A vector calculus for neural computation in the cerebellum.

Null space theory predicts that neurons generate spikes not only to produce behavior but also to prevent the undesirable effect of other neurons on behavior. In this work, we show that this competitive cancellation is essential for understanding computation in the cerebellum. In marmosets, we identified a vector for each Purkinje cell (P cell) along which its spikes displaced the eyes. Two spikes in two different P cells produced superposition of their vectors. In the resulting population activity, the spikes were canceled if their contributions were perpendicular to the intended movement. Mossy fibers provided a copy of the motor commands and the goal of the movement. Molecular layer interneurons transformed these inputs so that the P cell population predicted when the movement had reached the goal and should be stopped.