Optimization of energy and time predicts dynamic speeds for human walking

Article activity feed

1. 

   Version published to 10.7554/elife.81939 on eLife

   Feb 13, 2023
2. 

   eLife

   Nov 28, 2022

   Author Response

   Reviewer #1 (Public Review):

   The authors are trying to determine how time is valued by humans relative to energy expenditure during non-steady-state walking - this paper proposes a new cost function in an optimal control framework to predict features of walking bouts that start and stop at rest. This paper's innovation is the addition of a term proportional to the duration of the walking bout in addition to the conventional energetic term. Simulations are used to predict how this additional term affects optimal trajectories, and human subjects experiments are conducted to compare with simulation predictions.

   I think the paper's key strengths are its simulation and experimental studies, which I regard as cleverly-conceived and well-executed. I think the paper's key weakness is the connection between these two …

   Author Response

   Reviewer #1 (Public Review):

   The authors are trying to determine how time is valued by humans relative to energy expenditure during non-steady-state walking - this paper proposes a new cost function in an optimal control framework to predict features of walking bouts that start and stop at rest. This paper's innovation is the addition of a term proportional to the duration of the walking bout in addition to the conventional energetic term. Simulations are used to predict how this additional term affects optimal trajectories, and human subjects experiments are conducted to compare with simulation predictions.

   I think the paper's key strengths are its simulation and experimental studies, which I regard as cleverly-conceived and well-executed. I think the paper's key weakness is the connection between these two studies, which I regard as tenuous for reasons I will now discuss in detail.

   The Title asserts that "humans dynamically optimize walking speed to save energy and time". Directly substantiating this claim would require independently manipulating the (purported) energy and time cost of walking for human subjects, but these manipulations are not undertaken in the present study. What the Results actually report are two findings:

   1. (simulation) minimizing a linear combination of energy and time in an optimal control problem involving an inverted-pendulum model of walking bouts that (i) start and stop at rest and (ii) walk at constant speed yields a gently-rounded speed-vs-time profile (Fig 2A);

   1. (experiment) human subject walking bouts that started and stopped at rest had self-similar speed-vs-time profiles at several bout lengths after normalizing by the average duration and peak speed of each subject's bouts (Fig 4B).

   If the paper established a strong connection between (1.) and (2.), e.g. if speed-vs-time trajectories from the simulation predicted experimental results significantly better than other plausible models (such as the 'steady min-COT' and 'steady accel' models whose trajectories are shown in Fig 2A), this finding could be regarded as providing indirect evidence in support of the claim in the paper's Title. Personally, I would regard this reasoning as rather weak evidence - it would be more accurate to assert 'brief human walking bouts look like trajectories of an inverted-pendulum model that minimize a linear combination of energy and time' (of course this phrasing is too wordy to serve as a replacement Title -- I am just trying to convey what assertion I think can be directly substantiated by the evidence in the paper). But unfortunately, the connection between (1.) and (2.) is only discussed qualitatively, and the other plausible models introduced in the Results are not revisited in the Discussion. To my naive eye, the representative 'steady min-COT' trace in Fig 2A seems like a real contender with the 'Energy-Time' trace for explaining the experimental results in Fig 4, but this candidate is rejected at the end of the third-to-last paragraph in the 'Model Predictions' subsection of Results based on the vague rationale that is never revisited.

   We have addressed most of this comment above, but respond here regarding Fig. 4. The argument against steady min-COT should also point out the peak speed. The Results have been revised thus: “In contrast to the min-COT hypothesis, the human peak speeds increased with distance, many well below the min-COT speed of about 1.25 m/s. The human speed trajectories did not resemble the trapezoidal profiles of the steady min-COT hypothesis for all distances, nor the triangular profiles of steady acceleration.”

   An additional limitation of the approach not discussed in the manuscript is that a fixed step length was prescribed in the simulations. The 'Optimal control formulation' subsection in the Methods summarizes the results of a sensitivity analysis conducted by varying the fixed step length, but all results reported here impose a constant-step-length constraint on the optimal control problem. Although this is a reasonable modeling simplification for steady-state walking, it is less well-motivated for the walking bouts considered here that start and stop at rest. For instance, the representative trial from a human subject in Figure 8 clearly shows initiation and termination steps that differ in length from the intermediate steps (visually discernable via the slope of the dashed line interpolating the black dots). Presumably different trajectories would be produced by the model if the constant-step-length constraint were removed. It is unclear whether this change would significantly alter predictions from either the 'Energy-Time' or 'steady min-COT' model candidates, and I imagine that this change would entail substantial work that may be out of scope for the present paper, but I think it is important to discuss this limitation.

