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PLoS Comput Biol. 2010 July; 6(7): e1000856.

Published online 2010 July 15. doi: 10.1371/journal.pcbi.1000856

PMCID: PMC2904769

Jörn Diedrichsen, Editor^{}

University College London, United Kingdom

Conceived and designed the experiments: JBD JPC. Analyzed the data: JBD JJ. Contributed reagents/materials/analysis tools: JBD JPC. Wrote the paper: JBD JJ JPC. Did most of the actual computational modeling and generating of simulation data: JJ.

Received 2009 July 8; Accepted 2010 June 10.

Copyright Dingwell et al.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

This article has been cited by other articles in PMC.

It is widely accepted that humans and animals minimize energetic cost while walking. While such principles predict average behavior, they do not explain the *variability* observed in walking. For robust performance, walking movements must adapt at each step, not just on average. Here, we propose an analytical framework that reconciles issues of optimality, redundancy, and stochasticity. For human treadmill walking, we defined a goal function to formulate a precise mathematical definition of one possible control strategy: maintain constant speed at each stride. We recorded stride times and stride lengths from healthy subjects walking at five speeds. The specified goal function yielded a decomposition of stride-to-stride variations into new gait variables explicitly related to achieving the hypothesized strategy. Subjects exhibited greatly decreased variability for goal-relevant gait fluctuations directly related to achieving this strategy, but far greater variability for goal-irrelevant fluctuations. More importantly, humans immediately corrected goal-relevant deviations at each successive stride, while allowing goal-irrelevant deviations to persist across multiple strides. To demonstrate that this was not the only strategy people could have used to successfully accomplish the task, we created three surrogate data sets. Each tested a specific alternative hypothesis that subjects used a different strategy that made *no* reference to the hypothesized goal function. Humans did *not* adopt any of these viable alternative strategies. Finally, we developed a sequence of stochastic control models of stride-to-stride variability for walking, based on the Minimum Intervention Principle. We demonstrate that healthy humans are not precisely “optimal,” but instead consistently slightly *over*-correct small deviations in walking speed at each stride. Our results reveal a new governing principle for regulating stride-to-stride fluctuations in human walking that acts independently of, but in parallel with, minimizing energetic cost. Thus, humans exploit task redundancies to achieve robust control while minimizing effort and allowing potentially beneficial motor variability.

Existing principles used to explain how locomotion is controlled predict average, long-term behavior. However, neuromuscular noise continuously disrupts these movements, presenting a significant challenge for the nervous system. One possibility is that the nervous system must overcome all neuromuscular variability as a constraint limiting performance. Conversely, we show that humans walking on a treadmill exploit redundancy to adjust stepping movements at each stride and maintain performance. This strategy is not required by the task itself, but is predicted by appropriate stochastic control models. Thus, the nervous system simplifies control by strongly regulating goal-relevant fluctuations, while largely ignoring non-essential variations. Properly determining how stochasticity affects control is critical to developing biological models, since neuro-motor fluctuations are intrinsic to these systems. Our work unifies the perspectives of time series analysis researchers, motor coordination researchers, and motor control theorists by providing a single dynamical framework for studying variability in the context of goal-directedness.

Walking is an essential task most people take for granted every day. However, the neural systems that regulate walking perform many complex functions, especially when we walk in unpredictable environments. These systems continuously integrate multiple sensory inputs [1]–[4] and generate motor outputs to coordinate many muscles to achieve efficient, stable, and adaptable locomotion. Establishing the fundamental principles that guide this control is central to understanding how the central nervous system regulates walking.

The principal idea used to explain how humans and animals regulate walking has been energy cost [5]–[12]. At a given speed, humans choose an average step length and frequency that minimizes energy cost [7], [9], [10], [12]. Small changes in either average stride length or average stride time increase energy cost in humans similarly (Fig. 1, and Supplementary Text S1) [7]. These experimental findings have been supported by multiple computational models [9]–[11], [13], [14]. Such optimality principles have been a major focus for understanding the control of complex movements [15]–[20]. However, these optimization criteria have been used primarily to predict average behavior, not to explain the *variability* ubiquitously observed in movements like walking [21]–[24]. Understanding the nature of this variability may be critical to understanding how humans perform skilled movements [25]–[34]. Most optimization approaches do not address whether the nervous system must *overcome* all variability as a limiting constraint [16], [26], [29], [32], or instead exploits redundancy to *regulate* variability in ways that help maximize task performance [25], [27], [28], [34].

