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

A linking hypothesis for eyetracking and mousetracking in the visual world paradigm

For a linking hypothesis in the visual world paradigm to clearly accommodate existing findings and make unambiguous predictions, it needs to be computationally implemented in a fashion that transparently draws the causal connection between the activations of internal representations and the measured output of saccades and reaching movements. Quantitatively implemented linking hypotheses provide an opportunity to not only demonstrate an existence proof of that causal connection but also to test the fidelity of the measuring methods themselves. When a system of interest is measured one way (e.g., ballistic dichotomous outputs) or another way (e.g., smooth graded outputs), the apparent results can differ substantially. What is needed is one linking hypothesis that can produce both types of outputs. The localist attractor network simulation of spoken word recognition demonstrated here recreates eye and mouse movements that capture key findings in the visual world paradigm, and especially relies on one particularly powerful theoretical construct: feedback from the action-perception cycle. Visual feedback from the eye position enhancing the cognitive prominence of the fixated object allows the simulation to fit a wider range of findings, and points to predictions for new experiments. When that feedback is absent, the linking hypothesis simulation no longer fits human data as well. Future experiments, and improvements of this network simulation, are discussed.

Pink noise promotes sooner state transitions during bimanual coordination

The seemingly straightforward task of tying one's shoes requires a sophisticated interplay of joints, muscles, and neural pathways, posing a formidable challenge for researchers studying the intricacies of coordination. A widely accepted framework for measuring coordinated behavior is the Haken-Kelso-Bunz (HKB) model. However, a significant limitation of this model is its lack of accounting for the diverse variability structures inherent in the coordinated systems it frequently models. Variability is a pervasive phenomenon across various biological and physical systems, and it changes in healthy adults, older adults, and pathological populations. Here, we show, both empirically and with simulations, that manipulating the variability in coordinated movements significantly impacts the ability to change coordination patterns-a fundamental feature of the HKB model. Our results demonstrate that synchronized bimanual coordination, mirroring a state of healthy variability, instigates earlier transitions of coordinated movements compared to other variability conditions. This suggests a heightened adaptability when movements possess a healthy variability. We anticipate our study to show the necessity of adapting the HKB model to encompass variability, particularly in predictive applications such as neuroimaging, cognition, skill development, biomechanics, and beyond.

Visual feedback modulates the 1/f structure of movement amplitude time series

In our prior studies, human participants were required to generate long sequences of targeted hand movement when task difficulty varied between conditions, and where full vision of the hand and target was always available. The movement amplitude-that is, the actual distance travelled-for each movement was measured; consecutive movement amplitude values were formed into time series; then, the time series were submitted to spectral analysis. As task difficulty increased, there was a pink-to-white-noise shift in movement amplitude time-series structure. Those changes could be attributed to a difficulty-induced increase in the need to engage visual feedback processes, which maintain accurate guidance of the hand to the target. The current study was designed to provide a more direct test of the hypothesis that difficulty-induced increases in visual feedback processing modulate movement amplitude time-series structure. To that end, we examined cyclical aiming performance under four unique conditions created from the crossing of two index of difficulty (2 and 5 bits) and two visual feedback (visual feedback and no-visual feedback) conditions. That allowed us to examine how variations in visual feedback quality might influence difficulty-induced changes in time-series structure. In the visual feedback condition, we predicted that the increase in difficulty should result in a pink-to-white-noise shift in time-series structure. If that expected shift resulted from increased engagement of visual feedback processing, then in the no-visual feedback condition-where visual feedback processing was disabled-we should observe a strengthened pink-noise time-series structure that does not change with the increase in difficulty. The current results confirmed those predictions. That provides further support for the hypothesis that engagement of closed-loop visual feedback processing modulates movement amplitude time-series structure.

Temporal Structure in Sensorimotor Variability: A Stable Trait, But What For?

Human performance shows substantial endogenous variability over time, and this variability is a robust marker of individual differences. Of growing interest to psychologists is the realisation that variability is not fully random, but often exhibits temporal dependencies. However, their measurement and interpretation come with several controversies. Furthermore, their potential benefit for studying individual differences in healthy and clinical populations remains unclear. Here, we gather new and archival datasets featuring 11 sensorimotor and cognitive tasks across 526 participants, to examine individual differences in temporal structures. We first investigate intra-individual repeatability of the most common measures of temporal structures - to test their potential for capturing stable individual differences. Secondly, we examine inter-individual differences in these measures using: (1) task performance assessed from the same data, (2) meta-cognitive ratings of on-taskness from thought probes occasionally presented throughout the task, and (3) self-assessed attention-deficit related traits. Across all datasets, autocorrelation at lag 1 and Power Spectra Density slope showed high intra-individual repeatability across sessions and correlated with task performance. The Detrended Fluctuation Analysis slope showed the same pattern, but less reliably. The long-term component (d) of the ARFIMA(1,d,1) model showed poor repeatability and no correlation to performance. Overall, these measures failed to show external validity when correlated with either mean subjective attentional state or self-assessed traits between participants. Thus, some measures of serial dependencies may be stable individual traits, but their usefulness in capturing individual differences in other constructs typically associated with variability in performance seems limited. We conclude with comprehensive recommendations for researchers.

