Random stride intervals with memory

Application of vibration to the soles increases long-range correlations in the stride parameters during walking

Temporal fluctuations in the stride parameters during human walking exhibit long-range correlations, but these long-range correlations in the stride parameters decrease due to aging or neuromuscular diseases. These observations suggest that any quantified index of the long-range correlation can be regarded as an indicator of gait functionality. Considering the effect of task-relevant sensory feedback on augmenting human motor performance, we devised shoes with active insoles that could deliver noisy vibration to the soles of feet and assessed their efficacy in enhancing the long-range correlations in the stride parameters for healthy young adults. The vibration could be wirelessly controlled using a smartphone. The actuators, control unit, and battery in the devised shoes were light and embedded in the shoes. By virtue of this compactness, the shoes could be easily used for daily walking outside a laboratory. We performed walking experiments with 20 healthy adults and evaluated the effects of sub- and supra-threshold vibration on long-range correlations in stride interval and length. We performed detrended fluctuation analysis to quantify the long-range correlation of temporal changes in stride interval and length. We found that supra-threshold vibration, applied to the soles with the amplitude of 130 % of the sensory threshold, significantly increased the long-range correlations in stride interval and length by 10.3 % (p = 0.009) and 10.1 % (p = 0.021), respectively. On the other hand, sub-threshold vibration with the amplitude of 90 % of the sensory threshold had no significant effect. These results demonstrate that additional somatosensory feedback through barely detectable vibrations, which are supplied by compact shoes with active insoles, can enhance the indices of "healthy" complexity of locomotor function.

Assessing the Temporal Organization of Walking Variability: A Systematic Review and Consensus Guidelines on Detrended Fluctuation Analysis

Human physiological signals are inherently rhythmic and have a hallmark feature in that even distant intrasignal measurements are related to each other. This relationship is termed long-range correlation and has been recognized as an indicator of the optimal state of the observed physiological systems, among which the locomotor system. Loss of long-range correlations has been found as a result of aging as well as disease, which can be evaluated with detrended fluctuation analysis (DFA). Recently, DFA and the scaling exponent α have been employed for understanding the degeneration of temporal regulation of human walking biorhythms in, for example, Parkinson disease (PD). However, heterogeneous evidence on scaling exponent α values reported in the literature across different population groups has put into question what constitutes a healthy physiological pattern. Therefore, the purpose of this systematic review was to investigate the functional thresholds of scaling exponent α in young vs. older adults, as well as between patients with PD and age-matched asymptomatic controls. Aging and PD exhibited a negative effect size (i.e., led to decreased long-range correlations) of -0.20 and -0.53, respectively. Our meta-analysis based on 14 studies provides evidence that a mean scaling exponent α threshold of 0.86 [2 standard error (0.76, 0.96)] is able to optimally discriminate temporal organization of stride interval between young and old, whereas 0.82 (0.72, 0.92) differentiates patients with PD and age-matched asymptomatic controls. The optimal thresholds presented in this review together with the consensus guidelines for using DFA might allow a more sensitive and reliable application of this metric for understanding human walking physiology than has been achieved to date.

Which Gait Parameters and Walking Patterns Show the Significant Differences Between Parkinson's Disease and Healthy Participants?

This study investigated the difference in the gait of patients with Parkinson’s disease (PD), age-matched controls and young controls during three walking patterns. Experiments were conducted with 24 PD, 24 age-matched controls and 24 young controls, and four gait intervals were measured using inertial measurement units (IMU). Group differences between the mean and variance of the gait parameters (stride interval, stance interval, swing interval and double support interval) for the three groups were calculated and statistical significance was tested. The results showed that the variance in each of the four gait parameters of PD patients was significantly higher compared with the controls, irrespective of the three walking patterns. This study showed that the variance of any of the gait interval parameters obtained using IMU during any of the walking patterns could be used to differentiate between the gait of PD and control people.

