Recommendation for the Minimum Number of Steps to Analyze When Performing the Uncontrolled Manifold Analysis on Walking Data

Abstract

The uncontrolled manifold (UCM) analysis quantifies the extent to which co-variation among a set of variables facilitates consistent performance, by partitioning variance in those variables into two components then calculating their normalized difference (i.e., the synergy index). Although UCM-derived measures are thought to depend on the number of data points analyzed, the minimum number needed to reasonably approximate true values of these measures is unknown. For each of two performance variables related to mechanical stability of gait, we evaluated changes in UCM-derived measures when increasing the number of analyzed points, here steps. Fourteen older adults walked on a treadmill while motion capture tracked movement. For each subject, n steps (where n=2-99) were randomly sampled from the first 100, then used to calculate UCM-derived variables. For each subject, variables were expressed as a percent of the subject-specific value with n=100 and averaged across 50 simulations. For each n, 95% confidence intervals (CIs) were calculated from group data. The minimum number of steps to “reasonably approximate” a variables was defined as the value of n for which the lower CI was >90% of the value with n=100. Regardless of performance variable, reasonable approximations of the synergy index were attained with n=16 steps, whereas n= steps were needed for each of the variance components However, the differences between using 16 steps and 50 steps were small. Collecting 15-20 steps is recommended for a reasonable approximation of the synergy indices considered herein, particularly when data collection is constrained to a limited number of steps.

INTRODUCTION

The production of human movement inherently involves more degrees of freedom (DoFs) than required by constraints of the action (Bernstein, 1967), and the ability to exploit such redundancy may be important to maintaining a heathy motor system. Motor redundancy ensures that multiple solutions exist for a given motor problem, e.g., no single combination of muscle forces produces a given joint moment, which provides flexibility to the system (Latash, Mark L., 2018; Latash, M. L., 2012). The exploitation of motor redundancy may help to reduce variance in motor performance in the presence of system noise or varying initial conditions, and to facilitate performance of secondary actions and appropriate reactions to challenging circumstances (Hsu and Scholz, 2012; Latash, M. L. et al., 2007). The extent to which redundancy is exploited to accomplish a motor task can be evaluated using the uncontrolled manifold (UCM) analysis, which in essence quantifies coordination - i.e. how variations in all of the DoFs that contribute to performance (termed elemental variables) co-vary in order to produce consistent output in a performance variable.

As suggested in a 2010 review on UCM analysis: “when applying the UCM approach…the number of data points used in the analysis is an important consideration…Ideally the more data points…. the better” (Latash, M. L. et al., 2010). In the case of gait, every step is equivalent to a data points and the UCM analysis is performed across steps at comparable time-normalized points within the step. While more steps may be “better”, the number that can be captured during an experiment may be limited by the population or the experimental manipulation. An “informal analysis” from a single subject performing a reaching task suggested including “at least 20 trials for UCM analysis to increase the chances of having a stable (output) estimate” (Latash, M. L. et al., 2010), which is consistent with the number of steps analyzed in prior gait studies (Eckardt and Rosenblatt, 2018; Krishnan et al., 2013; Papi et al., 2015; Robert et al., 2009; Rosenblatt, N. J., Hurt et al., 2014; Verrel et al., 2010). However, whether 20 steps provides stable estimates of UCM measures during gait has yet to be tested. In light of the insights UCM analysis can offer into strategies used by non-impaired (Eckardt and Rosenblatt, 2018; Hsu et al., 2013; Kapur et al., 2010; Olafsdottir et al., 2007; Papi et al., 2015; Qu, 2012; Robert et al., 2009; Shim et al., 2004) and clinical populations (Black et al., 2007; Latash, M. L. et al., 2002; Papi et al., 2015; Park et al., 2012) to generate movement solutions during everyday activities, it is important know the number of observations needed to characterize these movements.