Supplementary Information

The online version contains supplementary material available at 10.1007/s42113-022-00162-1.

Early Warning Signals in Phase Space: Geometric Resilience Loss Indicators From Multiplex Cumulative Recurrence Networks

The detection of Early Warning Signals (EWS) of imminent phase transitions, such as sudden changes in symptom severity could be an important innovation in the treatment or prevention of disease or psychopathology. Recurrence-based analyses are known for their ability to detect differences in behavioral modes and order transitions in extremely noisy data. As a proof of principle, the present paper provides an example of a recurrence network based analysis strategy which can be implemented in a clinical setting in which data from an individual is continuously monitored for the purpose of making decisions about diagnosis and intervention. Specifically, it is demonstrated that measures based on the geometry of the phase space can serve as Early Warning Signals of imminent phase transitions. A publicly available multivariate time series is analyzed using so-called cumulative Recurrence Networks (cRN), which are recurrence networks with edges weighted by recurrence time and directed towards previously observed data points. The results are compared to previous analyses of the same data set, benefits, limitations and future directions of the analysis approach are discussed.

Dynamic Systems Approach in Sensorimotor Synchronization: Adaptation to Tempo Step-Change

This paper presents a dynamic systems model of a sensorimotor synchronization (SMS) task. An SMS task typically gives temporally discrete human responses to some temporally discrete stimuli. Here, a dynamic systems modeling approach is applied after converting the discrete events to regularly sampled time signals. To collect data for model parameter fitting, a previously published pilot study was expanded. Three human participants took part in an experiment: to tap a finger on a keyboard, following a metronome which changed tempo in steps. System identification was used to estimate the transfer function that represented the relationship between the stimulus and the step response signals, assuming a separate linear, time-invariant system for each tempo step. Different versions of model complexity were investigated. As a minimum, a second-order linear system with delay, two poles, and one zero was needed to model the most important features of the tempo step response by humans, while an additional third pole could give a somewhat better fit to the response data. The modeling results revealed the behavior of the system in two distinct regimes: tempo steps below and above the conscious awareness of tempo change, i.e., around 12% of the base tempo. For the tempo steps above this value, model parameters were derived as linear functions of step size for the group of three participants. The results were interpreted in the light of known facts from other fields like SMS, psychoacoustics and behavioral neuroscience.

Fractal properties and short-term correlations in motor control in cycling: influence of a cognitive challenge

Fluctuations in cyclic tasks periods is a known characteristic of human motor control. Specifically, long-range fractal fluctuations have been evidenced in the temporal structure of these variations in human locomotion and thought to be the outcome of a multicomponent physiologic system in which control is distributed across intricate cortical, spinal and neuromuscular regulation loops. Combined with long-range correlation analyses, short-range autocorrelations have proven their use to describe control distribution across central and motor components. We used relevant tools to characterize long- and short-range correlations in revolution time series during cycling on an ergometer in 19 healthy young adults. We evaluated the impact of introducing a cognitive task (PASAT) to assess the role of central structures in control organization. Autocorrelation function and detrending fluctuation analysis (DFA) demonstrated the presence of fractal scaling. PSD in the short range revealed a singular behavior which cannot be explained by the usual models of even-based and emergent timing. The main outcomes are that (1) timing in cycling is a fractal process, (2) this long-range fractal behavior increases in persistence with dual-task condition, which has not been previously observed, (3) short-range behavior is highly persistent and unaffected by dual-task. Relying on the inertia of the oscillator may be a way to distribute more control to the periphery, thereby allocating less resources to central process and better managing additional cognitive demands. This original behavior in cycling may explain the high short-range persistence unaffected by dual-task, and the increase in long-range persistence with dual-task.

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.

The Over-Extended Mind? Pink Noise and the Ethics of Interaction-Dominant Systems

There is a growing recognition within cognitive enhancement and neuroethics debates of the need for greater emphasis on cognitive artefacts. This paper aims to contribute to this broadening and expansion of the cognitive-enhancement and neuroethics debates by focusing on a particular form of relation or coupling between humans and cognitive artefacts: interaction-dominance. We argue that interaction-dominance as an emergent property of some human-cognitive artefact relations has important implications for understanding the attribution and distribution of causal and other forms of responsibility as well as agency relating to the actions of human-cognitive artefact couplings. Interaction-dominance is both indicated and constituted by the phenomenon of "pink noise". Understanding the role of noise in this regard will establish a necessary theoretical groundwork for approaching the ethical and political dimensions of relations between human cognition and digital cognitive artefacts. We argue that pink noise in this context plays a salient role in the practical, ethical, and political evaluation of coupling relations between humans and cognitive artefacts, and subsequently in the responsible innovation of cognitive artefacts and human-artefact interfaces.

