Many lines of evidence suggest that the cerebellum is involved in motor control. But what features of these movements are encoded by cerebellar neurons? For slow-tracking eye movements, the activity of Purkinje cells in the ventral paraflocculus of the cerebellum is known to be correlated with eye velocity and acceleration. Here we show that the complex temporal pattern of the firing frequency that occurs during the ocular following response elicited by movements of a large visual scene can be reconstructed by an inverse-dynamics representation, which uses the position, velocity and acceleration of eye movements. Further analysis reveals that the velocity and acceleration components can provide appropriate dynamic drive signals to ocular motor neurons, whereas the position component often has the wrong polarity. We conclude that these Purkinje cells primarily contribute dynamic command signals.