The purpose of this study was to evaluate the minimum number of steps needed to obtain a “reasonable approximation” of the “true” value of three UCM-derived measures – two variance components and the synergy index. Here we use the term “reasonable approximation” rather than “reliable” or “accurate”, which utilize specific methodologies. Specifically, we will “approximate” - i.e., obtain values that are ≥90% the “true value”, or the value obtained with a “large” number of steps - with “reasonable” certainty - i.e., 95% confidence.

METHODS

This study is a secondary analysis of data from fourteen healthy community-dwelling older (>65 years) adults (7 female; 71.4±6.8 years; 1.69±0.09 m; 67.4±7.5 kg) who participated in a larger study focused on the effects of obesity on fall risk. Only data from subjects with body mass index of 17.5-25.0 kg/m² is included. All subjects were screened for neuromusculoskeletal health (e.g., normal range of motion, no joint replacements or history of neurodegenerative disease) and provided written informed consent before participating in this study approved by the Rosalind Franklin University IRB.

Participants walked on a motorized treadmill (Motek; Amsterdam, Netherlands) for 10 minutes at a self-selected velocity, determined using a previously described approach (Rosenblatt, N. J. et al., 2014). An 8-camera motion capture system (Vicon; Oxford, UK) tracked the motions of passive reflective markers placed on body landmarks according to the full body plug in gait model. Marker data was processed using commercial software (Nexus; Oxford, UK) to obtain locations of the lower limb joint centers and segment lengths. Two performance variables related to mechanical stability of gait were then considered in the analysis. One variable, the mediolateral (ML) trajectory of the swing limb (Krishnan et al., 2013; Rosenblatt, NJ et al., 2015; Rosenblatt, N. J., Hurt et al., 2014) - defined as the mediolateral position of the swing limb ankle joint center relative to that of the stance limb (AJC_ML) - was chosen due to the importance of mediolateral foot placement in the control of mechanical stability while walking (Bauby and Kuo, 2000; Donelan et al., 2004) . We also considered the frontal plane position of whole body center of mass (CoM) relative to the stance limb ankle joint center (CoM_ML), given the importance of ML COM control in the maintenance of upright gait (Hurt et al., 2010; Papi et al., 2015). For both performance variables elemental variables were segment angles of the lower limbs and trunk. For the AJC_ML analysis, motion data was normalized from 0-100% corresponding to each left-leg swing; for CoM_ML analysis, motion data was normalized to each left-leg swing and ensuing double support. Normalized data was then entered into custom code (Matlab; Cambridge, MA) to calculate the UCM measures, using a four step process (Scholz and Schoner, 1999):

1) A geometric model was created to express the performance variables as functions of 7 or 9 elemental variables (Greek letters in equations 1 and 2), for AJC_ML and CoM_ML respectively (see Figure 1 for definitions of elemental variables). We used a previous described model for the former (Krishnan et al., 2013) and a similar model for the latter, with the addition of trunk mediolateral and frontal plane flexion. Anthropometric tables (Winter, 2005) were used to approximate the positions of the segmental CoMs relative to their ends (y_1,3,4,5 in Equation 2) and the magnitudes of segmental CoMs (m_1,3,4,5 in Equation 2). The terms L_1,−4 represent the lengths of model segments.

Figure 1. Definition of elemental variables used in the UCM analyses.

(left) sagittal plane view where white circles represent the hip and ankle joint centers as well as the center of the pelvis and the shoulders; (middle) frontal plane view with white circles representing similar locations, with the addition of a circle representing the knee joint center on the swing; (right) blank circles represent hip joint centers.

