Summation & Cancellation: The Cerebellum’s Vector Trick for Precise Saccades

When you make a saccade (a quick eye movement) toward a target and then need to correct mid-flight, how does the cerebellum compute which direction to nudge the eyes? Fakharian et al. (Science, 2025) show that Purkinje cells (P cells) implement a simple, but powerful, vector calculus that both generates corrections and cancels unwanted sideways pulls. Their experiments and analyses were performed in marmosets trained on a two-target saccade task. They trained marmosets on a random corrective-saccade task: a primary target appeared (random direction, fixed eccentricity) and then, shortly after, a secondary target appeared a short distance from the primary in a random direction. The cerebellum’s role was probed by measuring how the animal corrected the original saccade toward the secondary target.

The central idea is the potent vector. Each P cell has a preferred direction θᵢ (found from its climbing-fiber/complex-spike tuning) and a potency ρᵢ (how strongly that climbing input differentiates opposite directions). When a complex spike (CS) transiently suppresses a cell’s ongoing simple spikes (SSs), it produces a measurable, short displacement of the eye along that P cell’s preferred direction. That displacement defines a 2-D vector wᵢ whose direction is θᵢ and whose magnitude scales with ρᵢ. Practically, you can think of the population output as the weighted sum of these potent vectors.  

Two algebraic lessons emerge. First, superposition holds: simultaneous suppression of two P cells produces the vector sum wᵢ + wⱼ in eye displacement. Second, the population is organized so that components perpendicular to the intended movement cancel out. Individual P cells show modulation for many saccade directions, but because different cells’ potent vectors are arranged across the population, the perpendicular components sum to approximately zero while the parallel component remains — yielding a clean corrective command in the intended direction. This is a population null-space / projection property rendered explicit in vector form. 

How does the circuit produce this tidy cancellation? Recordings show that neurons are grouped into cliques (tightly interacting subsets). Mossy fibers supply two streams: a copy of the motor command (state) and the sensory goal (goal). Molecular-layer interneurons (MLIs) transform these inputs so that P-cell populations predict when the saccade should stop — the population output crosses zero at deceleration onset (burst → pause), precisely aligning stopping with the goal. Importantly, the goal signal—and thus the stop signal—appears only when the saccade is behaviorally relevant (reward-relevant trials). 

Why produce spikes that will be canceled? Fakharian et al. argue this enables robust, fast population dynamics: slow individual modulations can be placed in competitive (subtractive) relationships so the resulting population output is fast and temporally precise. It also creates reserves for learning or compensation (e.g., unilateral muscle weakness) should the mapping change.

In short: the cerebellum encodes corrective directions as potent vectors assigned to P cells; these vectors add linearly, cancel unwanted perpendicular components, and — via interneurons and separate sensory/motor inputs — yield a precise burst–pause population signal that times deceleration. The Science paper lays out the data, math, and circuitry for this neat vector calculus of eye control. 

Author of blog: Alex White

Reference: Fakharian, M. A., Shoup, A. M., Hage, P., Elseweifi, H. Y., & Shadmehr, R. (2025). A vector calculus for neural computation in the cerebellum. Science, 388(6749), 869-875.

https://doi.org/10.1126/science.adu6331