Hearing the Sound in the Brain: Influences of Different EEG References

If the scalp potential signals, the electroencephalogram (EEG), are due to neural "singers" in the brain, how could we listen to them with less distortion? One crucial point is that the data recording on the scalp should be faithful and accurate, thus the choice of reference electrode is a vital factor determining the faithfulness of the data. In this study, music on the scalp derived from data in the brain using three different reference electrodes were compared, including approximate zero reference-reference electrode standardization technique (REST), average reference (AR), and linked mastoids reference (LM). The classic music pieces in waveform format were used as simulated sources inside a head model, and they were forward calculated to scalp as standard potential recordings, i.e., waveform format music from the brain with true zero reference. Then these scalp music was re-referenced into REST, AR, and LM based data, and compared with the original forward data (true zero reference). For real data, the EEG recorded in an orthodontic pain control experiment were utilized for music generation with the three references, and the scale free index (SFI) of these music pieces were compared. The results showed that in the simulation for only one source, different references do not change the music/waveform; for two sources or more, REST provide the most faithful music/waveform to the original ones inside the brain, and the distortions caused by AR and LM were spatial locations of both source and scalp electrode dependent. The brainwave music from the real EEG data showed that REST and AR make the differences of SFI between two states more recognized and found the frontal is the main region that producing the music. In conclusion, REST can reconstruct the true signals approximately, and it can be used to help to listen to the true voice of the neural singers in the brain.

Correlated microtiming deviations in jazz and rock music

Musical rhythms performed by humans typically show temporal fluctuations. While they have been characterized in simple rhythmic tasks, it is an open question what is the nature of temporal fluctuations, when several musicians perform music jointly in all its natural complexity. To study such fluctuations in over 100 original jazz and rock/pop recordings played with and without metronome we developed a semi-automated workflow allowing the extraction of cymbal beat onsets with millisecond precision. Analyzing the inter-beat interval (IBI) time series revealed evidence for two long-range correlated processes characterized by power laws in the IBI power spectral densities. One process dominates on short timescales (t < 8 beats) and reflects microtiming variability in the generation of single beats. The other dominates on longer timescales and reflects slow tempo variations. Whereas the latter did not show differences between musical genres (jazz vs. rock/pop), the process on short timescales showed higher variability for jazz recordings, indicating that jazz makes stronger use of microtiming fluctuations within a measure than rock/pop. Our results elucidate principles of rhythmic performance and can inspire algorithms for artificial music generation. By studying microtiming fluctuations in original music recordings, we bridge the gap between minimalistic tapping paradigms and expressive rhythmic performances.

Classifying the variability in impact and active peak vertical ground reaction forces during running using DFA and ARFIMA models

The vertical ground reaction force (VGRF) during rear-foot striking running typically exhibits peaks referred to as the impact peak and the active peak; their timings and magnitudes have been implicated in injury. Identifying the structure of time-series can provide insight into associated control processes. The purpose here was to detect long-range correlations associated with the time from first contact to impact peak (TIP) and active peak (TAP); and the magnitudes of impact (IPM) and active peaks (APM) using a Detrended Fluctuation Analysis, and Auto-Regressive Fractionally Integrated Moving Average models. Twelve subjects performed an 8min trial at their preferred running speed on an instrumented treadmill. TIP, TAP; IPM, and APM were identified from the VGRF profile for each footfall. TIP and TAP time-series did not demonstrate long-range correlations, conversely IPM and APM time-series did. Short range correlations appeared as well as or instead of long range correlations for IPM. Conversely pure powerlaw behaviour was demonstrated in 11 of the 24 time series for APM, and long range dependencies along with short range correlations were present in a further 9 time series. It has been hypothesised that control mechanisms for IPM and APM are different, these results support this hypothesis.

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.

Fluctuations of hi-hat timing and dynamics in a virtuoso drum track of a popular music recording

Long-range correlated temporal fluctuations in the beats of musical rhythms are an inevitable consequence of human action. According to recent studies, such fluctuations also lead to a favored listening experience. The scaling laws of amplitude variations in rhythms, however, are widely unknown. Here we use highly sensitive onset detection and time series analysis to study the amplitude and temporal fluctuations of Jeff Porcaro’s one-handed hi-hat pattern in “I Keep Forgettin’”—one of the most renowned 16th note patterns in modern drumming. We show that fluctuations of hi-hat amplitudes and interbeat intervals (times between hits) have clear long-range correlations and short-range anticorrelations separated by a characteristic time scale. In addition, we detect subtle features in Porcaro’s drumming such as small drifts in the 16th note pulse and non-trivial periodic two-bar patterns in both hi-hat amplitudes and intervals. Through this investigation we introduce a step towards statistical studies of the 20th and 21st century music recordings in the framework of complex systems. Our analysis has direct applications to the development of drum machines and to drumming pedagogy.

