Re: Physical model of cochlear frequency discrimination

From: zigoteau (zigoteau_at_ausi.com)
Date: 07/17/04 

Date: 17 Jul 2004 09:32:33 -0700

zigoteau@ausi.com (zigoteau) wrote in message news:<9da9cba1.0407150849.2fd4b5ab@posting.google.com>...

Hi all again,

First: apologies, Tony - I've just realised that your surname is
Jefferies not Jeffs.

Many Google searches later, I'm getting up to speed with the cochlea.
I have found the following links useful:

http://www.geocities.com/medinotes/auditory_system.htm
http://www.iurc.montp.inserm.fr/cric/audition/english/corti/corti.htm
http://ctl.augie.edu/perry/ear/ear.htm

Rave continued:

I have realised from those websites that the stirrups and oval window
do not couple directly to the basilar membrane, but to the fluid on
one side. There are three compartments, separated by the two membranes
(basilar and Reissner's). The outer two compartments connect via the
apex.

As an electronics man, it looks to me a bit like a ribbon cable: the
hydraulic structure is balanced. It can be modelled electrically by a
ladder network. Due to its symmetry, it will have two modes: common
mode and differential mode. It's not clear whether there is some
clever mechanism ensuring that the common mode is not excited, but in
any case the organ of Corti will only be stimulated in response to the
differential mode component.

Tianying Ren seems to be wrong on something fairly basic. He shows the
distance from the base of the point of maximum basilar membrane
response as increasing with frequency (Figure 4 in his 2002 PNAS
paper: PDF file available on request). All web pages I have found
treat "base" as synonymous with "stirrup" and "oval window" - at the
big end of the cochlea - and agree with one another that the peak
response is closest to the stirrup at high frequencies, furthest from
the stirrup at low frequencies. The little end furthest from the
stirrup is the "apex". AFAICS, Ren's figure only makes sense if all
occurrences of the word "base" are replaced by "apex". Could someone
comment?

His plots of phase shift versus position for different frequencies
then make sense. The wave is propagating in from the big
end/stirrups/oval window. The wavevector decreases progressively,
reaching zero near the point of maximum response. (Here, I am not
making any judgement about the detailed lateral distribution of
pressure and velocity of the various membranes and in the fluid
compartments).

There is therefore a big difference with respect to my suggestion of
an electron spectrometer. In the case of electrons and microwaves in a
waveguide, high frequencies (energies) propagate, and low energies are
evanescent. In the ear in contrast, low frequencies "propagate", while
high frequencies are evanescent.

>From the Tianying Ren Figure 4, giving phase shift versus position at
two frequencies, it is possible to estimate group velocities, which
are of the order of 1 m.s^-1. The velocity of sound in water, 1.540
km/s, is so much higher than the group velocity that it is effectively
infinite. The velocity of sound is determined by the interplay of
density and compressibility. The common mode will have high velocity,
but the differential mode can have a much lower velocity as a result
of the elastic deformation of the basilar and Reissner's membranes.
For this mode the water is essentially incompressible, and just
provides inertia.

This suggests an equivalent ladder circuit in which the water is
represented by inductances along the two sides and capacitors in the
rungs. With such strong coupling to the fluid, it is hard to see how
the BM can show any resonant response by itself. However on the face
of it, this equivalent circuit would not give the required evanescent
behaviour at high frequencies

Franz, if by any chance you're reading this, do you happen to have, or
could you get access to, Lighthill's papers as PDF files you could
send me? It is starting to look as if the fluid is an essential part
of the mechanism, providing the inertial component. Maybe he has
already solved the problem (but in that case why didn't Ren cite
him?).

Cheers,

Zigoteau.

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