Effect of treadmill versus overground running on the structure of variability of stride timing

Gait timing dynamics of treadmill and overground running were compared. Nine trained runners ran treadmill and track trials at 80, 100, and 120% of preferred pace for 8 min. each. Stride time series were generated for each trial. To each series, detrended fluctuation analysis (DFA), power spectral density (PSD), and multiscale entropy (MSE) analysis were applied to infer the regime of control along the randomness-regularity axis. Compared to overground running, treadmill running exhibited a higher DFA and PSD scaling exponent, as well as lower entropy at non-preferred speeds. This indicates a more ordered control for treadmill running, especially at non-preferred speeds. The results suggest that the treadmill itself brings about greater constraints and requires increased voluntary control. Thus, the quantification of treadmill running gait dynamics does not necessarily reflect movement in overground settings.

The effects of rhythmic sensory cues on the temporal dynamics of human gait

Walking is a complex, rhythmic task performed by the locomotor system. However, natural gait rhythms can be influenced by metronomic auditory stimuli, a phenomenon of particular interest in neurological rehabilitation. In this paper, we examined the effects of aural, visual and tactile rhythmic cues on the temporal dynamics associated with human gait. Data were collected from fifteen healthy adults in two sessions. Each session consisted of five 15-minute trials. In the first trial of each session, participants walked at their preferred walking speed. In subsequent trials, participants were asked to walk to a metronomic beat, provided through visually, aurally, tactile or all three cues (simultaneously and in sync), the pace of which was set to the preferred walking speed of the first trial. Using the collected data, we extracted several parameters including: gait speed, mean stride interval, stride interval variability, scaling exponent and maximum Lyapunov exponent. The extracted parameters showed that rhythmic sensory cues affect the temporal dynamics of human gait. The auditory rhythmic cue had the greatest influence on the gait parameters, while the visual cue had no statistically significant effect on the scaling exponent. These results demonstrate that visual rhythmic cues could be considered as an alternative cueing modality in rehabilitation without concern of adversely altering the statistical persistence of walking.

Fractal physiology and the fractional calculus: a perspective

This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

Effect of treadmill walking on the stride interval dynamics of human gait

Metronomic walking has been found to diminish the statistical persistence intrinsic to the stride interval time series of human gait. Since treadmill walking (TW) possesses a similar form of external pacing, we proposed to study the disruptions in the natural neuromuscular rhythms of gait during TW. Treadmill walking is a widespread rehabilitative tool, however, its effect on an individual's stride dynamics is not well understood. To better elucidate potential effects, we tested the hypothesis that TW without handrails would diminish the statistical persistence in an individual's stride interval time series. The scaling exponent (alpha) was employed in this study as a measure of the statistical persistence of the stride interval time series. Sixteen able-bodied young adults (mean age: 23.3+/-3.3 years) were instructed to walk at a self-selected comfortable pace for 15 min in three different conditions in a randomized order: (1) overground walking, (2) TW without holding a handrail (NoRail) and (3) TW while holding a front handrail (Rail). The alpha did not differ significantly between the overground and NoRail conditions (P>0.5). However, the alpha of the Rail condition (alpha=0.92+/-0.10) differed significantly from both the overground (alpha=0.83+/-0.06; P<0.015) and NoRail conditions (alpha=0.82+/-0.08; P<0.01). In contrast, stride interval variability did not change between walking conditions (P>0.5). These findings indicate that comfortable-paced TW does not diminish the intrinsic stride dynamics of human gait.

Understanding the complexity of human gait dynamics

Time series of human gait stride intervals exhibit fractal and multifractal properties under several conditions. Records from subjects walking at normal, slow, and fast pace speed are analyzed to determine changes in the fractal scalings as a function of the stress condition of the system. Records from subjects with different age from children to elderly and patients suffering from neurodegenerative disease are analyzed to determine changes in the fractal scalings as a function of the physical maturation or degeneration of the system. A supercentral pattern generator model is presented to simulate the above two properties that are typically found in dynamical network performance: that is, how a dynamical network responds to stress and to evolution.

