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Cochlear Maps
Dear Dr. Reinhart Frosch and List,
I have not been following the various emails initiated by, and related
to yours, (at least not attentively). But one's own name attracts
one's attention, and hence I did attend to the one below.
In whatever way my 1961,1990 empirical-descriptive curve is now to be
categorized, the frequency-place data summarized in my 1990 paper
cannot be so neatly divided into (a) data from dead humans and (b) in
vivo data obtained at only high (> 100 dB SPL) levels in the various
other species considered.
The "passive" map displays the place x where a sine-tone of
given frequency f causes the strongest mammalian basilar-
membrane vibration in post-mortem experiments, or also in
in-vivo experiments with loud sine-tones (>100 dB SPL).
The map due to D.D. Greenwood (1990), which for human
cochleae follows an exponential function up to about
x = 28 mm, belongs to this "passive" category.
Of course, Bekesy's data are from dead cochleas, and were also obtained
at very high signal levels, so that "passive" or "passive-mode" data
may be an appropriate way to describe them. Not so for most of the in
vivo data, mis-characterized above as having been obtained at very hign
signal levels. Much of the in vivo frequency-place data in other
species summarized in 1990 were not obtained using tones greater than
100 dB, quite notably not in the case of Liberman's (1982) very full
set of data (or Kohlloffel's basal data - 1974,75) in the cat, where
single unit CF determinations (i.e. near-threshold measurements) were
related to fiber place of origin along the cochlea. In addition,
Muller's Mongolian gerbil data [(1996) in Hear Res, 94, 148-196] were
obtained in the same way as Liberman's. CFs determined at tuning curve
tips uniformly measured at signal levels over 100 dB SPL?
In the guinea pig probably the mechanical data (i.e. Wilson and
Johnstone's) were obtained at fairly high signal levels (and from
cochleas in an altered condition), so call them "passive", but not, I
think, Dallos' data (using CM). Moreover, as for the other GP data, we
are back to CFs: Russell and Sellick (inner haircell CFs), or
Robertson and Manley (spiral ganglion cell CFs).
Data from the chinchilla (Eldredge, CID data) and data of Stebbins and
Moody (1979- 88), relating to a species of old-world monkey, related
place of cochlear lesion to cut-off frequencies (a threshold curve
"shoulder" related to place of loss) and hence were obtained
differently than in the experiments above, but not at particularly high
levels.
It needs to be recognized by list members not working on cochlear
issues that most of the in vivo data collated in the 1990 paper cannot
be characterized correctly as obtained only, or even largely, at very
high signal levels, but rather the reverse. Most actually would fall
in the "Active-mode" data category.
Does that raise a problem? Curves of identical form appear to describe
data obtained at much lesser signal intensity levels than 100 dB SPL.
The curve is still empirical: one parameter (exponential coefficient)
can remain identical, a second nearly so, with only the third varying
to accommodate different frequency ranges. But is the curve now
"active", having been "passive" in respect to dead human data (and the
one "passive" set of GP data)?.
The same curve form would seem oddly categorized if called both
"passive" (when applied to dead humans and Keele guinea pigs) and
"active" (when applied to EPL cats and Sussex and Evanston guinea
pigs), i.e. depending just on the data or conditions applied to.
Perhaps, better apply the terms at this point only to data sets
themselves, depending on stimulus level used or cochlear physical
condition. But, it also seems that some data sets obtained at high
signal levels in the Keele guinea pigs appeared to agree with other
data sets obtained at low signal levels in other guinea pigs - an
oddness noted in a 1990 footnote. If that agreement were valid, what
do the terms "passive" and "active" mean in that context? This is not
to deny importance to efforts to use such terms and address these
issues; it is only perhaps to illustrate that a theoretician's lot may
not be a happy one if the data cupboard is opened. Some data may be
hard to reconcile with others - or with theoretical positions.
I hope this "clarification" of the in vivo data reviewed in 1990 will
be helpful - in trying to redirect attention to the methods used and
actual conditions under which each type of such data (in the field, not
just in that paper) have been obtained, however discouraging that may
be on occasion.
Cheers,
Donald D. Greenwood
On Sep 24, 2006, at 8:38 AM, reinifrosch@xxxxxxxxxx wrote:
Dear List,
A month ago I posted a formula for the dependence of the
stiffness S [spring constant per square mm] of the basilar
membrane on the distance x from the stapes:
S(x) = S(0) * [1 - x / (4d)]^4 . (1)
That formula can be rewritten as follows:
S(x) = S(0) * [1 - x / L]^4 . (2)
The length L in Eq. (2) is slightly greater than the length of
the basilar membrane. In humans that length is about 35 mm,
and L is proposed to be about 36 mm. S(0) is 1*10^9 N/m^3.
Eq. (2) is then found to be fairly close to the "resonance" map
of the human cochlea.
There are three cochlear frequency-versus-place maps which
differ distinctly from each other, namely the "passive", the
"active", and the "resonance" map.
The "passive" map displays the place x where a sine-tone of
given frequency f causes the strongest mammalian basilar-
membrane vibration in post-mortem experiments, or also in
in-vivo experiments with loud sine-tones (>100 dB SPL).
The map due to D.D. Greenwood (1990), which for human
cochleae follows an exponential function up to about
x = 28 mm, belongs to this "passive" category.
The "active" map shows the place x of strongest basilar-
membrane excitation in in-vivo experiments on healthy
cochleae with soft sine-tones.
The "resonance" map is defined by the following equation:
f_res = [1 / (2 pi)] * [S(x) / M]^(1/2) . (3)
In Eq. (3), M is the mass per square mm of the basilar
membrane and the attached cells; in humans, M is about
0.1 mg / mm^2.
With the help, e.g., of de Boer's chapter in the book
"The Cochlea" (Springer, 1996), it can be conjectured
that the theoretical resonance map based on Eqs. (2) and (3)
above may well be closer to observations than that based on
Eq. (3) and the exponential stiffness formula used by de Boer
(1996, with a parameter alfa of 3 cm^-1).
With many thanks for your patience,
Reinhart Frosch.
P.S. 1: The idea that "my" formula may work for the whole
cochlea came from a posting by Dick Lyon of August 25.
Unfortunately, I sent a fairly thoughtless reply on the same
day. Now I agree with all of Dick's comments.
P.S. 2: I still think that the new stiffness formula represents
an evolutionary advantage. For a given stapes velocity at low
frequency (50 to 300 Hz in the case studied), the liquid-
pressure wave after two thirds of the cochlear channel is more
intense if these two thirds obey Eq. (2) than if they obey a
suitably comparable exponential stiffness formula.
The difference, however, is not due to the fact that Eq. (2)
leads to accurate WKB approximations. The important feature
is the small relative decrease, per mm, of S(x) at small x.
I shall send details in a few days.
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 .