Dual Contribution to Amplification in the Mammalian Inner Ear

The inner ear achieves a wide dynamic range of responsiveness by mechanically amplifying weak sounds. The enormous mechanical gain reported for the mammalian cochlea, which exceeds a factor of 4000, poses a challenge for theory. Here we show how such a large gain can result from an interaction between amplification by low-gain hair bundles and a pressure wave: hair bundles can amplify both their displacement per locally applied pressure and the pressure wave itself. A recently proposed ratchet mechanism, in which hair-bundle forces do not feed back on the pressure wave, delineates the two effects. Our analytical calculations with a WKB approximation agree with numerical solutions.

Figure 1

The mammalian cochlea. Sound displaces the stapes (top left), producing a pressure difference $p_s$ across the basilar membrane (BM) that elicits a traveling wave of pressure difference and membrane displacement.

Figure 2

Cochlear pressure and global sensitivity of the basilar membrane from numerical solution of Eq. (1) (lines) and from the WKB approximation (circles) relative to $p_s$ for a frequency $f=8\mathrm{kHz}$. (a) Pressure (red or gray, active; black, passive). (b) Global sensitivity (red or gray, active; black, passive). The gain in global sensitivity exceeds the gain in pressure by the factor $|{\lambda }^{\mathrm{act}}/{\lambda }^{\mathrm{pass}}|=90$. The results in this figure and the following have been obtained using the two-mass model for the organ of Corti and parameters from Ref. 21; $\rho ={10}^3\mathrm{kg}·{\mathrm{m}}^{-3}$.

Figure 3

Nonlinearities at the 10 kHz resonant position. (a) Local sensitivity of hair-bundle displacement. Numerical solutions with the open probability given by Eq. (9) (red or light gray) agree with the approximation of Eq. (10) (blue or dark gray). (b) Global sensitivity for different sound frequencies. The response becomes linear as the frequency deviates from the resonant frequency.

Figure 4

Pressure and sensitivity for amplification through the ratchet mechanism; $f=200\mathrm{Hz}$. The data result from numerical solution of Eq. (1) (lines) and from the WKB approximation (circles). (a) Pressure (red or light gray, active; black, passive). (b) Gobal sensitivity of the hair bundles (blue or dark gray, active; green or gray, passive) and of the basilar membrane (red or light gray, active; black, passive).