Motor adaptation and generalization of reaching movements using motor primitives based on spatial coordinates

Abstract

The brain processes sensory and motor information in a wide range of coordinate systems, ranging from retinal coordinates in vision to body-centered coordinates in areas that control musculature. Here we focus on the coordinate system used in the motor cortex to guide actions and examine physiological and psychophysical evidence for an allocentric reference frame based on spatial coordinates. When the equations of motion governing reaching dynamics are expressed as spatial vectors, each term is a vector cross product between a limb-segment position and a velocity or acceleration. We extend this computational framework to motor adaptation, in which the cross-product terms form adaptive bases for canceling imposed perturbations. Coefficients of the velocity- and acceleration-dependent cross products are assumed to undergo plastic changes to compensate the force-field or visuomotor perturbations. Consistent with experimental findings, each of the cross products had a distinct reference frame, which predicted how an acquired remapping generalized to untrained location in the workspace. In response to force field or visual rotation, mainly the coefficients of the velocity- or acceleration-dependent cross products adapted, leading to transfer in an intrinsic or extrinsic reference frame, respectively. The model further predicted that remapping of visuomotor rotation should under- or overgeneralize in a distal or proximal workspace. The cross-product bases can explain the distinct patterns of generalization in visuomotor and force-field adaptation in a unified way, showing that kinematic and dynamic motor adaptation need not arise through separate neural substrates.

Keywords: computational model; force-field adaptation; generalization; motor control; motor cortex; proprioception; reference frames; visuomotor rotation.

Fig. 1.

Motor control in joint-angle and spatial coordinates. A link model of the arm represented in joint-angle space (A) and in spatial coordinates (B) is shown. Joint angles (θ₁, θ₂) represent a configuration of the two-link model in joint-angle space, and spatial vectors (X₁₀, X₂₀, X₂₁) are used for the spatial representation. C: visuomotor transformation proposed by the model. Task-oriented endpoint position and movement vectors are first determined, and those vectors in turn are decomposed into limb segment vectors. Vector cross products are then computed as intermediate variables, and joint torques or muscle tensions are finally generated as weighted sums of cross products. The readout coefficients in the computation of joint torques from cross products are determined in nonperturbed conditions according to Newtonian dynamics. This study posits that plasticity in the readout coefficients (i.e., weights from vector cross products to joint torques) provides adaptive changes for both dynamic and kinematic motor adaptation. See text for definition of terms.

Fig. 2.

Behavioral generalization after adapting to a viscous force field. Hand trajectories before adaptation (A), after adaptation (B), and after-effects to the viscous force field at the right workspace (C) are shown. Simulated model trajectories and desired minimum-jerk trajectories are solid and dashed lines, respectively (left) and are compared with the corresponding experimental trajectories (right). Eight directions of movement are color-coded. Generalization to hand trajectories in the left workspace when an intrinsic (D) or an extrinsic force field (E) was imposed after the linear readout model was trained at the right workspace. [The experimental figures are adapted from Figs. 9A, 9D, 13D, 15B, and 15A, respectively, from Shadmehr and Mussa-Ivaldi (1994), with permission.]

Fig. 3.

Generalization of intrinsic viscous force field to multiple workspace locations. The model learned the force field at the central location [(x, y) = (0, 40)], and 14 peripheral locations were tested with the intrinsic-based force field for generalization, which were positioned with one of three distances from the shoulder (30, 40 or 50 cm) and one of five directions (−30°, −15°, 0°, 15° or 30°). A: polar plots of directional errors at the training workspace (center, red) and the test workspaces (peripheral, black). Solid lines at each workspace represent directional errors for eight target directions, and dashed lines represent null directional errors for reference. They overlapped almost completely, indicating that there were little directional errors at all locations. The scale ticks along the radial axes indicate 15° directional errors (positive and negative values for clockwise and counterclockwise deviations, respectively). Hand trajectories at distal [(x, y) = (0 cm, 50 cm); B] and proximal [(x, y) = (0 cm, 50 cm); C] to the body, and left [(x, y) = (−25 cm, 35 cm); D] and right [(x, y) = (25 cm, 35 cm); E] from the center workspace are shown. The corresponding locations were indicated by the letters in A.

Fig. 4.

Generalization of extrinsic viscous force field to multiple workspace locations. A: polar plots of directional errors at the training workspace (center, red) and the test workspaces with the extrinsic-based force field (peripheral, black). Hand trajectories at distal [(x, y) = (0 cm, 50 cm); B] and proximal [(x, y) = (0 cm, 50 cm); C] to the body, and left [(x, y) = (−25 cm, 35 cm); D] and right [(x, y) = (25 cm, 35 cm); E] from the center workspace are shown. The same format was used as in Fig. 3.

Fig. 5.

Directional generalization of visuomotor rotation remapping. A: cursor from adapted movements (solid) and desired (dashed) trajectories at the trained workspace [(θ₁, θ₂) = (45°, 90°)]. The cursor trajectories were obtained by rotating the simulated hand trajectories by the imposed rotation so that the comparison with the desired trajectories was made straightforward. Cursor from movements adapted in the trained workshops (solid) and desired (dashed) trajectories in the left workspace [(θ₁, θ₂) = (90°, 90°); B] and in the right workspace [(θ₁, θ₂) = (0°, 90°); C] are shown.

Fig. 6.

Under- and over-generalization of rotated remapping for visuomotor rotation. A: cursor (solid) and desired (dashed) trajectories at the trained workspace [(x, y) = (0 cm, 40 cm)]. Under-generalization at a distal posture [(x, y) = (0 cm, 45 cm); B], and over-generalization at a proximal posture [(x, y) = (0 cm, 35 cm); C] are shown. D: the degree of generalization over the entire workspace. Color at each point in the workspace indicates the degree of counterrotation averaged over eight movement directions. The gray dashed line indicates iso-distance locations 40 cm from the shoulder. Values larger than 60° represent over-generalization, and values smaller than 60° represent under-generalization. A, B, and C indicate the locations of the starting hand position for A, B, and C, respectively.