Self-organized criticality, plasticity and sensorimotor coupling. Explorations with a neurorobotic model in a behavioural preference task

During the last two decades, analysis of 1/ƒ noise in cognitive science has led to a considerable progress in the way we understand the organization of our mental life. However, there is still a lack of specific models providing explanations of how 1/ƒ noise is generated in coupled brain-body-environment systems, since existing models and experiments typically target either externally observable behaviour or isolated neuronal systems but do not address the interplay between neuronal mechanisms and sensorimotor dynamics. We present a conceptual model of a minimal neurorobotic agent solving a behavioural task that makes it possible to relate mechanistic (neurodynamic) and behavioural levels of description. The model consists of a simulated robot controlled by a network of Kuramoto oscillators with homeostatic plasticity and the ability to develop behavioural preferences mediated by sensorimotor patterns. With only three oscillators, this simple model displays self-organized criticality in the form of robust 1/ƒ noise and a wide multifractal spectrum. We show that the emergence of self-organized criticality and 1/ƒ noise in our model is the result of three simultaneous conditions: a) non-linear interaction dynamics capable of generating stable collective patterns, b) internal plastic mechanisms modulating the sensorimotor flows, and c) strong sensorimotor coupling with the environment that induces transient metastable neurodynamic regimes. We carry out a number of experiments to show that both synaptic plasticity and strong sensorimotor coupling play a necessary role, as constituents of self-organized criticality, in the generation of 1/ƒ noise. The experiments also shown to be useful to test the robustness of 1/ƒ scaling comparing the results of different techniques. We finally discuss the role of conceptual models as mediators between nomothetic and mechanistic models and how they can inform future experimental research where self-organized critically includes sensorimotor coupling among the essential interaction-dominant process giving rise to 1/ƒ noise.

Embodiment of intersubjective time: relational dynamics as attractors in the temporal coordination of interpersonal behaviors and experiences

This paper addresses the issue of “being together,” and more specifically the issue of “being together in time.” We provide with an integrative framework that is inspired by phenomenology, the enactive approach and dynamical systems theories. To do so, we first define embodiment as a living and lived phenomenon that emerges from agent-world coupling. We then show that embodiment is essentially dynamical and therefore we describe experiential, behavioral and brain dynamics. Both lived temporality and the temporality of the living appear to be complex, multiscale phenomena. Next we discuss embodied dynamics in the context of interpersonal interactions, and briefly review the empirical literature on between-persons temporal coordination. Overall, we propose that being together in time emerges from the relational dynamics of embodied interactions and their flexible co-regulation.

Long-range correlation properties in timing of skilled piano performance: the influence of auditory feedback and deep brain stimulation

Unintentional timing deviations during musical performance can be conceived of as timing errors. However, recent research on humanizing computer-generated music has demonstrated that timing fluctuations that exhibit long-range temporal correlations (LRTC) are preferred by human listeners. This preference can be accounted for by the ubiquitous presence of LRTC in human tapping and rhythmic performances. Interestingly, the manifestation of LRTC in tapping behavior seems to be driven in a subject-specific manner by the LRTC properties of resting-state background cortical oscillatory activity. In this framework, the current study aimed to investigate whether propagation of timing deviations during the skilled, memorized piano performance (without metronome) of 17 professional pianists exhibits LRTC and whether the structure of the correlations is influenced by the presence or absence of auditory feedback. As an additional goal, we set out to investigate the influence of altering the dynamics along the cortico-basal-ganglia-thalamo-cortical network via deep brain stimulation (DBS) on the LRTC properties of musical performance. Specifically, we investigated temporal deviations during the skilled piano performance of a non-professional pianist who was treated with subthalamic-deep brain stimulation (STN-DBS) due to severe Parkinson's disease, with predominant tremor affecting his right upper extremity. In the tremor-affected right hand, the timing fluctuations of the performance exhibited random correlations with DBS OFF. By contrast, DBS restored long-range dependency in the temporal fluctuations, corresponding with the general motor improvement on DBS. Overall, the present investigations demonstrate the presence of LRTC in skilled piano performances, indicating that unintentional temporal deviations are correlated over a wide range of time scales. This phenomenon is stable after removal of the auditory feedback, but is altered by STN-DBS, which suggests that cortico-basal ganglia-thalamocortical circuits play a role in the modulation of the serial correlations of timing fluctuations exhibited in skilled musical performance.

Spectral convergence in tapping and physiological fluctuations: coupling and independence of 1/f noise in the central and autonomic nervous systems

When humans perform a response task or timing task repeatedly, fluctuations in measures of timing from one action to the next exhibit long-range correlations known as 1/f noise. The origins of 1/f noise in timing have been debated for over 20 years, with one common explanation serving as a default: humans are composed of physiological processes throughout the brain and body that operate over a wide range of timescales, and these processes combine to be expressed as a general source of 1/f noise. To test this explanation, the present study investigated the coupling vs. independence of 1/f noise in timing deviations, key-press durations, pupil dilations, and heartbeat intervals while tapping to an audiovisual metronome. All four dependent measures exhibited clear 1/f noise, regardless of whether tapping was synchronized or syncopated. 1/f spectra for timing deviations were found to match those for key-press durations on an individual basis, and 1/f spectra for pupil dilations matched those in heartbeat intervals. Results indicate a complex, multiscale relationship among 1/f noises arising from common sources, such as those arising from timing functions vs. those arising from autonomic nervous system (ANS) functions. Results also provide further evidence against the default hypothesis that 1/f noise in human timing is just the additive combination of processes throughout the brain and body. Our findings are better accommodated by theories of complexity matching that begin to formalize multiscale coordination as a foundation of human behavior.