   This is addressed elsewhere (Essential Revisions 2), but we explain more here. One of the parameter studies included step length increasing with speed according to the human preferred relationship. This is included in Fig. 3, and so we concluded that variable step lengths are not critical to the speed trajectories. A related assumption is that the energetic cost of modulating step length/frequency is small compared to the step-to-step transition cost. We believe that humans expend substantial energy for both costs, but that the overall cost of walking is still dominated by step-to-step transitions.

   With my concerns about the paper's framing and through-line noted as above, I want to emphasize that I regard the computational and empirical work reported here to be top-notch and potentially influential. In particular, the experimental study's use of inexpensive wearable sensors (as opposed to more conventional camera-based motion capture) is an excellent demonstration of efficient study design that other researchers may find instructive. To maximize potential impact, I encourage the authors to release their data, simulations, and details about their experimental apparatus (the first two I regard as essential for reproducibility - the third a selfless act of service to the scientific community).

   I think the most important point to emphasize is that the bulk of prior work on human walking has focused on steady-state movement - not because of the real-world relevance (since one study reports 50% of walking bouts in daily life are < 16 steps as summarized in Fig 1B), but rather because steady walking is a convenient behavior to study in the laboratory. Significantly, this paper advances both our theoretical and empirical understanding of the characteristics of non-steady-state walking.

   It is also significant to note the relationship between this study, where time was incorporated as an additive term in the cost of walking, with previous studies that incorporated time in a multiplicative discount in the cost of eye and arm movements. There is an emerging consensus that time plays a key role in the generation of movement across the body - future studies will discern whether and when additive or multiplicative effects dominate.

   We have acknowledged this in a brief sentence: “Indeed, we have found a similar valuation of time to explain how reaching durations and speed trajectories vary with reaching distance (Wong et al., 2021).” As an aside, in that reference we measured metabolic cost of cyclic arm reaching, combined it with a linear time cost, and predicted reaching durations vs. distance and bell-shaped hand speed trajectories. Others (Shadmehr et al. Curr Biol. 2016) have proposed multiplicative (hyperbolic) temporal discounting to explain durations, but the cost formulas are not dynamical, and cannot produce trajectories. We agree with reviewer’s point, but we think the evidence for hyperbolic discounting is not strong. Linear time costs are simpler and work at least as well. This is of great interest to us, but we didn’t discuss beyond the brief mention above, because we fear it is too far afield.

   Reviewer #2 (Public Review):

   This paper provides a novel approach to quantifying the tradeoff between energetic optimality during walking and the valuation of time to travel a given distance. Specifically, the authors investigated the relationships between walking speed trajectories, distance traveled, and the valuation of (completion) time. Time has been proposed as a potential factor influencing movement speed, but less is understood about how individuals balance energetic optimality and time constraints during walking. The authors used a simple, sagittal-plane walking model to test competing hypotheses about how individuals optimize gait speed from gait initiation to gait termination. Their approach extends literature in the space by identifying optimal gaits for shorter, partially non-steady speed walking bouts.

   The authors successfully evaluated three competing walking objectives (constant acceleration, minimum cost of transport at steady speed, and the energy-time objective), showing that the energy-time objective best matched experimental data in able-bodied adults. Although other candidate objectives may exist, the paper's findings provide a likely-generalizable explanation of how able-bodied humans select movement strategies that encompass studies of steady-speed walking.

   Overall, this paper provides a foundation for future studies testing the validity of the energy-time hypothesis for human gait speed selection in able-bodied and patient populations. Extensions of this work to patient populations may explain differences in walking speed during clinical assessments and provide insight into how individual differences in time valuation impact performance on assessments. For example, understanding whether physical capacity or time valuation (or something comparable) better explains individual differences in walking speed may suggest distinct approaches for improving walking speed.

   Strengths:

   The authors presented a compelling rationale for the tradeoffs between energetic optimality and time and their results provide strong support for a majority of their conclusions. In particular, significant reductions in the variance of experimental speed trajectories provides good support for the scaling of speeds across individuals and the plausibility of the energy-time hypothesis. Comparison to theoretical (model-based) reductions across difference time valuation (cT) parameters would further enhance confidence in the practical significance of the variance reductions. Further, while additional work is needed to determine the range of "normal" valuations of time, the authors present experimental ranges that appear reasonable and are well explained. The computational and analytical methods are rigorous and are supported by the literature. Overall, the paper's conclusions are consistent with experimental and computational results.

   The introduction of a model-based analytical approach to quantify the effects of time valuation of walking could generalize to test other cost functions, populations, or locomotion modes. Further, models of varying complexity could be implemented to test more individualized estimates of metabolic cost, ranging from 3D dynamic walking models (Faraji et al., Scientific Reports, 2018) or physiologically-detailed models (Falisse et al., Journal of The Royal Society Interface. 2019). The relatively simple set of analyses used in this paper is consistent with prior literature and should generalize across applications and populations.

   The authors justified simplifications in the analysis and addressed major limitations of the paper, such as using a fixed step length in model predictions, using a 2D model, and basing energy estimates on the mechanical work of a simple model. It is unlikely that the paper's conclusions would change given additional model complexity. For example, a 3D walking model would need to control frontal plane stability. However, in able-bodied adults, valuation of frontal-plane stability during normal walking would not likely alter the overall shape of the predicted speed profiles.

   Weaknesses:

   The primary weakness of this work is that alternative objectives may provide similar speed profiles and thus be plausible objectives for human movement. For example, the authors tested an objective minimizing the steady-speed cost of transport. This cost function is consistent with the literature, but (as predicted) unlikely to explain acceleration and deceleration during gait. An objective more comparable to the energy-time hypothesis would be to minimize the net energy cost over the entire bout, including accelerations and decelerations. This may produce results similar to the energy-time hypothesis. However, a more complex model that incorporates non-mechanical costs (e.g., cost of body weight support) may be needed to test such objectives. Therefore, the energy-time hypothesis should be considered in the context of a simple model that may be incapable of testing certain alternative hypotheses.

   We have addressed some of this comment in Essential Revisions 4.

   We are unsure what is meant by “net energy over the entire bout, including accelerations and decelerations.” Our hypothesis uses total (gross) energy over the entire bout, and already includes accelerations and decelerations. If “net” refers to the customary definition of metabolic energy minus resting, then it differs from our gross cost (Fig. 6A) only in the amount of constant offset, namely resting cost. Removing the offset is equivalent to a decrease in C_T. As shown in Fig. 3, this would reduce peak speeds magnitudes but not change the shape of the speed, peak speed, and duration patterns. There is also another interpretation where the cost of walking includes only net energy, and the cost of time includes the resting metabolic rate (Fig. 6C). This interpretation yields the same predictions, the only difference is whether resting rate is treated as an energy or a time cost. We have not made further changes, because we are unsure what the reviewer meant. The difference between net and total is at most one of degree, not of qualitatively different behavior.

   We do not address the proposed “cost of body weight support” because we are unsure of the definition. There is a hypothesis by Kram & Taylor (1990) that defines a metabolic cost rate proportional to body weight divided by ground contact time. It is unclear if this is what reviewer is referring to, so we did not include it in the manuscript. However, IF this is what reviewer means, we do not consider the Kram & Taylor (“K&T”) cost to be a viable hypothesis for computational models. It is a correlation observed from data, which is inadequate as a model, for several reasons. First, in a model optimization, it leads to absurd predictions, because metabolic cost could then be reduced simply by increasing stance (contact) time. A model could do so simply by walking with very long double support phases, or running with a very brief aerial phase, both of which people clearly do not do. In walking, extended double support durations result in much higher metabolic cost (Gordon et al., APMR 2009). Models must operate quite literally on whatever objective they are given, and here, a literal interpretation of K&T makes absurd predictions.

   Another issue with the K&T cost is that it is not mechanistic. A mechanistic model is concerned with the forces and work performed by an actuator such as muscle. Muscles experience forces far greater than body weight, not captured by the K&T cost. Of course, overall cost for animal locomotion is roughly proportional to body weight, but what a model needs is a cost associated with its control inputs, e.g. actuator forces.

