Cerebellar Delay Embedding
How fern-like branching structure samples delay coordinates from a 1D signal to reconstruct high-dimensional dynamics.
Implements Takens embedding: parallel fiber delays (5-15 ms) match optimal embedding parameters for motor signals.
The Problem
Motor control requires knowing system state, but the cerebellum only receives 1D signals (parallel fiber activity). How do you reconstruct high-D dynamics from a scalar?
Takens' Answer
Sample the signal at multiple delays: [y(t), y(t-τ), y(t-2τ), ...]. For deterministic systems, these delay coordinates reconstruct the original state space topology.
Why Fern-Like?
A branching structure naturally provides multiple delay taps at different distances. The Purkinje dendritic arbor samples parallel fibers across the full 5-15 ms delay range needed for motor frequencies.
What You're Seeing
Left panel: Multiple frequency components combine into a single scalar signal y(t). This is what mossy fibers deliver to the cerebellum.
Center panel: A fern-like structure samples this signal at different delay taps. Each branch tip represents y(t-nτ) for different n. The color shows signal phase at each tap.
Right panel: The reconstructed state space. Delay coordinates [y(t), y(t-τ), y(t-2τ)] form a trajectory that preserves the topology of the original high-D dynamics.
The key insight: Purkinje cell dendrites aren't just collecting inputs—they're implementing a computational algorithm. The 5-15 ms delay range matches τ* = 1/(4f) for 8-25 Hz motor error signals.
Why "Chair Flying" Works
Pilots mentally rehearse procedures ("chair flying") and it improves performance. Why? Because the cerebellum runs the same delay embedding on imagined motor commands. The forward model provides y(t) even without actual movement—the same Takens embedding runs, the same error corrections are computed, the same learning occurs. Mental rehearsal is real cerebellar computation.
Learn more:
The Cerebellum as Delay-Coordinate Sampler →