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# Cochlear maps.

```Dear List,

A month ago I posted a formula for the dependence of the
stiffness S [spring constant per square mm] of the basilar
membrane on the distance x from the stapes:

S(x) = S(0) * [1 - x / (4d)]^4 .              (1)

That formula can be rewritten as follows:

S(x) = S(0) * [1 - x / L]^4 .                 (2)

The length L in Eq. (2) is slightly greater than the length of
the basilar membrane. In humans that length is about 35 mm,
and L is proposed to be about 36 mm. S(0) is 1*10^9 N/m^3.
Eq. (2) is then found to be fairly close to the "resonance" map
of the human cochlea.

There are three cochlear frequency-versus-place maps which
differ distinctly from each other, namely the "passive", the
"active", and the "resonance" map.

The "passive" map displays the place x where a sine-tone of
given frequency f causes the strongest mammalian basilar-
membrane vibration in post-mortem experiments, or also in
in-vivo experiments with loud sine-tones (>100 dB SPL).
The map due to D.D. Greenwood (1990), which for human
cochleae follows an exponential function up to about
x = 28 mm, belongs to this "passive" category.

The "active" map shows the place x of strongest basilar-
membrane excitation in in-vivo experiments on healthy
cochleae with soft sine-tones.

The "resonance" map is defined by the following equation:

f_res = [1 / (2 pi)] * [S(x) / M]^(1/2) .            (3)

In Eq. (3), M is the mass per square mm of the basilar
membrane and the attached cells; in humans, M is about
0.1 mg / mm^2.

With the help, e.g., of de Boer's chapter in the book
"The Cochlea" (Springer, 1996), it can be conjectured
that the theoretical resonance map based on Eqs. (2) and (3)
above may well be closer to observations than that based on
Eq. (3) and the exponential stiffness formula used by de Boer
(1996, with a parameter alfa of 3 cm^-1).

With many thanks for your patience,

Reinhart Frosch.

P.S. 1: The idea that "my" formula may work for the whole
cochlea came from a posting by Dick Lyon of August 25.
Unfortunately, I sent a fairly thoughtless reply on the same
day. Now I agree with all of Dick's comments.

P.S. 2: I still think that the new stiffness formula represents
an evolutionary advantage. For a given stapes velocity at low
frequency (50 to 300 Hz in the case studied), the liquid-
pressure wave after two thirds of the cochlear channel is more
intense if these two thirds obey Eq. (2) than if they obey a
suitably comparable exponential stiffness formula.
The difference, however, is not due to the fact that Eq. (2)
leads to accurate WKB approximations. The important feature
is the small relative decrease, per mm, of S(x) at small x.
I shall send details in a few days.

Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx .

```