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Re: Wasn't v. Helmholtz right? (the helicotrema question)

-----Original Message-----
From: AUDITORY Research in Auditory Perception
[mailto:AUDITORY@LISTS.MCGILL.CA]On Behalf Of Jont Allen
Sent: Tuesday, 20 June 2000 12:49
Subject: Re: Wasn't v. Helmholtz right?

Jont Allen wrote:

>I dont understand.
>The normal assumption for the helicotrema is that its impedance
>is typically treated as an inductor, which at low frequencies, has a
>near-zero impedance. Are you suggesting it must be lower than one would get
>a large hole (i.e., the usual model)?

Dear Jont and List:

How 'low', how 'near-zero', and how 'large'? The lack of any figures in the
question highlights the fact that no-one knows what the acoustic impedance
of the helicotrema is. Yet it is article of faith of the traveling wave
theory that it is appreciable at acoustic frequencies (say 100 Hz and
above). It has to be (TW theorists assume), because we get movement of the
partition, and that has to be a sign of differential pressure at work. But
is that necessarily true? Maybe, the resonance theory says, the OHCs are
responding to common-mode pressure and causing the partition to move (in a
way that looks like it has been caused by differential pressure). After all,
as Wever, Lawrence and Bekesy (1954) pointed out, the graded delay in a bank
of resonators looks the same as a traveling wave.

We know that the helicotrema has a cross-sectional area of about 0.4 sq mm,
joining two galleries that each have a cross-sectional area of about 1.2 sq
mm (Bekesy, p 435). What cut-off frequency this represents acoustically is
something that has been, as still is, a matter of guess work. For what it is
worth, my guess is that a short pipe having a cross-sectional area about a
quarter of the sections it joins (each having effectively zero impedance)
must be very close to zero too.

Keep in mind that the volume velocity in the cochlea due to in and out
movement of the stapes is extremely small near threshold. The displacements
of the stapes are of atomic dimensions (sub nm) and the pressures (after
gain by the middle-ear transformer) are less than 1 mPa. My view is that for
such tiny pressures and flows, even at mid-frequencies, virtually all the
pressure must be short-circuited by the comparatively vast cross-section of
the helicotrema. That is, rather than treat it as an inductor, an
appropriate circuit element would be a wire link (that is, a short-circuit).

Of course, calculating the impedance of the helicotrema is not simple
because the hydrodynamics of the situation is complex. It was Bekesy himself
who cautioned that hydrodynamics was "a field in which plausible reasoning
has quite commonly led to incorrect results" and so applying our usual
hydrodynamical equations, with wavelengths in metres, to a structure the
size of a pea is a risky undertaking. I do not have expertise in
hydrodynamics, so I lack a rigorous model, but I do have a conceptual
picture of what might be going on in a cochlea where the helicotrema
presents a short-circuit.

As outlined in an earlier post, I think that below about 60 dB SPL, the OHC
just respond to the in-phase acoustic pressure (common-mode or excess
pressure) in the cochlea, generating ripples from the OHC cavity. At higher
levels, the tectorial membrane experiences acoustic forces across it which
cause the TM to move to and fro, this time launching ripples from the
vestibular lip. Now, the forces need not be differential pressures across
the galleries (although I cannot rule this out); I think that the forces
derive from the differing acoustic impedance of the TM compared to the
cochlear fluids. That is, the situation is like a sheet of metal suspended
in a large pool of water which experiences forces from sound reflected off
it; it is the differences in compressional wave speed that underlie the
force generation. The forces in the cochlea derive from 'beaming' and
'shadowing' of the acoustical signal as it is radiated from the oval window
and encounters the ribbon of the TM. The exact result must depend on the
detailed spiral geometry of the cochlea, and in particular on the strange
properties of the gelatinous tectorial membrane with its surface tension,
embedded fibres, and peculiar structural arrangement.

A working analogy may be the sealed-box loudspeaker to which a reflex port
is added and filled with a drone cone. The loudspeaker is the OW and the
drone cone is the RW. If we suspend a small, narrow strip of pressure
sensors inside the loudspeaker enclosure (with plenty of space surrounding
the strip), the analogy is complete. When the speaker is driven, the
pressure sensors detect the oscillating pressure, and react appropriately
(they may even move if we design the sensors that way). As we raise the
volume of the speakers, a point is reached where beaming and shadowing of
sound inside the enclosure becomes appreciable, and pressure differences
appear across the strip, which will cause the sensors to move to and fro.

Although an informal account, I hope this description shows that it is
possible to have a cochlear mechanics similar to what we have now without
relying on the impedance of the helicotrema at all (at least at low SPLs).
More importantly, it presents a resonance picture, not a traveling wave one.

Is it the correct picture?

Strengthening the notion that it is common-mode pressure (and not
differential pressure) that is essential for hearing, we have the classic
(but still germane) observations of Hardesty (Am. J. Anat. 8 [1908], 109)
who found in some specimens that the BM was resting on bone, while the rest
of the partition was perfectly well formed (and presumably functioning).
Hardesty mounts strong arguments for why the TM (and not the BM) should be
considered the resonant element, and I recommend his paper to this list. His
anatomical sketches of the TM are superb and, of relevance to my ripple
model, he notes in more than one place that the TM possesses a remarkable
surface tension that "its quality of adhesiveness is phenomenal. It is so
subject to surface tension that, it matters not how clean the needle point
may be, to touch the membrane is to have it stick". He describes the TM as
"most inconceivably delicate and flexible... and the readiness with which it
bends when touched or even agitated is beyond description." Comparing the TM
to the BM, he notes that "Reissner's and the basilar membrane appear as
boards compared with a strip of thin silk."

I follow Hardesty and believe that the TM is the resonant element in the
cochlea. I would be glad to hear of any other list members who have any
experience with the TM's properties.

Andrew Bell.

> -----Original Message----- in which Andrew Bell wrote:
> Dear Ben:
> Thankyou for your interest and your perceptive question.
> Yes, you're right: BM responses can be observed right down to near zero.
> What I meant to convey by my statement was that whole-scale up and down
> movement of the cochlear partition (like that which is supposed to happen
> under the traveling wave theory) does not begin until about 60 dB SPL.
> this level, there is insufficient differential pressure developed across
> partition for it to be pushed up and down. However, there is still
> sufficient common-mode pressure, a factor usually neglected in
> models, for the OHCs to detect and respond. Another way of saying this is
> that the acoustic impedance of the helicotrema is much lower than usually
> assumed, so that this hole effectively short-circuits the pressure
> difference across the partition. The common-mode pressure is simply the
> pressure built up uniformly throughout the cochlea by the inward movement
> the oval window (like pushing in the loudspeaker cone on a sealed-box
> speaker enclosure).