Others have sought to determine how muscles are organized into functional synergies to resolve the inherent redundancy of complex movements [35]–[37]. These efforts likewise characterize average behavior and so also provide few insights into movement variability. Conversely, redundancy gives rise to equifinality: i.e., there are typically an infinite number of ways to perform the same action [25], [38]. Equifinality permits individuals to perform complex tasks reliably and repeatedly while allowing variability in a movement's particulars. This is thought to facilitate adaptability in motor performance [25]. Recent researchers have addressed this issue experimentally using the geometry-based uncontrolled manifold (UCM) approach [39], [40]. A related concept, the minimum intervention principle (MIP) [27], [28], [41] ties these ideas to stochastic optimal control theory and provides a concrete computational framework for predicting precisely how trial-to-trial movement variability arises in redundant motor systems performing tasks with well prescribed goals [18], [27], [28], [41], [42].

During walking, humans need to adapt at *every* step (not just on average) to be able to respond to externally and/or internally generated perturbations [23], [43], [44]. While the neurophysiological mechanisms that enact these responses are well known [1]–[4], the fundamental principles governing adaptation *from stride to stride* remain unknown. Small stride-to-stride fluctuations in gait dynamics are typically assumed to reflect random noise. Indeed, there is ample evidence supporting multiple sensory and motor sources of physiological noise [31], [45]–[48]. However, stride-to-stride variations in gait cycle timing exhibit statistical persistence [22], [49], [50], which has been argued to be “indispensible” to healthy physiological function [51], [52]. Stride intervals become more uncorrelated (i.e., less persistent) in elderly subjects and patients with Huntington's disease [53], but not in patients with peripheral sensory loss [54]. Understanding how stride-to-stride control is enacted therefore requires quantifying not only average magnitudes of variations across strides, but also the specific temporal sequencing of those variations.

Here, we formulate goal functions [25] that give concrete mathematical form to hypotheses on the strategies used to achieve a given task. This provides a unifying framework for reconciling issues of optimality, redundancy, and stochasticity in human walking. Walking on a motor driven treadmill only requires that subjects do not “walk off” either the front or back end of the treadmill. While subjects must, over time, walk at the same average speed as the treadmill, variations in speed due to changes in stride length and/or stride time do occur and can be sustained over several consecutive strides [23], [24], [55], [56]. The main question addressed here is how do people *regulate* these variations?

We present a mathematical definition of a specific hypothesized task strategy [25], [57] with the goal to maintain constant walking speed *at each stride*. This yields a decomposition of stride-to-stride variations into new gait variables explicitly related to achieving this strategy. Time series analyses confirm that humans do indeed adopt this hypothesized strategy. We similarly analyze three alternative strategies that equally achieve the task requirements, but make *no* reference to the hypothesized goal function. Humans do *not* adopt any of these alternatives. Finally, we develop a sequence of stochastic optimal control models of stride-to-stride dynamics to determine if they replicate our observations. These models confirm that healthy humans do carefully regulate their movements explicitly to maintain constant speed at each stride. However, humans do not use strategies that are precisely “optimal” with respect to the employed cost functions, but instead slightly but consistently *over*-correct small deviations in walking speed from each stride to the next.

The primary task requirement for walking on a treadmill with belt speed *v* is to not walk off the treadmill. The net change in displacement, relative to the laboratory reference frame, for stride *n* is determined by the stride length, *L _{n}*, and stride time,

(1)

where the summation is the net displacement walked over *N* strides and *L _{TM}* is the length of the treadmill belt. A key observation is that

(2)

That is, subjects could attempt to maintain constant speed at each stride. This goal function is not a “constraint,” however, because *it is not required* by Eq. (1). It is instead only one possible movement strategy. The solid line in Fig. 2 defines a “Goal Equivalent Manifold” (GEM) [25] containing all [*T _{n}*,

