^{1}

^{*}

^{2}

^{2}

^{*}

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.

The authors have declared that no competing interests exist.

It is widely accepted that humans and animals minimize energetic cost while walking. While such principles predict average behavior, they do not explain the

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

The principal idea used to explain how humans and animals regulate walking has been energy cost

Contour lines represent iso-energy level curves for average energetic cost of transport: i.e., energy expenditure per distance walked per kg of body mass (cal/m/kg). The optimum (i.e., minimal) cost [_{Opt}_{Opt}_{Opt}_{Opt}

Others have sought to determine how muscles are organized into functional synergies to resolve the inherent redundancy of complex movements

During walking, humans need to adapt at

Here, we formulate goal functions

We present a mathematical definition of a specific hypothesized task strategy

The primary task requirement for walking on a treadmill with belt speed _{n}_{n}_{TM}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

(_{n}_{n}_{n}_{n}_{n}_{n}_{T}_{P}_{n}_{n}_{T}_{P}_{T}_{P}_{n}_{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 (_{T}_{P}_{n}_{n}_{T}_{P}_{T}_{P}_{T}_{P}_{T}_{P}

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 (_{n}_{n}_{n}_{n}_{n}

As expected, when subjects walked faster, they increased stride lengths (

Means (_{n}_{n}_{n}

Therefore, to quantify temporal correlations across consecutive strides, we computed scaling exponents,

Consistent with previous results _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

As expected _{n}_{n}

(_{net}

Plots of _{n}_{n}_{(1,16)} = 139.93; p = 2.53×10^{−9}; _{n}_{n}_{T}_{P}_{T}_{P}_{n}_{n}

(_{T}_{P}_{T}_{P}_{T}_{P}_{T}_{P}_{(1,16)} = 139.93; p = 2.53×10^{−9}. (_{T}_{P}_{T}_{P}_{(1,16)} = 368.21; p = 1.81×10^{−12}. Additionally, all subjects exhibited significant anti-persistence (95% confidence interval upper bounds all <½) for the goal-relevant _{P}

The _{T}_{P}_{T}_{P}_{(1,16)} = 368.21; p = 1.81×10^{−12}; _{P}_{P}_{T}

One obvious question is whether these observed dynamics represented the

The first alternative strategy was to choose a reference point, [^{*}, ^{*}] (e.g., _{n}_{n}_{n}_{n}

All error bars represent between-subject ±95% confidence intervals. By definition, these surrogates exhibited the same mean stride parameters (not shown) as the original walking data (_{n}_{n}_{n}_{T}_{P}_{T}_{P}_{(1,16)} = 2.614; p = 0.125) (Compare to _{T}_{P}_{(1,16)} = 0.413; p = 0.529) (Compare to

These surrogates exhibited approximately isotropic distributions (i.e., no obvious directionality) about [^{*}, ^{*}] within the [_{n}_{n}_{P}_{T}_{P}_{T}_{(1,16)} = 2.614; p = 0.125; _{P}_{T}_{(1,16)} = 0.413; p = 0.529;

_{n}_{n}_{n}_{n}_{n}_{n}

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) _{n}_{n}_{n}^{T}_{P}_{T}

By construction, this MIP model walked with nearly the same average stride parameters (_{n}_{n}_{n}_{n}_{n}_{n}_{T}_{(1,39)} = 6,076.51; p = 1.53×10^{−43}; _{(1,39)} = 1,969.18; p = 2.40×10^{−34}; _{P}_{T}

All error bars represent between-subject ±95% confidence intervals. In (_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{T}_{P}_{T}_{T}_{P}_{(1,39)} = 6,076.51; p = 1.53×10^{−43}). The MIP model exhibited much greater _{T}_{P}_{T}_{P}_{(1,39)} = 1,969.18; p = 2.40×10^{−34}). DFA exponents (_{T}_{P}_{P}

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 _{n}_{n}^{*}, ^{*}], was assumed to be equal to the mean stride time and stride length (

All error bars represent between-subject ±95% confidence intervals. In (_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{T}_{P}_{T}_{P}_{T}_{P}_{(1,39)} = 2,916.30; p = 1.55×10^{−37}). However, the variance ratio, σ(_{T}_{P}_{T}_{P}_{(1,39)} = 597.27; p = 7.61×10^{−25}). For _{T}_{P}_{P}

