Dear Peter and the list,
Thank you for your discussion of the basic fluid mechanics, I want to comment and ask a follow up question on this aspect of it, but first let me comment on your concluding remarks re compression waves. My understanding is that people advocating the compression wave as the stimulus accept the need for compressible elements in the outer hair cell and have put forward arguments for their existence (see Bell (2008) for one example I have recently read).
As you say the traveling wave comes out of the basic fluid dynamics equations and passive basilar membrane boundary conditions. The WKB approximation generally provides better insight into the form of the solution than numerical techniques such as integration of Greens functions and predicts a traveling wave in the passive cochlea. However, on one point at least I think that looking at the Greens function can give more insight: it allows us to discuss the effect of motion at one point on the basilar membrane on the pressure (force) at another. Nobili and Mammano (1993) FIG. A3 is a convenient figure to discuss this from. It clearly shows that the motion of a single point on the basilar membrane causes a similar pressure over a wide range of locations, but points in the near vicinity with an exaggerate pressure (due to the 'singular part').
If there were a fluid that did not have the singular part of the Greens function and the pressure exerted was constant spatially then each radial segment of the basilar membrane / organ of Corti complex would absorb or radiate real and reactive power to distant segments as easily as nearby. This suggests that the segments could be rearranged arbitrarily without affecting the behavior of the individual segments or the collection. In this thought experiment any traveling wave that exists seems to be merely a coincidence of the power requirements of each individual segment at a given instance.
However the slight spatial roll off of the Greens function combined with the effect of the singular part causes displacements of nearby segments to have a larger effect on each other than on distant segments. This ensures more effective power transfer between nearby segments and seems to be essential to a traveling wave that carries energy. Each segment still only acts by absorbing or radiating real and reactive power from each other via the fluid, but now there is a preference for power in the vicinity.
As to my question, surely the initial response of the basilar membrane to a pure tone is an important thing to consider on this topic? I realize that getting reliable data for a steady state pure tone is already challenging, but I think the cochlear response to the onset of a band limited signal would be very interesting. The prevalent view of the behavior of the traveling wave seems to be that during the first few cycles this band limited signal would propagate from the stapes towards the best place carrying energy. A plausible alternative is that energy radiated by the stapes would be stored and re-emitted by the ENTIRE basilar membrane, in differing proportions. The radial segments would start in phase with each other since each experiences a force at the same instant, but would quickly fall into their known phase relationship due to the fluid coupling. In this scenario energy transfer is though of as closer to a point to point transmission than a transfer with the traveling wave. (Although as discussed above there seems to be a preference to energy transfer in the vicinity).
I would like to hear your view on whether the alternative description of the energy transfer is compatible with the physics of the passive cochlea. If it is it seems to be at least compatible with the resonance theory. It would suggest that care should be taken in the importance placed on unwrapped phase plots as evidence of the traveling wave in frequency domain models and the WKB approximation, although I know that phase delay has support from other sources. I also acknowledge that the rapid fall off after the best place lends more support to the prevailing traveling wave view than the alternative interpretation, although this result might also come out in the maths for the alternative view. The time domain signal required to test this should be free of higher order harmonics, since these will stimulate the basilar membrane basal of the best place and thus are hard to generate.
In summary my current view is that the traveling wave observed in the physics of the passive cochlea is in part simply a feature of the arrangement of the radial segments, given that to some extent all the segments are coupled and can transfer energy point to point. A second important part is that local energy transfer is slightly assisted by the increased local coupling. Would you agree with this position or do you see a fundamental error in it? Studying the onset of pure tones seems to a good way to evaluate the relative importance of these effects.
I have deliberately only discussed the passive case to avoid extending the previous debate about the active mechanisms at work, although I know that they are the major focus of current work. I would however appreciate it if we could continue discussing this passive case, since it is adequate for understanding the question I pose. Hopefully if we come to an agreement on the passive case we can extend the discussion to the active elements.
All your thoughts are much appreciated and apologies for length,
Bell (2008) Andrew Bell, "The pipe and the pinwheel: Is pressure an effective stimulus for the 9 + 0 primary cilium?", Cell Biology International 32 (2008) 462-468
Nobili and Mammano (1993): R. Nobili and F. Mamano, "Biophysics of the cochlea: Linear approximation", J. Acoust. Soc. Am. Vol. 93, No. 6, June 1993
>>> On 2010/03/12 at 04:07 PM, in message
<20100312142543.815B989AA@xxxxxxxxxxxxxxxxxxxxxxx>, Peter van Hengel
> Dear list,
> With a background in fluid mechanics perhaps my perspective on the traveling
> wave helps the discussion.
> I don't think there is a question whether or not there is a traveling wave
> in the cochlea. Fluid mechanics dictates that there has to be one.
> The confusion comes - I think - from comparing the basilar memebrane with a
> string where the energy is passed on through the string and it is that same
> string which is showing the movement. In this respect the comparison with
> surface waves on water is much appropriate. The fluid-air interface is
> showing the movement, but it is the underlying fluid which passes
> on the motion. Imagine a pond surface covered with ducks. Imagine it to be
> covered so densely you cannot see the water surface. When the water is set
> in motion (not neccessarily at its
> surface), the ducks will move. This motion will look like a wave and I guess
> everyone would agree with the use of the term travelling wave in this case.
> The energy causing the
> ducks to move is not passed on from one duck to the other, but stems from
> the motion of
> the fluid.
> Likewise in the cochlea the BM motion is caused by motion of the fluid. The
> fact that we
> have fluid on both sides of the BM, whereas in the example we have fluid
> below and air on
> top can be shown (mathematically) to be of no consequence for the principle.
> Also the
> fact that in the example the restoring force acting on the ducks is gravity,
> whereas in
> the cochlea it is the BM stiffness does not affect this story.
> The main problem with the resonator/resonance theory (at least in the
> versions I know) is
> that the motion of neighbouring resonators is independent. In the example
> neighouring ducks can not move independently because their motion is linked
> through the motion of the underlying (continous) water.
> Complicating factor in the discussion is perhaps that in the cochlea, the
> restoring force being stiffness combined inevitably with mass, we
> automatically get resonators. So in my view it
> is not a question of resonance OR travelling wave. It has to be a bit of
> Fluid mechanics dictates that there is a travelling wave on the basilar
> membrane unless cochlear fluid is unlike any other fluid I know. The
> question that may remain is whether this wave motion is what causes the
> effective stimulation of haircells. But there should not be a question
> whether or not there is a traveling wave, even if it has not been shown
> definitively in measurements.
> The problem I see with a compression wave being the stimulus and the
> haircells acting as pressure sensors is that. This assumes that the
> haircells will be compressed by a pressure acting on them form the outside.
> However, the haircells are filled with fluid themselves and there will be no
> pressure difference between the inside and outside of the cell. This implies
> that the cell wil not deform and I do not quite see how the sensor would
> then operate. (But the fact that I don't see it does not mean it impossible,
> of course...).
> The references to texts already given by dr Frosch and others are excellent
> and I don't have much else to add.
> All the best,
> Peter van Hengel
UNIVERSITY OF CAPE TOWN
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