Fractal Analysis of Human Gait Variability via Stride Interval Time Series

On the number of steps required to measure the mean and variability of spatio-temporal parameters at preferred running speed

Preferred running speed is a relevant condition for measuring spatio-temporal parameters and their variability as it theoretically corresponds to an intensity where the energy cost, the biomechanical constraints and the pleasure are optimal. The objective of this study was to evaluate 1) the between-days reliability of spatio-temporal parameters measurements at preferred running speed and 2) the minimal number of steps required to obtain a representative measurement. Thirty-one recreational runners (age: 26 ± 5.5 years; running experience > 2 years) performed three sessions of treadmill running at preferred running speed separated by 24 h. Spatio-temporal parameters were stride, step, contact and flight times, cadence, step length, duty factor, vertical and leg stiffness. Mean, linear and non-linear measurements of spatio-temporal parameters were computed. Reliability was determined using intraclass coefficient correlation and his 95 % confidence interval lower band (95 %LB). Number of steps required for intra-session representativity were defined as the first number of steps from which all measurements are equal to the measurement at the longer time series length. For all spatio-temporal parameters, mean data showed excellent reliability (95 %LB > 0.90) but most spatio-temporal parameters linear and non-linear variability measurements showed questionable reliability (95 %LB < 0.79). The absolute speed variation at preferred running speed (minimal detectable change 0.8 km/h) may explain these results. Representative values required between 16 and 352 steps for mean value of spatio-temporal parameters, between 16 and 400 steps for linear measurements, and between 336 and 704 steps for non-linear measurements.

Reliability of running gait variability measures calculated from inertial measurement units

Changes to the variability within biomechanical signals may reflect a change in the health of the human system. However, for running gait variability measures calculated from wearable device data, it is unknown whether a between-day difference reflects a shift in system dynamics reflective of a change in human health or is a result of poor between-day reliability of the measurement device or the biomechanical signal. This study investigated the reliability of stride time and sacral acceleration variability measures calculated from inertial measurement units (IMUs). Nineteen runners completed six treadmill running trials on two occasions seven days apart. Stride time and sacral acceleration signals were obtained using IMUs. Stride time variability and complexity were calculated using the coefficient of variation (CV) and detrended fluctuation analysis (DFA), respectively. Sacral acceleration regularity was quantified using sample entropy with a range of input parameters m (vector length) and r (similarity threshold). Between-day reliability was assessed using the intraclass correlation coefficient (ICC), standard error of measurement (SEM) and minimum detectable change. Stride time CV displayed moderate relative reliability (ICC = 0.672), but with a large absolute minimum detectable change = 0.525 %, whilst stride time DFA-α displayed poor relative reliability (ICC = 0.457) and yielded large minimum detectable changes (≥ 0.208). Sample entropy displayed good relative reliability in mediolateral and resultant sacral acceleration signals for certain combinations of the parameters m and r, although again with large minimum detectable changes. Researchers should be cognisant of these reliability metrics when interpreting changes in running gait variability measures in clinical contexts.

Feature derivation and classification of auditory-motor coupling dynamics in healthy and neurologically impaired adults

The use of auditory stimuli in rehabilitation to target walking has been evidenced in persons with neurological conditions. The methodologies focus on the synchronisation of persons' steps to auditory stimuli showing that the type of stimuli and tempi significantly affect the synchronisation. However, the dynamic of the interaction over time between the motor system and the auditory stimuli, i.e., when steps are aligned (termed as locking) and not aligned (termed as unlocking) to the beat of the stimuli, remains unclear. Quantifying these dynamics would assist in the development of personalised rehabilitation. Nevertheless, it is currently challenging given the variability of responses per individual over time. We propose a methodological solution to quantify the dynamics of the step-to-beat coupling over time within an experimental paradigm where healthy (n = 7) and neurological impaired (n = 6) participants walk three minutes to music and metronomes at various tempi. We applied window partitioning within the time series to account for the changing pattern. To classify data into locked and unlocked events, features of fluctuation and trend were derived on which two statistical tests (circular statistical test and slope test) were done, respectively. Based on the ground truth, the performance of our proposed method yielded high accuracy (91%), precision (90%) and recall (97%). The standard deviation of the inter-step intervals was then modelled across the label and experimental factors. The proposed method is suitable for quantifying fine-grained observation of the dynamics of auditory-motor coupling in adult healthy and neurological impaired participants, with the potential of designing personalised rehabilitation.

