Kjerstin Torre

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.

Contemporary theories of 1/f noise in motor control

1/f noise has been discovered in a number of time series collected in psychological and behavioral experiments. This ubiquitous phenomenon has been ignored for a long time and classical models were not designed for accounting for these long-range correlations. The aim of this paper is to present and discuss contrasted theoretical perspectives on 1/f noise, in order to provide a comprehensive overview of current debates in this domain. In a first part, we propose a formal definition of the phenomenon of 1/f noise, and we present some commonly used methods for measuring long-range correlations in time series. In a second part, we develop a theoretical position that considers 1/f noise as the hallmark of system complexity. From this point of view, 1/f noise emerges from the coordination of the many elements that compose the system. In a third part, we present a theoretical counterpoint suggesting that 1/f noise could emerge from localized sources within the system. In conclusion, we try to draw some lines of reasoning for going beyond the opposition between these two approaches.

Theories and models for 1/fβ 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.

Transition from persistent to anti-persistent correlations in postural sway indicates velocity-based control.

The displacement of the center-of-pressure (COP) during quiet stance has often been accounted for by the control of COP position dynamics. In this paper, we discuss the conclusions drawn from previous analyses of COP dynamics using fractal-related methods. On the basis of some methodological clarification and the analysis of experimental data using stabilogram diffusion analysis, detrended fluctuation analysis, and an improved version of spectral analysis, we show that COP velocity is typically bounded between upper and lower limits. We argue that the hypothesis of an intermittent velocity-based control of posture is more relevant than position-based control. A simple model for COP velocity dynamics, based on a bounded correlated random walk, reproduces the main statistical signatures evidenced in the experimental series. The implications of these results are discussed.

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.

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.

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

Abstract1/fβ 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/fβ 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/fβ 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/fβ 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/fβ 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.

1/fβ 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/fβ noise. Diverging accounts of bimanual coordination are defended in the literature. Since the evidence for 1/fβ 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/fβ noise in DRP. None of these models was proven to generate the expected correlation structure.

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.

Long-range correlation properties in motor timing are individual and task specific

Abstract1/fβ noise represents a specific form of (long-range) correlations in a time series that is pervasive across many sensorimotor variables. Recent studies have shown that the precise properties of the correlations demonstrated by a group of test participants may vary as a function of experimental conditions or factors characterizing the group. Our purpose in the present study was to clarify whether long-range correlations affect sensorimotor performance generally or in a task-specific manner and whether each individual produces characteristic long-range correlations that are reliable across several runs of the same task. We analyzed the series of time intervals produced by 43 participants in two timing tasks: unimanual rhythmic tapping and circle drawing. We found that a participant’s 1/fβ properties in tapping were not related to the 1/fβ properties in circle drawing. However, within each task, individual differences were reliable, and a Cronbach’s alpha of .59 showed a high degree of within-subjects reproducibility of the long-range correlations. Thus, long-range correlations represent a consistent and distinctive characteristic of individuals performing a particular task, rather than a ubiquitous generic property of sensorimotor time series. The implications of these results are discussed from both a theoretical and a methodological perspective.