4.5 The cochlear amplifier

The most surprising fact about the mammalian cochlea is that it is not a passive instrument. The cells called outer hair cells, which we located in 4.1 sitting on the basilar membrane in three columns, do something no other cell type in the body does: they change length in response to voltage, on a cycle-by-cycle basis, at audio frequencies. They can shorten and lengthen at 20 kHz. This piezoelectric-like behavior is called electromotility, and it is the engine of the cochlear amplifier.

The mechanism, which was the subject of one of the most dramatic discoveries in 20th-century auditory biology, runs through a single protein. Prestin (named after presto, “fast”) is a membrane protein densely packed into the lateral walls of outer hair cells — roughly $10^7$ copies per cell, occupying a substantial fraction of the cell’s plasma membrane. Prestin’s job is mechanical. When the membrane potential of the OHC depolarizes, the protein’s conformational state shifts, the cell shortens. When it hyperpolarizes, prestin shifts the other way and the cell lengthens. The motion is so fast that it can match the highest audio frequencies the cochlea responds to.

What does this length-change do, mechanically? An OHC sits between the basilar membrane below and the reticular lamina (the apical surface of the organ of Corti) above. When it shortens or lengthens, it pulls the basilar membrane up or pushes it down — relative to the rest of the structure. In the right phase, this adds energy to the traveling wave on each cycle, compensating for the viscous damping that would otherwise dissipate it. In the wrong phase, it would extract energy. The OHC’s mechanical feedback is in the right phase by design (the geometry of the organ of Corti enforces it, broadly speaking), and the net effect is that the traveling wave’s peak grows larger and sharper than the passive physics from 4.3 allows.

Effective damping reduction

The cleanest way to think about this, mathematically, is as a reduction in the effective damping coefficient in the SHM equation from 4.2. Write the basilar-membrane equation with an active-feedback term that depends on the local velocity:

m(x)\, \ddot{\eta} + \bigl[b(x) - \beta(x, \dot{\eta})\bigr]\, \dot{\eta} + k(x)\, \eta = P(x, t),

where $\beta$ is the strength of the active feedback. At small amplitudes, $\beta$ is roughly constant, and the effective damping is $b_\text{eff} = b - \beta$ .

▶ Derivation: why the active feedback acts like negative damping

The viscous damping term in the SHM equation comes from forces that oppose motion — they are proportional to velocity, with a sign that resists. The outer hair cell, in contrast, generates a force that is in phase with the basilar-membrane velocity (or nearly so, given the geometry of how its electromotile contractions couple to membrane motion).

Take the SHM equation with both viscous damping $-b\dot\eta$ and an OHC-generated force $+F_\text{OHC}$ . If $F_\text{OHC}$ is proportional to velocity with a positive coefficient $\beta$ (in the same direction as motion):

F_\text{OHC} \approx \beta\, \dot\eta.

Then the total velocity-proportional force is $-b\dot\eta + \beta\dot\eta = -(b - \beta)\dot\eta$ . The equation becomes

m\,\ddot\eta + (b - \beta)\,\dot\eta + k\,\eta = P.

The effective damping coefficient is $b_\text{eff} = b - \beta$ . If $\beta$ is small, the effective damping is slightly reduced and $Q$ is slightly raised. If $\beta \to b$ , the effective damping vanishes and $Q \to \infty$ — the system becomes a perfect resonator. If $\beta > b$ , the effective damping is negative and the system spontaneously oscillates (the amplifier becomes an oscillator). Real cochleas are tuned to operate just shy of this instability — a regime called operation near the Hopf bifurcation in dynamical-systems language.

If $\beta$ approaches $b$ , the effective damping approaches zero, and the effective $Q$ approaches infinity — the system approaches the brink of spontaneous oscillation. Physicists recognize this as the Hopf bifurcation, the critical point at which a damped oscillator becomes a self-sustained one. The mammalian cochlea is believed to operate near this bifurcation, with the active feedback tuned just shy of triggering it.

At low input levels, the feedback wins by a hair, the effective $Q$ is enormous (50, 100, sometimes higher), and the cochlea is exquisitely sharp. At high levels, the saturation built into the OHC’s response shrinks $\beta$ , the effective damping recovers, and the system behaves more like the passive case. This is why hearing is most sharply tuned for quiet sounds and less tuned for loud ones.

The interactive below contrasts the passive and active tuning curves of a single basilar-membrane place. The dashed gray curve is the passive response (the same $Q \approx 4$ resonance from 4.2). The solid black curve is the active response, with a one-knob “active gain” parameter that can be slid from zero (passive) toward the bifurcation (where $Q$ goes to infinity). A second slider sets the input level — at high levels, the active gain saturates and the curve relaxes back toward passive. Watch what happens at the peak as you push the active gain up: the curve grows taller and narrower. Now drag the level slider up: the active sharpening disappears at high levels, exactly as in a real cochlea.

OHC active gain

0.85

input level (dB SPL)

40 dB

saturation factor: 0.98
effective gain: 0.83
peak amplification: 6.0×

Two consequences

Two consequences of this active process matter for what comes next.

First, the cochlear amplifier consumes energy. The outer hair cells need ATP (delivered by the stria vascularis, the metabolically active strip on the lateral wall of scala media that maintains the endocochlear potential) to do their work. Hearing is, biologically, an active sense. This is why severe hearing loss often has a metabolic component, and why cochlear hair cells are vulnerable to anything that compromises metabolism — ototoxic drugs, anoxia, aging.

Second, the active process is the reason the cochlea emits sound. A cochlea operating near the Hopf bifurcation, with active gain not perfectly balanced by damping, will occasionally produce its own faint vibrations outward — through the middle ear, out the ear canal, into the air, where they can be recorded with a sensitive microphone. These are called otoacoustic emissions, and they are routinely used in newborn hearing screening: shine a sound at the ear, listen for an echo that the cochlea adds, and you have non-invasive evidence that the outer hair cells are working. A cochlea that emits is a cochlea that is sharply tuned. A cochlea that does not emit is one whose amplifier is broken.

The OHCs sharpen the traveling wave; they do not transduce it. The cells that turn the sharpened mechanical signal into a neural one are the inner hair cells, sitting in their single column on the modiolar side of the organ of Corti. That is 4.6.