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.