Energy flow in the cochlea

Abstract

With moderate acoustic stimuli, measurements of basilar-membrane vibration (especially, those using a Mössbauer source attached to the membrane) demonstrate: 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;a substantial phase-lag in that response, and one which increases monotonically up to the characteristic frequency;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 wavenumberIn 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. 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-d...

Do Forward- and Backward-Traveling Waves Occur Within the Cochlea? Countering the Critique of Nobili et al.

“…Since before von Békésy first visualized the wavelike motion of the cochlear partition, the mechanical responses of the cochlea have been understood using concepts borrowed from the description of wave propagation in electrical transmission lines (e.g., Wegel and Lane 1924;Zwislocki-Mościcki 1948;Peterson and Bogert 1950;de Boer 1980;Lighthill 1981). In recent years, however, Nobili and colleagues have criticized wave-equation formulations of cochlear mechanics on the grounds that they fundamentally misrepresent the hydrodynamics of the cochlea (Nobili 2000;Nobili et al 1998Nobili et al , 2003a.…”

Panoramic Measurements of the Apex of the Cochlea

“…In contrast, when the wavelength is small compared with the depth of the scalae, a description in terms of "deep waves" is needed, a "short-wave model" must be invoked, and a onedimensional description is no longer valid. Lighthill (1981) provides the following criteria for deep waves, 2d/ Ͼ 1.5, and for shallow waves, 2d/Ͻ0.5, where d is the depth of the scalae. This depth is fairly constant in the apical turns of the cat cochlea covered by the present study: d Ϸ 0.5 mm (Wysocki, 2001).…”

Spontaneous Basilar-Membrane Oscillation (SBMO) and Coherent Reflection

“…We have almost always used a model of which the geometry is constant over the full length of the model. To incorporate Fshort_ waves is essential for physical-mathematical reasons (de Boer 1979;Lighthill 1981). Therefore, the model has been made three-dimensional so that it can accommodate Flong_ as well as Fshort_ waves -a threedimensional model (in which the BM is narrower than the width and the fluid displaced by the BM can thus move in three directions) is more realistic than a two-dimensional model.…”