How Gait Nonlinearities in Individuals Without Known Pathology Describe Metabolic Cost During Walking Using Artificial Neural Network and Multiple Linear Regression

This study uses Artificial Neural Networks (ANNs) and multiple linear regression (MLR) models to explore the relationship between gait dynamics and the metabolic cost. Six nonlinear metrics—Lyapunov Exponents based on Rosenstein’s algorithm (LyER), Detrended Fluctuation Analysis (DFA), the Approximate Entropy (ApEn), the correlation dimension (CD), the Sample Entropy (SpEn), and Lyapunov Exponents based on Wolf’s algorithm (LyEW)—were utilized to predict the metabolic cost during walking. Time series data from 10 subjects walking under 13 conditions, with and without hip exoskeletons, were analyzed. Six ANN models, each corresponding to a nonlinear metric, were trained using the Levenberg–Marquardt backpropagation algorithm and compared with MLR models. Performance was assessed based on the mean squared error (MSE) and correlation coefficients. ANN models outperformed MLR, with DFA and Lyapunov Exponent models showing higher R2 values, indicating stronger predictive accuracy. The results suggest that gait’s nonlinear characteristics significantly impact the metabolic cost, and ANNs are more effective for analyzing these dynamics than MLR models. The study emphasizes the potential of focusing on specific nonlinear gait variables to enhance assistive device optimization, particularly for hip exoskeletons. These findings support the development of personalized interventions that improve walking efficiency and reduce metabolic demands, offering insights into the design of advanced assistive technologies.

Running Gait Complexity During an Overground, Mass-Participation Five-Kilometre Run

Human locomotion contains innate variability which may provide health insights. Detrended fluctuation analysis (DFA) has been used to quantify the temporal structure of variability for treadmill running, although it has been less commonly applied to uncontrolled overground running. This study aimed to determine how running gait complexity changes in response to gradient and elapsed exercise duration during uncontrolled overground running. Sixty-eight participants completed an overground, mass-participation five-kilometre run (a parkrun). Stride times were recorded using an inertial measurement unit mounted on the distal shank. Data were divided into four consecutive intervals (uphill lap 1, downhill lap 1, uphill lap 2, downhill lap 2). The magnitude (SD) and structure (DFA) of stride time variability were compared across elapsed exercise duration and gradient using a repeated-measures ANOVA. Participants maintained consistent stride times throughout the run. Stride time DFA-α displayed a moderate decrease (d = |0.39| ± 0.13) during downhill running compared to uphill running. DFA-α did not change in response to elapsed exercise duration, although a greater stride time SD was found during the first section of lap 1 (d = |0.30| ± 0.12). These findings suggest that inter- and intra-run changes in gait complexity should be interpreted in the context of course elevation profiles before conclusions on human health are drawn.

Kinematic and neuromuscular characterization of cognitive involvement in gait control in healthy young adults

The signature of cognitive involvement in gait control has rarely been studied using both kinematic and neuromuscular features. The present study aimed to address this gap. Twenty-four healthy young adults walked on an instrumented treadmill in a virtual environment under two optic flow conditions: normal (NOF) and perturbed (POF, continuous mediolateral pseudorandom oscillations). Each condition was performed under single-task and dual-task conditions of increasing difficulty (1-, 2-, 3-back). Subjective mental workload (raw NASA-TLX), cognitive performance (mean reaction time and d-prime), kinematic (steadiness, variability, and complexity in the mediolateral and anteroposterior directions), and neuromuscular (duration and variability of motor primitives) control of gait were assessed. The cognitive performance and the number and composition of motor modules were unaffected by simultaneous walking, regardless of the optic flow condition. Kinematic and neuromuscular variability was greater under POF compared with NOF conditions. Young adults sought to counteract POF by rapidly correcting task-relevant gait fluctuations. The depletion of cognitive resources through dual-tasking led to reduced kinematic and neuromuscular variability and this occurred to the same extent regardless of simultaneous working memory (WM) load. Increasing WM load led to a prioritization of gait control in the mediolateral direction over the anteroposterior direction. The impact of POF on kinematic variability (step velocity) was reduced when a cognitive task was performed simultaneously, but this phenomenon was not modulated by WM load. Collectively, these results shed important light on how young adults adjust the processes involved in goal-directed locomotion when exposed to varying levels of task and environmental constraints.NEW & NOTEWORTHY The kinematic and neuromuscular signatures of cognitive involvement in gait control have rarely been studied jointly. We sought to address this issue using gait perturbation and dual-task paradigms. The protocol consisted of a fixed-speed treadmill walk to which visual and cognitive constraints were applied separately and together. The results revealed that young adults optimally regulated their gait to cope with these constraints by maintaining relatively stable muscle synergies and flexibly allocating attentional resources.

Evaluation of the optimal number of steps to obtain reliable running spatio-temporal parameters and their variability

BACKGROUND

Spatio-temporal running parameters and their variability help to determine a runner's running style. However, determining whether a change is due to the measurement or to a specific condition such as an injury is a matter of debate, as no recommendation on the number of steps required to obtain reliable assessments exists.

RESEARCH QUESTION

What is the optimal number of steps required to measure different spatio-temporal parameters and study their variability at different running speeds?

METHODS

Twenty-five runners performed three experimental sessions of three bouts of treadmill running at 8, 10 and 12 km/h separated by 24 h. We measured cadence, stride, step, contact and flight time. We calculated the duty factor and the leg stiffness index (Kleg). Mean spatio-temporal parameters and linear (coefficient of variation, standard deviation) and non-linear (Higuchi fractal index, α1 coefficient of detrended fluctuation analysis) analyses were computed for different numbers of steps. Relative reliability was determined using the intraclass coefficient correlation. The minimal number of steps which present a good reliability level was considered as the optimal number of steps for measurement. Absolute reliability was assessed by calculating minimal detectable change.