Levels of complexity in scale-invariant neural signals

Many physical and physiological signals exhibit complex scale-invariant features characterized by 1/f scaling and long-range power-law correlations, indicating a possibly common control mechanism. Specifically, it has been suggested that dynamical processes, influenced by inputs and feedback on multiple time scales, may be sufficient to give rise to 1/f scaling and scale invariance. Two examples of physiologic signals that are the output of hierarchical multiscale physiologic systems under neural control are the human heartbeat and human gait. Here we show that while both cardiac interbeat interval and gait interstride interval time series under healthy conditions have comparable 1/f scaling, they still may belong to different complexity classes. Our analysis of the multifractal scaling exponents of the fluctuations in these two signals demonstrates that in contrast to the multifractal behavior found in healthy heartbeat dynamics, gait time series exhibit less complex, close to monofractal behavior. Further, we find strong anticorrelations in the sign and close to random behavior for the magnitude of gait fluctuations at short and intermediate time scales, in contrast to weak anticorrelations in the sign and strong positive correlation for the magnitude of heartbeat interval fluctuations-suggesting that the neural mechanisms of cardiac and gait control exhibit different linear and nonlinear features. These findings are of interest because they underscore the limitations of traditional two-point correlation methods in fully characterizing physiological and physical dynamics. In addition, these results suggest that different mechanisms of control may be responsible for varying levels of complexity observed in physiological systems under neural regulation and in physical systems that possess similar 1/f scaling.

Control from an allometric perspective

Control of complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. I have attempted to emphasize the difference between homeostatic control and allometric control mechanisms. Homeostatic control is familiar and has as its basis a negative feedback character, which is both local and relatively fast. Allometric control, on the other hand, is a new concept that can take into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the system variable. Allometric control introduces the fractal character into otherwise featureless random time series to enhance the robustness of physiological networks by introducing the fractional calculus into the control of the networks.

Peripheral neuropathy does not alter the fractal dynamics of stride intervals of gait

The purpose of this study was to determine the effect (if any) of significant sensory loss on the long-range correlations normally observed in the stride intervals of human gait. Fourteen patients with severe peripheral neuropathy and 12 gender-, age-, height-, and weight-matched nondiabetic controls participated. Subjects walked around an approximately 200-m open-level walkway for 10 min at their comfortable pace. Continuous knee joint kinematics were recorded and used to calculate a stride interval time series for each subject. Power spectral density and detrended fluctuation analyses were used to determine whether these stride intervals exhibited long-range correlations. If the loss of long-range correlations indicates deterioration of the central control of gait, then changes in peripheral sensation should have no effect. If instead the loss of long-range correlations is a consequence of a general inability to regulate gait cycle timing, then a similar loss should occur in patients with peripheral locomotor disorders. Both power spectral density analyses and detrended fluctuation analyses showed that temporal correlations in the stride times of neuropathic and control subjects were statistically identical (P = 0.954 and P = 0.974, respectively), despite slower gait speeds (P = 0.008) and increased stride time variability (P = 0.036) among the neuropathy patients. All subjects in both groups exhibited long-range correlations. These findings demonstrate that the normal long-range correlation structure of stride intervals is unaltered by significant peripheral sensory loss. This further supports the hypothesis that the central nervous system is involved in the regulation of long-range correlations.

Wavelet analysis of scaling properties of gastric electrical activity

We present a novel approach to the analysis of fluctuations in human myoelectrical gastric activity measured noninvasively from the surface of the abdomen. The time intervals between successive maxima of the wavelet transformed quasi-periodic electrogastrographic waveform define the gastric rate variability (GRV) time series. By using the method of average wavelet coefficients, the statistical fluctuations in the GRV signal in healthy individuals are determined to scale in time. Such scaling was previously found in a variety of physiological phenomena, all of which support the hypothesis that physiological dynamics utilize fractal time series. We determine the scaling index in a cohort of 17 healthy individuals to be 0.80 ± 0.14, which compared with a set of surrogate data is found to be significant at the level P < 0.01. We also determined that the dynamical pattern, so evident in the spectrum of average wavelet coefficients of the GRV time series of healthy individuals, is significantly reduced in a cohort of systemic sclerosis patients having a scaling index 0.64 ± 0.17. These results imply that the long-term memory in GRV time series is significantly reduced from healthy individuals to those with systemic sclerosis. Consequently, this disease degrades the complexity of the underlying gastrointestinal control system and this degradation is manifest in the loss of scaling in the GRV time series.

