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Re: mechanical cochlear model

Dear Alain,

In my opinion it would be better to start a beginners' course
on cochlear mechanics with a treatment of surface waves
on the ocean. Their propagation velocity is much smaller
than the sound velocity in water. The water particles move on
closed (elliptical) trajectories in vertical planes. The liquid motion
is limited to a surface layer of thickness ~1/k, where k =
2pi/wavelength is the wave number. The wavelength of tsunamis
on a 3 km deep ocean is about 100 km, so in their case there 
is liquid motion down to the ocean floor.

In the case of the passive cochlear wave, the surface wave is
mass-loaded (mass = BM and cells attached to it), and the wave
is not gravity-driven, but spring-driven (springs = fibres of the BM),
so that a surface wave with liquid on both sides of the "surface"
(i.e., of the cochlear partition) is possible.

Etc. -- see the already mentioned Chapters 4 and 5 of the book
"The Cochlea", or also Chapter 2, "Cochlear Structure and Function",
by the late Graeme Yates, of the book "Hearing", B.C.J. Moore (Ed., 1995),
Academic Press, San Diego.

Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx .

----Ursprüngliche Nachricht----
Von: Alain.de.Cheveigne@xxxxxx
Datum: 06.03.2010 02:54
An: <AUDITORY@xxxxxxxxxxxxxxx>
Betreff: Re: mechanical cochlear model

Strangely I didn't see Jan's message on the list.  I agree with his 
comments.  The controversy seems partly due to people using the term 
"travelling wave" with different meanings.  The way I explain things 
to students is the following.  If this account is fundamentally 
flawed (as opposed to inaccurate) I'd be glad to know.  It goes like 

This is a simple explanation of cochlear selectivity.

We know that the cochlea is divided into two ducts separated by the 
basilar membrane,  surrounded by hard bone, and filled with fluid. 
Both ducts are closed at their base by flexible windows, the oval 
window for the upper duct (scala vestibuli), and the round window for 
the lower duct (scala tympani).  The ducts communicate at their apex 
via a hole, the helicotrema.

Our explanation assumes four ingredients.  First, the fluid within 
the cochlea is incompressible. Second, the bony walls that surround 
it are rigid.  Third, the fluid has mass.  Fourth, the basilar 
membrane is stiff at the base (where it is narrow) and compliant at 
the apex (where it is wide).

If the pressure in the outside air is slowly increased (relative to 
middle ear) the displacement of the ear drum, transmitted via the 
ossicles to the oval window, causes that window to flex inwards. 
Fluid flows up scala vestibuli, through the helicotrema, and down 
scala tympani, causing the round window to flex outwards. 
Acceleration being small, the effect of fluid inertia is negligible, 
and the pressure at all points in the ducts on both sides of the 
membrane is equal. The membrane does not move.

For faster vibrations, force is required to accelerate the column of 
fluid in the ducts, implying a pressure gradient along the duct and a 
pressure difference across the membrane.  Deformation of the membrane 
may then be an easier option than displacement of the fluid.  Whether 
deformation or displacement occurs depends on the balance between the 
inertia of  the  fluid to move on one hand, and the stiffness of the 
membrane to bend on the other.  The terms of this balance depends on 
frequency of vibration and place along the cochlea.

At the base of the cochlea stiffness is maximal, and so deformation 
requires a large trans-membrane pressure. This occurs only with the 
relatively large accelerations involved in vibration at high 
frequencies.  Hence the base of the cochlea responds only to high 
frequencies.   For lower frequencies the inertia/compliance balance 
occurs where the membrane is less stiff, ie apically.  Thus the apex 
responds to lower frequencies.

This explanation is a rough first-order explanation. A more complete 
explanation would involve detailed modeling of the mechanics of the 
basilar and tectorial membranes, of the fluid mechanics of the fluid 
in both ducts, and of the electrical and chemical phenomena that 
interact with cochlear mechanics at the outer hair cells.  In 
particular, we know that the moving parts of the inner ear are 
immersed in fluid and therefore resonances should be very much damped 
(the "underwater piano" metaphor). To compensate for damping and 
produce the exquisitely fine tuning and high sensitivity of the 
cochlea requires a form of "negative damping" by which energy is 
injected into the system to compensate for dissipation. This is 
ensured by active processes involving the outer hair cells, that 
interact with cochlear mechanics under control from the central 
nervous system.

Is this account reasonably correct?  For example if the balance of 
membrane compliance and fluid inertia did not play a role I'd 
consider it to be flawed.