When subjects adapt their reaching movements in the setting of a systematic force or visual perturbation, generalization of adaptation can be assessed psychophysically in two ways: by testing untrained locations in the work space at the end of adaptation (slow postadaptation generalization) or by determining the influence of an error on the next trial during adaptation (fast trial-by-trial generalization). These two measures of generalization have been widely used in psychophysical studies, but the reason that they might differ has not been addressed explicitly. Our goal was to develop a computational framework for determining when a two-state model is justified by the data and to explore the implications of these two types of generalization for neural representations of movements. We first investigated, for single-target learning, how well standard statistical model selection procedures can discriminate two-process models from single-process models when learning and retention coefficients were systematically varied. We then built a two-state model for multitarget learning and showed that if an adaptation process is indeed two-rate, then the postadaptation generalization approach primarily probes the slow process, whereas the trial-by-trial generalization approach is most informative about the fast process. The fast process, due to its strong sensitivity to trial error, contributes predominantly to trial-by-trial generalization, whereas the strong retention of the slow system contributes predominantly to postadaptation generalization. Thus, when adaptation can be shown to be two-rate, the two measures of generalization may probe different brain representations of movement direction.

We investigated the differences between two well-known optimization principles for understanding movement planning: the minimum variance (MV) model of Harris and Wolpert (1998) and the minimum torque change (MTC) model of Uno, Kawato, and Suzuki (1989). Both models accurately describe the properties of human reaching movements in ordinary situations (e.g., nearly straight paths and bell-shaped velocity profiles). However, we found that the two models can make very different predictions when external forces are applied or when the movement duration is increased. We considered a second-order linear system for the motor plant that has been used previously to simulate eye movements and single-joint arm movements and were able to derive analytical solutions based on the MV and MTC assumptions. With the linear plant, the MTC model predicts that the movement velocity profile should always be symmetrical, independent of the external forces and movement duration. In contrast, the MV model strongly depends on the movement duration and the system's degree of stability; the latter in turn depends on the total forces. The MV model thus predicts a skewed velocity profile under many circumstances. For example, it predicts that the peak location should be skewed toward the end of the movement when the movement duration is increased in the absence of any elastic force. It also predicts that with appropriate viscous and elastic forces applied to increase system stability, the velocity profile should be skewed toward the beginning of the movement. The velocity profiles predicted by the MV model can even show oscillations when the plant becomes highly oscillatory. Our analytical and simulation results suggest specific experiments for testing the validity of the two models.