The hypothesized GEM exists prior to, and independent of, any specific control policy people might adopt to regulate their stepping movements. To determine if humans adopt a strategy that explicitly recognizes this GEM, we defined deviations tangent (*δ _{T}*) and perpendicular (

To test GEMs of different location/orientation, subjects walked on a motorized treadmill at each of 5 constant speeds, from 80% to 120% of their preferred walking speed (PWS). Time series of stride times (*T _{n}*), stride lengths (

As expected, when subjects walked faster, they increased stride lengths (Fig. 3A), decreased stride times (Fig. 3B), and increased stride speeds (Fig. 3C). Stride length variability (Fig. 3D) increased slightly at speeds faster and slower than PWS, while stride time variability (Fig. 3E) increased at slower walking speeds, and stride speed variability (Fig. 3F) increased at faster walking speeds. However, standard deviations only quantify the average magnitude of differences across all strides, regardless of temporal order. They yield no information about how each stride affects subsequent strides.

Therefore, to quantify temporal correlations across consecutive strides, we computed scaling exponents, *α*, using Detrended Fluctuation Analysis (DFA) [22], [49], [51], [52] (see Methods). *α*>½ indicates statistical *persistence*: deviations in one direction are more likely to be followed by deviations in the same direction. *α*<½ implies *anti*-persistence: deviations in one direction are more likely to be followed by deviations in the opposite direction. *α*=½ indicates uncorrelated noise: all deviations are equally likely to be followed by deviations in either direction. In the context of control, statistical persistence (*α*>½) is interpreted as indicating variables that are *not* tightly regulated. Conversely, variables that are tightly regulated are expected to exhibit either uncorrelated or anti-persistent fluctuations (*α*≤~½).

Consistent with previous results [22], [50], [54], *T _{n}* and

As expected [23], [24], [55], [56], subjects did “drift” forward and backward (Eq. 1) over time along the treadmill belt (Fig. 4A). Most of these drifting movements remained contained within approximately the middle one third of the treadmill belt (Fig. 4B). This suggested that subjects adopted a more “conservative” walking strategy than actually *required* by the inequality constraint of Eq. (1). However, these movements also exhibited a *high* degree of statistical persistence (~1.25<*α*<~1.55) at all walking speeds (Fig. 4C). Thus, deviations in absolute position along the treadmill belt were allowed to persist even more so than deviations in either *T _{n}* or

Plots of *L _{n}* versus

The *δ _{T}* and

One obvious question is whether these observed dynamics represented the *only* viable strategy subjects could have used. Rejecting this possibility requires only that we identify at least *one* alternative strategy that still satisfied the fundamental task requirements (Eq. 1), but was completely “ignorant” of the proposed GEM defined by Eq. 2. Here, we present *three* such alternatives using “surrogate” data [60], [61] that each represent the output of a particular type of *data-based model* of the observed stride-to-stride dynamics. Each surrogate model directly tested a specific null hypothesis that subjects could have successfully completed the treadmill walking task (i.e., satisfied Eq. 1) using a strategy that made absolutely *no* reference to the GEM.

The first alternative strategy was to choose a reference point, [*T*
^{*}, *L*
^{*}] (e.g., Fig. 1), on the GEM and maintain sufficiently small variance about this point to satisfy Eq. (1). Here, “control” would consist entirely of suppressing variability in both *L _{n}* and

These surrogates exhibited approximately isotropic distributions (i.e., no obvious directionality) about [*T*
^{*}, *L*
^{*}] within the [*T _{n}*,

Fig. 6 demonstrates unequivocally that the strategy subjects used (Fig. 5) was not the only successful strategy they could have adopted. They could have adopted a control policy that equally achieved the task requirement defined by Eq. 1 without using the GEM-based control strategy defined by Eq. 2. We also used surrogate data techniques to test two additional model hypotheses of how subjects might have controlled their stride-to-stride dynamics. We tested a second alternative strategy that also regulated *T _{n}* and

To obtain more definitive conclusions about the underlying control policies used, we first hypothesized that subjects controlled their movements based on the minimum intervention principle (MIP) [27], [28], [41], [42]. We created a model “walker” (see Methods), where a two-dimensional state variable, **x**
* _{n}*=[

