Vestibular balance: the semicircular canal as a torsion pendulum

Each semicircular canal is an overdamped torsion pendulum: endolymph inertia against cupula stiffness and viscous drag. Over the behavioural band 0.1–6 Hz the canal integrates angular acceleration, so its output encodes angular velocity (band flatness 0.018). The vestibulo-ocular reflex stabilises gaze with gain about 1.0. Canal dynamics are classical.

The canal is modelled as a heavily damped second-order (torsion-pendulum) system; across 0.1–6 Hz its transfer function is flat in angular velocity (flatness ratio 0.018), so afferent firing reports head angular velocity rather than acceleration. The VOR gain is about 1.0 in healthy young adults. These are classical fluid-mechanics anchors [V-arith]/[L] (Van Egmond–Groen–Jongkees 1949), documented and linked, not re-derived from R19.

A semicircular canal is a fluid-filled torus closed by an elastic cupula. Head rotation drives the endolymph against the cupula, and the system behaves as an overdamped torsion pendulum: inertia versus stiffness versus viscous drag, with damping dominating across the behavioural band. Three canals on each side, roughly orthogonal, sense rotation about all axes.

Why the canal reports velocity, not acceleration

Because it is overdamped, the canal effectively integrates angular acceleration over 0.1–6 Hz, so its transfer function is flat in angular velocity (flatness ratio 0.018). The afferent signal therefore reports how fast the head is turning, not how hard — which is exactly the quantity the brain needs to hold the eyes and body steady.

The vestibulo-ocular reflex closes the loop

The velocity signal drives compensatory eye movements that hold gaze stable during head motion; in healthy young adults this vestibulo-ocular reflex (VOR) runs at a gain near 1.0. Otoliths complement the canals by sensing linear acceleration and gravity, so the labyrinth as a whole reports both rotation and translation.

Classical, reproduced by arithmetic [V-arith]. Canal fluid mechanics and the velocity-band behaviour are cited classical anchors (Steinhausen 1933; Van Egmond–Groen–Jongkees 1949). Vestibular balance is also correctly the one node with no single master gene (see organ emergence), so it carries no γ — a circuit-level property represented at its proper level.

When otoconia dislodge into a canal, the same mechanics produce a false angular-velocity signal — benign paroxysmal positional vertigo (BPPV) — which is why repositioning the particles at the source is curative, as set out in the therapy program.