RESULTS

To assess the mean values of spatio-temporal running parameters, between 16 and 150 steps were required. We were unable to obtain an optimal number of steps for cadence, stride and step-time variabilities for all speeds. For the linear analyses, we deduced the optimal number of steps for Kleg and the contact time (around 350 steps). Non-linear analyses measurements required between 350 and 540 steps, depending on the parameter.

SIGNIFICANCE

Researchers and clinicians should optimize experimental conditions (number of steps and running speed) depending on the parameter or the variability analysis targeted. Future studies must use absolute reliability metrics to report changes in response to a specific condition with no bias due to measurement error.

The Agreement between Wearable Sensors and Force Plates for the Analysis of Stride Time Variability

The variability and regularity of stride time may help identify individuals at a greater risk of injury during military load carriage. Wearable sensors could provide a cost-effective, portable solution for recording these measures, but establishing their validity is necessary. This study aimed to determine the agreement of several measures of stride time variability across five wearable sensors (Opal APDM, Vicon Blue Trident, Axivity, Plantiga, Xsens DOT) and force plates during military load carriage. Nineteen Australian Army trainee soldiers (age: 24.8 ± 5.3 years, height: 1.77 ± 0.09 m, body mass: 79.5 ± 15.2 kg, service: 1.7 ± 1.7 years) completed three 12-min walking trials on an instrumented treadmill at 5.5 km/h, carrying 23 kg of an external load. Simultaneously, 512 stride time intervals were identified from treadmill-embedded force plates and each sensor where linear (standard deviation and coefficient of variation) and non-linear (detrended fluctuation analysis and sample entropy) measures were obtained. Sensor and force plate agreement was evaluated using Pearson's r and intraclass correlation coefficients. All sensors had at least moderate agreement (ICC > 0.5) and a strong positive correlation (r > 0.5). These results suggest wearable devices could be employed to quantify linear and non-linear measures of stride time variability during military load carriage.

Effects of walking speeds on lower extremity kinematic synergy in toe vertical position control: An experimental study

BACKGROUND

This study aimed to investigate whether lower limb joints mutually compensate for each other, resulting in motor synergy that suppresses toe vertical position fluctuation, and whether walking speeds affect lower limb synergy.

METHODS

Seventeen male university students walked at slow (0.85 ± 0.04 m/s), medium (1.43 ± 0.05 m/s) and fast (1.99 ± 0.06 m/s) speeds on a 15-m walkway while lower limb kinematic data were collected. Uncontrolled manifold analysis was used to quantify the strength of synergy. Two-way (speed × phase) repeated-measures analysis of variance was used to analyze all dependent variables.

RESULTS

A significant speed-by-phase interaction was observed in the synergy index (SI) (P  < .001). At slow walking speeds, subjects had greater SI during mid-swing (P  < .001), while at fast walking speeds, they had greater SI during early-swing (P  < .001). During the entire swing phase, fast walking exhibited lower SI values than medium (P  = .005) and slow walking (P  = .027).

CONCLUSION

Kinematic synergy plays a crucial role in controlling toe vertical position during the swing phase, and fast walking exhibits less synergy than medium and slow walking. These findings contribute to a better understanding of the role of kinematic synergy in gait stability and have implications for the development of interventions aimed at improving gait stability and reducing the risk of falls.

Biomechanical Assessment Methods Used in Chronic Stroke: A Scoping Review of Non-Linear Approaches

Non-linear and dynamic systems analysis of human movement has recently become increasingly widespread with the intention of better reflecting how complexity affects the adaptability of motor systems, especially after a stroke. The main objective of this scoping review was to summarize the non-linear measures used in the analysis of kinetic, kinematic, and EMG data of human movement after stroke. PRISMA-ScR guidelines were followed, establishing the eligibility criteria, the population, the concept, and the contextual framework. The examined studies were published between 1 January 2013 and 12 April 2023, in English or Portuguese, and were indexed in the databases selected for this research: PubMed®, Web of Science®, Institute of Electrical and Electronics Engineers®, Science Direct® and Google Scholar®. In total, 14 of the 763 articles met the inclusion criteria. The non-linear measures identified included entropy (n = 11), fractal analysis (n = 1), the short-term local divergence exponent (n = 1), the maximum Floquet multiplier (n = 1), and the Lyapunov exponent (n = 1). These studies focused on different motor tasks: reaching to grasp (n = 2), reaching to point (n = 1), arm tracking (n = 2), elbow flexion (n = 5), elbow extension (n = 1), wrist and finger extension upward (lifting) (n = 1), knee extension (n = 1), and walking (n = 4). When studying the complexity of human movement in chronic post-stroke adults, entropy measures, particularly sample entropy, were preferred. Kinematic assessment was mainly performed using motion capture systems, with a focus on joint angles of the upper limbs.

Optimizing Wearable Device and Testing Parameters to Monitor Running-Stride Long-Range Correlations for Fatigue Management in Field Settings

PURPOSE

There are important methodological considerations for translating wearable-based gait-monitoring data to field settings. This study investigated different devices' sampling rates, signal lengths, and testing frequencies for athlete monitoring using dynamical systems variables.