By construction, this MIP model walked with nearly the same average stride parameters (Fig. 7A) and stride speed (*S _{n}*) standard deviations (Fig. 7B) as humans. However, the MIP model exhibited substantially greater variability in both

However, the MIP model did not incorporate any additional physiological and/or biomechanical constraints. Because human legs have finite length, they cannot take extremely long steps easily. Because they have inertia, they cannot easily move extremely fast. Likewise, the MIP model incorporated no capacity to minimize energy cost [5]–[12]. Each of these factors would act to constrain the choices of *L _{n}* and

By construction, this POP model also walked with nearly the same average stride parameters (Fig. 8A) and variability (Fig. 8B) as humans. Likewise, this model exhibited statistical persistence (*α*>½) for both *L _{n}* and

The MIP and POP models both optimally corrected deviations away from the GEM at the next stride. Thus, the *δ _{P}* fluctuations in each case (Figs. 7G, ,8G)8G) reflected nearly uncorrelated white noise (

By construction, this OVC model walked with nearly the same average stride parameters (Fig. 9A), stride variability (Fig. 9B), and statistical persistence for both *T _{n}* and

This study set out to determine how humans regulate stride-to-stride variations in treadmill walking. We specifically sought to determine if the nervous system always overcomes all variability as a fundamental performance limitation [16], [26], [29], [32], or if it instead exploits redundancy to *selectively* regulate the effects of variability and enhance task performance [25], [27], [28]. We demonstrate that formulating mathematical hypotheses on specific strategies (e.g., Eq. 2) used to achieve task requirements (e.g., Eq. 1) can reconcile issues of optimality, redundancy, and stochasticity in human walking. Our results reveal a new governing principle for regulating stride-to-stride fluctuations in human walking that acts *independently* of, but in parallel with, the principle of minimizing energy cost [5]–[12].

We hypothesized that humans walking on a treadmill would adopt a specific strategy [25], [57] to maintain constant speed at each consecutive stride (Eq. 2), something *not* absolutely required to complete this task. This yielded a decomposition of stride-to-stride variations into new gait variables (*δ _{P}* and

Beyond the five alternative control strategies clearly rejected by our results (Figs. 6–
88 and Supplementary Text S2), other plausible alternatives were considered. One seemingly reasonable strategy might be to try to stay at a fixed location on the treadmill. Such absolute position control would necessitate regulating *d _{net}*(

Minimizing energy cost has been the primary explanation for how humans and animals regulate walking [5]–[12]. This criterion predicts the presence of a single optimal operating point, [*T _{Opt}*,

Our findings, however, remain compatible with the idea that humans also try to minimize energy cost while walking. The failure of the MIP model (Fig. 7) to capture the experimentally observed gait dynamics demonstrates that humans do not *only* minimize deviations away from the GEM. The POP model (Fig. 8), is precisely compatible with adding the secondary goal of minimizing energy cost. For the average walking speed modeled (*v*=1.21.m/s), we computed a POP of [*T*
^{*}, *L*
^{*}]=[1.105 s, 1.337 m]. Mechanical walking models of Minetti [9] and Kuo [10] predict similar energetically optimal POPs of [*T _{Opt}*,

Humans also consistently *over*-corrected *δ _{P}* deviations (Fig. 5D). Our OVC model (Fig. 9) provides one possible explanation: that humans use sub-optimal control to correct stride-to-stride deviations. In the model, anti-persistence in

The principal contribution of our work is thus to demonstrate that considerations other than minimizing energy cost help determine [*T _{n}*,

The nervous system appears to estimate both motor errors and the sources of those errors to guide continued adaptation [30], [31], [33]. The neural structures involved in decision making may even deliberately insert noise into the process to enhance adaptation [64], [65]. Exposing humans to tasks that share similar structural characteristics but vary randomly may even help facilitate the ability to generalize to novel tasks [33]. Similar capacities were recently demonstrated even in highly-learned (i.e., “crystallized”) adult bird song [66], where residual variability in this skill represented “meaningful motor exploration” to enhance continued learning and performance optimization [31], [66], [67]. Our findings suggest that similar purposeful motor exploration occurs in the highly-learned task of human walking.

