Complex Upper-Limb Movements Are Generated by Combining Motor Primitives that Scale with the Movement Size

The hand trajectory of motion during the performance of one-dimensional point-to-point movements has been shown to be marked by motor primitives with a bell-shaped velocity profile. Researchers have investigated if motor primitives with the same shape mark also complex upper-limb movements. They have done so by analyzing the magnitude of the hand trajectory velocity vector. This approach has failed to identify motor primitives with a bell-shaped velocity profile as the basic elements underlying the generation of complex upper-limb movements. In this study, we examined upper-limb movements by analyzing instead the movement components defined according to a Cartesian coordinate system with axes oriented in the medio-lateral, antero-posterior, and vertical directions. To our surprise, we found out that a broad set of complex upper-limb movements can be modeled as a combination of motor primitives with a bell-shaped velocity profile defined according to the axes of the above-defined coordinate system. Most notably, we discovered that these motor primitives scale with the size of movement according to a power law. These results provide a novel key to the interpretation of brain and muscle synergy studies suggesting that human subjects use a scale-invariant encoding of movement patterns when performing upper-limb movements.


Robustness to Additive Noise of the Movement Element Decomposition Method
To assess the robustness to additive noise of the movement element decomposition method, we generated simulated time series with elements marked by a velocity profile v(. ) determined according to In this equation, represents time, the duration of the movement element, and the displacement associated with the movement element. We simulated three sets of 100 time-series corresponding to unidimensional movement trajectories consisting of 30 movement elements obeying Equation S1. The velocity profiles were scaled using a "normalized" scaling exponent (i.e. =1). The value of the displacement was generated randomly with uniform distribution between a minimum value and a maximum value . The minimum displacement was the same for all the sets of simulated time-series ( = 5 ). The maximum displacement was different for each set as follows: = 2 for the set of simulations whose results are shown in Figure S1, = 1 for those shown in Figure S2, and = 0.2 for those shown in Figure S3. The simulated sampling rate was equal to 120 Hz, namely the same as the sampling rate of the actual recordings obtained using the camera-based motion capture system utilized in the study. We randomly positioned the movement elements in time, with the constraint that movement elements would not overlap in time. We then added uniformly distributed random noise of amplitude ranging from 1E-4 to 0.5 m to assess the robustness of the movement element decomposition method to additive noise.
Noise levels up to 1 mm in amplitude did not significantly affect the estimation of the relationship between the mean of the absolute value of the velocity of the movement elements and the corresponding displacement (panel A of each figure), the number of movement elements identified by the algorithm (panel B), and the peaks detected for each movement element (panel C). These results show that the proposed technique is suitable to process recordings carried out using a camera-based motion capture system, which are typically affected by noise of amplitude smaller than 1 mm.

