Re: mechanical cochlear model

["Richard F. Lyon" <DickLyon@xxxxxxx>]

Alain, your "simple account" was "reasonably correct" whether you 
mentioned the traveling wave or not, so I agreed with you.

I admit I don't see why you don't explicitly identify that account 
with traveling waves.  Sinusoids in such a system are locally 
described as cos(ometa*t - k*x), with some varying amplitude, where 
the relations between omega and k locally obey the same kind of 
dispersion relation (more or less) as gravity waves on water.

The real issue seems to be about where the energy is, and how it 
propagates and gets detected.  The fast pressure wave is another 
physical mode that really exists, but its wavelengths are all very 
large compared to the cochlea, so the pressure due to this mode can 
be viewed as equal throughout the cochlea; pushing on the stapes 
immediately pushes out at the round window.  But this mode doesn't 
carry much energy, as the middle ear leverage isn't nearly enough to 
couple efficiently to it; there's also no efficient way to get energy 
back out of this mode into a place and a form for which detectors are 
known; the fast compression wave therefore carries pressure, but not 
much energy.  The traveling wave mode, on the other hand, involves 
much larger fluid displacements at the same pressures; it's a 
differential mode between the scalae, propagated by the spring of the 
BM instead of by fluid compression.  And the energy converges on the 
BM as the wave slows down and the wavelength gets short, focusing the 
energy into a small layer with large displacements at the BM.  There, 
the hair cells, which evolved to detect fluid motion via cilia 
bending, are well positioned to respond.

A wave has three kinds of delays:  phase, group, and wave-front.  In 
typical models, the wave-front delay is essentially zero, and the 
response latency can be made to approach zero as the level gets very 
high.  The group delay depends on the damping, including negative 
damping effects, and so varies a lot with level.  The phase delay is 
in between, around a cycle and half, and pretty stable.  Pretty much 
everything about it is as Lighthill described, in terms of energy 
flow, except that there's not much BM mass so it's not really 
significantly resonant, except for very high frequencies very near 
the base where the accelerations are very high.  And except for some 
positive feedback from outer hair cells that modifies the dispersion 
relation, providing active gain instead of loss over some range of 
wavelengths.

If it walks like a wave, and quacks like a wave, and transports 
energy like a wave, why not call it a wave?

Dick