Human locomotion, even over a level smooth surface, is a complex task requiring the coordination of a large number of muscles acting over a collective of joints. The neuromuscular system faces the challenge of controlling the center of mass against large gravitational and forward momentum forces when, much of the time, the body is supported by only one limb. Yet, despite this challenging control problem, the gait cycle is subject to a very low level of variability. This has traditionally been taken to indicate that human gait patterns reflect a very repeatable kinematic pattern (19), with the increases in the amount of variability assumed to be indicative of problems in control.
Gait cycle variability is increasingly being examined to better understand the mechanics and control of human locomotion. Although there have been a significant number of studies focusing on the amount of variability of gait cycle parameters (e.g., stride interval, step frequency, ground reaction force), in many cases, these studies have yielded contradictory results. There is a lack of comparison of different gait parameters within studies, and often, some sort of constraint to locomotion is used to isolate the gait parameter of interest. Furthermore, most studies of gait variability have examined only the amount and not the time-dependent structure of variability. It is becoming increasingly apparent that global distributional measures of the amount of variability are insufficient for adequate assessment of gait variability (e.g., (6)).
Traditionally, variability in human movement has been treated as noise superimposed upon a signal, where the signal is the intended movement and the variability is random white noise about this intended movement (11). The focus of this approach has been to quantify the amount of variation (noise) associated with the movement property of interest. Typically, variability is indexed either by the standard deviation (in absolute terms) or, in relative terms, by the coefficient of variation (standard deviation divided by mean). In terms of gait, increased variability has been linked to an increased risk of falling in the elderly, but the underlying mechanisms of this relation have yet to be fully elucidated.
There is a growing body of literature showing that the cycle-to-cycle variation seen in a wide variety of physiological systems is nontrivial and may offer insight into the organization and control of these systems (e.g., (3)). For example, the strength of long-range correlations in the interbeat interval of the heart has been shown to be a powerful predictor of mortality among patients with chronic congestive heart failure. Similarly, the strength of long-range correlations in the stride interval decreases significantly with both aging and disease (6). Such findings indicate that the structure of stride-to-stride variability can offer insight into the control of the locomotor system, and a means to quantify age-related and pathological alterations in the control system.
In this article, we show that the variability of gait in human walking and running is not random but exhibits self-similarity that is dependent on the speed of locomotion. This pattern of fractal structure to variability is important because it suggests a different class of model construct in locomotion than the traditional view of noise-driven gait variability.
SELF-SIMILARITY IN THE STRIDE INTERVAL OF HUMAN WALKING
Fluctuations in the gait cycle are not random. Rather, they are self-similar and exhibit long-range dependence, such that, any given stride interval is dependent on the stride interval at remote previous times (7). The early work in this area has shown that the strength of these correlations is dependent on several factors, including neuromuscular health, age (6), and possibly, gait speed (7). In this review, we show the influence of the speed of locomotion on the dynamical structure of intraindividual variability of human walking and running. We give emphasis to the time-dependent processes of the fractal dynamics of gait variability and the method of detrended fluctuation analysis (DFA) that is useful for detecting long-term memory processes in nonstationary signals.
DFA is a nonlinear technique created by Peng and colleagues (12) for the calculation of long-range correlations in physiological time series. DFA is a modified random walk analysis that makes use of the fact that a long range-correlated time series can be mapped to a self-similar process by integration (7). Figure 1 shows a sample stride interval time series and the associated measure through DFA of the long-range correlation.
In DFA, the integrated time series is examined using a windowing process, the log of the variance for a given observation window (F(n)) is plotted against the log of the window size (n), and the linear slope is calculated, yielding an "alpha" value (α). White noise corresponds to an α of 0.5 or a spectral slope of 0; pink noise corresponds to an α of 1 or a spectral slope of 1. An α value between 0.5 and 1 is indicative of long-range correlations such that any given stride interval is dependent on the stride interval at remote previous times and that the dependence of stride intervals decays in a power law, fractal-like manner with time. Alpha values in this range also indicate that fluctuations are self similar, in that fluctuations at one time scale are statistically similar to fluctuations at all other time scales.
