the intended ITIs. In general, tapping frequency affected force level, inter-tap variability, model parameters, and muscle co-activation. Parameters for the condition with a decreasing ITI were comparable to those found in the constant frequency conditions. That is, although tapping with an intentional drift is different from constant tapping and more difficult to perform, the timing properties of both forms of tapping are remarkably similar and described well by the Wing and Kristofferson model.

Distinct ways of timing movements in bimanual coordination tasks: contribution of serial correlation analysis and implications for modeling

Bimanual coordination dynamics have been conceived as the outcome of a global coordinative system, and coordination stability properties and theories of underlying processes have often been generalized over various bimanual tasks. In unimanual timing tasks it has been shown that different timing processes are involved according to tasks, yielding distinctive correlation properties in the within-hand temporal patterns. In this study we compare unimanual with bimanual, tapping with oscillation, and self-paced with externally paced tasks, and we analyze the correlation properties of temporal patterns at both the component level and the coordinative level. Results show that the distinctive signatures of event-based versus emergent, and self-paced versus synchronization timing control known from unimanual tasks persist in the corresponding bimanual coordination tasks. Accordingly, we argue that these different timing processes, and related temporal patterns at the component level, constitute a task-dependent background on which coordination builds. One direct implication of these results is that the bimanual coordination paradigm should be considered multifaceted and not governed by some unitary generic principle. We discuss the need to assess the relationship between temporal patterns at the component level and the collective level, and to integrate serial (long-range) correlation properties into bimanual coordination models. Finally, we test whether the architectures of current bimanual coordination models can account for the experimentally observed serial correlations.

Unraveling the finding of 1/f beta noise in self-paced and synchronized tapping: a unifying mechanistic model

1/f (beta) noise has been revealed in both self-paced and synchronized tapping sequences, without being consistently taken into consideration for the modeling of underlying timing mechanisms. In this study we characterize variability, short-range, and long-range correlation properties of asynchronies and inter-tap intervals collected in a synchronization tapping experiment, attesting statistically the presence of 1/f (beta) noise in asynchronies. We verify that the linear phase correction model of synchronization tapping in its original formulation cannot account for the empirical long-range correlation properties. On the basis of previous accounts of 1/f (beta) noise in the literature on self-paced tapping, we propose an extension of the original synchronization model by modeling the timekeeping process as a source of 1/f (beta) fluctuations. Simulations show that this '1/f-AR synchronization model' accounts for the statistical properties of empirical series, including long-range correlations, and provides an unifying mechanistic account of 1/f (beta) noise in self-paced and synchronization tapping. This account opens the original synchronization framework to further investigations of timing mechanisms with regard to the serial correlation properties in performed time intervals.

Detrended windowed (lag one) autocorrelation: a new method for distinguishing between event-based and emergent timing

The aim of this study was to test different methods for distinguishing between two known timing processes involved in human rhythmic behaviours. We examined the implementation of two approaches used in the literature: the high-frequency slope of the power spectrum and the lag one value of the autocorrelation function, ACF(1). We developed another method based on the Wing and Kristofferson (1973a) model and the predicted negative ACF(1) for event-based series: the detrended windowed (lag one) autocorrelation (DWA). We compared the reliability and performance of these three methods on simulation and experimental series. DWA gave the best results, and guidelines are given for its appropriate use for identifying underlying timing processes.

Fractal models for event-based and dynamical timers

Some recent papers proposed to distinguish between event-based and emergent timing. Event-based timing is conceived as prescribed by events produced by a central clock, and seems to be used in discrete tasks (e.g., finger tapping). Emergent or dynamical timing refers to the exploitation of the dynamical properties of effectors, and is typically used in continuous tasks (e.g., circle drawing). The analysis of period series suggested that both timing control processes possess fractal properties, characterized by self-similarity and long-range dependence. The aim of this article is to present two models that produce period series presenting the statistical properties previously evidenced in discrete and continuous rhythmic tasks. The first one is an adaptation of the classical activation/threshold models, including a plateau-like evolution of the threshold over time. The second one is a hybrid limit-cycle model, including a time-dependent linear stiffness parameter. Both models reproduced satisfactorily the spectral signatures of event-based and dynamical timing processes, respectively. The models also produced auto-correlation functions similar to those experimentally observed. Using ARFIMA modeling we show that these simulated series possess fractal properties. We suggest in conclusion some possible extensions of this modeling approach, to account for the effects of metronomic pacing, or to analyze bimanual coordination.