METHODS

Secondary analysis of previous wearables data (N = 10 runners) from a 5-week intensive training intervention investigated impacts of sampling rate (100-2000 Hz) and signal length (100-300 strides) on detection of gait changes caused by intensive training. Primary analysis of data from 13 separate runners during 1 week of field-based testing determined day-to-day stability of outcomes using single-session data and mean data from 2 sessions. Stride-interval long-range correlation coefficient α from detrended fluctuation analysis was the gait outcome variable.

RESULTS

Stride-interval α reduced at 100- and 200- versus 300- to 2000-Hz sampling rates (mean difference: -.02 to -.08; P ≤ .045) and at 100- compared to 200- to 300-stride signal lengths (mean difference: -.05 to -.07; P < .010). Effects of intensive training were detected at 100, 200, and 400 to 2000 Hz (P ≤ .043) but not 300 Hz (P = .069). Within-athlete α variability was lower using 2-session mean versus single-session data (smallest detectable change: .13 and .22, respectively).

CONCLUSIONS

Detecting altered gait following intensive training was possible using 200 to 300 strides and a 100-Hz sampling rate, although 100 and 200 Hz underestimated α compared to higher rates. Using 2-session mean data lowers smallest detectable change values by nearly half compared to single-session data. Coaches, runners, and researchers can use these findings to integrate wearable-device gait monitoring into practice using dynamic systems variables.

Comparison of the effects of rhythmic vibrotactile stimulations and rhythmic auditory stimulations on Parkinson's disease patients' gait variability: a pilot study

INTRODUCTION

Parkinson's disease patients' gait is characterized by shorter step length, reduced gait velocity and deterioration of temporal organization of stride duration variability (modified Long Range Autocorrelations). The objective of this study was to compare effects of rhythmic auditory stimulations (RAS) and Rhythmic Vibrotactile Stimulations (RVS) on Parkinson's disease patients' gait.

METHODS

Ten Parkinson's disease patients performed three walking conditions lasting 5-7 min each: control condition (CC), RAS condition and RVS condition. Inertial measurement units were used to assess spatiotemporal gait parameters. Stride duration variability was assessed in terms of magnitude using coefficient of variation and in terms of temporal organization (i.e., Long Range Autocorrelations computation) using the evenly spaced averaged Detrended Fluctuation Analysis (α-DFA exponent).

RESULTS

Gait velocity was significantly higher during RAS condition than during CC (Cohen's d = 0.52) and similar to RVS condition (Cohen's d = 0.17). Cadence was significantly higher during RAS (Cohen's d = 0.77) and RVS (Cohens' d = 0.56) conditions than during CC. Concerning variability, no difference was found either for mean coefficient of variation or mean α-DFA between conditions. However, a great variability of individual results between the RAS and the RVS conditions is to be noted concerning α-DFA.

CONCLUSIONS

RAS and RVS improved similarly PD patients' spatiotemporal gait parameters, without modifying stride duration variability in terms of magnitude and temporal organization at group level. Future studies should evaluate the relevant parameters for administering the right cueing type for the right patient.

TRIAL REGISTRATION

ClinicalTrial.gov registration number NCT05790759, date of registration: 16/03/2023, retrospectively registered.

Multifractal detrended fluctuation analysis of insole pressure sensor data to diagnose vestibular system disorders

The vestibular system (VS) is a sensory system that has a vital function in human life by serving to maintain balance. In this study, multifractal detrended fluctuation analysis (MFDFA) is applied to insole pressure sensor data collected from subjects in order to extract features to identify diseases related to VS dysfunction. We use the multifractal spectrum width as the feature to distinguish between healthy and diseased people. It is observed that multifractal behavior is more dominant and thus the spectrum is wider for healthy subjects, where we explain the reason as the long-range correlations of the small and large fluctuations of the time series for this group. We directly process the instantaneous pressure values to extract features in contrast to studies in the literature where gait analysis is based on investigation of gait dynamics (stride time, stance time, etc.) requiring long walking time. Thus, as the main innovation of this work, we detrend the data to give meaningful information even for a relatively short walk. Extracted feature set was input to fundamental classification algorithms where the Support-Vector-Machine (SVM) performed best with an average accuracy of 98.2% for the binary classification as healthy or suffering. This study is a substantial part of a big project where we finally aim to identify the specific VS disease that causes balance disorder and also determine the stage of the disease, if any. Within this scope, the achieved performance gives high motivation to work more deeply on the issue.

A guide to Whittle maximum likelihood estimator in MATLAB

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

A Scoping Review of the Validity and Reliability of Smartphone Accelerometers When Collecting Kinematic Gait Data

The aim of this scoping review is to evaluate and summarize the existing literature that considers the validity and/or reliability of smartphone accelerometer applications when compared to 'gold standard' kinematic data collection (for example, motion capture). An electronic keyword search was performed on three databases to identify appropriate research. This research was then examined for details of measures and methodology and general study characteristics to identify related themes. No restrictions were placed on the date of publication, type of smartphone, or participant demographics. In total, 21 papers were reviewed to synthesize themes and approaches used and to identify future research priorities. The validity and reliability of smartphone-based accelerometry data have been assessed against motion capture, pressure walkways, and IMUs as 'gold standard' technology and they have been found to be accurate and reliable. This suggests that smartphone accelerometers can provide a cheap and accurate alternative to gather kinematic data, which can be used in ecologically valid environments to potentially increase diversity in research participation. However, some studies suggest that body placement may affect the accuracy of the result, and that position data correlate better than actual acceleration values, which should be considered in any future implementation of smartphone technology. Future research comparing different capture frequencies and resulting noise, and different walking surfaces, would be useful.

