Subject: Re: Traveling waves or resonance? From: "Richard F. Lyon" <DickLyon(at)ACM.ORG> Date: Tue, 19 Oct 2004 23:10:36 -0700Following 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 http://www.pnas.org/cgi/doi/10.1073/pnas.262663699 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 cochlea. 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 dB SPL. 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.