Linking Motor Learning to Function Approximation: Learning in an Unlearnable Force Field

Part of Advances in Neural Information Processing Systems 14 (NIPS 2001)

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O. Donchin, Reza Shadmehr


Reaching movements require the brain to generate motor com- mands 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. Us- ing a framework from function approximation, we argue that the sequence of errors should reflect 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 ele- ments 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. 1 Introduction

It appears that in constructing the motor commands to guide the arm toward a target, the brain relies on an internal model (IM) of the dynamics of the task that it learns through practice [1]. The IM is presumably a system that transforms a desired limb trajectory in sensory coordinates to motor commands. The motor commands in turn create the complex activation of muscles necessary to cause action. A major issue in motor control is to infer characteristics of the IM from the actions of subjects. Recently, we took a first step toward mathematically characterizing the IM's rep- resentation in the brain [2]. We analyzed the sequence of errors made by subjects on successive movements as they reached to targets while holding a robotic arm. The robot produced a force field and subjects learned to compensate for the field (presumably by constructing an IM) and eventually produced straight movements within the field. Our analysis sought to draw conclusions about the structure of the IM from the sequence of errors generated by the subjects. For instance, in a

velocity-dependent force field (such as the fields we use), the IM must be able to encode velocity in order to anticipate the upcoming force. We hoped that the e#ect of errors in one direction on subsequent movements in other directions would give information about the width of the elements which the IM used in encoding velocity. For example, if the basis elements were narrow, then movements in a given direction would result in little or no change in performance in neighboring directions. Wide basis elements would mean appropriately larger e#ects. We hypothesized that an estimate of the width of the basis elements could be cal- culated by fitting the time sequence of errors to a set of equations representing a dynamical system. The dynamical system assumed that error in a movement resulted from a di#erence between the IM's approximation and the actual environ- ment, an assumption that has recently been corroborated [3]. The error in turn changed the IM, a#ecting subsequent movements: