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

A guide to Whittle maximum likelihood estimator in MATLAB

The assessment of physiological complexity via the estimation of monofractal exponents or multifractal spectra of biological signals is a recent field of research that allows detection of relevant and original information for health, learning, or autonomy preservation. This tutorial aims at introducing Whittle's maximum likelihood estimator (MLE) that estimates the monofractal exponent of time series. After introducing Whittle's maximum likelihood estimator and presenting each of the steps leading to the construction of the algorithm, this tutorial discusses the performance of this estimator by comparing it to the widely used detrended fluctuation analysis (DFA). The objective of this tutorial is to propose to the reader an alternative monofractal estimation method, which has the advantage of being simple to implement, and whose high accuracy allows the analysis of shorter time series than those classically used with other monofractal analysis methods.

The effect of perceptual-motor continuity compatibility on the temporal control of continuous and discontinuous self-paced rhythmic movements

One of the questions yet to be fully understood is to what extent the properties of the sensory and the movement information interact to facilitate sensorimotor integration. In this study, we examined the relative contribution of the continuity compatibility between motor goals and their sensory outcomes in timing variability. The variability of inter-response intervals was measured in a synchronization-continuation paradigm. Participants performed two repetitive movement tasks whereby they drew circles either using continuous or discontinuous self-paced movements while receiving discrete or continuous auditory feedback. The results demonstrated that the effect of perceptual-motor continuity compatibility may be limited in self-paced auditory-motor synchronization as timing variability was not significantly influenced by the continuity of the feedback or the continuity compatibility between feedback and the movement produced. In addition, results suggested that the presence of salient perceptual events marking the completion of the time intervals elicited a common timing process in both continuous and discontinuous circle drawing, regardless of the continuity of the auditory feedback. These findings open a new line of investigation into the role of the discriminability and reliability of the event-based information in determining the nature of the timing mechanisms engaged in continuous and discontinuous self-paced rhythmic movements.

Interpersonal Synchronization Processes in Discrete and Continuous Tasks

Three frameworks have been proposed to account for interpersonal synchronization: The information processing approach argues that synchronization is achieved by mutual adaptation, the coordination dynamics perspective supposes a continuous coupling between systems, and complexity matching suggests a global, multi-scale interaction. We hypothesized that the relevancy of these models was related to the nature of the performed tasks. 10 dyads performed synchronized tapping and synchronized forearm oscillations, in two conditions: full (participants had full information about their partner), and digital (information was limited to discrete auditory signals). Results shows that whatever the task and the available information, synchronization was dominated by a discrete mutual adaptation. These results question the relevancy of the coordination dynamics perspective in interpersonal coordination.

Delayed feedback embedded in perception-action coordination cycles results in anticipation behavior during synchronized rhythmic action: A dynamical systems approach

Dancing and playing music require people to coordinate actions with auditory rhythms. In laboratory perception-action coordination tasks, people are asked to synchronize taps with a metronome. When synchronizing with a metronome, people tend to anticipate stimulus onsets, tapping slightly before the stimulus. The anticipation tendency increases with longer stimulus periods of up to 3500ms, but is less pronounced in trained individuals like musicians compared to non-musicians. Furthermore, external factors influence the timing of tapping. These factors include the presence of auditory feedback from one’s own taps, the presence of a partner performing coordinated joint tapping, and transmission latencies (TLs) between coordinating partners. Phenomena like the anticipation tendency can be explained by delay-coupled systems, which may be inherent to the sensorimotor system during perception-action coordination. Here we tested whether a dynamical systems model based on this hypothesis reproduces observed patterns of human synchronization. We simulated behavior with a model consisting of an oscillator receiving its own delayed activity as input. Three simulation experiments were conducted using previously-published behavioral data from 1) simple tapping, 2) two-person alternating beat-tapping, and 3) two-person alternating rhythm-clapping in the presence of a range of constant auditory TLs. In Experiment 1, our model replicated the larger anticipation observed for longer stimulus intervals and adjusting the amplitude of the delayed feedback reproduced the difference between musicians and non-musicians. In Experiment 2, by connecting two models we replicated the smaller anticipation observed in human joint tapping with bi-directional auditory feedback compared to joint tapping without feedback. In Experiment 3, we varied TLs between two models alternately receiving signals from one another. Results showed reciprocal lags at points of alternation, consistent with behavioral patterns. Overall, our model explains various anticipatory behaviors, and has potential to inform theories of adaptive human synchronization.

