**Donchin O, Shadmehr R (2002) Linking motor learning to function
approximation: Learning in an unlearnable force field. **__Advances in Neural
Information Processing Systems__, Dietterich T. G., Becker S., Ghahramani Z.
(eds), MIT Press, Cambridge,
MA, vol. 14, pp. 197-203.

**Abstract** 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 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 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.

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