Movement Elements Are Defined According to the Anatomical Planes
Previous work showed that simple one-dimensional point-to-point movements are marked by a velocity profile with shape that obeys Equation S1. However, previous investigators failed to identify movement elements with such velocity profile when they analyzed complex upper-limb movements. We argue that this is the case because previous studies attempted to identify movement elements by analyzing the magnitude of the velocity vector associated with the trajectory of movement. Figure S4 shows examples of the velocity profiles that we estimated from data collected while subjects performed two of the tasks chosen for the study, i.e. drawing an ellipse (panel A) and drawing a spiral (panel B). The figure shows both the magnitude of the velocity vector and the velocity along the x-and y-components of the movement trajectory, where x and y are the axes of a Cartesian coordinate system oriented according to the anatomical planes in the medio-lateral and antero-posterior directions, respectively.
It is worth emphasizing that movement elements of shape consistent with Equation S1 are only apparent when one observes the velocity trajectories along the x-and y-components of movement. In contrast, the magnitude of the velocity vector displays multiple peaks that are difficult to relate to the peaks of the movement elements along the x and y axes. A simple relationship between the magnitude of the velocity vector and the velocity trajectory along the x-and y-components of movement would be apparent only if the movement elements along the x and y axes happened to be synchronous, i.e. if they occurred at the same time. However, this can only be possible for linear movements, but not for curvilinear movements. Besides, estimating the magnitude of the velocity vector requires combining the velocity of the x-and y-components of the movement trajectory via a quadratic transformation. This makes it challenging to decompose the movement velocity trajectory in its components without constraining the algorithm to search for solutions marked by two distinct components along the x and y axes of movement.
The movement elements that we discovered by analyzing the x-and y-components of the movement trajectory were found to be consistent across all the tasks that we tested in the study, irrespective of the fact that the task consisted of a linear or a curvilinear movement. The only exception to this rule was the data collected during the performance of the one-dimensional movements without targets. Table S1 shows the differences observed across tasks in how closely the movement elements identified for each task matched the theoretical velocity profile shown above (Equation S1). Specifically, we compared the correlation coefficients between the theoretical velocity profile and the experimental data using the Friedman ANOVA test as reported in the main manuscript. Furthermore, we performed Conover post-hoc tests for dependent samples to compare the correlation coefficients across tasks. The post-hoc tests showed consistent differences between the correlation coefficient values for the onedimensional movements without targets vs. all other tasks. A few other differences were identified among the correlation coefficient values for the other tasks. However, the magnitude of such differences was small as shown in Figure 6A of the main manuscript where average correlation coefficients ranging from 0.77 to 0.89 are shown, thus indicating that the movement elements match well the velocity profiles predicted by Equation S1 .
Interestingly, not only the movement elements were shown to be very similar across tasks, but also they were shown to be very similar across axes. In other words, the shape of the movement elements was the same for the x-and y-components of the movement trajectory for planar movements and for all three axes of movement (i.e. the x-, y-, and z-components of movement) for three-dimensional movements. Table S2, which summarizes the results of the Wilcoxon tests performed to compare the velocity profiles of the x and y components of movement. Similarly, Table S3 shows the results of the analysis of three-dimensional movements obtained by performing Friedman ANOVA tests performed to compare the velocity profiles of the movement elements identified by analyzing separately the x-, y-, and z-components of movement. In the few cases in which a statistically significant difference was identified, the magnitude of such difference was very modest.

An Optimization Process Underlies the Generation of Upper-Limb Movements
Our experimental observations unraveled an optimization process underlying the generation of upperlimb movements. In fact, not only the movement elements that we derived using the proposed method displayed the predicted shape shown in Equation S1, but the identified movement elements also scaled with the size of movement according to Equation 6 of the main manuscript, which is reported below: v ̅ = D 2/3 60 1/3 K 1/6 (S2) In this equation, ̅ is the mean velocity of movement, is the displacement associated with each movement element, and is a constant whose value determines the relevance of the smoothness of movement compared to the cost of time as captured by the duration of the movement element as shown below:  Interestingly, whereas the optimization principle utilized by all subjects to generate the movement elements appears to be consistent across subjects and results in a smooth movement output, the approach to the trade-off between the cost of time and the smoothness of movement (i.e. Equation S3) seems to vary significantly across subjects. Interestingly, the table suggests that different optimization trade-off approaches are adopted by different subjects. In other words, while some subjects appear to prefer achieving the tasks in a short time period (i.e. they prioritize minimizing the cost of time), others appear to prefer achieving a smooth movement and hence they weigh less the cost of time.

Movement Elements as Motor Primitives
We look upon the movement elements discovered in this study as motor primitives underlying the generation of a broad range of upper-limb motor tasks. Interestingly, the characteristics of the identified movement elements suggest that complex upper-limb movements consist of elements aimed to reach for intermediate target points along the axes of the trajectory of movement. This observation is consistent with the way infants learn how to perform arm reaching movements. It is also consistent with recent literature that has shown that brain activity associated with a motor task such as writing is invariant to the scale of movement. In this context, the movement elements discovered in the study might be considered the kinematic basis of the scale-invariant encoding of movement patterns. Figure S1. Results