$${\mathbf{AJC}}_{\mathbf{ML}}=L_1\cos \alpha \sin {\theta }_1+L_2\cos \beta \cos {\theta }_2+L_3\cos \gamma \sin {\theta }_3+L_4\cos \gamma \sin {\theta }_4$$ (Equation 1)

$$\begin{gathered} {\mathbf{COM}}_{\mathbf{ML}}=\frac{1}{m_1+m_5+m_3+m_4}\ast [m_1\ast y_1\ast L_1\cos \alpha \sin {\theta }_1+\cdots \\ {m}_5\ast (L_1\cos \alpha \sin {\theta }_1+0.5\ast L_2\cos \beta \cos {\theta }_2+y_5\ast L_5\cos \epsilon \cos {\theta }_5)+\cdots \\ {m}_3\ast (L_1\cos \alpha \sin {\theta }_1+L_2\cos \beta \cos {\theta }_2+y_3\ast L_3\cos \gamma \sin {\theta }_3)+\cdots \\ {m}_4\ast (L_1\cos \alpha \sin {\theta }_1+L_2\cos \beta \cos {\theta }_2+y_3\ast L_3\cos \gamma \sin {\theta }_3+y_4\ast L_4\cos \gamma \sin {\theta }_4)] \end{gathered}$$ (Equation 2)

2) A Jacobian matrix was derived from the model to relate changes in elemental variables to changes in the performance variable. At every percent of the normalized gait cycle, the Jacobian was evaluated at the mean values of the elemental variables. The null space of the evaluated Jacobian defined the UCM.

3) For each left-step, deviation vectors were independently calculated at every percent of swing as the difference between the elemental variables at that point and their respective means. These vectors were projected onto the UCM and a space orthogonal to it. The squared length of the projected vectors in each space, relative to the number of DoFs in that space, was averaged across multiple steps to define two variance components (V_UCM and V_ORT, respectively).

4) The synergy index was calculated at each percent as the difference between V_UCM and V_ORT, relative to the total variance. The index was then z-transformed (Robert et al., 2009). All outcomes were averaged across 0-100%

For each participant, we ran the four-step process within a larger Monte Carlo simulation to generate subject-specific curves of the synergy index, V_UCM and V_ORT as a function of number of steps. This was done to visually verify that the curves plateaued after some “large” number of steps (N_max). Within the simulation, we performed the following actions: 1) set N_max= 100; 2) randomly select n=2 steps from the first N_max steps; 3) calculate UCM-derived variables for n steps using the four-step process; 4) repeat actions 2 and 3 over 50 replications (above which limited improvements in simulation accuracy were expected) (Efron and Tibshirani, 1986); 5) calculate an average value across replications for the synergy index _z, V_UCM and V_ORT; 6) increase n to n+1 and repeat actions 2-5 until n= N_max; 7) visually identify a plateau in the synergy index, V_UCM and V_ORT curves; if none exists then increase N_max and repeat actions 1-7.

After determining an appropriate N_max, the values for the synergy index, V_UCM and V_ORT obtained within a given replication, for a given n, were expressed as percentages of the values attained with N_max. For each subject these percentages were then averaged across all 50 simulation replications. From these subject-specific averages, at each n we calculated 95% confidence intervals (CIs) then calculated the grand mean curves by averaging subject-specific curves. The minimum number of steps needed to provide a “reasonable approximation” of a measure was defined as the lowest value of n, above which all lower CIs were ≥90% of the grand mean values.

RESULTS

In general, regardless of performance variable, the curves for each of the UCM-related variables as a function of number of steps in the analysis plateaued prior to n=100 (Figure 2). Therefore we set N_max = 100 for all ensuing analyses when determining the minimum n needed to approximate “plateau” values (Figure 1).

Figure 2. UCM-related variables for two performance variables plotted as a function of number of steps (n) included in the UCM analysis.

For each n, 50 Monte Carlo repetitions were conducted and the output was averaged across repetitions. Each curve represents averaged result for a given subject. All curves plateau at n<100.

Regardless of the performance variable, the synergy index sharply increased for small step counts then leveled out (Figure 3); with 16 steps in the analysis, the 95% CIs for the synergy index _z were ≥90% of the values obtained when 100 steps were analyzed. (Figure 2). Thus 16 steps provides a reasonable approximation of the synergy index.

Figure 3. Grand-mean curves of UCM-related variables as a percent of the value with N_max=100, for two performance variables plotted as a function of number of steps (n).