Synchronization in human musical rhythms and mutually interacting complex systems

Though the music produced by an ensemble is influenced by multiple factors, including musical genre, musician skill, and individual interpretation, rhythmic synchronization is at the foundation of musical interaction. Here, we study the statistical nature of the mutual interaction between two humans synchronizing rhythms. We find that the interbeat intervals of both laypeople and professional musicians exhibit scale-free (power law) cross-correlations. Surprisingly, the next beat to be played by one person is dependent on the entire history of the other person's interbeat intervals on timescales up to several minutes. To understand this finding, we propose a general stochastic model for mutually interacting complex systems, which suggests a physiologically motivated explanation for the occurrence of scale-free cross-correlations. We show that the observed long-term memory phenomenon in rhythmic synchronization can be imitated by fractal coupling of separately recorded or synthesized audio tracks and thus applied in electronic music. Though this study provides an understanding of fundamental characteristics of timing and synchronization at the interbrain level, the mutually interacting complex systems model may also be applied to study the dynamics of other complex systems where scale-free cross-correlations have been observed, including econophysics, physiological time series, and collective behavior of animal flocks.

A Review of Theoretical Perspectives in Cognitive Science on the Presence of 1/f Scaling in Coordinated Physiological and Cognitive Processes

Time series of human performances present fluctuations around a mean value. These fluctuations are typically considered as insignificant, and attributable to random noise. Over recent decades, it became clear that temporal fluctuations possess interesting properties, however, one of which the property of fractal 1/f scaling. 1/f scaling indicates that a measured process extends over a wide range of timescales, suggesting an assembly over multiple scales simultaneously. This paper reviews neurological, physiological, and cognitive studies that corroborate the claim that 1/f scaling is most clearly present in healthy, well-coordinated activities. Prominent hypotheses about the origins of 1/f scaling are confronted with these reviewed studies. It is concluded that 1/f scaling in living systems appears to reflect their genuine complex nature, rather than constituting a coincidental side-effect. The consequences of fractal dynamics extending from the small spatial and temporal scales (e.g., neurons) to the larger scales of human behavior and cognition, are vast, and impact the way in which relevant research questions may be approached. Rather than focusing on specialized isolable subsystems, using additive linear methodologies, nonlinear dynamics, more elegantly so, imply a complex systems methodology, thereby exploiting, rather than rejecting, mathematical concepts that enable describing large sets of natural phenomena.

Movement variability near goal equivalent manifolds: fluctuations, control, and model-based analysis

Fluctuations in the repeated performance of human movements have been the subject of intense scrutiny because they are generally believed to contain important information about the function and health of the neuromotor system. A variety of approaches has been brought to bear to study these fluctuations. However it is frequently difficult to understand how to synthesize different perspectives to give a coherent picture. Here, we describe a conceptual framework for the experimental study of motor variability that helps to unify geometrical methods, which focus on the role of motor redundancy, with dynamical methods that characterize the error-correcting processes regulating the performance of skilled tasks. We describe how goal functions, which mathematically specify the task strategy being employed, together with ideas from the control of redundant systems, allow one to formulate simple, experimentally testable dynamical models of inter-trial fluctuations. After reviewing the basic theory, we present a list of five general hypotheses on the structure of fluctuations that can be expected in repeated trials of goal-directed tasks. We review recent experimental applications of this general approach, and show how it can be used to precisely characterize the error-correcting control used by human subjects.

Modulation of internal estimates of gravity during and after prolonged roll-tilts

Perceived direction of gravity, as assessed by the subjective visual vertical (SVV), shows roll-angle dependent errors that drift over time and a bias upon return to upright. According to Bayesian observer theory, the estimated direction of gravity is derived from the posterior probability distribution by combining sensory input and prior knowledge about earth-vertical in a statistically optimal fashion. Here we aimed to further characterize the stability of SVV during and after prolonged roll-tilts. Specifically we asked whether the post-tilt bias is related to the drift pattern while roll-tilted. Twenty-nine healthy human subjects (23-56 yo) repetitively adjusted a luminous arrow to the SVV over periods of 5 min while upright, roll-tilted (± 45°, ± 90°), and immediately after returning to upright. Significant (p<0.05) drifts (median absolute drift-amplitude: 10°/5 min) were found in 71% (± 45°) and 78% (± 90°) of runs. At ± 90° roll-tilt significant increases in absolute adjustment errors were more likely (76%), whereas significant increases (56%) and decreases (44%) were about equally frequent at ± 45°. When returning to upright, an initial bias towards the previous roll-position followed by significant exponential decay (median time-constant: 71 sec) was noted in 47% of all runs (all subjects pooled). No significant correlations were found between the drift pattern during and immediately after prolonged roll-tilt. We conclude that the SVV is not stable during and after prolonged roll-tilt and that the direction and magnitude of drift are individually distinct and roll-angle-dependent. Likely sensory and central adaptation and random-walk processes contribute to drift while roll-tilted. Lack of correlation between the drift and the post-tilt bias suggests that it is not the inaccuracy of the SVV estimate while tilted that determines post-tilt bias, but rather the previous head-roll orientation relative to gravity. We therefore favor central adaptation, most likely a shift in prior knowledge towards the previous roll orientation, to explain the post-tilt bias.