Fractional Langevin model of gait variability

The stride interval in healthy human gait fluctuates from step to step in a random manner and scaling of the interstride interval time series motivated previous investigators to conclude that this time series is fractal. Early studies suggested that gait is a monofractal process, but more recent work indicates the time series is weakly multifractal. Herein we present additional evidence for the weakly multifractal nature of gait. We use the stride interval time series obtained from ten healthy adults walking at a normal relaxed pace for approximately fifteen minutes each as our data set. A fractional Langevin equation is constructed to model the underlying motor control system in which the order of the fractional derivative is itself a stochastic quantity. Using this model we find the fractal dimension for each of the ten data sets to be in agreement with earlier analyses. However, with the present model we are able to draw additional conclusions regarding the nature of the control system guiding walking. The analysis presented herein suggests that the observed scaling in interstride interval data may not be due to long-term memory alone, but may, in fact, be due partly to the statistics.

Reliability of the long-range power-law correlations obtained from the bilateral stride intervals in asymptomatic volunteers whilst treadmill walking

Stride intervals measured during steady-state walking are irregular. These stride interval fluctuations are not random but exhibit long-range power-law correlation (alpha) such that a given stride interval is 'influenced' by earlier variations in the stride intervals. To estimate alpha, one requires a minute long sequence of right or left side stride interval data. However, to obtain a reliable alpha point estimate, the minimal stride sequence length is unknown. Additionally, it is unknown if the right and left side alpha are equivalent. In this study, the within-day and the right and left side reliabilities of alpha point estimates were examined in 23 volunteers performing three 8-min treadmill walks. In addition, eight volunteers were retested on three additional days to estimate between-day reliability. The standard error of measurement (S.E.M.) and the within- and between-day intraclass correlation (ICC) values, and their 95% confidence intervals, each calculated using the combined right and left leg 8-min alpha estimates were acceptable [0.047 (0.044-0.051); 0.914 (0.882-0.932) and 0.769 (0.689-0.815), respectively]. The left alpha (0.688 +/- 0.93) was greater than the right alpha (0.664 +/- 0.094), albeit this finding was underpowered (0.55). The alpha point estimates obtained from the full 8-min walks provided minimal S.E.M. and maximal within- and between-day ICCs. However, the minimal S.E.M. was statistically indistinguishable from the 6- and 7-min walk durations and all of the within-day and between-day ICCs were similar except for the 3- and 8-min between-day ICCs. This study suggests that data from four 3 min, three 6 min or two 8 min walk duration trials provide reliable alpha point estimates from a short series of short treadmill walks.

Managing variability in the summary and comparison of gait data

Variability in quantitative gait data arises from many potential sources, including natural temporal dynamics of neuromotor control, pathologies of the neurological or musculoskeletal systems, the effects of aging, as well as variations in the external environment, assistive devices, instrumentation or data collection methodologies. In light of this variability, unidimensional, cycle-based gait variables such as stride period should be viewed as random variables and prototypical single-cycle kinematic or kinetic curves ought to be considered as random functions of time. Within this framework, we exemplify some practical solutions to a number of commonly encountered analytical challenges in dealing with gait variability. On the topic of univariate gait variables, robust estimation is proposed as a means of coping with contaminated gait data, and the summary of non-normally distributed gait data is demonstrated by way of empirical examples. On the summary of gait curves, we discuss methods to manage undesirable phase variation and non-robust spread estimates. To overcome the limitations of conventional comparisons among curve landmarks or parameters, we propose as a viable alternative, the combination of curve registration, robust estimation, and formal statistical testing of curves as coherent units. On the basis of these discussions, we provide heuristic guidelines for the summary of gait variables and the comparison of gait curves.