   We have also examined the K&T hypothesis in previous publications. In Schroeder & Kuo (Plos Comp Biol 2021), we used a simple model of running that minimizes an energetic cost dominated by mechanical work. Even though the model has no cost similar to K&T, its predicted metabolic cost is correlated with the K&T cost. Correlation does not imply causation, which is known in this model.

   We have also examined the K&T hypothesis in experimental data. In Riddick & Kuo (Sci Rep 2022), we examined human data and found that there are many variables that correlate quite well with metabolic cost, including the K&T correlate. We use human data to show how mechanical work could explain metabolic cost, and even if it does, the K&T cost appears as a correlate. In our interpretation, both model and data that experience an energetic cost proportional to mechanical work may have a number of variables correlated to energy cost. Those correlates need not have any causal influence.

   There are, of course, many similar correlates that could be or have been proposed to explain the metabolic cost of running. Most such correlates are not operational enough to work in a model, and it is also difficult to predict what a reader might consider plausible, even if we do not.

   We agree with this statement: “the energy-time hypothesis should be considered in the context of a simple model that may be incapable of testing certain alternative hypotheses.” In fact, in Discussion of limitations we listed other potential factors (e.g. forced leg motion, stability, 3D motion), and stated “We did not explore more complex models here, but would expect similar predictions to result if similar, pendulum-like principles of work and energetic cost apply.” We had also cited other models that include such factors and are compatible with the step-to-step transition concept. Finally, we already stated, “the Energy-Time hypothesis should be regarded as a subset of the many factors that should govern human actions, rendered here in a simple but quantitative form.” We believe this is already aligned with reviewer’s comment.

   An experimental design involving an intervention to perturb the valuation of time would provide stronger support for time being a critical factor influencing gait speed trajectories. The authors noted this limitation as an area of future work.

   While the results are compelling, the limited sample size and description of participants limit the obvious generalizability of the results. Older adults tend to have higher metabolic costs of walking than younger adults, which may alter the predicted time-energy relationships (Mian OS, et al., Acta physiologica. 2006). As noted in the introduction, differences in walking speeds have been observed in different living environments. General information on where participants lived (city, small town, etc...) may provide readers with insight into the generalizability of the paper's conclusions. Additionally, the experimental results figures show group-level trends, but individual-specific trends and the existence of exceptional cases are unclear.

   We wish to defend the “limited sample size.” The present sample size was (in our opinion) sufficient to test the hypothesis, and we have reported confidence intervals and other statistics where appropriate. (As always, it is up to the individual reader to decide whether they are convinced or not.) It is true that the data may be insufficient for other purposes, but we cannot anticipate or address all other purposes.

   We appreciate the relevant connection to aging. We have added to Discussion, “We do not know whether that family [of trajectories] also applies to older adults, who prefer slower steady speeds and expend more energy to walk the same speed (Malatesta, 2003). Perhaps an age-related valuation of time might explain some of the differences in speed.”

   We agree about the participants, and have added “Subjects were recruited from the community surrounding the University of Calgary; the city has a moderately affluent population of about 1.4 M, with a developed Western culture.”

   No specific reviewer recommendation was made about individual-specific trends, but there are several indicators already included in the manuscript. First, all trials from all subjects are shown in Fig. 4A. Any truly exceptional cases should be visible as substantial deviations from the group. Second, the normalization by peak speed in Fig. 4B shows how individuals tend to be fairly consistent in their preferred speeds, in that shorter and longer bouts of an individual are consistent with each other, even if some walk faster than others. Third, this observation is analyzed more quantitatively by the reduction in standard deviations with normalization (Results). Fourth, we will provide a data repository with all the data, to allow readers to inspect individuals more carefully (Data availability statement).

   The authors' interpretation of clinical utility is vague and should be interpreted with caution. A simple pendulum-based walking model is unlikely to generalize to patient populations, whose gait energetics may involve greater positive and negative mechanical work (Farris et al., 2015; Holt et al., 2000). Additionally, the proposed analytical framework based on mechanical work as a proxy for the metabolic cost may not generalize to patient populations who have heterogeneous musculotendon properties and increased co-contraction (e.g., children with cerebral palsy; Ries et al., 2018). Consequently, the valuation of time for an individual could be incorrectly estimated if the estimates of metabolic cost were inaccurate. Therefore, as the authors noted for their able-bodied participants, more precise measures of metabolic rates will be critical for translating this work into clinical settings.

