Finding the impedance of the organ of Corti

Measurements of the nonlinear response of the basilar membrane to a pure tone are shown to have a simple form for moderate membrane velocities:    where the response $V$ is the velocity of the membrane at measurement position x, $Vu$ is the umbo velocity, f is the frequency of the stimulus, and $fc(x)$ is the local characteristic frequency. The frequency dependence of the functions $ν(x,f)$ and $Ṽ(x,f)$ is determined from the data, and $ν(x,f)$ and $ln Ṽ(x,f)$ are shown to be analytic functions in the lower half of the complex frequency plane, with $Re{ν(x,f)}$ a monotonically increasing function of f at fixed x. The linear limit of basilar membrane motion is characterized by a transfer function $T(x,f)=(Ṽ/V1)ν/(1−ν),$ estimated by extrapolating $V(x,f;Vu)/Vu$ to a small membrane velocity $V1.$ $T(x,f)$ and $ln T(x,f)$ are shown to be analytic functions in the lower half of the complex frequency plane. The inverse of the amplitude of the transfer function, which has both a deep dip at $f≈fc(x)$ and a broad shoulder at lower frequencies, bears a striking resemblance to the neural threshold tuning curve. The functional form of $T(x,f)$ is used to deduce the equation governing the motion of a section of the organ of Corti. Each section acts like a negatively damped harmonic oscillator stabilized at time t by a feedback force proportional to the velocity at the previous time $t−τ.$ The time delay τ is proportional to the oscillator period $[τ=1.75/fc(x)].$ Like a laser, the organ of Corti pumps energy into harmonic traveling waves. Unlike the laser, the direction of energy flow abruptly reverses as the traveling wave approaches the point of maximum membrane velocity $[fc(x)≈f].$ All accumulated wave energy is then pumped back into a small section of the organ of Corti where transduction presumably occurs. Outer hair cells are conjectured to be active elements contributing to the negative damping and feedback of the cochlear amplifier.