The cochlear Hopf amplifier: parameter-free cube-root compression
The cochlear amplifier is an active oscillator poised at a Hopf bifurcation. In normal form dz/dt=(µ+iω₀)z−β|z|²z+Fe^{iω₀t}, the critical point µ=0 forces response R=(F/β)^{1/3} — cube-root compression with exponent exactly 0.3333, parameter-free. Small-signal gain rises from 1 far below to ~464 at criticality. Prestin (SLC26A5) supplies the active force.
Sitting an outer-hair-cell oscillator exactly at the Hopf bifurcation (µ=0) yields a compressive nonlinearity with exponent 1/3 that contains no fitted parameter — it follows from the normal form alone. The simulated small-signal gain rises monotonically toward criticality (1 → 10 → 99 → 464 as µ→0), and the 1/3 exponent matches the cited ~0.3–0.5 basilar-membrane compression [L]. The exponent is [V]; how close a real cell sits to µ=0 is an empirical question [H], not tuned.
A passive cochlea would be too insensitive and too broadly tuned to explain human hearing. The resolution is an active amplifier in each outer hair cell, modelled as an oscillator held right at the edge of spontaneous oscillation — a Hopf bifurcation. Operating at that edge buys the largest possible gain for faint sounds together with sharp frequency tuning.
Why is the cochlear compression parameter-free?
Writing the amplifier in Hopf normal form, dz/dt=(µ+iω₀)z−β|z|²z+Fe^{iω₀t}, and setting the control parameter µ=0 gives a forced response R=(F/β)^{1/3}. The compression exponent is exactly 0.3333 and does not depend on β, which is merely a unit — so the cube-root is a structural prediction of sitting at the bifurcation, with nothing fitted.
| control parameter µ | small-signal gain |
|---|---|
| −1 | 1 |
| −0.1 | 10 |
| −0.01 | 99 |
| 0 (critical) | 464 |
As µ approaches zero the small-signal gain climbs from 1 to about 464. This is why the cochlea operates near criticality: maximal gain for faint sounds plus sharp tuning, with the cube-root compression then handling the enormous dynamic range of audible intensities — from a whisper to a jet engine on one set of cells.
Prestin is the active force
The somatic motor protein prestin (SLC26A5) drives the outer-hair-cell length changes that supply energy to the oscillator — the physical realisation of the forcing term. Loss of prestin or of the outer hair cells removes the active force.