A vector calculus for neural computation in the cerebellum

When a neuron modulates its firing rate during a movement, we tend to assume that it is contributing to control of that movement. However, null space theory makes the counter-intuitive prediction that neurons often generate spikes not to cause behavior, but to prevent the effects that other neurons would have on behavior. What is missing is a direct way to test this theory in the brain. Here, we found that in the marmoset cerebellum, spike-triggered averaging identified a potent vector unique to each Purkinje cell (P-cell) along which the spikes of that cell displaced the eyes. Yet, the P-cells were active not just for saccades along their potent vector, but for all saccades. Simultaneous recordings revealed that two spikes in two different P-cells produced superposition of their potent vectors. The result was a population activity in which the spikes were canceled if their contributions were perpendicular to the intended movement. But why was one neuron producing spikes that would be canceled by another neuron? To answer this question, we recorded from the mossy fiber inputs and the interneurons in the molecular layer. We found that the mossy fibers provided a copy of the motor commands as well as the sensory goal of the movement, then the interneurons transformed these inputs so that the P-cells as a population predicted when the movement had reached the goal and should be stopped. Because this output had faster dynamics than was present in the individual neurons, the cerebellum placed the cells in subtractive competition with each other such that the downstream effects of spiking in one neuron were partially or completely eliminated because of the spiking in another neuron.