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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Cochlear maps.*From*: "reinifrosch@xxxxxxxxxx" <reinifrosch@xxxxxxxxxx>*Date*: Sun, 24 Sep 2006 17:38:10 +0200*Delivery-date*: Sun Sep 24 11:51:51 2006*List-help*: <mailto:LISTSERV@LISTS.MCGILL.CA?body=INFO AUDITORY>*List-owner*: <mailto:AUDITORY-request@LISTS.MCGILL.CA>*List-subscribe*: <mailto:AUDITORY-subscribe-request@LISTS.MCGILL.CA>*List-unsubscribe*: <mailto:AUDITORY-unsubscribe-request@LISTS.MCGILL.CA>*Reply-to*: reinifrosch@xxxxxxxxxx*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

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 .

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