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

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.

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.

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.

Aging induced loss of complexity and dedifferentiation: consequences for coordination dynamics within and between brain, muscular and behavioral levels

Growing evidence demonstrates that aging not only leads to structural and functional alterations of individual components of the neuro-musculo-skeletal system (NMSS) but also results in a systemic re-organization of interactions within and between the different levels and functional domains. Understanding the principles that drive the dynamics of these re-organizations is an important challenge for aging research. The present Hypothesis and Theory paper is a contribution in this direction. We propose that age-related declines in brain and behavior that have been characterized in the literature as dedifferentiation and the loss of complexity (LOC) are: (i) synonymous; and (ii) integrated. We argue that a causal link between the aforementioned phenomena exists, evident in the dynamic changes occurring in the aging NMSS. Through models and methods provided by a dynamical systems approach to coordination processes in complex living systems, we: (i) formalize operational hypotheses about the general principles of changes in cross-level and cross-domain interactions during aging; and (ii) develop a theory of the aging NMSS based on the combination of the frameworks of coordination dynamics (CD), dedifferentiation, and LOC. Finally, we provide operational predictions in the study of aging at neural, muscular, and behavioral levels, which lead to testable hypotheses and an experimental agenda to explore the link between CD, LOC and dedifferentiation within and between these different levels.

Degeneracy and long-range correlations

Degeneracy is a ubiquitous property of complex adaptive systems, which refers to the ability of structurally different components to perform the same function in some conditions and different functions in other conditions. Here, we suppose a causal link between the level of degeneracy in the system and the strength of long-range correlations in its behavior. In a numerical experiment, we manipulated degeneracy through the number of networks available in a model composed of a chain of correlated networks over which a series of random jumps are performed. Results showed that correlations in the outcome series increased with the number of available networks, and that a minimal threshold of degeneracy was required to generate long-range correlations. We conclude that degeneracy could underlie the presence of long-range correlations in the outcome series produced by complex systems. In turn, we suggest that quantifying long-range correlations could allow to assess the level of degeneracy of the system. Degeneracy affords a maybe more intuitive way than former hypotheses for understanding the effects of complexity on essential properties such as robustness and adaptability.

Identifying multiplicative interactions between temporal scales of human movement variability

Conventional scaling analyses of human movement variability such as detrended fluctuation analyses assume that the movement variable can be decomposed into scale-dependent variation. However, these conventional scaling analyses are insensitive to multiplicative interactions within the movement variable. Multiplicative interactions refer to couplings between the scale-dependent variations across multiple scales that generate intermittent changes in the human movement variable. The mathematical concept for intermittent variability generated by multiplicative interactions is called multifractal variability. Multifractal variability is numerically defined by a spectrum of scaling exponents (i.e., a multifractal spectrum) that can be an important feature of coordinated movements and, consequently, relevant when aiming to identify movement disorders. In the current study, a new method is introduced based on detrended fluctuation analysis that can identify the multifractal spectrum from the temporal variation of local scaling exponents. The influence of multiplicative interactions on the local scaling exponents is tested by a Monte Carlo surrogate test. The methods are validated on multiplicative cascading processes with known multiplicative interactions. The application of the new methods is subsequently illustrated on an example of centre of pressure variations during quiet and relaxed standing. The results show that multiplicative interactions are present during periods with large movements of the center of gravity, where the movements of the centre of gravity and centre of pressure couple into coordinative structures. Further application and interpretation of the developed method for the study of human movement variability are discussed.

A tutorial introduction to adaptive fractal analysis

The authors present a tutorial description of adaptive fractal analysis (AFA). AFA utilizes an adaptive detrending algorithm to extract globally smooth trend signals from the data and then analyzes the scaling of the residuals to the fit as a function of the time scale at which the fit is computed. The authors present applications to synthetic mathematical signals to verify the accuracy of AFA and demonstrate the basic steps of the analysis. The authors then present results from applying AFA to time series from a cognitive psychology experiment on repeated estimation of durations of time to illustrate some of the complexities of real-world data. AFA shows promise in dealing with many types of signals, but like any fractal analysis method there are special challenges and considerations to take into account, such as determining the presence of linear scaling regions.

Relative roughness: an index for testing the suitability of the monofractal model

Fractal analyses have become very popular and have been applied on a wide variety of empirical time series. The application of these methods supposes that the monofractal framework can offer a suitable model for the analyzed series. However, this model takes into account a quite specific kind of fluctuations, and we consider that fractal analyses have been often applied to series that were completely outside of its relevance. The problem is that fractal methods can be applied to all types of series, and they always give a result, that one can then erroneously interpret in the context of the monofractal framework. We propose in this paper an easily computable index, the relative roughness (RR), defined as the ratio between local and global variances, that allows to test for the applicability of fractal analyses. We show that RR is confined within a limited range (between 1.21 and 0.12, approximately) for long-range correlated series. We propose some examples of empirical series that have been recently analyzed using fractal methods, but, with respect to their RR, should not have been considered in the monofractal model. An acceptable level of RR, however, is a necessary but not sufficient condition for considering series as long-range correlated. Specific methods should be used in complement for testing for the effective presence of long-range correlations in empirical series.

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.

Motor imagery may incorporate trial-to-trial error

The authors tested for 1/f noise in motor imagery (MI). Participants pointed and imagined pointing to a single target (Experiment 1), to targets of varied size (Experiment 2), and switched between pointing and grasping (Experiment 3). Experiment 1 showed comparable patterns of serial correlation in actual and imagined movement. Experiment 2 suggested increased correlation for MI and performance with increased task difficulty, perhaps reflecting adaptation to a more complex environment. Experiment 3 suggested a parallel decrease in correlation with task switching, perhaps reflecting discontinuity of mental set. Although present results do not conclusively reveal 1/f fluctuation, the emergent patterns suggest that MI could incorporate trial-to-trial error across a range of constraints.

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.

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.

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.

Fractal dynamics of human gait: a reassessment of the 1996 data of Hausdorff et al

We propose in this paper a reassessment of the original data of Hausdorff et al. (Hausdorff JM, Purdon PL, Peng C-K, Ladin Z, Wei JY, Goldberger AR. J Appl Physiol 80: 1448-1457, 1996). We confirm, using autoregressive fractionally integrated moving average modeling, the presence of genuine fractal correlations in stride interval series in self-paced conditions. In contrast with the conclusions of the authors, we show that correlations did not disappear in metronomic conditions. The series of stride intervals presented antipersistent correlations, and 1/f fluctuations were evidenced in the asynchronies to the metronome. We show that the super central pattern generator model (West B, Scafetta N. Phys Rev E Stat Nonlin Soft Matter Phys 67: 051917, 2003) allows accounting for the experimentally observed correlations in both self-paced and metronomic conditions, by the simple setting of the coupling strength parameter. We conclude that 1/f fluctuations in gait are not overridden by supraspinal influences when walking is paced by a metronome. The source of 1/f noise is still at work in this condition, but expressed differently under the influence of a continuous coupling process.

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.

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.