When navigating a busy sidewalk, people coordinate their behavior in an orderly manner. Other activities require people to carefully synchronize periodic actions, as in a group rowing or marching. When individuals tap in synchrony with a metronome, their taps tend to anticipate the metronome. Experiments have revealed that factors like musical expertise, the presence of a synchronizing partner, auditory feedback, and the sound travel time, all systematically affect the tendency to anticipate. While researchers have hypothesized a number of potential mechanisms for such anticipatory behavior, none have successfully accounted for all of the effects. Previous research on coupled physical systems has shown that when one system receives input from a second system, plus its own delayed signal as input, this causes system 1 to anticipate system 2. We hypothesize that the tendency to anticipate is the result of delayed communication between neurons. Our work demonstrates the ability of delay-coupled physical systems to capture human anticipation and the effect of external factors in the anticipation tendency. Our model supports the theory that delayed communication within the nervous system is crucial to understanding anticipatory coordinative behavior.

Origins of 1/f noise in human music performance from short-range autocorrelations related to rhythmic structures

1/f fluctuations have been described in numerous physical and biological processes. This noise structure describes an inverse relationship between the intensity and frequency of events in a time series (for example reflected in power spectra), and is believed to indicate long-range dependence, whereby events at one time point influence events many observations later. 1/f has been identified in rhythmic behaviors, such as music, and is typically attributed to long-range correlations. However short-range dependence in musical performance is a well-established finding and past research has suggested that 1/f can arise from multiple continuing short-range processes. We tested this possibility using simulations and time-series modeling, complemented by traditional analyses using power spectra and detrended fluctuation analysis (as often adopted more recently). Our results show that 1/f-type fluctuations in musical contexts may be explained by short-range models involving multiple time lags, and the temporal ranges in which rhythmic hierarchies are expressed are apt to create these fluctuations through such short-range autocorrelations. We also analyzed gait, heartbeat, and resting-state EEG data, demonstrating the coexistence of multiple short-range processes and 1/f fluctuation in a variety of phenomena. This suggests that 1/f fluctuation might not indicate long-range correlations, and points to its likely origins in musical rhythm and related structures.

Cerebellar Role in Predictive Control of Eye Velocity Initiation and Termination

Predictive motor control is essential to achieve rapid and precise motor action in all vertebrates. Visuomotor transformations have been a popular model system to study the underlying neural mechanisms, in particular, the role of the cerebellum in both predictive and gain adaptations. In all species, large-field visual motion produces an involuntary conjugate ocular movement facilitating gaze stabilization called the optokinetic response. Gain adaptation can be induced by prolonged optokinetic visual stimulation; and if the visual stimulation is temporally periodic, predictive behavior emerges. Two predictive timing components were identifiable in this behavior. The first was prediction of stimulus initiation (when to move) and the other was stimulus termination (when to stop). We designed visual training that allowed us to evaluate initiation and termination independently that included the recording of cerebellar activity followed by acute and chronic cerebellar removal in goldfish of both sexes. We found that initiation and termination predictions were present in the cerebellum and more robust than conflicting visual sensory signals. Each prediction could be acquired independently, and both the acquisition and maintenance of each component were cerebellar-dependent. Subsequent analysis of the neuronal connectivity strongly supports the hypothesis that the acquired eye velocity behaviors were dependent on feedforward velocity buildup signals from the brainstem, but the adaptive timing mechanism itself originates within the circuitry of the cerebellum.SIGNIFICANCE STATEMENT Predictive and rapid motor control is essential in our daily life, such as in the playing of musical instruments or sports. The current work evaluates timing of a visuomotor behavior shown to be similar in humans as well as goldfish. Given the latter species' known brainstem cerebellar neuronal connectivity and experimental advantage, it was possible to demonstrate the cerebellum to be necessary for acquisition and maintenance of both the initiation and termination components of when to move and to stop. All evidence in this study points to the adaptive predictive control site to lie within the cerebellar circuitry.