   We agree, and did not intend to say that clinical populations must walk the same way, rather that the Normal patterns could be used as a basis of comparison. To make this clearer, we have amended the Discussion of clinical implications (new text emphasized): “it may be possible to predict the duration and steady speed for another distance, referenced from a universal family of walking trajectories. We have identified one such family that applies to healthy individuals with pendulum-like gait. Of course, some clinical conditions might be manifested by a deviance from that family, perhaps in the acceleration or deceleration phases, or in how the trajectories vary with distance. If quantified, such deviance might prove clinically useful… the characterization of distance-dependent speed trajectories can potentially provide more information than available from steady speed alone.”

   We agree that the valuation of time can be inaccurate if the metabolic cost is inaccurate. That is why we did not make a precise estimate of the valuation. We have amended the text to help clarify that our rough estimates are based on previous data. In addition, our general scientific intent is to reveal behavioral sensitivities, for example of walking duration to bout distance, as opposed to absolute numerical quantities.

   Read the original source
3. 

   eLife

   Nov 28, 2022

   eLife assessment

   This valuable study presents a new optimal control cost framework to predict features of walking bouts, adding a cost function term proportional to the duration of the walking bout in addition to the conventional energetic term. While predicted optimal trajectories from simulations qualitatively matched walking data from human subjects, the evidence supporting these claims is incomplete, as some methodological choices raise questions about the strength of the authors' claims.

   Read the original source
4. 

   eLife

   Nov 28, 2022

   Reviewer #1 (Public Review):

   The authors are trying to determine how time is valued by humans relative to energy expenditure during non-steady-state walking - this paper proposes a new cost function in an optimal control framework to predict features of walking bouts that start and stop at rest. This paper's innovation is the addition of a term proportional to the duration of the walking bout in addition to the conventional energetic term. Simulations are used to predict how this additional term affects optimal trajectories, and human subjects experiments are conducted to compare with simulation predictions.

   I think the paper's key strengths are its simulation and experimental studies, which I regard as cleverly-conceived and well-executed. I think the paper's key weakness is the connection between these two studies, which I regard as …

   Reviewer #1 (Public Review):

   The authors are trying to determine how time is valued by humans relative to energy expenditure during non-steady-state walking - this paper proposes a new cost function in an optimal control framework to predict features of walking bouts that start and stop at rest. This paper's innovation is the addition of a term proportional to the duration of the walking bout in addition to the conventional energetic term. Simulations are used to predict how this additional term affects optimal trajectories, and human subjects experiments are conducted to compare with simulation predictions.

   I think the paper's key strengths are its simulation and experimental studies, which I regard as cleverly-conceived and well-executed. I think the paper's key weakness is the connection between these two studies, which I regard as tenuous for reasons I will now discuss in detail.

   The Title asserts that "humans dynamically optimize walking speed to save energy and time". Directly substantiating this claim would require independently manipulating the (purported) energy and time cost of walking for human subjects, but these manipulations are not undertaken in the present study. What the Results actually report are two findings:
   1. (simulation) minimizing a linear combination of energy and time in an optimal control problem involving an inverted-pendulum model of walking bouts that (i) start and stop at rest and (ii) walk at constant speed yields a gently-rounded speed-vs-time profile (Fig 2A);
   2. (experiment) human subject walking bouts that started and stopped at rest had self-similar speed-vs-time profiles at several bout lengths after normalizing by the average duration and peak speed of each subject's bouts (Fig 4B).
   If the paper established a strong connection between (1.) and (2.), e.g. if speed-vs-time trajectories from the simulation predicted experimental results significantly better than other plausible models (such as the 'steady min-COT' and 'steady accel' models whose trajectories are shown in Fig 2A), this finding could be regarded as providing indirect evidence in support of the claim in the paper's Title. Personally, I would regard this reasoning as rather weak evidence - it would be more accurate to assert 'brief human walking bouts look like trajectories of an inverted-pendulum model that minimize a linear combination of energy and time' (of course this phrasing is too wordy to serve as a replacement Title -- I am just trying to convey what assertion I think can be directly substantiated by the evidence in the paper). But unfortunately, the connection between (1.) and (2.) is only discussed qualitatively, and the other plausible models introduced in the Results are not revisited in the Discussion. To my naive eye, the representative 'steady min-COT' trace in Fig 2A seems like a real contender with the 'Energy-Time' trace for explaining the experimental results in Fig 4, but this candidate is rejected at the end of the third-to-last paragraph in the 'Model Predictions' subsection of Results based on the vague rationale that is never revisited.