Statistical persistence and timing characteristics of repetitive circle drawing in children with ASD

OBJECTIVE

Standardized tests to assess movement difficulties in autism spectrum disorder (ASD) evaluate motor output, but do not provide information about underlying dynamics. The objective of this research is to study the statistical persistence and temporal dynamics of a circle drawing task in children with ASD.

METHODS

For this study 15 children diagnosed with ASD, aged 4-8 years, were asked to draw circles under various conditions using a computerized tablet. We then assessed fractal dynamics and global temporal dynamics (mean and coefficient of variation) and compared these quantities to those of typically developing (TD) controls.

RESULTS

No difference in statistical persistence was found between children with ASD and TD children. Temporal measures showed increased variability in the ASD population in the discontinuous task.

CONCLUSION

Results support the hypothesis that children with ASD have an intact ability to consistently produce continuous movements, but increased variability in production of discontinuous movements.

The role of the circadian system in fractal neurophysiological control

Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system's role in fractal regulation.

Does sample rate introduce an artifact in spectral analysis of continuous processes?

Spectral analysis is a widely used method to estimate 1/f^α noise in behavioral and physiological data series. The aim of this paper is to achieve a more solid appreciation for the effects of periodic sampling on the outcomes of spectral analysis. It is shown that spectral analysis is biased by the choice of sample rate because denser sampling comes with lower amplitude fluctuations at the highest frequencies. Here we introduce an analytical strategy that compensates for this effect by focusing on a fixed amount, rather than a fixed percentage of the lowest frequencies in a power spectrum. Using this strategy, estimates of the degree of 1/f^α noise become robust against sample rate conversion and more sensitive overall. Altogether, the present contribution may shed new light on known discrepancies in the psychological literature on 1/f^α noise, and may provide a means to achieve a more solid framework for 1/f^α noise in continuous processes.

Differential effects of visual feedback on subjective visual vertical accuracy and precision

The brain constructs an internal estimate of the gravitational vertical by integrating multiple sensory signals. In darkness, systematic head-roll dependent errors in verticality estimates, as measured by the subjective visual vertical (SVV), occur. We hypothesized that visual feedback after each trial results in increased accuracy, as physiological adjustment errors (A-/E-effect) are likely based on central computational mechanisms and investigated whether such improvements were related to adaptational shifts of perceived vertical or to a higher cognitive strategy. We asked 12 healthy human subjects to adjust a luminous arrow to vertical in various head-roll positions (0 to 120deg right-ear down, 15deg steps). After each adjustment visual feedback was provided (lights on, display of previous adjustment and of an earth-vertical cross). Control trials consisted of SVV adjustments without feedback. At head-roll angles with the largest A-effect (90, 105, and 120deg), errors were reduced significantly (p<0.001) by visual feedback, i.e. roll under-compensation decreased, while precision of SVV was not significantly (p>0.05) influenced. In seven subjects an additional session with two consecutive blocks (first with, then without visual feedback) was completed at 90, 105 and 120deg head-roll. In these positions the error-reduction by the previous visual feedback block remained significant over the consecutive 18-24 min (post-feedback block), i.e., was still significantly (p<0.002) different from the control trials. Eleven out of 12 subjects reported having consciously added a bias to their perceived vertical based on visual feedback in order to minimize errors. We conclude that improvements of SVV accuracy by visual feedback, which remained effective after removal of feedback for ≥18 min, rather resulted from a cognitive strategy than by adapting the internal estimate of the gravitational vertical. The mechanisms behind the SVV therefore, remained stable, which is also supported by the fact that SVV precision - depending mostly on otolith input - was not affected by visual feedback.

Temporal constancy of perceived direction of gravity assessed by visual line adjustments

Here we investigated how well internal estimates of direction of gravity are preserved over time and if the subjective visual vertical (SVV) and horizontal (SVH) can be used inter-changeably. Fourteen human subjects repetitively aligned a luminous line to SVV, SVH or subjective visual oblique (± 45°) over 5 min in otherwise complete darkness and also in dim light. Both accuracy (i.e., the degree of veracity as reflected by the median adjustment error) and precision (i.e., the degree of reproducability as reflected by the trial-to-trial variability) of adjustments along the principle axes were significantly higher than along the oblique axes. Orthogonality was only preserved in a minority of subjects. Adjustments were significantly different between SVV vs. SVH (7/14 subjects) and between ±45° vs. -45° (12/14) in darkness and in 6/14 and 14/14 subjects, respectively, in dim light. In darkness, significant drifts over 5min were observed in a majority of trials (33/56). Both accuracy and precision were higher if more time was taken to make the adjustment. These results introduce important caveats when interpreting studies related to graviception. The test re-test reliability of SVV and SVH can be influenced by drift of the internal estimate of gravity. Based on spectral density analysis we found a noise pattern consistent with 1/fβ noise, indicating that at least part of the trial-to-trial dynamics observed in our experiments is due to the dependence of the serial adjustments over time. Furthermore, using results from the SVV and SVH inter-changeably may be misleading as many subjects do not show orthogonality. The poor fidelity of perceived ± 45° indicates that the brain has limited ability to estimate oblique angles.