It has been widely argued that statistically persistent fluctuations are a critical marker of “healthy” physiological function [51], [52] and that uncorrelated or anti-persistent fluctuations are a sign of disease or pathology [51]–[53]. The present results strongly refute this interpretation. The subjects tested here clearly cannot be simultaneously both “healthy” (according to *α*(*δ _{T}*)) and “unhealthy” (according to

One question is whether the theoretical framework developed here will generalize to other contexts. During unconstrained overground walking [50], humans exhibited strong statistical persistence for *T _{n}* and

All participants provided written informed consent, as approved by the University of Texas Institutional Review Board.

Seventeen young healthy adults (12M/5F, age 18–28, height 1.73±0.09 m, body mass 71.11±9.86 kg), participated. Subjects were screened to exclude anyone who reported any history of orthopedic problems, recent lower extremity injuries, any visible gait anomalies, or were taking medications that may have influenced their gait.

Subjects walked on a level motor-driven treadmill (Desmo S model, Woodway USA, Waukesha WI) while wearing comfortable walking shoes and a safety harness (Protecta International, Houston TX) that allowed natural arm swing [44]. First, preferred self-selected walking speed (PWS) was determined [23]. Subjects reported the limits of their PWS while the treadmill was slowly accelerated and then decelerated three times. These upper and lower limits were averaged to determine PWS [23]. Following a 2-minute rest, subjects completed two 5-minute walking trials at each of five speeds (80, 90, 100, 110 and 120% of PWS), presented in pseudo-random order [44]. Subjects rested at least 2 minutes between each trial to prevent fatigue. Subjects were instructed to look ahead and avoid extraneous movements while walking. Data from 1 trial from each of 4 subjects (i.e., 2.35% of all 170 trials collected) were discarded due to poor data quality. For the remaining 166 trials, an average of 272±25 total strides (range: 213–334) were analyzed.

Five 14-mm retro-reflective markers were mounted to each shoe (heads of the 2^{nd} phalanx and 5^{th} metatarsal, dorsum of the foot, inferior to the fibula, and calcaneous). The movements of these markers were recorded using an 8-camera Vicon 612 motion capture system (Oxford Metrics, UK). All data were processed using MATLAB 7.04 (Mathworks, Natick MA). Brief gaps in the raw kinematic recordings were filled using rigid-body assumptions. Marker trajectories were low-pass filtered with a zero-lag Butterworth filter at a cutoff frequency of 10 Hz. A heel strike was defined as the point where the heel marker of the forward foot was at its most forward point during each gait cycle.

For the present analyses, the relevant walking dynamics were entirely captured by the impact Poincaré [58], [59] section defined by the [*T _{n}*,

*T _{n}* and

We defined a specific operating point on each GEM as and , and defined new coordinates centered at this operating point, and . We then performed a linear coordinate transformation to define the deviations along the GEM, *δ _{T}*, and perpendicular to the GEM,

(3)

Standard deviations and DFA scaling exponents (*α*, see Supplementary Text S4) were computed across all strides for each *δ _{T}* and

Three types of surrogate time series [60], [61] were generated and analyzed. First, *randomly shuffled* surrogates (Fig. 6) were generated for each trial by independently shuffling each original *T _{n}* and

Second, *phase-randomized* surrogates [43], [60], [61] were generated separately for the original *T _{n}* and

Third, for each trial *paired* randomly shuffled surrogates were generated simultaneously by randomly shuffling both *T _{n}* and

All surrogates were constrained so they did not “walk off” the treadmill (i.e., *all* surrogates satisfied Eq. 1). This was easily verified by computing the net cumulative distance (*d _{net}*) each surrogate time series would have walked relative to the treadmill at each stride,

(4)

where *d*=0 represents the center of the treadmill belt. We then extracted the maximum forward [max(*d _{net}*)], and backward [min(

For each surrogate, we then computed a new stride speed (*S _{n}*) time series by dividing the surrogate

The stride-to-stride dynamics on the treadmill were modeled as a discrete map:

(5)

where was the state for the current stride *n*, was the corresponding state for the next stride, and was a vector of control inputs. *I* was the 2×2 identity matrix. *G* was a 2×2 diagonal matrix with diagonal elements *g*
_{1} and *g*
_{2} denoting additional gains, each set initially to 1 and used *only* as a convenient means to tune the system away from optimality (see Supplementary Text S3). *N* was a 2×2 diagonal multiplicative (i.e., motor output) noise matrix with nonzero diagonal elements. **η** was a 2×1 vector of additive (i.e., sensory and/or perceptual) noise. Non-zero elements of *N* and **η** were taken to be independent, Gaussian random variables with mean zero and standard deviation σ* _{k}* (see Supplementary Text S3).