   An additional limitation of the approach not discussed in the manuscript is that a fixed step length was prescribed in the simulations. The 'Optimal control formulation' subsection in the Methods summarizes the results of a sensitivity analysis conducted by varying the fixed step length, but all results reported here impose a constant-step-length constraint on the optimal control problem. Although this is a reasonable modeling simplification for steady-state walking, it is less well-motivated for the walking bouts considered here that start and stop at rest. For instance, the representative trial from a human subject in Figure 8 clearly shows initiation and termination steps that differ in length from the intermediate steps (visually discernable via the slope of the dashed line interpolating the black dots). Presumably different trajectories would be produced by the model if the constant-step-length constraint were removed. It is unclear whether this change would significantly alter predictions from either the 'Energy-Time' or 'steady min-COT' model candidates, and I imagine that this change would entail substantial work that may be out of scope for the present paper, but I think it is important to discuss this limitation.

   With my concerns about the paper's framing and through-line noted as above, I want to emphasize that I regard the computational and empirical work reported here to be top-notch and potentially influential. In particular, the experimental study's use of inexpensive wearable sensors (as opposed to more conventional camera-based motion capture) is an excellent demonstration of efficient study design that other researchers may find instructive. To maximize potential impact, I encourage the authors to release their data, simulations, and details about their experimental apparatus (the first two I regard as essential for reproducibility - the third a selfless act of service to the scientific community).

   I think the most important point to emphasize is that the bulk of prior work on human walking has focused on steady-state movement - not because of the real-world relevance (since one study reports 50% of walking bouts in daily life are < 16 steps as summarized in Fig 1B), but rather because steady walking is a convenient behavior to study in the laboratory. Significantly, this paper advances both our theoretical and empirical understanding of the characteristics of non-steady-state walking.

   It is also significant to note the relationship between this study, where time was incorporated as an additive term in the cost of walking, with previous studies that incorporated time in a multiplicative discount in the cost of eye and arm movements. There is an emerging consensus that time plays a key role in the generation of movement across the body - future studies will discern whether and when additive or multiplicative effects dominate.

   Read the original source
5. 

   eLife

   Nov 28, 2022

   Reviewer #2 (Public Review):

   This paper provides a novel approach to quantifying the tradeoff between energetic optimality during walking and the valuation of time to travel a given distance. Specifically, the authors investigated the relationships between walking speed trajectories, distance traveled, and the valuation of (completion) time. Time has been proposed as a potential factor influencing movement speed, but less is understood about how individuals balance energetic optimality and time constraints during walking. The authors used a simple, sagittal-plane walking model to test competing hypotheses about how individuals optimize gait speed from gait initiation to gait termination. Their approach extends literature in the space by identifying optimal gaits for shorter, partially non-steady speed walking bouts.

   The authors …

   Reviewer #2 (Public Review):

   This paper provides a novel approach to quantifying the tradeoff between energetic optimality during walking and the valuation of time to travel a given distance. Specifically, the authors investigated the relationships between walking speed trajectories, distance traveled, and the valuation of (completion) time. Time has been proposed as a potential factor influencing movement speed, but less is understood about how individuals balance energetic optimality and time constraints during walking. The authors used a simple, sagittal-plane walking model to test competing hypotheses about how individuals optimize gait speed from gait initiation to gait termination. Their approach extends literature in the space by identifying optimal gaits for shorter, partially non-steady speed walking bouts.

   The authors successfully evaluated three competing walking objectives (constant acceleration, minimum cost of transport at steady speed, and the energy-time objective), showing that the energy-time objective best matched experimental data in able-bodied adults. Although other candidate objectives may exist, the paper's findings provide a likely-generalizable explanation of how able-bodied humans select movement strategies that encompass studies of steady-speed walking.

