The cerebellum computes frequency dynamics for motions with numerical precision and cross-individual uniformity

Cross-individual variability is considered the essence of biology, preventing precise mathematical descriptions of biological motion^1-7 like the physics law of motion. Here we report that the cerebellum shapes motor kinematics by encoding dynamic motor frequencies with remarkable numerical precision and cross-individual uniformity. Using in-vivo electrophysiology and optogenetics in mice, we confirmed that deep cerebellar neurons encoded frequencies via populational tuning of neuronal firing probabilities, creating cerebellar oscillations and motions with matched frequencies. The mechanism was consistently presented in self-generated rhythmic and non-rhythmic motions triggered by a vibrational platform, or skilled tongue movements of licking in all tested mice with cross-individual uniformity. The precision and uniformity allowed us to engineer complex motor kinematics with designed frequencies. We further validated the frequency-coding function of the human cerebellum using cerebellar electroencephalography recordings and alternating-current stimulation during voluntary tapping tasks. Our findings reveal a cerebellar algorithm for motor kinematics with precision and uniformity, the mathematical foundation for brain-computer interface for motor control.