How stable is perceived direction of gravity over extended periods in darkness?

Previous studies reported linear drift of perceived vertical for brief (≤10 min) observation periods. Here, we repeated estimates of direction of gravity up to 60 min to evaluate whether the drift is sustained, shows saturation or even reverses over time. Fifteen healthy human subjects repetitively adjusted a luminous line along subjective visual vertical (SVV) and horizontal (SVH) over periods of 5 min (constituting one block). We obtained seven blocks within 60 min in each subject for SVV and SVH. In between the first six blocks, subjects remained in darkness for 5 min each, whereas the lights were briefly turned on before block 7. We noted significantly (p < 0.05) increased errors in perceived direction of gravity by block 2 (SVV) and 3 (SVH). These increases disappeared after turning on the lights before block 7. Focusing on blocks 2-6, significant drift started from similar offset positions and pointed to the same direction in a majority of runs in 9/15 (SVV) and 11/15 (SVH) subjects. When pooling data from all blocks, orthogonality of errors was lost in all subjects. Trial-to-trial variability remained stable over the seven runs for SVV and SVH. Only when pooling all runs, precision was significantly (p < 0.05) higher for the SVH. Our findings suggest that perceived direction of gravity continues to fluctuate over extended recording periods with individuals showing unique patterns of direction-specific drift while variability remains stable. As subjects were upright during the entire experiment and as drift persisted over several blocks, sensory adaptation seems unlikely. We therefore favor a central origin of this kind of drift.

Measuring fractality

WHEN INVESTIGATING FRACTAL PHENOMENA, THE FOLLOWING QUESTIONS ARE FUNDAMENTAL FOR THE APPLIED RESEARCHER: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators ([Formula: see text] power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series.

Interactive rhythmic auditory stimulation reinstates natural 1/f timing in gait of Parkinson's patients

Parkinson's disease (PD) and basal ganglia dysfunction impair movement timing, which leads to gait instability and falls. Parkinsonian gait consists of random, disconnected stride times—rather than the 1/f structure observed in healthy gait—and this randomness of stride times (low fractal scaling) predicts falling. Walking with fixed-tempo Rhythmic Auditory Stimulation (RAS) can improve many aspects of gait timing; however, it lowers fractal scaling (away from healthy 1/f structure) and requires attention. Here we show that interactive rhythmic auditory stimulation reestablishes healthy gait dynamics in PD patients. In the experiment, PD patients and healthy participants walked with a) no auditory stimulation, b) fixed-tempo RAS, and c) interactive rhythmic auditory stimulation. The interactive system used foot sensors and nonlinear oscillators to track and mutually entrain with the human's step timing. Patients consistently synchronized with the interactive system, their fractal scaling returned to levels of healthy participants, and their gait felt more stable to them. Patients and healthy participants rarely synchronized with fixed-tempo RAS, and when they did synchronize their fractal scaling declined from healthy 1/f levels. Five minutes after removing the interactive rhythmic stimulation, the PD patients' gait retained high fractal scaling, suggesting that the interaction stabilized the internal rhythm generating system and reintegrated timing networks. The experiment demonstrates that complex interaction is important in the (re)emergence of 1/f structure in human behavior and that interactive rhythmic auditory stimulation is a promising therapeutic tool for improving gait of PD patients.

1/f Noise in human color vision: the role of S-cone signals

We examine the functional role of S-cone signals on reaction time (RT) variability in human color vision. Stimuli were selected along red-green and blue-yellow cardinal directions and at random directions in the isoluminant plane of the color space. Trial-to-trial RT variability was not statistically independent but correlated across experimental conditions and exhibited 1/f noise spectra with an exponent close to unity in most of the cases. Regarding contrast coding, 1/f noise for random chromatic stimuli at isoluminance was similar to that for achromatic stimuli, thus suggesting that S-cone signals reduce variability of higher order color mechanisms. If we regard spatial coding, the effect of S-cone density in the retina on RT variability was investigated. The magnitude of 1/f noise at 16 min of arc (S-cone free zone) was higher than at 90 min of arc in the blue-yellow channel, and it was similar for the red-green channel. The results suggest that S-cone signals are beneficial and they modulate 1/f noise spectra at postreceptoral stages. The implications related to random multiplicative processes as a possible source of 1/f noise and the optimal information processing in color vision are discussed.

