^{1}

^{*}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

Edited by: Pietro Avanzini, University of Parma, Italy

Reviewed by: William M. Land, The University of Texas at San Antonio, United States; David Sherwood, University of Colorado Boulder, United States

This article was submitted to Movement Science and Sport Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In sequential tasks, a partial reuse of former motor plans results in a persistence in the former posture (termed hysteresis). The

When reaching for a cup of coffee, we are unaware of the series of sensorimotor transformations that are necessary to translate the intended perceptual effect of grasping the cup into a muscle activation pattern that guides our hand to its final position. Due to these transformations, the creation of each motor plan is associated with a cognitive cost. In a repetitive motor task, we therefore do not create a new motor plan for each movement, but instead reuse (and modify) the previous plan (

For example, if participants have to open a column of drawers in a descending sequence, they adopt a pronated initial posture for the highest drawer and persist in a more pronated posture throughout the sequence. In an ascending sequence, they start in a supinated posture at the lowest drawer and subsequently persist in a more supinated posture (

Motor hysteresis can be formally explained by dynamic systems theory (

Instead, the cognitive cost of movement planning seems to be the cause of the hysteresis effect. To reduce the planning cost, previous motor plans are partially reused (

Model of the ^{p} (|

The

A potential shortcoming of the

Interestingly, all mentioned drawer studies also found a sigmoid relationship between drawer (height) and posture (

To test these assumptions, we asked participants to execute a drawer opening task. Pro/supination of the hand was measured as the dependent variable. In randomized sequences of drawers, postures are unaffected by the previous posture (

In ordered (a/descending) sequences of drawers, each motor plan is created from the previous motor plan. We hypothesize that a sigmoid model (1) with a constant percentage of motor plan reuse (2) between subsequent drawers should capture a major fraction of the variance of “drawer” (sigmoid optimal posture), of “sequence” (hysteresis effect), and of the interaction of “sequence” × “drawer” (higher hysteresis effect at the central drawers, 3).

If the percentage of reuse was constant across drawers, the size of the hysteresis effect should increase if the spatial difference and, thus, the difference in optimal posture, between subsequent drawers increased. To test this, we asked participants to execute an ordered (a/descending) drawer opening task, but to skip every second drawer. If the percentage of reuse depended only on the cognitive and mechanical cost of the task, we expected a larger hysteresis effect in the skipped task compared to the normal task (4). This should be reflected by an interaction of “task” × “sequence.”

If the model is adequate for randomized, ordered, and skipped tasks, one can reasonably assume that it can be used to measure the percentages of motor plan reuse and novel planning in any sequential task. For example, it could be applied (1) to test how proposed hemispheric differences in cognitive (

The proposed modeling approach is imperative for sequential posture selection tasks. The percentage of reuse in hand path planning tasks (

Thirty-one students [19 female, 12 male, age 24.5 ± 3.3 (SD) years] from Bielefeld University participated in the experiment in exchange for either course credit or 5€. Twenty-one participants were right handed (handedness score 0.97 ± 0.07), four left handed (handedness score −0.80 ± 0.08), and six ambidextrous (handedness score 0.02 ± 0.28) according to the revised Edinburgh inventory (

The apparatus used was a tall metal frame (222 cm high, 40 cm wide, and 30 cm deep) with nine wooden shelves (

Retroreflective markers were attached to four bony landmarks on the right arm of the participant: the most cranial point of the

The center of drawer #7 was aligned with the height of the shoulder joint center. Drawer spacing was set to a quarter arm length. The participant was positioned with the shoulder joint center one arm length in front of the setup and a third arm length to the left of the drawer centers. The participant’s position was marked by two strips of black tape: point of the toes and median plane of the body.

The experiment consisted of three tasks. Each task contained up to eight sequences of trials. A single trial was defined as the opening and closing of one drawer. Each trial started from an initial position, with the palm of the hand touching the thigh. The participant had to (1) raise the arm to the drawer, (2) grasp the handle with a five-finger grip, (3) fully open the drawer, (4) close the drawer, and (5) return to the initial position.

In Task 1, the participant performed four randomized sequences of the nine drawers (4 repetitions

In Task 2 and Task 3, the participants performed eight ordered sequences of trials, respectively: four ascending, four descending. Sequence order was alternated and counterbalanced across participants. The experimenter did not announce drawer numbers, but only the order of the sequence (“from top to bottom”/“from bottom to top”). The participant executed the trials of each sequence on their own.

In Task 2, a single sequence consisted of all nine drawers (2 sequences

Each participant conducted Task 1 first, to get accustomed to the experiment. Postures in randomized tasks are unaffected by motor hysteresis (

Before each task, the position of the participant in front of the apparatus was checked based on the floor marks. Participants had a resting period of 30 s between each sequence and of 2 min between each task. The entire experiment lasted approximately 45 min.

Movement data were recorded by a Vicon MX (Vicon Motion Systems, Oxford, United Kingdom) motion capture system. Marker trajectories were reconstructed in Vicon Nexus 1.8.5, labeled manually, and exported to MATLAB (2008b, The MathWorks, Natick, MA, United States) for data analysis. The laboratory’s coordinate system was defined with the

To identify the moment of drawer grasp for each trial, the

For the calculation of α, the wrist axis was projected onto the drawer face (_{x} and v_{z}, the pro/supination angle α was calculated with the four-quadrant inverse tangent function of MATLAB. It was zero when the back of the hand pointed directly to the right (

Previous studies’ results suggest that the pro/supination angle is a sigmoid function of drawer height (

A simple, five-parameter model was developed to capture the pattern of results found in a sequential drawer task.

Four parameters are required to model the pro/supination angle as a function of the drawer number (

Parameters of the mathematical model. Four parameters describe the optimal pro/supination angle as a sigmoid function of drawer (height): its

The fifth parameter is a fixed

The

In our continuous posture selection task, the percentages of reuse and novel planning are reflected in the modification of the pro/supination angle between subsequent drawers. In

The model parameters were fitted to the measured data of individual participants using a least squares optimization algorithm (

In the randomized task, grasp angles were unaffected by motor hysteresis and, thus, should reflect the optimal grasp angle (

In an ordered task, the overall variance of the data is dominated by the main effect of “drawer.” Thus, for our proposed model to be valid for the ordered task, it should not only capture a major fraction of the overall variance, but also a major fraction of the hysteresis-specific variance, i.e., of the factor “sequence” and of the interaction of “sequence” × “drawer.” To this end, sums of squares are calculated for the model data and the measured data of each participant individually.

If a good fit is achieved, the fifth parameter returns the percentage of motor plan reuse in the ordered task. In theory, all model parameters could be modified arbitrarily by the optimization algorithm. The

Based on the percentage of reuse calculated from the ordered task, the size of the hysteresis effect in the skipped ordered task is predicted. In the skipped ordered task, the distance between subsequent drawers is larger than in the ordered task. Therefore, the size of the hysteresis effect should be larger. The hysteresis effect size in the skipped ordered task is compared to the effect size at the equivalent drawers (#1, #3, #5, #7, and #9) in the ordered task.

To estimate the impact of “drawer” on the total variance in the ordered task and to compare the size of the hysteresis effect in the ordered and the skipped ordered task, repeated measures analyses of variance (rmANOVAs) were calculated on the pro/supination angles. The four repetitions for each condition were averaged to reduce variance. The rmANOVAs were calculated in SPSS (22, IBM Corp., Armonk, NY, United States) to apply the Greenhouse–Geisser correction to the

The four-parameter model was applied to the randomized data of Task 1. The model captured 99.3 ± 0.6% (SD) of the total variance and reached an RMSE of 3.7 ± 2.1° (

For the four model parameters, the fitting algorithm returned the following values:

After confirming the validity of the sigmoid fit with the randomized data, the five-parameter model was applied to the ordered data of Task 2. The model captured 98.6 ± 1.6% of the total variance and reached an RMSE of 4.8 ± 1.7° (

To test whether the total variance was dominated by the factor “drawer,” a 2 (sequence) × 9 (drawer) rmANOVA was calculated on the measured pro/supination angles, with “sequence” and “drawer” as within-subject factors. The main effect of “sequence,” ^{2} = 0.003, the main effect of “drawer,” ^{2} = 0.857, and the interaction of “sequence” × “drawer,” ^{2} = 0.001, were significant (

To evaluate the model fit with respect to the motor hysteresis effect only, we calculated the combined fraction of variance of “sequence” and “sequence” × “drawer.” The model captured only 46.8 ± 38.2% of this variance. A closer inspection of individual fits revealed that a number of participants exhibited no effect of “sequence” and that the interaction of “sequence” × “drawer” in reality reflected random noise on the grasp angles instead of a modulation of the hysteresis effect across drawers.

We therefore restricted the analysis to participants with an actual hysteresis effect. To this end, we calculated the ratio between the variance of “sequence” and the variance of “sequence” × “drawer.” The top 15 participants were classified as “hysteresis,” the bottom 15 participants were classified as “noise.” In the “noise” group, only 16.0 ± 28.8% of the combined variance was captured. In the “hysteresis” group, the model captured 76.0 ± 18.0% of the combined variance. If a hysteresis effect was present, the proposed model achieved a decent fit.

The model parameters were similar to those of the randomized task:

Using the model parameters from the ordered task, we predicted the size of the hysteresis effect in the skipped ordered task (Task 3). Due to the larger differences between the previous and the optimal posture, we expected a larger hysteresis effect in the skipped ordered task. The model predicted an average difference of 10.3° between the ascending and descending sequences in the skipped ordered task, as opposed to an average difference of 5.6° at drawers #1, #3, #5, #7, and #9 in the ordered task (

A 2 (task) × 2 (sequence) rmANOVA was calculated on the pro/supination angles measured in both tasks to verify this prediction, with “task” and “sequence” as within-subject variables. The grasp angles at drawers #1, #3, #5, #7, and #9 were averaged. We expected a significant interaction of “task” × “sequence” and a larger hysteresis effect in the skipped ordered task. The main effect of “sequence,” ^{2} = 0.025, and the interaction of “task” × “sequence,” ^{2} = 0.002 were significant. The size of the hysteresis effect differed depending on the task. Effect direction, however, was in contrast to the model prediction: in the skipped ordered task, the size of the hysteresis effect decreased to 3.3° (

The five-parameter model was applied to the skipped ordered data. The model captured 99.2 ± 1.0% of the total variance and reached an RMSE of 3.6 ± 1.5° (

In the current study, we asked whether, in a sequential drawer opening task, (1) the optimal grasp angle was a sigmoid function of drawer (height), (2) the percentage of motor plan reuse was constant across all drawers, and whether (3) such a constant percentage of reuse across all drawers resulted in a larger hysteresis effect at the central and a smaller hysteresis effect at the peripheral drawers. To this end, participants had to execute three sequential drawer opening tasks: a randomized task, an ordered (a/descending) task where every drawer was opened, and an ordered task where every second drawer was opened. The pro/supination angle of the hand was measured as the dependent variable.

A simple model was created to capture the variance of the pro/supination angle in all three tasks. The model assumes that optimal grasp angle is a sigmoid function of drawer (height), as has been implied in previous drawer studies (

For our first assumption to hold true, the reduced (four-parameter, no

As the sigmoid function is a good estimate of the optimal grasp angle, the parameters returned by the fitting algorithm can be used to interpret the grasping behavior. A

The pro/supination angles in the ordered task (Task 2) follow the pattern of results found in previous drawer studies (

As the total variance was dominated by the main effect of “drawer,” the model should also explain a major fraction of the combined variance of the factor “sequence” and of the interaction “sequence” × “drawer,” the two variance components affected by hysteresis. Of this combined variance, the model only captured 46.8 ± 38.2%. A closer inspection of individual participants showed that a number of participants had no main effect of “sequence,” i.e., they exhibited no hysteresis effect. The tendency to reuse the previous motor plan seems to differ strongly across participants, as is reflected by the high standard deviation [15.6 ± 13.0 (SD)] found for the

In participants with low/no hysteresis effect, the interaction of “sequence” × “drawer” did not reflect a modulation of hysteresis effect size across drawers, but instead reflected random noise, which could not be captured by the model. If the analysis was restricted to participants

None of the four parameters for the sigmoid (optimal grasp) function differed from their randomized counterparts, which supports the notion that optimal grasp angles are stable across experiments and reflect behavioral data and not just abstract fitting parameters. The additional fifth parameter, the

This finding has important implications for the future study of cost-optimization.

If the percentage of reuse was constant and depended only on the cognitive and mechanical cost of the movement, the size of the hysteresis effect should increase with the difference in optimal posture between subsequent drawers. We therefore expected a larger hysteresis effect in the skipped ordered task (Task 3). Results showed a significant interaction of “task” × “sequence” between Tasks 2 and 3, indicating that the size of the hysteresis effect differed depending on the task. However, in contrast to our prediction, the hysteresis effect actually decreased in the skipped ordered task. While this result is in contrast to the prediction, it does not invalidate the model, but instead shows that the assumption of a constant percentage of reuse across different tasks was flawed.

When applied to the skipped ordered data of Task 3, the five-parameter model captured 99.2 ± 1.0% of the total variance and, thus, was able to replicate the measured pattern of results almost perfectly. The first four parameters were similar to the ordered task (Task 2), lending further support to the idea that a sigmoid optimality function is a stable characteristic of vertical reaching. The fifth parameter, the

In conclusion, we showed that a simple model with a sigmoid optimal grasp angle function and a fixed percentage of reuse could capture a major fraction of the variance in a randomized and an ordered sequential drawer opening task. This finding supports the notion that (1) the optimal pro/supination angle is a sigmoid function of drawer (height), (2) the percentage of motor plan reuse is constant across drawers, and (3) a constant percentage of reuse results in a larger hysteresis effect at the central drawers. A smaller percentage of reuse was found in the skipped ordered task. This indicates that motor planning is not only affected by the cognitive and mechanical cost of a movement, but by other factors (e.g., task context) as well.

While the proposed model in the current study was applied to single, well-established hysteresis task, it should be applicable to arbitrary sequential tasks and provide a sensitive tool to measure changes in the percentage of reuse under different experimental manipulations. Especially in sequential posture selection tasks, where hysteresis effects blend with an optimal posture that differs from trial to trial, the current model offers a straightforward approach to isolate the hysteresis effect and to estimate the amount of novel motor planning.

The datasets generated for this study are available on request to the corresponding author.

This study was carried out in accordance with the recommendations of the Bielefeld University ethics committee and the guidelines of the DGPs. All subjects gave written informed consent before the experiment in accordance with the Declaration of Helsinki. The protocol was approved by the Bielefeld University ethics committee.

CS and TS conceived the idea and contributed to the final version of the manuscript. CS planned and carried out the experiments and the simulations, and conducted the statistical analyses.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.