Nonadditive Entropy Application to Detrended Force Sensor Data to Indicate Balance Disorder of Patients with Vestibular System Dysfunction

The healthy function of the vestibular system (VS) is of vital importance for individuals to carry out their daily activities independently and safely. This study carries out Tsallis entropy (TE)-based analysis on insole force sensor data in order to extract features to differentiate between healthy and VS-diseased individuals. Using a specifically developed algorithm, we detrend the acquired data to examine the fluctuation around the trend curve in order to consider the individual's walking habit and thus increase the accuracy in diagnosis. It is observed that the TE value increases for diseased people as an indicator of the problem of maintaining balance. As one of the main contributions of this study, in contrast to studies in the literature that focus on gait dynamics requiring extensive walking time, we directly process the instantaneous pressure values, enabling a significant reduction in the data acquisition period. The extracted feature set is then inputted into fundamental classification algorithms, with support vector machine (SVM) demonstrating the highest performance, achieving an average accuracy of 95%. This study constitutes a significant step in a larger project aiming to identify the specific VS disease together with its stage. The performance achieved in this study provides a strong motivation to further explore this topic.

Indoor running temporal variability for different running speeds, treadmill inclinations, and three different estimation strategies

Inertial measurement units (IMU) constitute a light and cost-effective alternative to gold-standard measurement systems in the assessment of running temporal variables. IMU data collected on 20 runners running at different speeds (80, 90, 100, 110 and 120% of preferred running speed) and treadmill inclination (±2, ±5, and ±8%) were used here to predict the following temporal variables: stride frequency, duty factor, and two indices of running variability such as the detrended fluctuation analysis alpha (DFA-α) and the Higuchi's D (HG-D). Three different estimation methodologies were compared: 1) a gold-standard optoelectronic device (which provided the reference values), 2) IMU placed on the runner's feet, 3) a single IMU on the runner's thorax used in conjunction with a machine learning algorithm with a short 2-second or a long 120-second window as input. A two-way ANOVA was used to test the presence of significant (p<0.05) differences due to the running condition or to the estimation methodology. The findings of this study suggest that using both IMU configurations for estimating stride frequency can be effective and comparable to the gold-standard. Additionally, the results indicate that the use of a single IMU on the thorax with a machine learning algorithm can lead to more accurate estimates of duty factor than the strategy of the IMU on the feet. However, caution should be exercised when using these techniques to measure running variability indices. Estimating DFA-α from a short 2-second time window was possible only in level running but not in downhill running and it could not accurately estimate HG-D across all running conditions. By taking a long 120-second window a machine learning algorithm could improve the accuracy in the estimation of DFA-α in all running conditions. By taking these factors into account, researchers and practitioners can make informed decisions about the use of IMU technology in measuring running biomechanics.

The Application of non-linear methods to quantify changes to movement dynamics during running: A scoping review

The aim of this scoping review was to evaluate research approaches that quantify changes to non-linear movement dynamics during running in response to fatigue, different speeds, and fitness levels. PubMed and Scopus were used to identify appropriate research articles. After the selection of eligible studies, study details and participant characteristics were extracted and tabulated to identify methodologies and findings. Twenty-seven articles were included in the final analysis. To evaluate non-linearities in the time series, a range of approaches were identified including motion capture, accelerometery, and foot switches. Common methods of analysis included measures of fractal scaling, entropy, and local dynamic stability. Conflicting findings were evident when studies examined non-linear features in fatigued states when compared to non-fatigued. More pronounced alterations to movement dynamics are evident when running speed is changed markedly. Greater fitness levels resulted in more stable and predictable running patterns. The mechanisms by which these changes are underpinned require further examination. These could include the physiological demand of running, biomechanical constraints of the runner, and the attentional demands of the task. Moreover, the practical implications are yet to be elucidated. This review has identified gaps in the literature which should be addressed for further understanding of the field.

Four Methods to Distinguish between Fractal Dimensions in Time Series through Recurrence Quantification Analysis

Fractal properties in time series of human behavior and physiology are quite ubiquitous, and several methods to capture such properties have been proposed in the past decades. Fractal properties are marked by similarities in statistical characteristics over time and space, and it has been suggested that such properties can be well-captured through recurrence quantification analysis. However, no methods to capture fractal fluctuations by means of recurrence-based methods have been developed yet. The present paper takes this suggestion as a point of departure to propose and test several approaches to quantifying fractal fluctuations in synthetic and empirical time-series data using recurrence-based analysis. We show that such measures can be extracted based on recurrence plots, and contrast the different approaches in terms of their accuracy and range of applicability.