For each subject, for each n, 50 Monte Carlo repetitions were performed and the outputs were expressed relative to the values with n=N_max. Normalized valued were then averaged across the 50 repetitions. Each curve represents the grand mean of the normalized curves. Error bars are 95% CI obtained from the subject-specific normalized curves. The number of steps that provide a “reasonable approximation” of values at N_max is noted by the arrow

In contrast, with 16 steps in the analysis, we could be 95% confident that the values of the variance components were 78-81% of the values with n=100 (Figure 2). We required n=49 steps we obtained “reasonable approximations” for both variance components for each performance variable (Figure 2). The average absolute change in the variance components for n=16 compared to n= N_max varied from 3.4e-5 to 6.4e-5, depending on the component and performance variable (Figure 1).

DISCUSSION

The purpose of this study was to evaluate the minimum number of steps needed to obtain a “reasonable approximation” for three UCM-derived measures. Regardless of performance variable, 16 steps was sufficient to obtain a “reasonable approximation” of the synergy index _z. While nearly 50 steps were needed to reasonably approximate the variance components, the two proportionally changed from n=16 to n= 50, explaining the lower number of steps for the synergy index. Importantly, previous results regarding variance components from studies using n<50 (Krishnan et al., 2013; Papi et al., 2015; Robert et al., 2009; Rosenblatt, N. J., Hurt et al., 2014; Verrel et al., 2010) should not be dismissed. While reported values from these studies may not represent the “true values” (i.e. those obtained with N_max), the difference between reported and “true” values is expected to be relatively small; average absolute changes in variance components with n=16 vs. n=100 were an order of magnitude less than the smallest subject-specific values for these same variables (1e-5 vs. 1e-4, respectively). Because the magnitudes of the variance components are small relative to the synergy index small deviation from N_max may manifest as larger percentage differences. As there is no reason to believe differences between reported and “true” values should depend on experimental conditions, previous reports of between-condition effects would be expected to persist even with more data points. However additional work is needed to demonstrate this.

Several factors may limit generalizability of the current findings. This study utilized treadmill walking, which could limit generalizability to overground conditions, although foot placement and CoM motion relative to the foot may be independent of walking modality (Rosenblatt, N. J. and Grabiner, 2010). Moreover, in the current study, the 95% CI for the synergy index _z for AJC_ML with n=16 overlaps with previous data from overground walking (Krishnan et al., 2013) (95% CIs: 1.22-1.47 vs. 1.23-1.51, respectively). Results from the current study may not generalize to other performance variables, although findings were similar for AJC_ML and COM_ML. Generalizability across performance variables (and tasks) is not trivial in light of the wide variety of tasks to which the UCM analysis has been applied (Black et al., 2007; Eckardt and Rosenblatt, 2018; Hsu et al., 2013; Kapur et al., 2010; Latash, M. L. et al., 2002; Olafsdottir et al., 2007; Papi et al., 2015; Park et al., 2012; Qu, 2012; Robert et al., 2009; Shim et al., 2004). Given that thresholds for “reasonable approximations” depend on intersubject variability, results may differ with a different sampling of subjects. While variance component curves show considerable intersubject variability (Figure 1) their shapes, and thus values at a given n relative to N_max (used to calculate thresholds) tend to be consistent across subjects.

The ability to obtain a “reasonable approximation” for UCM-derived measures using a relatively small number of steps is important if these measures are to be employed within clinical settings where time, space and patient mobility may be constrained. However, it is first critical to evaluate reliability using interclass coefficients, to establish the functional implication of smaller/larger synergy index during gait and relatedly and to estimate the minimum clinical important difference. Once accomplished, clinical implementation may be possible; similar measures have demonstrated sensitivity to functional changes in patient population (Falaki et al., 2016; Lewis et al., 2016) .

In conclusion, the individual components used to calculate the synergy index required 50 steps to provide “reasonable approximations”, however the differences between using 15-20 steps and 50 steps were small (i.e. 20%) but may be important. The collection of 15-20 steps results in a reasonable approximation of the synergy index related to frontal plane foot placement and COM control, which is advantageous when the data collections are constrained to a limited number of steps.