Physical model of cochlear frequency discrimination

From: zigoteau (zigoteau_at_ausi.com)
Date: 07/15/04 

Date: 15 Jul 2004 09:49:38 -0700

Hi all

Tony Jeffs asked a question a couple of months ago about the mechanism
for frequency discrimination of the basilar membrane of the cochlea. I
suggested then that there was a model for this in the Schroedinger
equation for an electron moving in a constant electric field F. In the
present posting, I want to analyse the frequency discrimination
capability of the model, and the discrepancies between it and
measurements on the ear. Schroedinger's equation in this case is:

h^2/8/pi^2/m*d2psi/dx2 = (hf-eEx)psi

Note that E here is the electric field, not the electron energy. The
latter is given by hf

The Airy function Ai(x) satisfies Ai''(x) = xAi(x) and converges to
zero at ħinfty. Its peak value of ~0.5 occurs at x~-1. Its value drops
to half that at ~0.5 and ~-20. Hence the "3dB resolving width" of the
Airy function can be considered to be around 20. There are 10 cycles
between the 3dB points.

The solution of the Schroedinger equation is

psi(x) = Ai[(x-hf/eE)/(h^2/8/pi^2/m/e/E)^(1/3)]

The 3dB points of the response due to frequency f are located at
hf/eE+0.5*(h^2/8/pi^2/m/e/E)^(1/3) and
hf/eE-20*(h^2/8/pi^2/m/e/E)^(1/3) . The next highest frequency f'
which can be resolved is given by

f' = f + 20.5*(eE/h)*(h^2/8/pi^2/m/e/E)^(1/3) i.e. delta f
proportional to E^(2/3)

Hence the frequency discrimination of this system can be made
arbitrarily fine by using sufficiently low electric fields. There is
the usual Heisenberg-type trade-off between frequency discrimination
and response time, with a delay significantly longer than the
theoretical minimum, but it's perhaps liveable with.

The actual value for the frequency discrimination is irrelevant for
the human ear. The important point is that this mechanism is capable
of separating up the different frequencies in a signal to any
specified resolution.

In the ear, the dependent variable would not be psi but the
displacement z at position x and time t of the basilar membrane and/or
organ of Corti. The essential part of the mechanism is that any given
frequency propagates along the membrane up to a certain point, beyond
which it is evanescent. The amplitude peaks at the junction between
these two regions.

In the Schroedinger/Airy case I have given, at any given point, low
frequencies are evanescent and high frequencies propagate. This is
also true for e.g. microwave propagation down a waveguide. In a
frequency-selective system based on either principle, the signal input
is at the low-frequency end. AFAICT this is also the way it is done
in the ear.

I have looked at lots of pictures of the cochlea, with the organ of
Corti and the basilar membrane, and it still doesn't quite make sense.
On Tony Jeff's web page www.members.aol.com/tonyjeffs/text/dia.htm it
looks like there is a cantilever gallery sticking halfway out into the
cochlear fluid. There's therefore a node at the wall and an antinode
at the edge. I think this shows the desired frequency behaviour,
cutting off at frequencies below the first cantilever resonance. The
second resonance of a simple cantilever is nine times higher in
frequency.

However in www2.sfu.ca/sonic-studio/handbook/Cochlea.html it looks as
if perhaps there's a membrane going all the way across the tympanic
canal. What is Reissner's membrane doing? Enlightenment in terms of
physics, please, someone?

In the Schroedinger/Airy model, the wavevector is nonzero to start
with, and decreases progressively, reaching zero just beyond the
resonance point. The measurements of gerbils' ears by Tianying Ren
[Proc. Nat. Acad. Sci. 99 (2002) 17101] are not quite in accordance
with this idea, because the phase shift per unit length, i.e. the
wavevector, appears first to become nonzero, and then increase
progressively, past the point of greatest response. This does not make
a great deal of sense, and perhaps someone can enlighten me. Maybe I
have misunderstood his figures, or maybe the travelling wave
hypothesis is not correct.

Cheers,

Zigoteau.