The state update equation (Eq. 5) is intended to model only the discrete-time *inter*-stride walking dynamics. That is, it represents a simple model of the control processes that regulate noise-induced fluctuations away from perfect performance by adjusting *T _{n}* and

The controller was modeled as an unbiased stochastic optimal single-step controller with direct error feedback. This controller design was based on the Minimum Intervention Principle (MIP) [27], [28], but modified to incorporate a preferred operating point (POP) for the controller along the GEM. Accordingly, the cost function took the form:

(6)

The first term, *αe*
^{2}, depended on the definition of the goal-level error for the task [25]. For treadmill walking, we assumed the controller's strategy was to maintain constant speed at each stride, *L _{n}*/

The objective of the controller was to minimize *C* in a probabilistic sense across each trial. That is, we did not minimize the cost itself function directly, but rather its expected value, . The optimal control inputs *u*
_{1} and *u*
_{2} were then determined by solving a classic quadratic optimal control problem with an equality constraint. This process yielded optimal control inputs obtained analytically as a function of the current state, **x**
* _{n}* (see Supplementary Text S3 for details).

The optimal, strictly MIP controller (Fig. 7) was implemented as follows. First, we set *β*=0 so the cost function, Eq. (6), depended *only* on the goal-level error *e*. This strict MIP controller only corrected *δ _{P}* deviations off of the GEM (Fig. 2). When the state,

The optimal POP controller (Fig. 8) was implemented as follows. To drive the states to a preferred operating point, [*T ^{*}*,

To match our human data in terms of the anti-persistent DFA exponents in the *δ _{P}* time-series (Fig. 5D), we implemented the

It is important to note that for each model, no explicit or rigorous attempts were made to find “best fits” to our experimental data. For example, we could adjust model parameters to fit different values for the means and SD's of different stride variables to try to more closely replicate the data of any of our individual subjects. However, our overall results were insensitive to the precise parameter values: i.e., the contrasts in the fundamental qualitative features of each of these models will remain the same.

For all three model configurations, we generated 20 simulations of 500 walking strides each to represent a single simulated “average” subject. Model outputs consisted of stride time (*T _{n}*) and stride length (

All statistical tests were performed in Minitab 15 (Minitab, Inc., State College, PA). For all dependent measures, we computed between-subject means and ±95% confidence intervals at each walking speed. Where appropriate (Figs. 3, ,4C,4C, 6A–B, and 7A–B), linear or quadratic trends across speeds were computed using standard least squares regression [23]. The standard deviations and DFA *α* exponents computed from the experimental (Fig. 5C–D) and surrogate (Figs. 6F–G and 7F–G) data sets were subjected to a 3-factor (Direction×Speed×Subject) mixed-effects, repeated measures, general linear model analysis of variance (ANOVA). Direction (*δ _{T}* vs.

Extended description of the construction of Figure 1.

(0.30 MB PDF)

Click here for additional data file.^{(291K, pdf)}

Additional surrogate data analyses and results.

(0.44 MB PDF)

Click here for additional data file.^{(427K, pdf)}

Derivation of the GEM-based inter-stride optimal controller for treadmill walking.

(0.26 MB PDF)

Click here for additional data file.^{(249K, pdf)}

Extended description of the detrended fluctuation analysis algorithm.

(0.27 MB PDF)

Click here for additional data file.^{(267K, pdf)}

The authors thank Dr. Hyun Gu Kang and Dr. Deanna H. Gates for their assistance with data collection and initial processing.

The authors have declared that no competing interests exist.

Partial funding was provided by a Biomedical Engineering Research Grant (grant # RG-02-0354) from the Whitaker Foundation (JBD), by US National Institutes of Health grants 1-R03-HD058942-01 and 1-R21-EB007638-01A1 (JBD), and by National Science Foundation grant 0625764 (JPC). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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