An analysis of the basal basilar membrane response to tones
A comparison of BM displacement and Neural excitation patterns
basal to CF
J. B. Allen1
AT&T Labs-Research
Florham Park, NJ 07932-0971
Abstract
Abstract
We provide an analysis of the basal response of basilar membrane (BM)
and cilia response for low frequency tonal stimulation.
In the base, the BM impedance is K0 e-2ax/i w
while the scala impedance is i wrc A(x).
Thus the pressure is very close to that in a ridged-walled
box, namely P(x) = i wrc (L-x) ust, since
the BM impedance is much greater than the scala impedance.
It follows from Hooke's Law that the BM displacement
must vary in an exponential manner with place:
x(x,w) = - w2 rc (L-x) ust e2ax/K0.
Hair cells are known to be displacement detectors (Hudspeth and Corey, 1977).
Above 1 kHz Dallos has found that the inner hair
cell (IHC) responds to the shear displacement of the sub-tectorial
space. Direct measurements of the neural
population by Kim et al., as well as transformations of populations
of neural tuning curves from frequency to place (Allen, 1991),
have shown that the basal neural response
is independent of x. These results are consistent with
two-tone suppression (Fahey and Allen, 1979)
(and upward spread of masking) data which show that
the low frequency suppression threshold is
approximately independent of frequency.
There are two possible explanations for the above contradiction.
Either the inner and outer hair cell sensitivity must vary as
e-2ax, or the micromechanics of the tectorial membrane must
transform the BM displacement shear signal from exponential to a constant
place dependence. The evidence for a gradient in cilia sensitivity is in the
direction of increased sensitivity for lower frequencies,
because the cilia increase in length with place. This
leaves us with the option that the TM to cilia stiffness
ratio must vary as e-2ax to compensate for the e2ax
dependence of the BM displacement.
A simple and natural solution to all these requirements is to assume that the
linear component of the partition stiffness is dominated by the
TM radial stiffness. Model results support these conclusions.
Transmission line model
- The 1D cochlear model
- The scala impedance: Zp = i wr/ A
- The cochlear partition impedance:
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Zp(x,w) = K0 e-2ax/i w + R0 + i wM |
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The place to frequency map
- The cochlear map describes the location of the frequency
maximum of the tuning curve along the cochlear partition
- From basic theory fcf = [Ö(K(x)/M)]/2 p = fmaxe-ax
- The space constant a = -2.31 for the Cat
may be computed from the slope (3 mm/oct)
|
2 = e-a 0.3 ® a = -log(2)/0.3 (cm-1) |
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Neural excitation pattern
- Neural tuning curves along with the cochlear map allow us to
estimate neural excitation patterns
- These frequency domain data were transformed to
the place domain using Liberman's cochlear map
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fcf = 456 [ 102.1(1-x/L) -0.8 ] |
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Stiffness dominated tail region
- For each pure tone stimulus, the partition impedance is compliance
dominated
from the stapes to Xz(f) (i.e., a few mm basal to the CF)
-
Hooke's Law
relates partition
pressure P(x) and displacement D(x)
Cochlear Pressure for a tone
- From the WKB solution method, the spatial pressure distribution
of a tone stimulus in the base of the cochlea is given by
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æ
Ö
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c(x)/c(0)
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e-iwòx = 0x dx/c(x) |
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| |
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where c(x) = Ö{K(x) A/r} is the local wave speed
- Conclusion:
- The partition pressure magnitude decays as
- The partition displacement magnitude increases as
- Since the inner haircell is a displacement detector above about 1 kHz
- For the cat (where 2 = ea 0.3), the cilia (neural) response should grow as
- This gives a partition displacement growth of 3 dB/mm
Estimation of basal EP slope
- Neural excitation pattern estimated from FTC
- Slopes S1, S2, and S3 (dB/mm)
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|
| CF | S1 | S2 | S3 |
| kHz | | SLOPE* (dB/mm) | |
| 5.0 | 9.3 | 32.7 | -66.1 |
| 4.0 | 5.0 | 26.3 | -69.3 |
| 2.0 | 1.3 | 15.2 | -34.5 |
| 1.0 | 1.2 | 17.4 | -25.6 |
| 0.5 | 0.3 | 14.8 | -34.5 |
| 0.25 | 0.3 | 17.1 | -11.0 |
* Mult by 3 mm/oct to convert to dB/oct
- FINAL CONCLUSION:
- There must be a transduction filter H(x,f) to account
for the slope difference of 3 dB/mm for D(x,f) and 0.3-1.3
dB/mm for the cilia EP C(x,f)
A natural solution to the transduction filter problem
- I propose that the partition stiffness is dominated by the
tectorial membrane stiffness Kt(x)
- The partition impedance is:
Kp = [(Kt Kc)/( Kt + Kc)] + Kbm
- Assume: Kc >> Kt µ e-2ax and Kbm » Kt
- It follows that
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fz(x) º |
1
2 p
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| _____
ÖKt/Mt
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= fcf(x)/Ö2 |
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and at 0.5 kHz
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H º |
C
D
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= Kt/Kc » eax/10-3ax/2 |
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Footnotes:
1 jba@research.att.com; TEL: 973/360-8545
File translated from TEX by TTH, version 1.96.
On 20 Feb 1999, 07:24.