Otolith Illusion — Hair-Cell Population Coding (2D)

Head / Vehicle Motion

Head tilt 0°

Forward linear acceleration 0.00 g

Gel viscosity (otolith time constant) 0.40 s

Live Readouts

Specific force |f|

1.00 g

Peak hair-cell firing

0.0%

Actual tilt

0.0°

Perceived tilt (decoded)

0.0°

Illusion error (perceived − actual)

0.0°

How it works (2D population-coding model)

This 2D companion models the same physical illusion as the 3D utricle scene, but represents the sensory transduction stage itself as a population of directionally-tuned hair cells spread all the way around the macula, instead of a single 1-D shear axis. Real otolith afferents are known to be cosine-tuned (Fernandez & Goldberg): a hair cell whose bundle points along preferred direction φ fires in proportion to how much the specific-force vector projects onto φ, rectified (a bundle only depolarises for deflection toward its excitatory side):

f = a − g      (specific force, identical to the 3D model)
r(φ) = max(0, f·(cos φ, sin φ) / g₀)     for φ spread over the full circle
τ · dx/dt = f − x                        (same first-order viscous-gel lag)

The brain's estimate of "which way is down" is then reconstructed from this whole firing pattern by a population-vector decode — summing every active cell's preferred-direction vector, weighted by its firing rate:

P = Σ r(φᵢ)·(cos φᵢ, sin φᵢ)
θ_perceived = atan2(Pₓ, P_y)

Because gravity and linear acceleration enter f identically, the population's firing pattern — and therefore the decoded direction — cannot distinguish a tilted head from an accelerating one. This reproduces the exact same somatogravic illusion as the 3D model (a level head under steady 0.35 g reads back as ≈19° of false pitch), but the mechanism shown here is the actual sensory-coding scheme rather than a single shear number, and a rolling strip chart makes the lag between the true and perceived angle directly visible over time — something the 3D scene's static readouts don't show.

Tilt slider — kinematic head tilt, applied instantly.
Acceleration slider — forward linear acceleration in g, added into the specific-force vector.
Gel viscosity — sets the lag time constant τ of the sensed (lagged) population response versus the instantaneous (unlagged) target, shown as two separate arrows on the ring.
Aircraft Takeoff — ramps acceleration 0→0.35 g with tilt held at 0°, reproducing the classic in-flight illusion.

Sensed (lagged, decoded) Instantaneous target (unlagged) True vertical reference