When Coordinating Finger Tapping to a Variable Beat the Variability Scaling Structure of the Movement and the Cortical BOLD Signal are Both Entrained to the Auditory Stimuli

Rhythmic actions are characterizable as a repeating invariant pattern of movement together with variability taking the form of cycle-to-cycle fluctuations. Variability in behavioral measures is atypically random, and often exhibits serial temporal dependencies and statistical self-similarity in the scaling of variability magnitudes across timescales. Self-similar (i.e. fractal) variability scaling is evident in measures of both brain and behavior. Variability scaling structure can be quantified via the scaling exponent (α) from detrended fluctuation analysis (DFA). Here we study the task of coordinating thumb-finger tapping to the beats of constructed auditory stimuli. We test the hypothesis that variability scaling evident in tap-to-tap intervals as well as in the fluctuations of cortical hemodynamics will become entrained to (i.e. drawn toward) manipulated changes in the variability scaling of a stimulus's beat-to-beat intervals. Consistent with this hypothesis, manipulated changes of the exponent α of the experimental stimuli produced corresponding changes in the exponent α of both tap-to-tap intervals and cortical hemodynamics. The changes in hemodynamics were observed in both motor and sensorimotor cortical areas in the contralateral hemisphere. These results were observed only for the longer timescales of the detrended fluctuation analysis used to measure the exponent α. These findings suggest that complex auditory stimuli engage both brain and behavior at the level of variability scaling structures.

Effects of visual and auditory guidance on bimanual coordination complexity

Perceptual guidance of movement with simple visual or temporal information can facilitate performance of difficult coordination patterns. Guidance may override coordination constraints that usually limit stability of bimanual coordination to only in-phase and anti-phase. Movement dynamics, however, might not have the same characteristics with and without perceptual guidance. Do visual and auditory guidance produce qualitatively different dynamical organization of movement? An anti-phase wrist flexion and extension coordination task was performed under no specific perceptual guidance, under temporal guidance with a metronome, and under visual guidance with a Lissajous plot. For the time series of amplitudes, periods and relative phases, temporal correlations were measured with Detrended Fluctuation Analysis and complexity levels were measured with multiscale entropy. Temporal correlations of amplitudes and relative phases deviated from the typical 1/f variation towards more random variation under visual guidance. The same was observed for the series of periods under temporal guidance. Complexity levels for all time series were lower in visual guidance, but higher for periods under temporal guidance. Perceptual simplification of the task's goal may produce enhancement of performance, but it is accompanied by changes in the details of movement organization that may be relevant to explain dependence and poor retention after practice under guidance.

Keeping the Beat: A Large Sample Study of Bouncing and Clapping to Music

The vast majority of humans move in time with a musical beat. This behaviour has been mostly studied through finger-tapping synchronization. Here, we evaluate naturalistic synchronization responses to music-bouncing and clapping-in 100 university students. Their ability to match the period of their bounces and claps to those of a metronome and musical clips varying in beat saliency was assessed. In general, clapping was better synchronized with the beat than bouncing, suggesting that the choice of a specific movement type is an important factor to consider in the study of sensorimotor synchronization processes. Performance improved as a function of beat saliency, indicating that beat abstraction plays a significant role in synchronization. Fourteen percent of the population exhibited marked difficulties with matching the beat. Yet, at a group level, poor synchronizers showed similar sensitivity to movement type and beat saliency as normal synchronizers. These results suggest the presence of quantitative rather than qualitative variations when losing the beat.

Experimental control of scaling behavior: what is not fractal?

