The Cerebellum as Delay-Coordinate Sampler: A Takens Embedding Interpretation of Parallel Fiber Architecture

Why do cerebellar parallel fibers have systematically varying conduction delays? Existing theories (Marr-Albus-Ito) explain what the cerebellum computes but treat delays as anatomical side-effects. We propose that parallel fiber delays implement Takens delay embedding—reconstructing high-dimensional motor error dynamics from low-dimensional sensory input.

The delays (5–15 ms per Wyatt 2005; 10–23 ms per Braitenberg/Llinás) match optimal embedding parameters for motor error signals (8–25 Hz). This quantitative match suggests delay structure is functionally tuned, not incidental.

Mapping to biology: The scalar signal y(t) is mossy fiber activity (proprioceptive/sensory input). The cerebellum delay-embeds the prediction error (actual sensory − predicted sensory), explaining why it needs sensory input, not just efference copy.

On ferns (a footnote): Purkinje arbors resemble ferns, but for different reasons. Ferns: fractal geometry as byproduct of low-D GRN. Purkinje: fractal-like geometry functionally tuned for delay sampling. Developmental evidence supports tuning: Purkinje arbors develop postnatally through activity-dependent refinement.

Robotic control application (IP filed): The delay-line bank architecture has direct applications to robotic motor control—state reconstruction from partial observations, graceful degradation under sensor noise/delay variation.