   Overall, this paper provides a foundation for future studies testing the validity of the energy-time hypothesis for human gait speed selection in able-bodied and patient populations. Extensions of this work to patient populations may explain differences in walking speed during clinical assessments and provide insight into how individual differences in time valuation impact performance on assessments. For example, understanding whether physical capacity or time valuation (or something comparable) better explains individual differences in walking speed may suggest distinct approaches for improving walking speed.

   Strengths:
   The authors presented a compelling rationale for the tradeoffs between energetic optimality and time and their results provide strong support for a majority of their conclusions. In particular, significant reductions in the variance of experimental speed trajectories provides good support for the scaling of speeds across individuals and the plausibility of the energy-time hypothesis. Comparison to theoretical (model-based) reductions across difference time valuation (cT) parameters would further enhance confidence in the practical significance of the variance reductions. Further, while additional work is needed to determine the range of "normal" valuations of time, the authors present experimental ranges that appear reasonable and are well explained. The computational and analytical methods are rigorous and are supported by the literature. Overall, the paper's conclusions are consistent with experimental and computational results.

   The introduction of a model-based analytical approach to quantify the effects of time valuation of walking could generalize to test other cost functions, populations, or locomotion modes. Further, models of varying complexity could be implemented to test more individualized estimates of metabolic cost, ranging from 3D dynamic walking models (Faraji et al., Scientific Reports, 2018) or physiologically-detailed models (Falisse et al., Journal of The Royal Society Interface. 2019). The relatively simple set of analyses used in this paper is consistent with prior literature and should generalize across applications and populations.

   The authors justified simplifications in the analysis and addressed major limitations of the paper, such as using a fixed step length in model predictions, using a 2D model, and basing energy estimates on the mechanical work of a simple model. It is unlikely that the paper's conclusions would change given additional model complexity. For example, a 3D walking model would need to control frontal plane stability. However, in able-bodied adults, valuation of frontal-plane stability during normal walking would not likely alter the overall shape of the predicted speed profiles.

   Weaknesses:
   The primary weakness of this work is that alternative objectives may provide similar speed profiles and thus be plausible objectives for human movement. For example, the authors tested an objective minimizing the steady-speed cost of transport. This cost function is consistent with the literature, but (as predicted) unlikely to explain acceleration and deceleration during gait. An objective more comparable to the energy-time hypothesis would be to minimize the net energy cost over the entire bout, including accelerations and decelerations. This may produce results similar to the energy-time hypothesis. However, a more complex model that incorporates non-mechanical costs (e.g., cost of body weight support) may be needed to test such objectives. Therefore, the energy-time hypothesis should be considered in the context of a simple model that may be incapable of testing certain alternative hypotheses.

   An experimental design involving an intervention to perturb the valuation of time would provide stronger support for time being a critical factor influencing gait speed trajectories. The authors noted this limitation as an area of future work.

   While the results are compelling, the limited sample size and description of participants limit the obvious generalizability of the results. Older adults tend to have higher metabolic costs of walking than younger adults, which may alter the predicted time-energy relationships (Mian OS, et al., Acta physiologica. 2006). As noted in the introduction, differences in walking speeds have been observed in different living environments. General information on where participants lived (city, small town, etc...) may provide readers with insight into the generalizability of the paper's conclusions. Additionally, the experimental results figures show group-level trends, but individual-specific trends and the existence of exceptional cases are unclear.

   The authors' interpretation of clinical utility is vague and should be interpreted with caution. A simple pendulum-based walking model is unlikely to generalize to patient populations, whose gait energetics may involve greater positive and negative mechanical work (Farris et al., 2015; Holt et al., 2000). Additionally, the proposed analytical framework based on mechanical work as a proxy for the metabolic cost may not generalize to patient populations who have heterogeneous musculotendon properties and increased co-contraction (e.g., children with cerebral palsy; Ries et al., 2018). Consequently, the valuation of time for an individual could be incorrectly estimated if the estimates of metabolic cost were inaccurate. Therefore, as the authors noted for their able-bodied participants, more precise measures of metabolic rates will be critical for translating this work into clinical settings.

   Read the original source
6. 

   Version published to 10.1101/2022.07.15.500158v3 on bioRxiv

   Nov 26, 2022
7. 

   Version published to 10.1101/2022.07.15.500158v2 on bioRxiv

   Aug 13, 2022
8. 

   Version published to 10.1101/2022.07.15.500158v1 on bioRxiv

   Jul 18, 2022