The cerebellum shapes motions by encoding motor frequencies with precision and cross-individual uniformity

Understanding brain behaviour encoding or designing neuroprosthetics requires identifying precise, consistent neural algorithms across individuals. However, cerebral microstructures and activities are individually variable, posing challenges for identifying precise codes. Here, despite cerebral variability, 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 confirm that deep cerebellar neurons encode frequencies using populational tuning of neuronal firing probabilities, creating cerebellar oscillations and motions with matched frequencies. The mechanism is 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 validate 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 a brain-computer interface for motor control.