1/f (beta) fluctuations in bimanual coordination: an additional challenge for modeling

We analyzed the correlation structure of discrete relative phase (DRP) series in bimanual in-phase and anti-phase coordination by associating a number of fractal methods and using discrete rather than continuous relative phase measurement. ARFIMA/ARMA modeling provided statistical evidence for the presence of long-range correlation, and the series were unambiguously characterized as 1/f (beta) noise. Diverging accounts of bimanual coordination are defended in the literature. Since the evidence for 1/f (beta) noise provides new insight into the properties of stability in coordination, it should be considered as an empirical criterion for determining which mechanisms are likely to be engaged in bimanual coordination models. We discussed some implications for studying the neural basis of coordination, and we tested the performance of three current models in accounting for 1/f (beta) noise in DRP. None of these models was proven to generate the expected correlation structure.

Transcranial magnetic stimulation in a finger-tapping task separates motor from timing mechanisms and induces frequency doubling

We study the interplay between motor programs and their timing in the brain by using precise pulses of transcranial magnetic stimulation (TMS) applied to the primary motor cortex. The movement of the finger performing a tapping task is periodically perturbed in synchronization with a metronome. TMS perturbation can profoundly affect both the finger trajectory and its kinematics, but the tapping accuracy itself is surprisingly not affected. The motion of the finger during the TMS perturbation can be categorized into two abnormal behaviors that subjects were unaware of: a doubling of the frequency of the tap and a stalling of the finger for half the period. More stalls occurred as the tapping frequency increased. In addition, an enhancement of the velocity of the finger on its way up was observed. We conclude that the timing process involved in controlling the tapping movement is separate from the motor processes in charge of execution of the motor commands. We speculate that the TMS is causing a release of the motor plan ahead of time into activation mode. The observed doubles and stalls are then the result of an indirect interaction in the brain, making use of an existing motor plan to correct the preactivation and obtain the temporal goal of keeping the beat.

Detection of long-range dependence and estimation of fractal exponents through ARFIMA modelling

We evaluate the performance of autoregressive, fractionally integrated, moving average (ARFIMA) modelling for detecting long-range dependence and estimating fractal exponents. More specifically, we test the procedure proposed by Wagenmakers, Farrell, and Ratcliff, and compare the results obtained with the Akaike information criterion (AIC) and the Bayes information criterion (BIC). The present studies show that ARFIMA modelling is able to adequately detect long-range dependence in simulated fractal series. Conversely, this method tends to produce a non-negligible rate of false detections in pure autoregressive and moving average (ARMA) series. Generally, ARFIMA modelling has a bias favouring the detection of long-range dependence. AIC and BIC gave dissimilar results, due to the different weights attributed by the two criteria to accuracy and parsimony. Finally, ARFIMA modelling provides good estimates of fractal exponents, and could adequately complement classical methods, such as spectral analysis, detrended fluctuation analysis or rescaled range analysis.

Poor penmanship in children correlates with abnormal rhythmic tapping: a broad functional temporal impairment

Timing is crucial for proficient motor tasks; temporal impairments may lead to dysfunctional motor activities. Although much research has been dedicated to the study of movement timing, clinical examination often overlooks temporal impairment of motor activity. The authors hypothesize that some children have a global temporal impairment leading to dysfunctional motor skills. This article checks whether temporal abnormalities detected on a simple tapping task correlate with temporal dysfunction during complex motor skills such as handwriting. Twenty-three school-aged children, 8-14 years (11.1 +/- 1.3 years), underwent tests to assess finger tapping and cursive handwriting. Handwriting samples were rated by experienced teachers. Children with abnormal tapping had lower handwriting rating scores. Temporal features were similar in both tasks; variability on the tapping test correlated with handwriting variability. Temporal variability was not significantly higher for children with poor penmanship as a whole but rather specific to the subgroup of children with a tapping abnormality. Poor penmanship could be attributed in certain children to global temporal impairment reflected as variable finger tapping and handwriting. Evaluation of dysfunctional motor performance should include temporal aspects, and further studies are needed to better delineate and address treatment of "dysrhythmia."

