CHAPTER M 4 (Model)
Cochlear fine-tuning: a surface acoustic wave resonator
4.2 Basic description of the model
4.3 Parallels between SAW devices and the cochlea
4.4 Prima facie validity
4.5 What is the wave propagation mode?
This chapter introduces the idea that the cochlea analyses sound in a way
resembling electronic surface acoustic wave (SAW) devices. The outer hair cells
(OHCs) of our cochlea typically occur in three distinct rows1, and it is suggested the
rows act like the interdigital electrodes of a single-port SAW resonator. Frequency
analysis occurs through sympathetic resonance of a graded bank of these resonators.
This active process relies on the motor properties of OHCs: as well as being sensors,
OHCs undergo cycle-by-cycle length changes in response to stimulation. The SAW
analogy works if the motility of one row of cells can be communicated to a
neighbouring row, allowing positive feedback between one cell’s body (an effector)
and another’s stereocilia (a sensor). If the wavelength of the disturbance were
comparable to the row spacing then inter-row reverberation could occur. A candidate
wave possessing the required properties has been identified – a ‘squirting wave’
known in ultrasonics – and this wave, operating in the subtectorial space between the
tectorial membrane and the plateau of Corti, is described in Chapter R5.
1 As we will see in Chapter R6, occasional extra rows are not a problem.
4.1 Introduction
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Standing-wave resonance between the rows could thus provide amplification
and high Q, characteristics underlying the ‘cochlear amplifier’ – the device whose
action is evident to auditory science but the identity of which has not yet been
established. Also emerging naturally from such a local interaction on the partition are
spontaneous, evoked, and distortion-product emissions (Chapter R7).
The SAW mechanism, as outlined in this chapter, may provide a second filter
for a primary traveling wave stimulus, and this cannot be ruled out without further
experiments. The advantage of taking this approach is conformity with the standard
model. The drive of this thesis, however, is towards a simple resonance theory of
hearing along the lines of Helmholtz, in which case we require that the input to the
SAW is the fast pressure wave stimulus. The advantage here is simplification of
cochlear mechanics. A later chapter (Chapter D8) strengthens the supposition about
pressure sensitivity.
4.1 Introduction
Earlier chapters have focused on the cochlea’s behaviour as an active
transducer, a necessary consideration flowing from Kemp’s seminal discovery2 of
sound emerging from the ear. An essential element of the auditory organ is therefore
a ‘cochlear amplifier’ whose action improves gain and tuning. If the gain is excessive
at some frequency, the cochlea will spontaneously oscillate, a possible source of
‘spontaneous otoacoustic emissions’ – soft, pure tones – that may be detected with a
microphone placed in the ear canal. Most human ears continuously emit such tones at
frequencies of 1–4 kHz and with bandwidths ranging down to 1 Hz or less. A
detailed review3 of these developments was given by Robinette and Glattke in 2002.
What sort of biological structure could produce such pure tones? The outer
hair cells (OHCs) must be intimately involved, for these sensing cells are active,
possessing a property known as ‘electromotility’. When an audio-frequency voltage
2 Kemp, D. T. (1979). Evidence of mechanical nonlinearity and frequency selective wave amplification in the cochlea. Arch. Otorhinolaryngol. 224: 37-45. 3 Robinette, M. S. and T. J. Glattke (2002). Otoacoustic Emissions: Clinical Applications. (Thieme: New York).
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is applied to an isolated cell, it synchronously changes length4. Nevertheless, how the
cochlear amplifier harnesses electromotility is unknown. This thesis offers a possible
solution.
The standard cochlear model assumes that stimulation of hair cells occurs by
a hydrodynamic traveling wave moving along the partition. The difficulty has been
to understand how the response of such an intrinsically broad-tuned system can be
sharpened to give the fine frequency resolution it displays5. This chapter puts forward
a proposal whereby the unique structure of the sensing surface of the cochlea can
achieve narrow-band frequency analysis. It conjectures that the cochlear amplifier is
based upon the cooperative activity of neighbouring OHCs, which work together like
the interdigital electrodes of the SAW resonator6,7.
4.2 Basic description of the model
The SAW model builds on a remarkable fact: in all higher animals, including
humans, OHCs lie in three (or more) well-defined rows (Fig. 4.1), hitherto
unexplained.
SAW devices are signal-processing modules in which finger-like electrodes
are interleaved on the surface of a piezoelectric substrate to create slowly
propagating electromechanical ripples of wavelength equal to the periodicity of the
interdigital electrodes8,9. The minimum number of electrode fingers is three
(Fig. 4.2), an arrangement giving the widest bandwidth response10.