By construction, this POP model also walked with nearly the same average stride parameters (_{n}_{n}_{n}_{P}_{T}_{P}_{(1,39)} = 2,916.30; p = 1.55×10^{−37}; _{T}_{P}_{(1,39)} = 597.27; p = 7.61×10^{−25}; _{T}_{P}

The MIP and POP models both optimally corrected deviations away from the GEM at the next stride. Thus, the _{P}_{P}_{P}_{P}

All error bars represent between-subject ±95% confidence intervals. In (_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{T}_{P}_{P}_{T}_{P}_{(1,39)} = 1,736.81; p = 2.49×10^{−33}). The variance _{T}_{P}_{T}_{P}_{(1,39)} = 713.02; p = 3.15×10^{−26}). Deviations along the GEM (_{T}_{P}_{P}

By construction, this OVC model walked with nearly the same average stride parameters (_{n}_{n}_{n}_{P}

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

We hypothesized that humans walking on a treadmill would adopt a specific strategy _{P}_{T}_{P}_{T}_{P}

Beyond the five alternative control strategies clearly rejected by our results (_{net}_{net}_{P}_{P}_{P}_{P}_{P}_{P}_{T}_{T}_{net}_{net}_{net}

Minimizing energy cost has been the primary explanation for how humans and animals regulate walking _{Opt}_{Opt}_{n}_{n}_{Opt}_{Opt}_{Opt}_{Opt}_{Opt}_{Opt}_{n}_{n}_{n}_{n}_{P}_{T}

Our findings, however, remain compatible with the idea that humans also try to minimize energy cost while walking. The failure of the MIP model (^{*}, ^{*}] = [1.105 s, 1.337 m]. Mechanical walking models of Minetti _{Opt}_{Opt}_{Opt}_{Opt}^{*}, ^{*}] of actual humans

Humans also consistently _{P}_{P}

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

The nervous system appears to estimate both motor errors and the sources of those errors to guide continued adaptation

It has been widely argued that statistically persistent fluctuations are a critical marker of “healthy” physiological function _{T}_{P}

One question is whether the theoretical framework developed here will generalize to other contexts. During unconstrained overground walking _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

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

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 _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}

_{n}_{n}

We defined a specific operating point on each GEM as _{T}_{P}_{T}_{P}

Three types of surrogate time series _{n}_{n}_{n}_{n}

Second, _{n}_{n}

Third, for each trial _{n}_{n}

All surrogates were constrained so they did not “walk off” the treadmill (i.e., _{net}_{net}_{net}_{net}_{net}

For each surrogate, we then computed a new stride speed (_{n}_{n}_{n}

The stride-to-stride dynamics on the treadmill were modeled as a discrete map:_{1} and _{2} denoting additional gains, each set initially to 1 and used _{k}

The state update equation (Eq. 5) is intended to model only the discrete-time _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{+1} = _{n}_{n}_{n}

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) ^{2}, depended on the definition of the goal-level error for the task _{n}_{n}^{2}, penalized the distance, _{n+1}^{*}, ^{*}]. The last two terms in Eq. (6) were effort penalty terms where _{1}, _{2}]^{T}

The objective of the controller was to minimize _{1} and _{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, _{n}

The optimal, strictly MIP controller (_{P}_{n}_{1} = σ_{3} = 0.017 and σ_{2} = σ_{4} = 0.010 (see Supplementary

The optimal POP controller (^{*}^{*}^{*}^{*}^{*}_{P}

To match our human data in terms of the anti-persistent DFA exponents in the _{P}_{P}_{1} = g_{2} = 1.24. We retained the same preferred operating point, [^{*}^{*}^{*}^{*}^{*}_{T}_{P}

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 (_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{T}_{P}_{T}_{P}

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 (_{T}_{P}_{T}_{P}

Extended description of the construction of

(0.30 MB PDF)

Additional surrogate data analyses and results.

(0.44 MB PDF)

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

(0.26 MB PDF)

Extended description of the detrended fluctuation analysis algorithm.

(0.27 MB PDF)

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