Adiabatic Invariant of Center-of-Mass Motion during Walking as a Dynamical Stability Constraint on Stride Interval Variability and Predictability

Simple Summary

Human walking exhibits properties of both stability and variability. On the one hand, the variability of the interval of time between heel strikes is autocorrelated, i.e., not randomly organized. On the other hand, walking is highly stereotyped and arguments from general mechanics suggest that the stability of gait can be assessed according to invariant properties. This study aims at proposing one of those invariants. Participants walked for 10 min at a natural pace, with and without a metronome indicating participants’ preferred step frequency. In both cases, we use different parameters to assess both the variability and stability of walking. We verify a known result: the metronome strongly alters the variability of the motion. However, despite the large variability changes, our proposed adiabatic invariant is preserved in both conditions, demonstrating the stability of gait. It appears as though our model reveals dynamical constraints that are “hidden” beyond apparent walking variability.

Abstract

Human walking exhibits properties of global stability, and local dynamic variability, predictability, and complexity. Global stability is typically assessed by quantifying the whole-body center-of-mass motion while local dynamic variability, predictability, and complexity are assessed using the stride interval. Recent arguments from general mechanics suggest that the global stability of gait can be assessed with adiabatic invariants, i.e., quantities that remain approximately constant, even under slow external changes. Twenty-five young healthy participants walked for 10 min at a comfortable pace, with and without a metronome indicating preferred step frequency. Stride interval variability was assessed by computing the coefficient of variation, predictability using the Hurst exponent, and complexity via the fractal dimension and sample entropy. Global stability of gait was assessed using the adiabatic invariant computed from averaged kinetic energy value related to whole-body center-of-mass vertical displacement. We show that the metronome alters the stride interval variability and predictability, from autocorrelated dynamics to almost random dynamics. However, despite these large local variability and predictability changes, the adiabatic invariant is preserved in both conditions, showing the global stability of gait. Thus, the adiabatic invariant theory reveals dynamical global stability constraints that are “hidden” behind apparent local walking variability and predictability.

Individuals with a COVID-19 history exhibit asymmetric gait patterns despite full recovery

COVID-19 is a multisystem infectious disease affecting the body systems. Its neurologic complications include -but are not limited to headache, loss of smell, encephalitis, and cerebrovascular accidents. Even though gait analysis is an objective measure of the neuro-motor system and may provide significant information about the pathophysiology of specific diseases, no studies have investigated the gait characteristics in adults after full recovery from COVID-19. This was a cross-sectional, controlled study that included 12 individuals (mean age, 23.0 ± 4.1 years) with mild-to-moderate COVID-19 history (COVD) and 20 sedentary controls (CONT; mean age, 24.0 ± 3.6 years). Gait was evaluated using inertial sensors on a motorized treadmill. Spatial-temporal gait parameters and gait symmetry were calculated by using at least 512 consecutive steps for each participant. The effect-size analyses were utilized to interpret the impact of the results. Spatial-temporal gait characteristics were comparable between the two groups. The COVD group showed more asymmetrical gait patterns than the CONT group in the double support duration symmetry (p = 0.042), single support duration symmetry (p = 0.006), loading response duration symmetry (p = 0.042), and pre-swing duration symmetry (p = 0.018). The effect size analyses of the differences showed large effects (d = 0.68-0.831). Individuals with a history of mild-to-moderate COVID-19 showed more asymmetrical gait patterns than individuals without a disease history. Regardless of its severity, the multifaceted long-term effects of COVID-19 need to be examined and the scope of clinical follow-up should be detailed.

Influence of Groundwater Depth on Pile–Soil Mechanical Properties and Fractal Characteristics under Cyclic Loading

The analysis of the behavior of soil and foundations when the piles in offshore areas are subjected to long-term lateral loading (wind) is one of the major problems associated with the smooth operation of superstructure. The strength of the pile-soil system is influenced by variations in the water content of the soil. At present, there are no studies carried out analyzing the mechanical and deformational behavior of both the material of the laterally loaded piles and soil with groundwater level as a variable. In this paper, a series of 1-g model tests were conducted to explore the lateral behavior of both soil and monopile under unidirectional cyclic loading, based on the foundation of an offshore wind turbine near the island. The influence of underground water level and cyclic load magnitude on the performance of the pile–soil system was analyzed. To visualize the movements of soil particles during the experimental process, particle image velocimetry (PIV) was used to record the soil displacement field under various cyclic loading conditions. The relationship curves between pile top displacement and cyclic steps, as well as the relationship curves between cyclic stiffness and cyclic steps, were displayed. Combined with fractal theory, the fractal dimension of each curve was calculated to evaluate the sensitivity of the pile–soil interaction system. The results showed that cyclic loading conditions and groundwater depth are the main factors affecting the pile–soil interaction. The cyclic stiffness of the soil increased in all test groups as loading progressed; however, an increase in the cyclic load magnitude decreased the initial and cyclic stiffness. The initial and cyclic stiffness of dry soil was higher than that of saturated soil, but less than that of unsaturated soil. The ability of the unsaturated soil to limit the lateral displacement of the pile decreased as the depth of the groundwater level dropped. The greater the fluctuation of the pile top displacement, the larger the fractal dimension of each relationship curve, with a variation interval of roughly 1.24–1.38. The average increment of the cumulative pile top displacement between each cycle step following the cyclic loading was positively correlated with fractal dimension. Based on the PIV results, the changes in the pile–soil system were predominantly focused in the early stages of the experiment, and the short-term effects of lateral cyclic loading are greater than the long-term effects. In addition, this research was limited to a single soil layer. The pile–soil interaction under layered soil is investigated, and the results will be used in more complex ground conditions in the future.