4 Brownell, W. E., et al. (1985). Evoked mechanical responses of isolated cochlear outer hair cells. Science 227: 194-196. 5 Patuzzi, R. B. (1996). Cochlear micromechanics and macromechanics. In: The Cochlea, edited by P. Dallos et al. (Springer: New York), 186−257. 6 Campbell, C. K. (1998). Surface Acoustic Wave Devices for Mobile and Wireless Communications. (Academic Press: San Diego). 7 Ballantine, D. S., et al. (1997). Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications. (Academic: San Diego). 8 Bell, D. T. and R. C. Li (1976). Surface-acoustic-wave resonators. Proc. IEEE 64: 711-721. 9 Smith, W. R. (1981). Circuit-model analysis and design of interdigital transducers for surface acoustic wave devices. In: Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic: New York), 15, 99-189. 10 Campbell (1998), p. 501.
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Fig. 4.1. Three rows of outer hair cells (bottom) and one row of inner hair cells (top) in the cochlea of a rabbit. Spacing between OHC rows is about 15 µm. [From Counter et al. (1993)11 with permission of S. A. Counter (Karolinska Institute) and Elsevier Science]
Fig. 4.2. The simplest interdigital electrode has three fingers.
The SAW resonator (Fig. 4.3) has a topology in which the electromechanical
waves on the surface are arranged in a feedback loop between two sets of electrodes
– a ‘two-port’ system, normally operated as a delay line, in which one set of
electrodes launches the ripples and a similar set some distance away detects them.
11 Counter, S. A. (1993). Electromagnetic stimulation of the auditory system: effects and side-effects. Scand. Audiol. Suppl. 37: 1-32.
+–
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Fig. 4.3. Schematic of surface acoustic wave delay line (after Maines and Paige, 1976). Electromechanical ripples are generated and detected by two sets of interdigital electrodes on a piezoelectric substrate to give a delay line. Driving and sensing functions can be combined into a single set of electrodes with resonance between the fingers.
SAW modules are used whenever a number of cycles of signal need to be
stored and operated on. Feeding the output of the second set of electrodes back to the
input set creates a highly tuned resonance (Q ≈ 103–104) typically in the megahertz
range. Resonance can also occur when the two sets of electrodes are merged into a
‘single-port’ resonator12; in this degenerate case, ripples now reverberate back and
forth between the fingers instead of between the two electrode sets. The present
hypothesis is that audiofrequency resonances arise in the cochlea in a similar way
(Fig. 4.4).
–
+
–Fig. 4.4. The rows of outer hair cells are assumed to act like the 3 fingers of an interdigital transducer.
12 Bell and Li (1976); Campbell (1998), Ch. 11.3.
surface acoustic wave
electrical input
absorber
highly polished piezoelectric crystal
detector output
interdigital electrode
roughened and/or waxed back surface
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The model is most easily conveyed by reference to Fig. 4.5, which shows a
cross-section of the cochlear partition. The key components are the three rows of
OHCs overlain by the soft gel13 of the tectorial membrane (TM) in which the tips of
the hair-like stereocilia are embedded. Forced oscillation of an OHC could send out
waves that, if sufficiently slow, will have wavelengths comparable to the OHC row
spacing.
Fig. 4.5. Because outer hair cells are both sensors and effectors, ripples could reverberate between them to produce a standing wave. Double-headed arrows indicate contraction/ elongation of cells (electromotility) in response to a common stimulus.
As described later on (§M 4.5), a literature search into wave propagation at
liquid–solid interfaces revealed a prime candidate for such a wave: a slow, highly
dispersive wave known in ultrasonics as a symmetric Lloyd–Redwood wave or
‘squirting’ wave14. These waves can arise when a thin fluid layer is sandwiched
between two deformable plates, as occurs in the cochlea (Fig. 4.5). Because squirting
waves rely on the interaction between the mass of the fluid and the elastic restoring
force of the plates, they are characteristically slow – measured in the cochlear case in
millimetres per second.
Squirting waves provide a ready basis for positive mechanical feedback and
amplification in the partition (Chapter R5). The proximity of motors – OHC bodies –
to sensors – OHC stereocilia – invites feedback, and if the phase delay of the wave
13 Steel, K. P. (1983). The tectorial membrane of mammals. Hear. Res. 9: 327–359. 14 Bell, A. and N. H. Fletcher (2004). The cochlear amplifier as a standing wave: "squirting" waves between rows of outer hair cells? J. Acoust. Soc. Am. 116: 1016-1024.
(reticular lamina)
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reaches 360º, oscillation between the rows of OHCs will occur. Modeling
(Chapter R6) shows that after applying an input signal to the rows of OHCs, a
standing wave appears between the outermost rows (OHC1 and OHC3). The
standing wave can occur either in a whole wavelength mode (if the polarity of the
input to all rows is the same) or half-wavelength mode (if the polarity of OHC1 is the
inverse of that of OHC3). At the same time, a progressive, but attenuating, wave will
move towards the inner hair cells (IHCs), the ear’s detectors. We now have a SAW
resonator amplifying an input signal and passing it to a detector.
