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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: 

"
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. 

Alain




Dear Jan and list:

Your intuitions are close to the mark, I think. The problem with the
traveling wave model is that the maths tends to obscure rather than
clarify what is going on.

The TW model describes what's happening in passive structures at high
sound pressure levels. But for active systems and low SPLs, it doesn't
seem to work so well. A live cochlea functions very differently to a dead
one.

I hope some of the references I provided help you find a believable
explanation of what is going on. The cochlea is tiny and hidden, and we
need logic and intuition - and the remarkable window provided by
otoacoustic emissions - to work out the truth of the matter.


Andrew.


Andrew Bell
Research School of Biology
Australian National University
Canberra, ACT 0200


On Thu, March 4, 2010 8:29 pm, Jan Schnupp wrote:

 Dear Andrew and List


 here you have been putting your finger on something that I have never
 properly understood (and which I secretly suspected the large majority of
 hearing researchers haven't really understood either. You off us two
 analogies: a piano being thumped, and a surface wave on a pond. To
 my mind it seems that the cochlea *must* be a lot closer to the first of
 those model than the second one: Firstly it seems to me that the cochlea,
 being completely filled with fluid and encased in a rigid shell, has no
 surface along which a wave could propagate, and you'd have a hard time
 making a propagating pond surface wave that has an amplitude envelope like
 that seen on the BM. The BM is really very unlike the surface of any pond
 I have ever seen. Secondly, pressure
 waves propagate very fast in fluid (1200 m/s or so), and the cochlea is
 rather small, so when the stapes "thumps" the cochlea, the resulting
 pressure gradients ought to be available to set things in motion almost
 instantaneously and simultaneously along the whole length of the BM. Or am
 I wrong about this? The only way I could ever
 understand the travelling wave is by realizing that simultaneously
 activated filters tuned to different frequencies will go out of phase to
 produce, as you say, something that looks like a travelling wave. Is that
 not good enough? I have great difficulty believing that the lion's share
 of mechanical energy travels along the cochlea by virtue of the a more
 basal part of the BM dragging up and then pushing down a more apical part,
 given how hard the fluids above and below the BM would oppose such a
 movement. I can more easily imagine the mechanical energy mostly
 propagating though the fluid column and dragging the BM along, so we end
 up with spring (BM segments) mass (fluid column) resonators. So to my
 mind, there is a lot more thumped piano in the cochlea than there is pond
 surface, but I don't claim to be an expert on cochlear mechanics, and if I
 have got it all wrong then I'd really like to know, ideally with an
 *accessible* reference, why I am wrong.
 I do have to give lectures on the cochlea and would hate to spread
 misunderstanding, but I don't think I can get up in front of my class and
 tell them "imagine the travelling wave like the ripples on a pond". If the
 students are like me (always a dangerous assumption, I know) then they'll
 find that analogy more confusing than enlightening.

 Many thanks in advance for any insights, and best regards,


 Jan


...


 Dr Jan Schnupp
 University of Oxford
 Dept. of Physiology, Anatomy and Genetics
 Sherrington Building - Parks Road
 Oxford OX1 3PT - UK
 +44-1865-272513
 www.oxfordhearing.com