DFA has shown that structure is present in the fluctuations of the stride interval (where stride interval is defined as the time between the heel strike of one foot and the successive heel strike of the same foot). During walking in healthy young adults, α falls between 0.5 and 1.0. For example, during 6-min walking trials, the average value for the DFA scaling exponent for young adults walking at their preferred walking speed was 0.87 (6). To differentiate statistically between a long-range scaling process and a process without correlations, Hausdorff and colleagues (7) generated surrogate data sets by randomly shuffling the original time series. In this way, the distributional statistics (i.e., mean, standard deviation, and higher moments) are the same for both the original time series and the corresponding surrogate time series; however, the sequential ordering was destroyed, as illustrated in Figure 2 (7).
West and Griffin (16) confirmed the findings of Hausdorff and colleagues (7) in healthy young individuals. Instead of using force sensors and heel strike to calculate the interstride interval, West and Griffin (16) used a goniometer and generated the time series using successive maximal positive extensions of the same knee. The time series were analyzed using relative dispersion (R) and data aggregation, such that the relative dispersion (or CV) was calculated over successively larger numbers of data points (n). The logged values of R and n were plotted, and slope was calculated. The value of the slopes were in agreement with the α values found by Hausdorff and colleagues (7), confirming the presence of long-range correlations in the stride interval of human walking.
When individuals are required to constrain their step frequency by walking in time to a metronome, the long-range correlations in the stride interval break down (α ≈ 0.5), such that the time series approximates uncorrelated noise (7). This finding supports the hypothesis that higher cortical centers modulate the fluctuations of gait variability. Interestingly, Terrier and colleagues (15) showed that, during metronomic walking, only the long-range correlations of the step frequency time series became anticorrelated; those of the step length were unaffected by the timing constraint.
In unconstrained overground walking, there is no explicit requirement to walk with a specific time interval between strides. However, when walking in time to a metronome, the time series of stride intervals is, in effect, a time series of interresponse intervals. It is well-established that there is a negative covariation between adjacent time intervals during repetitive response tasks (17,18). As such, it should be anticipated that the long-range correlations in the stride interval time series break down when people are required to walk in time to a metronome. Because there is no explicit constraint on the step length in the study of Terrier et al. (15), the long-range correlations of these time series are unaltered. One would expect that had the step or stride length been constrained, a similar breakdown in long-range correlations would occur in the time series of this variable.
Use of a treadmill to collect stride interval data may influence α. Although mechanically there is no theoretical difference between overground and treadmill locomotion, walking on a treadmill does stabilize the locomotor output (4). It is possible that the constantly driven speed of the treadmill, the unchanging surface, and optic flow influence the scaling behavior of gait cycle fluctuations; however, it has been shown that α is unchanged for patients with peripheral neuropathy (5) (i.e., people with reduced afferent input). Although future research should include a direct comparison of overground to treadmill locomotion, the values for α reported in our studies (9,10) are within the range of those reported for overground locomotion (e.g., (6,7,15)).
Variables Influencing the Size of α
The long-range correlations in walking data are significantly lower in elderly (α = 0.68) as compared with young adults (α = 0.87), and lower still in Huntington disease patients (α = 0.60) (6). Conversely, in the stride interval of young children, the long-range correlations decay more slowly with time compared with young adults (8). Thus, across the life span, there is a general trend for a decrease in long-range correlations in walking. The mechanisms for these age-related changes are likely to be independent, given that the direction of the shift in size of the scaling exponent is constant with increasing age, whereas there is a U-shaped function for walking ability with age. One modeling approach suggests that the decrease in strength of long-range correlations seen from childhood to adulthood is related to an increase in neuronal connectivity, whereas the decrease in strength associated with aging and disease is related to the unavailability of some neuronal centers (2).
With a view to confirming the robustness of the long-range correlations in the stride interval, Hausdorff and colleagues (7) examined the fluctuations of unconstrained walking at slow, normal, and fast walking paces. Figure 2 shows the essence of these preliminary speed-dependent findings (7). The average α for these respective speeds was 0.98, 0.90, and 0.97. The difference in α with speed was significant, suggesting that there may be systematic changes in the size of α with walking speed.
SPEED-DEPENDENT SELF-SIMILARITY IN THE VARIABILITY OF LOCOMOTION
We have conducted a set of experiments examining the influence of walking and running at preferred versus nonpreferred walking (9) and running (10) speeds on the self-similarity of variability in walking and running (Jordan, K. unpublished manuscript/observations, 2006). The experiments each had young adult female subjects walking and/or running on a treadmill with a force plate embedded. This experimental setup allowed the recording and derivation of a set of gait parameters as a function of walking and running speed. Before the experiment, each subject's preferred walking and running speeds were determined through calibrating the belt speed to the subject's perception of the preferred speed for each gait. In the first two experiments, the experimental speed conditions were set to ±10% and 20% of the preferred speed for the respective gait pattern.