Effects of accuracy feedback on fractal characteristics of time estimation

The current experiment investigated the effect of visual accuracy feedback on the structure of variability of time interval estimates in the continuation tapping paradigm. Participants were asked to repeatedly estimate a 1-s interval for a prolonged period of time by tapping their index finger. In some conditions, participants received accuracy feedback after every estimate, whereas in other conditions, no feedback was given. Also, the likelihood of receiving visual feedback was manipulated by adjusting the tolerance band around the 1-s target interval so that feedback was displayed only if the temporal estimate deviated from the target interval by more than 50, 100, or 200 ms respectively. We analyzed the structure of variability of the inter-tap intervals with fractal and multifractal methods that allow for a quantification of complex long-range correlation patterns in the timing performance. Our results indicate that feedback changes the long-range correlation structure of time estimates: Increased amounts of feedback lead to a decrease in fractal long-range correlations, as well to a decrease in the magnitude of local fluctuations in the performance. The multifractal characteristics of the time estimates were not impacted by the presence of accuracy feedback. Nevertheless, most of the data sets show significant multifractal signatures. We interpret these findings as showing that feedback acts to constrain and possibly reorganize timing performance. Implications for mechanistic and complex systems-based theories of timing behavior are discussed.

Exploring the fundamental dynamics of error-based motor learning using a stationary predictive-saccade task

The maintenance of movement accuracy uses prior performance errors to correct future motor plans; this motor-learning process ensures that movements remain quick and accurate. The control of predictive saccades, in which anticipatory movements are made to future targets before visual stimulus information becomes available, serves as an ideal paradigm to analyze how the motor system utilizes prior errors to drive movements to a desired goal. Predictive saccades constitute a stationary process (the mean and to a rough approximation the variability of the data do not vary over time, unlike a typical motor adaptation paradigm). This enables us to study inter-trial correlations, both on a trial-by-trial basis and across long blocks of trials. Saccade errors are found to be corrected on a trial-by-trial basis in a direction-specific manner (the next saccade made in the same direction will reflect a correction for errors made on the current saccade). Additionally, there is evidence for a second, modulating process that exhibits long memory. That is, performance information, as measured via inter-trial correlations, is strongly retained across a large number of saccades (about 100 trials). Together, this evidence indicates that the dynamics of motor learning exhibit complexities that must be carefully considered, as they cannot be fully described with current state-space (ARMA) modeling efforts.

Kinematic variability, fractal dynamics and local dynamic stability of treadmill walking

Background

Motorized treadmills are widely used in research or in clinical therapy. Small kinematics, kinetics and energetics changes induced by Treadmill Walking (TW) as compared to Overground Walking (OW) have been reported in literature. The purpose of the present study was to characterize the differences between OW and TW in terms of stride-to-stride variability. Classical (Standard Deviation, SD) and non-linear (fractal dynamics, local dynamic stability) methods were used. In addition, the correlations between the different variability indexes were analyzed.

Methods

Twenty healthy subjects performed 10 min TW and OW in a random sequence. A triaxial accelerometer recorded trunk accelerations. Kinematic variability was computed as the average SD (MeanSD) of acceleration patterns among standardized strides. Fractal dynamics (scaling exponent α) was assessed by Detrended Fluctuation Analysis (DFA) of stride intervals. Short-term and long-term dynamic stability were estimated by computing the maximal Lyapunov exponents of acceleration signals.

Results

TW did not modify kinematic gait variability as compared to OW (multivariate T², p = 0.87). Conversely, TW significantly modified fractal dynamics (t-test, p = 0.01), and both short and long term local dynamic stability (T² p = 0.0002). No relationship was observed between variability indexes with the exception of significant negative correlation between MeanSD and dynamic stability in TW (3 × 6 canonical correlation, r = 0.94).

Conclusions

Treadmill induced a less correlated pattern in the stride intervals and increased gait stability, but did not modify kinematic variability in healthy subjects. This could be due to changes in perceptual information induced by treadmill walking that would affect locomotor control of the gait and hence specifically alter non-linear dependencies among consecutive strides. Consequently, the type of walking (i.e. treadmill or overground) is important to consider in each protocol design.

Re-interpreting detrended fluctuation analyses of stride-to-stride variability in human walking

Detrended fluctuation analyses (DFA) have been widely used to quantify stride-to-stride temporal correlations in human walking. However, significant questions remain about how to properly interpret these statistical properties physiologically. Here, we propose a simpler and more parsimonious interpretation than previously suggested. Seventeen young healthy adults walked on a motorized treadmill at each of 5 speeds. Time series of consecutive stride lengths (SL) and stride times (ST) were recorded. Time series of stride speeds were computed as SS=SL/ST. SL and ST exhibited strong statistical persistence (α≫0.5). However, SS consistently exhibited slightly anti-persistent (α<0.5) dynamics. We created three surrogate data sets to directly test specific hypotheses about possible control processes that might have generated these time series. Subjects did not choose consecutive SL and ST according to either independently uncorrelated or statistically independent auto-regressive moving-average (ARMA) processes. However, cross-correlated surrogates, which preserved both the auto-correlation and cross-correlation properties of the original SL and ST time series successfully replicated the means, standard deviations, and (within computational limits) DFA α exponents of all relevant gait variables. These results suggested that subjects controlled their movements according to a two-dimensional ARMA process that specifically sought to minimize stride-to-stride variations in walking speed (SS). This interpretation fully agrees with experimental findings and also with the basic definitions of statistical persistence and anti-persistence. Our findings emphasize the necessity of interpreting DFA α exponents within the context of the control processes involved and the inherent biomechanical and neuro-motor redundancies available.

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.