Gait variability: methods, modeling and meaning

The study of gait variability, the stride-to-stride fluctuations in walking, offers a complementary way of quantifying locomotion and its changes with aging and disease as well as a means of monitoring the effects of therapeutic interventions and rehabilitation. Previous work has suggested that measures of gait variability may be more closely related to falls, a serious consequence of many gait disorders, than are measures based on the mean values of other walking parameters. The Current JNER series presents nine reports on the results of recent investigations into gait variability. One novel method for collecting unconstrained, ambulatory data is reviewed, and a primer on analysis methods is presented along with a heuristic approach to summarizing variability measures. In addition, the first studies of gait variability in animal models of neurodegenerative disease are described, as is a mathematical model of human walking that characterizes certain complex (multifractal) features of the motor control's pattern generator. Another investigation demonstrates that, whereas both healthy older controls and patients with a higher-level gait disorder walk more slowly in reduced lighting, only the latter's stride variability increases. Studies of the effects of dual tasks suggest that the regulation of the stride-to-stride fluctuations in stride width and stride time may be influenced by attention loading and may require cognitive input. Finally, a report of gait variability in over 500 subjects, probably the largest study of this kind, suggests how step width variability may relate to fall risk. Together, these studies provide new insights into the factors that regulate the stride-to-stride fluctuations in walking and pave the way for expanded research into the control of gait and the practical application of measures of gait variability in the clinical setting.

The independently fractal nature of respiration and heart rate during exercise under normobaric and hyperbaric conditions

To test the hypothesis that the fractal character of breathing and heart rate are independent, inter-breath intervals (IBI) and R-R intervals (RRI) were measured during rest and two levels of exercise at 1 and 2.8 ATA in a hyperbaric chamber in 18 male and female subjects (ages 19-74 years). Both RRI and IBI showed fractal properties. Fractal dimensions (D) for IBI were (mean +/- S.D.) 1.33 +/- 0.11, 1.29 +/- 0.12, 1.19 +/- 0.16 (rest, light and heavy exercise at 1ATA); 1.33 +/- 0.13, 1.25 +/- 0.13, 1.18 +/- 0.14 (same conditions at 2.8 ATA). Corresponding D for RRI were 1.19 +/- 0.11, 1.05 +/- 0.07 and 1.02 +/- 0.05 (1ATA); 1.20 +/- 0.10, 1.03 +/- 0.04 and 1.01 +/- 0.02 (2.8 ATA). The fractal dimension of each variable decreased with exercise and was unaffected by hyperbaric exposure. These two systems were not cross-correlated under any of the six conditions. During rest and light and moderate exercise at 1 and 2.8 ATA the results are consistent with heart rate variability and breathing rate variability being mutually independent of one another.

Nonlinear dynamical model of human gait

We present a nonlinear dynamical model of the human gait control system in a variety of gait regimes. The stride-interval time series in normal human gait is characterized by slightly multifractal fluctuations. The fractal nature of the fluctuations becomes more pronounced under both an increase and decrease in the average gait. Moreover, the long-range memory in these fluctuations is lost when the gait is keyed on a metronome. Human locomotion is controlled by a network of neurons capable of producing a correlated syncopated output. The central nervous system is coupled to the motocontrol system, and together they control the locomotion of the gait cycle itself. The metronomic gait is simulated by a forced nonlinear oscillator with a periodic external force associated with the conscious act of walking in a particular way.

Automatic stride interval extraction from long, highly variable and noisy gait timing signals

This paper presents a probabilistic algorithm for automatically extracting the stride interval time series from long, highly variable and noisy two-state timing signals. Long interstride temporal records are of particular interest in nonlinear dynamical analysis of gait. The proposed method consists of probabilistic estimation and extraction followed by post-extraction filtering. With noisy timing signals from 10 children with Spastic Diplegia, no statistical differences in the numbers of extracted strides (p=0.94), the mean stride intervals (p=0.55) and the scaling exponents (p=0.94) (a measure of temporal heterogeneity) were found between series extracted by hand and by the probabilistic algorithm. The method is robust to noise and violations of normality. Results support the use of probabilistic extraction as an alternative to laborious manual extraction.