Rhythmic performance during a whole body movement: dynamic analysis of force-time curves

The purpose of this study was to investigate rhythmic performance during two-legged hopping in place. In particular, it was tested whether (a) timing control is independent of force control, (b) a dynamic timer model explains rhythmic performance, and (c) it is a force related parameter that carries the timing information. Eleven participants performed two-legged hopping at their preferred hopping frequency (PHF) and at two hopping frequencies set by an external rhythmic stimulus as lower (LHF) and higher (HHF) than their PHF, respectively. A force plate was used to record the ground reaction force (GRF) time curves during two-legged hopping. The primary temporal and force related parameters determined from the GRF-time curves were the durations of the cycle of movement (t(cycle)), of the contact phase (t(contact)), of the flight phase (t(flight)), the magnitude of peak force (Fz(peak)) and the rate of peak force development (RFD). Control of t(cycle) was independent of force control as shown by the non-significant correlations between t(cycle) and the force parameters of the GRF-time curve. Lag 1 autocorrelations of t(cycle) were not significant in any of the HF, thereby a dynamic timer model is considered to explain the timing of t(cycle) during two-legged hopping. RFD varied more than any other GRF-time curve parameter, exhibited consistent significant strong correlations with the GRF-time curve parameters and significant negative lag 1 autocorrelations in PHF, thus, it was highlighted as the potent timing control parameter. Finally, we provide a practical application for the optimization of rhythmic performance.

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.

Testing Bayesian Models of Human Coincidence Timing

A sensorimotor control task often requires an accurate estimation of the timing of the arrival of an external target (e.g., when hitting a pitched ball). Conventional studies of human timing processes have ignored the stochastic features of target timing: e.g., the speed of the pitched ball is not generally constant, but is variable. Interestingly, based on Bayesian theory, it has been recently shown that the human sensorimotor system achieves the optimal estimation by integrating sensory information with prior knowledge of the probabilistic structure of the target variation. In this study, we tested whether Bayesian integration is also implemented while performing a coincidence- timing type of sensorimotor task by manipulating the trial-by-trial variability (i.e., the prior distribution) of the target timing. As a result, within several hundred trials of learning, subjects were able to generate systematic timing behavior according to the width of the prior distribution, as predicted by the optimal Bayesian model. Considering the previous studies showing that the human sensorimotor system uses Bayesian integration in spacing and force-grading tasks, our result indicates that Bayesian integration is fundamental to all aspects of human sensorimotor control. Moreover, it was noteworthy that the subjects could adjust their behavior both when the prior distribution was switched from wide to narrow and vice versa, although the adjustment was slower in the former case. Based on a comparison with observations in a previous study, we discuss the flexibility and adaptability of Bayesian sensorimotor learning.

Point process model of 1/f noise vs a sum of Lorentzians

We present a simple point process model of 1/f(beta) noise, covering different values of the exponent beta . The signal of the model consists of pulses or events. The interpulse, interevent, interarrival, recurrence, or waiting times of the signal are described by the general Langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power-law distribution. Our model is free from the requirement of a wide distribution of relaxation times and from the power-law forms of the pulses. It contains only one relaxation rate and yields 1/f(beta) spectra in a wide range of frequencies. We obtain explicit expressions for the power spectra and present numerical illustrations of the model. Further we analyze the relation of the point process model of 1/f noise with the Bernamont-Surdin-McWhorter model, representing the signals as a sum of the uncorrelated components. We show that the point process model is complementary to the model based on the sum of signals with a wide-range distribution of the relaxation times. In contrast to the Gaussian distribution of the signal intensity of the sum of the uncorrelated components, the point process exhibits asymptotically a power-law distribution of the signal intensity. The developed multiplicative point process model of 1/f(beta)noise may be used for modeling and analysis of stochastic processes in different systems with the power-law distribution of the intensity of pulsing signals.