Dimensionality of joint torques and muscle patterns for reaching

Abstract

Muscle activities underlying many motor behaviors can be generated by a small number of basic activation patterns with specific features shared across movement conditions. Such low-dimensionality suggests that the central nervous system (CNS) relies on a modular organization to simplify control. However, the relationship between the dimensionality of muscle patterns and that of joint torques is not fixed, because of redundancy and non-linearity in mapping the former into the latter, and needs to be investigated. We compared the torques acting at four arm joints during fast reaching movements in different directions in the frontal and sagittal planes and the underlying muscle patterns. The dimensionality of the non-gravitational components of torques and muscle patterns in the spatial, temporal, and spatiotemporal domains was estimated by multidimensional decomposition techniques. The spatial organization of torques was captured by two or three generators, indicating that not all the available coordination patterns are employed by the CNS. A single temporal generator with a biphasic profile was identified, generalizing previous observations on a single plane. The number of spatiotemporal generators was equal to the product of the spatial and temporal dimensionalities and their organization was essentially synchronous. Muscle pattern dimensionalities were higher than torques dimensionalities but also higher than the minimum imposed by the inherent non-negativity of muscle activations. The spatiotemporal dimensionality of the muscle patterns was lower than the product of their spatial and temporal dimensionality, indicating the existence of specific asynchronous coordination patterns. Thus, the larger dimensionalities of the muscle patterns may be required for CNS to overcome the non-linearities of the musculoskeletal system and to flexibly generate endpoint trajectories with simple kinematic features using a limited number of building blocks.

Keywords: EMGs; human subjects; inverse dynamics; joint torques; modularity; muscle synergies; reaching movements; temporal components.

Figure 1

Joint angle definition for the arm model. The four joint angles included in the model (shoulder adduction, shoulder flexion, shoulder external rotation, and elbow flexion) are illustrated by a sequence of postures in space of a two-link arm. The four rotation axes are indicated by colored arrows and correspond to the z-axes of the four reference frames defined according to the D-H notation (see text) and labeled z₀–z₃.

Figure 2

Types of multidimensional decomposition of data from different task conditions. Data collected from D channels (4 in this schematic illustration, represented by different patterns) over T time samples (6, represented by different color saturations) in a single task condition are represented by a grid of squares. Different task conditions are represented with different background colors. A spatial decomposition is obtained by factorizing the data matrix obtained by stacking the data from individual conditions horizontally (i.e., matching their spatial—channels—dimension) into a matrix of N (3) spatial generators (D rows and N columns) times a matrix of time-and condition-dependent coefficients. A temporal decomposition is obtained by factorizing the transpose of data matrix obtained by stacking the data from individual conditions vertically (i.e., matching their temporal dimension) into a matrix of N (3) temporal generators (T rows and N columns) times a matrix of channel- and condition-dependent coefficients. Finally, a spatiotemporal decomposition is obtained by arranging all the data samples of each condition into a column and factorizing the resulting matrix into a matrix of N (3) spatiotemporal generators (D × T rows and N columns) times a matrix of condition-dependent coefficients.

Figure 3

Example of endpoint speed, velocity, joint angles and torques. Example of endpoint trajectories, end-point speed profiles, joint angles, joint angular velocities, gravitational (light gray) and dynamic (dark gray) torques for eight center-out movements in the frontal plane of subject 1 (mean across repetitions of each movement). Vertical dashed lines represent the times of movement onset and movement end. Shaded areas around gravitational and dynamic torque profiles represent ±1 SD around the mean.

Figure 4

Example of coordination between pairs of dynamic torques. Each scatter plot illustrated the dynamic torques for a pair of joints recorded in an interval of 250 ms around the time of movement onset for 8 center-out movements in the frontal plane of subject 1.

Figure 5

Spatial decomposition of dynamic torques. (A) R² curve for subject 1 obtained by spatial decomposition using PCA. (B) Three spatial generators selected for subject 1. (C) Example of the reconstructions of the dynamic torques for six movement conditions of subject 1 obtained with the generators illustrated in panel (B) (shaded area: original data, thick line: reconstructed data, bottom: time-varying combination coefficients).

Figure 6

Temporal decomposition of dynamic torques. (A) R² curve for subject 1 obtained by temporal decomposition using PCA. (B) The single temporal generators selected for subject 1. (C) Example of the reconstructions of the dynamic torques for six movement conditions of subject 1 obtained with the generator illustrated in panel (B) (shaded area: original data, thick line: reconstructed data, bottom: joint-specific weights).

Figure 7

Spatiotemporal decomposition of dynamic torques. (A) R² curve for subject 1 obtained by spatiotemporal decomposition using PCA. (B) Three spatiotemporal generators selected for subject 1. (C) Example of the reconstructions of the dynamic torques for six movement conditions of subject 1 obtained with the generator illustrated in panel (B) (shaded area: original data, thick line: reconstructed data, bottom: combination coefficients represented by the height of the rectangle containing the temporal profile of each generators averaged over joints).

Figure 8

Spatial decomposition of phasic muscle patterns. (A) R² curve for subject 1 obtained by spatial decomposition using NMF. (B) Four spatial generators selected for subject 1. (C) Example of the reconstructions of the muscle patterns for six movement conditions of subject 1 obtained with the generators illustrated in panel (B) (shaded area: original data, thick line: reconstructed data, bottom: time-varying combination coefficients).

Figure 9

Temporal decomposition of phasic muscle patterns. (A) R² curve for subject 1 obtained by temporal decomposition using NMF. (B) The four temporal generators selected for subject 1. (C) Example of the reconstructions of muscle patterns for six movement conditions of subject 1 obtained with the generator illustrated in panel (B) (shaded area: original data, thick line: reconstructed data, bottom: muscle-specific weights of each generator).

Figure 10

Spatiotemporal decomposition of phasic muscle patterns. (A) R² curve for subject 1 obtained by spatiotemporal decomposition using NMF. (B) Seven spatiotemporal generators selected for subject 1. (C) Example of the reconstructions of the muscle patters for six movement conditions of subject 1 obtained with the generator illustrated in panel (B) (shaded area: original data, thick line: reconstructed data, bottom: combination coefficients represented by the height of the rectangle containing the temporal profile of each generators averaged over muscles).