A vector calculus for neural computation in the cerebellum

Null space theory suggests that neurons often produce spikes not to drive behavior, but to nullify other neurons' effects on behavior. What has been missing is a direct way to test this idea in the brain. We found that in the marmoset cerebellum, each Purkinje cell (P-cell) had a unique vector that moved the eyes along a specific direction. Interestingly, P-cells fired during all saccades, not just those aligned with their vector. Simultaneous recordings showed that spikes in different cells combined their vectors, conforming to superposition, creating population activity where contributions perpendicular to the intended action were present but canceled. Mossy fibers conveyed motor commands and movement goals, while interneurons transformed these signals, enabling P-cells to collectively signal movement termination. Thus, the cerebellum placed neurons in competition, canceling the downstream impact of one cell’s spike by another’s, producing a fast-changing output via a population of slow neurons.