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Re: mechanical cochlear model
Dear Peter, Matt and the list,
Thank you for your responses.
Peter, I like your phrase "all section together 'agree' on a pattern of motion" because the fluid coupling of the sections does not allow for individual motion. This leads to some complications in time domain models due to the 'strong fluild-structure interaction' that your thesis addressed.
However, I would argue in the reverse direction to your last post, as follows:
If the resonators were (sufficiently) independent the situation is like a filterbank, ie a Helmholtz cochlea. Douglas Creelman's post gives a mechanical example of this, where vertical strings with a weight are suspended from a single bar. Although as he says in this case a 'traveling wave' is generated it is a function of the arrangement of the weights, and not a property of the weights interacting. (I.e. we can rearrange the weights and completely destroy the traveling wave.)
The situation described above is identical to a cochlear filled with a hypothetical fluid that exerts a similar force on a section independent of its spatial location.
I would then argue that the force in the real cochlear fluid, or even the incompressible inviscid case (which seems to be widely accepted by both sides of the traveling wave - resonance discussion for general conditions), does vary with spatial location. This spatial variation is much less than in the force that sets up a traveling wave on a guitar string for example. This seems to imply that some of the conditions for the 'guitar string ' style of traveling wave are also present.
Arranging the argument in this manner avoids assuming a traveling wave up front.
Based on the Green's function, it seems to me that the conditions in the cochlea do not lend themselves to either the 'filterbank' or 'guitar string' scenario only, but the behavior must be a combination of the two. Which is dominant would depend on the spatial variation of the force, a small variation favors the 'filterbank' and a large variation favors the 'guitar string'.
So far I am talking about the behavior of the membrane. In your answers (in particular to Martin) you comment "fluid motion implies a traveling wave". I completely agree with this statement in so far as we are talking about a small region of the fluid. Equations of the form v_x(x,y) = A(sin(kx-wt)sin(ky-wt)), v_x(x,y) = B(cos(kx-wt)cos(ky-wt)) and variants are a solution to the Laplace equation (in 2d, w = \omega for compactness). So certainly the Laplace equation requires that the local solution in a region has the form of a traveling wave. However, even in the passive cochlea the wave number k is not constant. Surely some active mechanism could be proposed such that the wave number varies so radically that the fluid motion cannot be described as a traveling wave across any reasonable length?
Matt, thanks for your comments. So far I have only tested my models with inputs of the form u(t)*sin(wt) which contains higher harmonics. In my results I do see the high frequencies responding to the input first and what looks like a traveling wave in a passive cochlea. Whether this is an artifact of the higher harmonics, or a true traveling wave and alternatively whether it is more consistent with a bank of resonators taking a few cycles to reach their peak amplitude I cannot say without looking at the data more closely. For the model I am using the WKB method predicts a traveling wave in steady state so the only thing to look for is whether there is evidence from the transient behavior to discuss. Personally, I am trying to suspend interpretation on this question until I have an opportunity to run some of these tests, but I believe that the behavior is a combination of the two, which is possibly similar to Peter's view?).
UNIVERSITY OF CAPE TOWN
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