The mean group α value as a function of the percentage of preferred (a) walking and (b) running speed is shown in Figure 3 (9,10). The figure shows the speed-dependent α value for the stride interval, step interval, stride length, step length, and impulse. In both walking and running, a shallow U-shaped function is apparent for each variable with the mean centered close to the preferred speed. These findings confirm those of Hausdorff et al. (7) and extend them by showing a U-shaped speed-dependent function for α in several gait variables for both the walking and running gaits. The data also indicate that the long-range correlations are stronger over the cycle stride than the single step.
Gait asymmetry is present in the gait cycle of healthy adults, and this seems to be related to limb dominance (13). In subjects with dominant right limb, the left limb was associated with the function of control, whereas the main function of the right limb was associated with propulsion. Thus, the difference between strength of long-range correlations in the stride versus step interval may be related to a possible division of labor between limbs regarding propulsion and control. This highlights the need to examine the influence of limb dominance and gait asymmetry in terms of long-range correlations.
A third experiment examined walking and running at speeds at and around preferred gait transition speeds, where subjects walk at high speeds where running typically occurs and vice versa. Figure 4 shows the values of DFA for stride interval over the three experiments on walking and running (Jordan, K., unpublished manuscript/observations, 2006). The average α values for all participants are plotted against the average speeds corresponding to the percentage of preferred walking (walk), preferred running (run), walk-run (fast walk), and run-walk (slow run) transition speeds. Figure 4 further highlights that long-range correlations are influenced by speed of locomotion. A U-shaped pattern of change with speed was observed for α of the stride interval of both waking and running, with the minimum falling at or close to the preferred speed of locomotion. This pattern of findings supports the prediction that preferred speeds of locomotion have weaker long-range correlations compared with speeds faster and slower than preferred. The reduced strength of long-range correlations in a statistical sense indicates that any given stride is less influenced or dependent upon all preceding strides (7). As such, one interpretation of the relatively smaller α values at preferred speeds is that any given stride is less constrained by those that precede it and, hence, more readily adaptable than at speeds faster and slower than preferred. This finding provides additional evidence for the special dynamics of preferred oscillations in human movement.
For a healthy heart, α value is typically close to 1 (3). In contrast, during walking, the size of α in young children is close to 1, decreases to approximately 0.75 for healthy young adults, and approaches 0.5 in aging and diseased populations. If we assume that control of locomotion is optimized in healthy young adults, we can conclude that an α value of 1, on average, is not ideal for the stride interval of human locomotion. Rather, it seems that α value close to 1 for the stride interval is indicative of a lack of exploration of the full range of gait dynamics. This may be a result of an immature central nervous system (2) or a result of certain external constraints that preclude exploration of this space, such as faster or slower than preferred speeds. Given the differences between the largely autonomic processes responsible for control of heart rate rhythm and the largely voluntary processes responsible for control of gait rhythm, it is not surprising that the fractal nature of these system outputs is different.
There is a significant cognitive cost associated with walking at nonpreferred speeds (1) that may be related to the increasing size of α at nonpreferred speeds of locomotion. Humans can walk at speeds they would prefer to run at and vice versa, and horses can be trained to adopt nonpreferred gait patterns over a range of speeds without exhibiting any obvious instabilities. In contrast, decerebrate cats locomoting near typical transition speeds are unable to stabilize their gait pattern and alternate between walking and trotting (14). Thus, in the mature neurologically intact animal, there are central mechanisms in place to stabilize a given locomotor pattern under nonpreferred conditions. The previously discussed studies of metronomic walking, Huntington disease, and peripheral neuropathy suggest that the long-range correlations observed in healthy unconstrained gait are of central origin. The speed-dependent changes in long-range correlations in humans may result in part from the increasing influence of central control associated with stabilizing locomotion at nonpreferred speeds.
The sequential structure of the variability of the gait cycle is not, as traditionally assumed, white Gaussian noise. Our studies on walking and running show the speed-dependent nature of the self-similarity in the variability of gait. The preferred gait speed seems to be an anchoring point that has the greatest potential for adaptability in the dynamics of the stride and step of a locomotory mode as revealed by the lower α value. A challenge for the future is that there are many kinds of models that can produce self-similar fractal dynamics. Our studies suggest that the speed of locomotion should be fundamental to the development of these models in gait dynamics.
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