Persistence and anti-persistence in treadmill walking

BACKGROUND

Strong, long-range persistent correlations in stride time (ST) and length (SL) are the fundamental traits of treadmill gait. Our recent work showed that the ST and SL time series' statistical properties originated from the superposition of large-scale trends and small-scale fluctuations (residuals). Trends served as the control manifolds about which ST and SL fluctuated.

RESEARCH QUESTION

Do random changes in treadmill belt speed affect the trend properties and ST/SL scaling exponents?

METHODS

We used Multivariate Adaptive Regression Splines (MARS) to determine gait trends during a walk on a treadmill whose belt speed was perturbed by a strong random noise (coefficient of variation was equal to 0.075, 0.1, and 0.13 for treadmill speed 0.8 m/s, 1.2 m/s, and 1.6 m/s, respectively). Then, we calculated the ST/SL scaling exponents of the experimental time series and the corresponding MARS residuals with the madogram estimator.

RESULTS

Except for the ST at the lowest treadmill speed, the normalized trend duration was at least two times greater than that for the unperturbed walk. The Cauchy distribution scale parameter, which served as a measure of the width of SL and ST trend slope distributions, was at v=1.2m/s, almost 50% and 25% smaller than the unperturbed values. The differences were even greater at v=1.6 m/s: 73% and 83%. Apart from ST at v=0.8m/s, the ST/SL scaling indices were close to 0.5. For all speeds, the ST and SL MARS residuals were strongly anti-persistent. At v=1.2m/s, the corresponding scaling exponents were equal to 0.37±0.10 and 0.25±0.09.

SIGNIFICANCE

At normal and moderate treadmill speeds, in the presence of random belt speed perturbations, strongly anti-persistent fluctuations about gentle, persistent trends can lead to weak persistence/antipersistence of ST/SL time series.

Gait analysis under the lens of statistical physics

Human gait is a fundamental activity, essential for the survival of the individual, and an emergent property of the interactions between complex physical and cognitive processes. Gait is altered in many situations, due both to external constraints, as e.g. paced walk, and to physical and neurological pathologies. Its study is therefore important as a way of improving the quality of life of patients, but also as a door to understanding the inner working of the human nervous system. In this review we explore how four statistical physics concepts have been used to characterise normal and pathological gait: entropy, maximum Lyapunov exponent, multi-fractal analysis and irreversibility. Beyond some basic definitions, we present the main results that have been obtained in this field, as well as a discussion of the main limitations researchers have dealt and will have to deal with. We finally conclude with some biomedical considerations and avenues for further development.

Age-Related Changes in Standing Balance in Preschoolers Using Traditional and Nonlinear Methods

Considerable disagreement exists on the linearity of the development of standing balance in children. This study aimed to use different traditional and nonlinear methods to investigate age-related changes in standing balance in preschoolers. A sample of 118 preschoolers took part in this study. A force platform was used to record the center of pressure during standing balance over 15 s in three conditions: eyes open, eyes closed, and/or head extended backward. Detrended fluctuation analysis (DFA), recurrence quantification analysis (RQA), and traditional measures were used to evaluate standing balance. The main results are as follows: (1) Higher range and SD in the anterior-posterior (AP) direction were observed for 5-year-old than for 4-year-old children, while higher DFA coefficient (at shorter time scales) and higher determinism and laminarity in the AP direction were found for 5-year-old children compared to 3- and 4-year-old children; and (2) as sensory conditions became more challenging, all traditional measures increased and DFA coefficients (at shorter and longer time scales) decreased in the AP and mediolateral directions, while determinism and laminarity significantly declined in the AP direction. In conclusion, although increased postural sway, 5-year-old preschool children's balance performance improved, and their control strategy changed significantly compared with the younger preschoolers. Sensory perturbation (eye closure and/or head extension) changed preschoolers' balance performance and control strategy. Moreover, both traditional and nonlinear methods provided complementary information on the control of standing balance in preschoolers.

Influence of Autocorrelated Rhythmic Auditory Stimulations on Parkinson's Disease Gait Variability: Comparison With Other Auditory Rhythm Variabilities and Perspectives

Parkinson's Disease patients suffer from gait impairments such as reduced gait speed, shortened step length, and deterioration of the temporal organization of stride duration variability (i.e., breakdown in Long-Range Autocorrelations). The aim of this study was to compare the effects on Parkinson's Disease patients' gait of three Rhythmic Auditory Stimulations (RAS), each structured with a different rhythm variability (isochronous, random, and autocorrelated). Nine Parkinson's Disease patients performed four walking conditions of 10-15 min each: Control Condition (CC), Isochronous RAS (IRAS), Random RAS (RRAS), and Autocorrelated RAS (ARAS). Accelerometers were used to assess gait speed, cadence, step length, temporal organization (i.e., Long-Range Autocorrelations computation), and magnitude (i.e., coefficient of variation) of stride duration variability on 512 gait cycles. Long-Range Autocorrelations were assessed using the evenly spaced averaged Detrended Fluctuation Analysis (α-DFA exponent). Spatiotemporal gait parameters and coefficient of variation were not modified by the RAS. Long-Range Autocorrelations were present in all patients during CC and ARAS although all RAS conditions altered them. The α-DFA exponents were significantly lower during IRAS and RRAS than during CC, exhibiting anti-correlations during IRAS in seven patients. α-DFA during ARAS was the closest to the α-DFA during CC and within normative data of healthy subjects. In conclusion, Isochronous RAS modify patients' Long-Range Autocorrelations and the use of Autocorrelated RAS allows to maintain an acceptable level of Long-Range Autocorrelations for Parkinson's Disease patients' gait.

Benefits of nonlinear analysis indices of walking stride interval in the evaluation of neurodegenerative diseases

Indices characterising the long-range temporal structure of walking stride interval (SI) variability such as Hurst exponent (H) and fractal dimension (D) may be used in addition to indices measuring the amount of variability like the coefficient of variation (CV). We assess the added value of the former indices in a clinical neurological context. Our aim is to demonstrate that they provide a clinical significance in aging and in frequent neurodegenerative diseases such as Parkinson's disease, Huntington, and amyotrophic lateral sclerosis. Indices assessing the temporal structure of variability are mainly dependent on SI time series length and algorithms used, making quantitative comparisons between different studies difficult or even impossible. Here, we recompute these indices from available SI time series, either from our lab or from online databases. More precisely, we recompute CV, H, and D in a unified way. The average SI is also added to the measured parameters. We confirm that variability indices are relevant indicators of aging process and neurodegenerative diseases. While CV is sensitive to aging process and pathology, it does not discriminate between specific neurodegenerative diseases. H, which measures predictability of SI, significantly decreases with age but increases in patients suffering from amyotrophic lateral sclerosis. D, catching complexity of SI, is correlated with total functional capacity in patients with Huntington's disease. We conclude that the computation of H complements the clinical diagnosis of walking in patients with neurodegenerative diseases and we recommend it as a relevant supplement to classical CV or averaged SI. Since H and D indices did not lead to the same observations, suggesting the multi-fractal nature of SI dynamics, we recommend to open clinical gait analysis to the evaluation of more parameters.

Immersive Virtual Reality to Restore Natural Long-Range Autocorrelations in Parkinson's Disease Patients' Gait During Treadmill Walking

Effects of treadmill walking on Parkinson’s disease (PD) patients’ spatiotemporal gait parameters and stride duration variability, in terms of magnitude [coefficient of variation (CV)] and temporal organization [long range autocorrelations (LRA)], are known. Conversely, effects on PD gait of adding an optic flow during treadmill walking using a virtual reality headset, to get closer to an ecological walk, is unknown. This pilot study aimed to compare PD gait during three conditions: Overground Walking (OW), Treadmill Walking (TW), and immersive Virtual Reality on Treadmill Walking (iVRTW). Ten PD patients completed the three conditions at a comfortable speed. iVRTW consisted in walking at the same speed as TW while wearing a virtual reality headset reproducing an optic flow. Gait parameters assessed were: speed, step length, cadence, magnitude (CV) and temporal organization (evenly spaced averaged Detrended Fluctuation Analysis, α exponent) of stride duration variability. Motion sickness was assessed after TW and iVRTW using the Simulator Sickness Questionnaire (SSQ). Step length was greater (p = 0.008) and cadence lower (p = 0.009) during iVRTW compared to TW while CV was similar (p = 0.177). α exponent was similar during OW (0.77 ± 0.07) and iVRTW (0.76 ± 0.09) (p = 0.553). During TW, α exponent (0.85 ± 0.07) was higher than during OW (p = 0.039) and iVRTW (p = 0.016). SSQ was similar between TW and iVRTW (p = 0.809). iVRTW is tolerable, could optimize TW effects on spatiotemporal parameters while not increasing CV in PD. Furthermore, iVRTW could help to capture the natural LRA of PD gait in laboratory settings and could potentially be a challenging second step in PD gait rehabilitation.

Stochastic Resonance Reduces Sway and Gait Variability in Individuals With Unilateral Transtibial Amputation: A Pilot Study

Sub-threshold (imperceptible) vibration, applied to parts of the body, impacts how people move and perceive our world. Could this idea help someone who has lost part of their limb? Sub-threshold vibration was applied to the thigh of the affected limb of 20 people with unilateral transtibial amputation. Vibration conditions tested included two noise structures: pink and white. Center of pressure (COP) excursion (range and root-mean-square displacements) during quiet standing, and speed and spatial stride measures (mean and standard deviations of step length and width) during walking were assessed. Pink noise vibration decreased COP displacements in standing, and white noise vibration decreased sound limb step length standard deviation in walking. Sub-threshold vibration positively impacted aspects of both posture and gait; however, different noise structures had different effects. The current study represents foundational work in understanding the potential benefits of incorporating stochastic resonance as an intervention for individuals with amputation.