Journal of Fluid Mechanics



Energy flow in the cochlea


James  Lighthill a1
a1 University College London

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Abstract

With moderate acoustic stimuli, measurements of basilar-membrane vibration (especially, those using a Mössbauer source attached to the membrane) demonstrate:

  1. a high degree of asymmetry, in that the response to a pure tone falls extremely sharply above the characteristic frequency, although much more gradually below it;
  2. a substantial phase-lag in that response, and one which increases monotonically up to the characteristic frequency;
  3. a response to a ‘click’ in the form of a delayed ‘ringing’ oscillation at the characteristic frequency, which persists for around 20 cycles.
This paper uses energy-flow considerations to identify which features in a mathematical model of cochlear mechanics are necessary if it is to reproduce these experimental findings.

The response (iii) demands a travelling-wave model which incorporates an only lightly damped resonance. Admittedly, waveguide systems including resonance are described in classical applied physics. However, a classical waveguide resonance reflects a travelling wave, thus converting it into a standing wave devoid of the substantial phase-lag (ii); and produces a low-frequency cutoff instead of the high-frequency cutoff (i).

By contrast, another general type of travelling-wave system with resonance has become known more recently; initially, in a quite different context (physics of the atmosphere). This is described as critical-layer resonance, or else (because the resonance absorbs energy) critical-layer absorption. It yields a high-frequency cutoff; but, above all, it is characterized by the properties of the energy flow velocity. This falls to zero very steeply as the point of resonance is approached; so that wave energy flow is retarded drastically, giving any light damping which is present an unlimited time in which to dissipate that energy.

Existing mathematical models of cochlear mechanics, whether using one-, two- or three-dimensional representations of cochlear geometry, are analysed from this standpoint. All are found to have been successful (if only light damping is incorporated, as (iii) requires) when and only when they incorporate critical-layer absorption. This resolves the paradox of why certain grossly unrealistic one-dimensional models can give a good prediction of cochlear response; it is because they incorporate the one essential feature of critical-layer absorption.

At any point in a physical system, the high-frequency limit of energy flow velocity is the slope of the graph of frequency against wavenumber 1 at that point. In the cochlea, this is a good approximation at frequencies above about 1 kHz; and, even at much lower frequencies, remains good for wavenumbers above about 0·2 mm−1 (which excludes only a relatively unimportant region near the base).

Frequency of vibration at any point can vary with wavenumber either because stiffness or inertia varies with wavenumber. However, we find that models incorporating a wavenumber-dependent membrane stiffness must be abandoned because they fail to give critical-layer absorption; this is why their predictions (when realistically light damping is used) have been unsuccessful. Similarly, models neglecting the inertia of the cochlear partition must be rejected.

One-dimensional modelling becomes physically unrealistic for wavenumbers above about 0.7 mni-1, and the error increases with wavenumber. The main trouble is that a one-dimensional theory makes the effective inertia ‘flatten out’ to its limiting value (inertia of the cochlear partition alone) too rapidly as wavenumber increases. Fortunately, a two-dimensional, or even a three-dimensional, model can readily be used to calculate a more realistic, and significantly more gradual, ‘flattening out’ of this inertia. All of the models give a fair representation of the experimental data, because they all predict critical-layer absorption. However, the more realistic two- or three-dimensional models must be preferred. These retard the wave energy flow still more, thus facilitating its absorption by even a very modest level of damping. The paper indicates many other features of these models.

The analysis described above is preceded by a discussion of waves generated a t the oval window. They necessarily include:

  1. the already-mentioned travelling wave, or ‘slow wave’, in which the speed of energy flow falls from around 100 m s−l at the base to zero at the position of resonance;
  2. a pure sound wave, or ‘fast wave’, travelling a t 1400 m s-1, with reflection at the apex which makes i t into a standing wave. Half of the rate of working of the stapes footplate against the oval window is communicated as an energy flow a t this much higher speed down the scala vestibuli, across the cochlear partition and back up the scala tympani to the round window, whence it becomes part of the slow general apical progress of the travelling wave; a progress which, as described above, comes to a halt altogether just in front of the position of resonance.

Mathematical detail is avoided in the discussion of cochlear energy flow in the main part of the paper (§ 1–10), but a variety of relevant mathematical analysis is given in appendices A–E. These include, also, new comments about the functions of the tunnel of Corti (appendix A) and the helicotrema (appendix C).

(Published Online April 20 2006)



Footnotes

1 In any travelling wave, the wavenumber is the rate of change of phase with distance; for example, it is 2π/λ in a sine wave of length λ.



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