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. Editor's summary: Over the past decade, null space theory has become one of the principal frameworks with which neuronal recordings are analyzed. However, a direct way to test the idea in the brain has been lacking. Fakharian et al. investigated population coding in the marmoset cerebellum (see the Perspective by Churchland and Sawtell). Purkinje cells exhibited saccade-evoked simple spike activity patterns that were most effectively activated along a potent vector. Despite this tuning specificity, most Purkinje cells fired simple spikes during all saccades. Comparison of simple spike activity revealed that individual Purkinje cells canceled each other out in the nonpotent vector compared with the potent vector. This agrees with the principles of null space computation theory. Mossy fiber inputs interacting through interneurons enabled Purkinje cell populations to predict key kinematic aspects of goal-directed actions. —Peter Stern [ABSTRACT FROM AUTHOR]