Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model. A number of invariant features of multijoint planar reaching movements have been observed in measured hand trajectories. These features include roughly straight hand paths and bell-shaped speed profiles where the trajectory curvatures between transverse and radial movements have been found to be different. For quantitative and statistical investigations, we obtained a large amount of trajectory data within a wide range of the workspace in the horizontal and sagittal planes (400 trajectories for each subject). A pair of movements within the horizontal and sagittal planes was set to be equivalent in the elbow and shoulder flexion/extension. The trajectory curvatures of the corresponding pair in these planes were almost the same. Moreover, these curvatures can be accurately reproduced with a linear regression from the summation of rotations in the elbow and shoulder joints. This means that trajectory curvatures systematically depend on the movement location and direction represented in the intrinsic body coordinates. We then examined the following four candidates as planning spaces and the four corresponding computational models for trajectory planning. The candidates were as follows: the minimum hand jerk model in an extrinsic-kinematic space, the minimum angle jerk model in an intrinsic-kinematic space, the minimum torque change model in an intrinsic-dynamic-mechanical space, and the minimum commanded torque change model in an intrinsic-dynamic-neural space. The minimum commanded torque change model, which is proposed here as a computable version of the minimum motor command change model, reproduced actual trajectories best for curvature, position, velocity, acceleration, and torque. The model's prediction that the longer the duration of the movement the larger the trajectory curvature was also confirmed. Movements passing through via-points in the horizontal plane were also measured, and they converged to those predicted by the minimum commanded torque change model with training. Our results indicated that the brain may plan, and learn to plan, the optimal trajectory in the intrinsic coordinates considering arm and muscle dynamics and using representations for motor commands controlling muscle tensions.

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There is an infinity of impedance parameter values, and thus different co-contraction levels, that can produce similar movement kinematics from which the CNS must select one. Although signal-dependent noise (SDN) predicts larger motor-command variability during higher co-contraction, the relationship between impedance and task performance is not theoretically obvious and thus was examined here. Subjects made goal-directed, single-joint elbow movements to either move naturally to different target sizes or voluntarily co-contract at different levels. Stiffness was estimated as the weighted summation of rectified EMG signals through the index of muscle co-contraction around the joint (IMCJ) proposed previously. When subjects made movements to targets of different sizes, IMCJ increased with the accuracy requirements, leading to reduced endpoint deviations. Therefore without the need for great accuracy, subjects accepted worse performance with lower co-contraction. When subjects were asked to increase co-contraction, the variability of EMG and torque both increased, suggesting that noise in the neuromotor command increased with muscle activation. In contrast, the final positional error was smallest for the highest IMCJ level. Although co-contraction increases the motor-command noise, the effect of this noise on the task performance is reduced. Subjects were able to regulate their impedance and control endpoint variance as the task requirements changed, and they did not voluntarily select the high impedance that generated the minimum endpoint error. These data contradict predictions of the SDN-based theory, which postulates minimization of only endpoint variance and thus require its revision.

The learning process of reaching movements was examined under novel environments whose kinematic and dynamic properties were altered. We used a kinematic transformation (visuomotor rotation), a dynamic transformation (viscous curl field), and a combination of these transformations. When the subjects learned the combined transformation, reaching errors were smaller if the subject first learned the separate kinematic and dynamic transformations. Reaching errors under the kinematic (but not the dynamic) transformation were smaller if subjects first learned the combined transformation. These results suggest that the brain learns multiple internal models to compensate for each transformation and has some ability to combine and decompose these internal models as called for by the occasion.

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