Linking motor learning to function approximation: Learning in an unlearnable force field

Reaching movements require the brain to generate motor commands that rely on an internal model of the task's dynamics. Here we consider the errors that subjects make early in their reaching trajectories to various targets as they learn an internal model. Using a framework from function approximation, we argue that the sequence of errors should reect the process of gradient descent. If so, then the sequence of errors should obey hidden state transitions of a simple dynamical system. Fitting the system to human data, we find a surprisingly good fit accounting for 98% of the variance. This allows us to draw tentative conclusions about the basis elements used by the brain in transforming sensory space to motor commands. To test the robustness of the results, we estimate the shape of the basis elements under two conditions: in a traditional learning paradigm with a consistent force field, and in a random sequence of force fields where learning is not possible. Remarkably, we find that the basis remains invariant.