The list of psychological processes thought to exhibit fractal behavior is growing. Although some might argue that the seeming ubiquity of fractal patterns illustrates their significance, unchecked growth of that list jeopardizes their relevance. It is important to identify when a single behavior is and is not fractal in order to make meaningful conclusions about the processes underlying those patterns. The hypothesis tested in the present experiment is that fractal patterns reflect the enactment of control. Participants performed two steering tasks: steering on a straight track and steering on a circular track. Although each task could be accomplished by holding the steering wheel at a constant angle, steering around a curve may require more constant control, at least from a psychological standpoint. Results showed that evidence for fractal behavior was strongest for the circular track; straight tracks showed evidence of two scaling regions. We argue from those results that, going forward, the goal of the fractal literature should be to bring scaling behavior under experimental control.

Parameter Estimation of Linear Sensorimotor Synchronization Models: Phase Correction, Period Correction, and Ensemble Synchronization

Linear models have been used in several contexts to study the mechanisms that underpin sensorimotor synchronization. Given that their parameters are often linked to psychological processes such as phase correction and period correction, the fit of the parameters to experimental data is an important practical question. We present a unified method for parameter estimation of linear sensorimotor synchronization models that extends available techniques and enhances their usability. This method enables reliable and efficient analysis of experimental data for single subject and multi-person synchronization. In a previous paper (Jacoby et al., 2015), we showed how to significantly reduce the estimation error and eliminate the bias of parameter estimation methods by adding a simple and empirically justified constraint on the parameter space. By applying this constraint in conjunction with the tools of matrix algebra, we here develop a novel method for estimating the parameters of most linear models described in the literature. Through extensive simulations, we demonstrate that our method reliably and efficiently recovers the parameters of two influential linear models: Vorberg and Wing (1996), and Schulze et al. (2005), together with their multi-person generalization to ensemble synchronization. We discuss how our method can be applied to include the study of individual differences in sensorimotor synchronization ability, for example, in clinical populations and ensemble musicians.

Learning, motor skill, and long-range correlations

Long-range correlations have been evidenced in a number of experiments, generally using overlearned and overpracticed tasks. The authors hypothesized that long-range correlation could represent the byproduct of learning. They analyzed the series of periods produced by a group of expert and a group of novices during prolonged trials on a ski simulator. Results showed a very low variability in expert's series, as compared to novices. Fractal analyses showed that fluctuations were significantly more structured and correlated in experts. These results suggest that learning could be conceived as the progressive installation of complexity in the system.

Sensorimotor synchronization: a review of recent research (2006-2012)

Sensorimotor synchronization (SMS) is the coordination of rhythmic movement with an external rhythm, ranging from finger tapping in time with a metronome to musical ensemble performance. An earlier review (Repp, 2005) covered tapping studies; two additional reviews (Repp, 2006a, b) focused on music performance and on rate limits of SMS, respectively. The present article supplements and extends these earlier reviews by surveying more recent research in what appears to be a burgeoning field. The article comprises four parts, dealing with (1) conventional tapping studies, (2) other forms of moving in synchrony with external rhythms (including dance and nonhuman animals' synchronization abilities), (3) interpersonal synchronization (including musical ensemble performance), and (4) the neuroscience of SMS. It is evident that much new knowledge about SMS has been acquired in the last 7 years.

Continuity of visual and auditory rhythms influences sensorimotor coordination

People often coordinate their movement with visual and auditory environmental rhythms. Previous research showed better performances when coordinating with auditory compared to visual stimuli, and with bimodal compared to unimodal stimuli. However, these results have been demonstrated with discrete rhythms and it is possible that such effects depend on the continuity of the stimulus rhythms (i.e., whether they are discrete or continuous). The aim of the current study was to investigate the influence of the continuity of visual and auditory rhythms on sensorimotor coordination. We examined the dynamics of synchronized oscillations of a wrist pendulum with auditory and visual rhythms at different frequencies, which were either unimodal or bimodal and discrete or continuous. Specifically, the stimuli used were a light flash, a fading light, a short tone and a frequency-modulated tone. The results demonstrate that the continuity of the stimulus rhythms strongly influences visual and auditory motor coordination. Participants' movement led continuous stimuli and followed discrete stimuli. Asymmetries between the half-cycles of the movement in term of duration and nonlinearity of the trajectory occurred with slower discrete rhythms. Furthermore, the results show that the differences of performance between visual and auditory modalities depend on the continuity of the stimulus rhythms as indicated by movements closer to the instructed coordination for the auditory modality when coordinating with discrete stimuli. The results also indicate that visual and auditory rhythms are integrated together in order to better coordinate irrespective of their continuity, as indicated by less variable coordination closer to the instructed pattern. Generally, the findings have important implications for understanding how we coordinate our movements with visual and auditory environmental rhythms in everyday life.

