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Re: Traveling waves or resonance?

Following up on this topic, and in spite of epiphenomenal
intervening "unsubscribe me" requests, here is:

A personal and biased review by Dick Lyon of
"Longitudinal pattern of basilar membrane vibration in the
sensitive cochlea" by Tianying Ren; PNAS 2002 99: 17101-17106;
published online before print as

Thanks to Martin Braun for pointing it out, I have read this lovely
paper from several years ago on a direct observation of traveling
waves on the basilar membrane using laser interferometry.  I'd like
to consider how the data relate to models of tuning and active wave
propagation in the cochlea, and to cochlear signal processing.

One of the features of the data that is strikingly nice, but perhaps
not obvious to the uninitiated observer, is the smoothly compressive
nonlinear growth of response with SPL.  See Figures 1A, 1C, and 2.
Figure 2 is most striking.  Subplots A through E are driven with 20
dB increments of power at a fixed frequency.  Although the input
levels are therefore in ratios of 10, the plots of BM velocity have
axis labels of 10, 50, 200, 1000, 3000, in ratios of 5, 4, 5, and 3,
suggesting that the system's "gain" is decreasing by about a factor
of 2 (6 dB) or more for each 20 dB increase in input level.  No
linear model can do anything like this, and to me this "automatic
gain control" aspect of cochlear mechanics has always been one of the
most important but under-rated signal-processing functions of the

What's harder to see, but can be sort of seen and extrapolated from
Fig. 1A, is that if you look basal to the place of maximum response
by a distance of 400 microns or so, the growth of response with input
level is much more nearly linear.  At the 2200 micron location, the
response to levels 40 and 90 dB SPL are about 40 dB apart (factor of
100, not quite the linear difference of 50 dB), whereas at the 2600
micron place they are not much over 20 dB apart.  This means that
most of the variable gain of wave propagation is localized to this
400 micron region leading up to the peak response.  The 400 microns
is about one wavelength (2 pi of phase shift) of the traveling wave.
The region of variable gain, or variable damping, is perhaps about
twice that long in total, from 1.5 wavelengths before the peak to 0.5
wavelength after, without sharp boundaries.

Of course, these observations are also the flip side of the
observation that the place of maximum response shifts basalward as
the SPL increases, as is most explicit in Fig. 2, where the place of
maximum velocity response moves from about 2650 microns at low level
to about 2450 at high level, or about a half wavelength.  At high
levels, the wave damps out sooner; the absolute response may be
higher everywhere, linearly higher near the base, and only very
little higher past the best place where it rapidly goes to zero.

According to Fig. 3A, the responses at 2200 microns (to the same 16
kHz tone) are about equal in healthy and postmortem cochleae, but at
2600 microns differ by nearly 40 dB.  Therefore it is clear that the
variable gain, the compressive response, is due to a living active
amplification process that the traveling wave undergoes.  For the
postmortem cochlea, the place of best response appears to be a little
basal of 2200, probably corresponding to a much higher level than 90

Ren comments in several places about the "sharp tuning", "sharp peak
in magnitude transfer functions" and "spatially restricted
vibration", and asks "Considering that the restricted longitudinal
extent of BM vibration is the spatial representation of the sharp
tuning, the question of how spatially restricted vibration occurs in
the sensitive cochlea seems to be as important as how the cochlea
achieves the sharp tuning."  To me this seems a little peculiar on
two counts: (1) those two equally important questions are really the
same question; and (2) the tuning is not really all that sharp, nor
the region of vibration all that restricted, relative to what is
often described as "sharp", and is entirely in line with what
conventional traveling wave models can do.  Where's the mystery?

Figure 4C shows the response as a function of frequency, or "transfer
function" for two places, 2300 and 2750 microns, at 60 dB SPL, where
the maximum responses are ellicited by 18 kHz and 13 kHz
respectively.  These frequencies are in the ratio 1.38, or almost 1/2
octave, so we can associate about 350 microns of distance with about
1/3 octave of CF change, or about a "critical band".  These curves
for places with CFs nearly 1/2 octave apart cross each other about 15
dB down from their peaks, and have tip-to-tail ratios (relative to a
low frequency tail taken about an octave below the CF) of only about
15 dB or so.  The width at 10 dB down from the peak is about 3 to 4
kHz, for a "Q10" of only about 4 to 5.  That's all good if Q10 of 5
is what is meant by "sharp"; but to me that's only "moderately sharp"
(not much sharper than a "critical band"), compared to the "very
sharp" we sometimes see discused when a "Q10" is measured from a
threshold tuning curve, giving values of 10, 20, and higher.  Higher
"Q10" numbers such as 10 or so might be measured from the same system
if it were measured with a threhold tuning curve, or iso-response
curve (e.g. iso-velocity), as opposed to the iso-power curves of Fig.
4C.  I wonder if Ren would be able to construct such curves from data
he has.

The sharpest aspect of the curves, which is likely what mediates
sharp frequency discrimination, is the rapid falloff of response
above CF.

Ren's statement that "In sensitive cochleae, the cochlear partition
vibration at a given location shows a maximum response to a stimulus
at the CF, falls off quickly at frequencies above or below the CF,
and forms a sharp peak in magnitude transfer functions" has been
interpreted by some as saying that the response is fairly "symmetric"
about CF.  But the data do not show that, as Fig. 4C makes most
clear.  There is definitely a long response "tail" toward low
frequencies, corresponding to vibrations basal of the best place, and
a sharp high-frequency cutoff, correponding to high damping of the
traveling wave beyond the best place.

Ren's data don't show any novel or surprising phenomena, but are a
beautiful confirmation, over almost an octave range of places, of
what has been observed with single-point measurements in the past.
Other interpretations are possible, but to me this paper very clearly
supports the traveling wave with variable damping, implementing an
automatic gain control function in the cochlear hydrodynamic system,
using active un-damping at low levels, and possibly active damping
at very high levels.