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Re: mechanical cochlear model
Rhode's 1971 data showed about 3-4 cycles of phase accumulation to the best place, independent of amplitude. I believe that is strong evidence of a traveling wave that takes 3-4 cycles to arrive, and I believe from discussions directly with Bill in 2000-2003 that this is also how he interpreted his measurements.
Regarding the stiffness change not being sufficient to explain 3 orders of magnitude of frequency, I believe that argument is based on the assumption of constant membrane mass (what I called in my thesis, the constant-mass scaling assumption, which does indeed imply 6 orders of magnitude of change of stiffness from base to apex, and the damping decreases by 3 orders of magnitude). But the widening of the basilar membrane is more consistent with an increasing-mass assumption, in which the stiffness decreases, the mass increases, and the damping remains constant. Neither is probably exactly correct, but it shows that having all the change in the stiffness is not necessary, and the widening of the basilar membrane is consistent with both a mass increase and a stiffness decrease.
The original discussion began around a tutorial objective - how to explain the basic idea of cochlea operation to a beginner, with a mechanical model that could demonstrate the wave propagation in real time. My preferred way is to describe the 2D model (much as Alain eloquently did, although I usually use a picture), and show the following animated figure for the 2D model, which is sufficient to illustrate the main behaviors of the passive model:
from my thesis (page 19, Figure 2.8), which shows the long-wave, short-wave, and cutoff behaviours, all with smooth transitions from one to the other, showing both membrane displacement and fluid pressure in the 2D duct. And then to explain that the 2D passive model can explain the passive behavior at high sound levels - the active processes of the outer hair cells are necessary to explain the extra gain and sharpness found at low sound levels, as in Rhode's 1971 data. Of course, the fluid pressure figure could be flipped over and inverted to illustrate more explicitly the behavior of the upper and lower ducts.
Finally, I note that the animation above is really just a visualization of one of the curves from Steele and Taber, 1979, "Comparison of WKB and finite difference calculations for a two-dimensional cochlear model", JASA 65, 1001-1006. I replotted those curves here:
along with my mode-coupling correction that fixes up the incorrect behavior in the cut-off region - a second wave mode is needed to explain the change of slope and occasional notch which is observed after the best place. I note that the combination of the plausible 2D simulation model that can be animated, with the Liouville-Green (or WKB) analytic model is a powerful combination for both personal understanding and for tutorial explanation to beginners.
The mode-coupling solution that reconciles the simulation model and the LG (or WKB) model in all regions, is published in my 1992 thesis, available online:
and also was published in
L. Watts, "The mode-coupling Liouville-Green approximation for a two-dimensional cochlear model", Nov. 2000, JASA 108, pp. 2266-2271.
From: AUDITORY - Research in Auditory Perception [mailto:AUDITORY@xxxxxxxxxxxxxxx] On Behalf Of Matt Flax
Sent: Monday, March 08, 2010 2:16 PM
Subject: Re: [AUDITORY] mechanical cochlear model
The concept that the passive inner ear mechanism does not conduct energy
as a fast pressure wave (with long wavelength) may be true - however not
everyone agrees on this point. I believe that you are talking about the
coupling between the outer ear and the inner ear below - and thus the
passive action of the hearing system.
The concept that the active inner ear mechanism is an active travelling
wave seems a little impossible (from what people have discussed
previously - experimental evidence and theory) for any location other
then the base of the inner ear. I stand on record as the first for
claiming that the base is dominated by the active travelling wave and
the mid/apex is dominated by the active long pressure wave (my
compression wave cochlear-amplifier definition).
On Mon, 2010-03-08 at 07:52 -0800, Richard F. Lyon wrote:
> Alain, your "simple account" was "reasonably correct" whether you
> mentioned the traveling wave or not, so I agreed with you.
> I admit I don't see why you don't explicitly identify that account
> with traveling waves. Sinusoids in such a system are locally
> described as cos(ometa*t - k*x), with some varying amplitude, where
> the relations between omega and k locally obey the same kind of
> dispersion relation (more or less) as gravity waves on water.
> The real issue seems to be about where the energy is, and how it
> propagates and gets detected. The fast pressure wave is another
> physical mode that really exists, but its wavelengths are all very
> large compared to the cochlea, so the pressure due to this mode can
> be viewed as equal throughout the cochlea; pushing on the stapes
> immediately pushes out at the round window. But this mode doesn't
> carry much energy, as the middle ear leverage isn't nearly enough to
> couple efficiently to it; there's also no efficient way to get energy
> back out of this mode into a place and a form for which detectors are
> known; the fast compression wave therefore carries pressure, but not
> much energy. The traveling wave mode, on the other hand, involves
> much larger fluid displacements at the same pressures; it's a
> differential mode between the scalae, propagated by the spring of the
> BM instead of by fluid compression. And the energy converges on the
> BM as the wave slows down and the wavelength gets short, focusing the
> energy into a small layer with large displacements at the BM. There,
> the hair cells, which evolved to detect fluid motion via cilia
> bending, are well positioned to respond.
> A wave has three kinds of delays: phase, group, and wave-front. In
> typical models, the wave-front delay is essentially zero, and the
> response latency can be made to approach zero as the level gets very
> high. The group delay depends on the damping, including negative
> damping effects, and so varies a lot with level. The phase delay is
> in between, around a cycle and half, and pretty stable. Pretty much
> everything about it is as Lighthill described, in terms of energy
> flow, except that there's not much BM mass so it's not really
> significantly resonant, except for very high frequencies very near
> the base where the accelerations are very high. And except for some
> positive feedback from outer hair cells that modifies the dispersion
> relation, providing active gain instead of loss over some range of
> If it walks like a wave, and quacks like a wave, and transports
> energy like a wave, why not call it a wave?