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.

Event-based and emergent timing: dichotomy or continuum? A reply to Repp and Steinman (2010)

B. H. Repp and S. R. Steiman (2010) suggested that event-based and emergent timing, usually conceived as mutually exclusive modes of timing, could in fact coexist in a single activity. According to this point of view, rhythmic activities could exploit mixtures of control modes, in which the relative importance of event-based and emergent components could depend on task characteristics. This point of view, in the opinion of the authors of the present article, corresponds to a fundamental misunderstanding of the theoretical basis of the event-based and emergent distinction, and is not supported by any experimental evidence. However, they present some new results that could support new lines of reasoning for the future developments of research in this domain.

Time dependence of coupling in frequency-scaled bimanual coordination

Prior research has shown that fluctuations in the relative phase of bimanual coordination do not reflect a white Gaussian noise process. The present study furthered the examination of time-dependent properties in bimanual coordination by comparing the magnitude of relative phase variability and the degree of effector independence within the time domain. The original Kelso (1984) [10] bimanual frequency-scaling protocol was reproduced in which phase transitions from antiphase to in-phase were induced with increasing movement frequency. The results showed that as movement frequency was scaled-up the amount of relative phase variability increased and the effector movements became more dependent prior to the transition. This is consistent with previous modeling showing that stronger effector coupling can prevent the occurrence of phase transitions when long range correlations in relative phase are present. It appears that, as movement frequency is scaled up, increases in effector coupling strength minimize loss of pattern stability and delay the onset of phase transitions.

Comfortable synchronization of cyclic drawing movements with a metronome

Continuous circle drawing is considered a paragon of emergent timing, whereas the timing of finger tapping is said to be event-based. Synchronization with a metronome, however, must to some extent be event-based for both types of movement. Because the target events in the movement trajectory are more poorly defined in circle drawing than in tapping, circle drawing shows more variable asynchronies with a metronome than does tapping. One factor that may have contributed to high variability in past studies is that circle size, drawing direction, and target point were prescribed and perhaps outside the comfort range. In the present study, participants were free to choose most comfortable settings of these parameters for two continuously drawn shapes, circles and infinity signs, while synchronizing with a regular or intermittently perturbed metronome at four different tempi. Results showed that preferred circle sizes were generally smaller than in previous studies but tended to increase as tempo decreased. Synchronization results were similar for circles and infinity signs, and similar to earlier results for circles drawn within a fixed template (Repp & Steinman, 2010). Comparison with tapping data still showed drawing to exhibit much greater variability and persistence of asynchronies as well as slower phase correction in response to phase shifts in the metronome. With comfort level ruled out as a factor, these differences can now be attributed more confidently to differences in event definition and/or movement dynamics.

The correlation structure of relative phase variability influences the occurrence of phase transition in coordination

The coordination dynamics framework has presumed that fluctuations in relative phase series produced in bimanual coordination are random. However, results from recent studies have shown that relative phase series contain 1/f(beta) noise (persistent long-range correlations) instead. Using an incremental protocol in line with the paradigmatic bimanual coordination framework, the author shows that the movement frequencies at which individuals spontaneously switch from anti-phase to in-phase coordination are significantly correlated with the intensity of long-range correlations but not with the amplitude of baseline fluctuations in relative phase. This finding illustrates the tangible relationship between present theoretical perspectives and accumulating evidence for 1/f(beta) noise. The author underscores the heuristic potential of systematic efforts to bridge the gap between present theories and the pervasive findings of 1/f(beta) noise.

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.

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.

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.