This scheme meets Gold’s prescription for a ‘regenerative receiver’ in the
cochlea where, to avoid compromising signal-to-noise ratio, positive feedback is
used to amplify a signal before its detection. Later (Gold, 1987), he drew an analogy
to the functioning of an ‘underwater piano’: only by introducing a sensor onto each
string and supplying positive feedback could such a viscously loaded instrument be
made to operate, and this is what the SAW model achieves. The SAW resonator can
be identified with the cochlear amplifier and would explain spontaneous emissions
and other active aspects of cochlear mechanics.
The arrangement also fits the description by Hudspeth15 of a distributed
amplifier. Reinterpreting Hudspeth, the separate OHCs could be likened to a
pendulum clock (a distributed mechanical amplifier in which a pendulum is driven
by a coiled spring through an escapement); thus, their oscillation frequencies would
be determined by one physical property (the length of the SAW cavity, equivalent to
the length of a pendulum), while the timed release of energy (from the OHC motor –
the escapement) overcomes damping and sustains activity.
The key assumptions of the SAW hypothesis are set out below.
1. The OHCs and their surroundings have properties conducive to the
propagation of a slow wave – probably a squirting wave – communicating the
motion of one row to the next.
2. The speed of the waves varies from base to apex in a systematic way,
thereby supplying the cochlea’s tonotopic tuning. Halfway along the cochlea, where
the spacing between OHC1 and OHC3 is about 30 µm and a characteristic frequency
(CF) of 1 kHz is typical, the wave will need to have a propagation speed of
15 Hudspeth, A. J. (1997). Mechanical amplification of stimuli by hair cells. Curr. Opin. Neurobiol. 7: 480-486.
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30 µm/ms (= 30 mm/s) to establish whole-wavelength resonance. Near the base, with
a CF of say 10 kHz and spacing is 20 µm, the necessary speed is 200 mm/s; in
contrast, near the apex where CF may be 0.1 kHz and the spacing 50 µm, the
corresponding speed would fall to 5 mm/s. Chapter R5 shows that the dispersive
properties of a squirting wave allow the human cochlea to be tuned over its full
frequency range (20–20 000 Hz) based largely on the spacing of OHC rows.
3. Rows 1 and 3 respond with the same phase to a stimulus, while row 2
probably responds in anti-phase. Modeling in Chapter R6 shows two likely modes: a
full wavelength mode, like the first overtone of an organ pipe open at both ends, in
which the response of OHC2 is in antiphase to OHC1 and OHC3; and a half-
wavelength mode, like the fundamental mode of the open-ended organ pipe, in which
the responses of OHC1 are in antiphase to OHC3 (and OHC2 is a displacement
node)16. Evidence for bi-phasic activity in OHCs is presented in §R 9.1.
The result of the interaction is a standing wave, a mechanical cochlear
amplifier. That is, IHCs respond to wave energy delivered to them by a squirting
wave generated by the combined activity of the outer hair cells.
4.3 Parallels between SAW devices and the cochlea
There are strong analogies between a SAW resonator and the anatomy of the
cochlea. Comparison is aided by reference to Fig. 4.5 and anatomical texts17.
1. The three rows of OHCs are the interdigital electrodes. It is significant that
the required minimum number of fingers for an electronic SAW device is three, and
in all vertebrate animals there are three or more OHC rows18. Additional rows,
sometimes present, would supply extra gain.
2. Wave energy propagating on the surface of a SAW resonator can be
absorbed or reflected by impedance discontinuities, and, when required, this is
16 A dynamic display of organ pipe modes can be found at http://www.cord.edu/dept/physics/p128/lecture99_35.html 17 Such as Slepecky, N. B. (1996). Structure of the mammalian cochlea. In: The Cochlea, edited by P. Dallos et al. (Springer: New York), 44−129. 18 Bredberg, G. (1968). Cellular pattern and nerve supply of the human organ of Corti. Acta Otolaryngol. Suppl. 236: 1-135.
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normally achieved by etching grooves or placing strips of material on the surface of
the device19. The TM possesses Hensen’s stripe20, a rounded feature located above the
IHC stereocilia (see Fig. 4.5) which may act as a mechanical impedance
discontinuity: this might redirect wave energy emerging from the OHC cavity and
send it to this detector.
3. Energy escaping the OHC cavity towards the outer edge of the TM is a
potential problem and needs to be reflected so as to re-enter the cavity with
appropriate phase delay. At the outer edge of the TM another aggregation of material
is found, a rounded thickening known as the marginal band21, which may act this way
(Fig. 4.5).
4. To absorb and disperse unwanted bulk propagation modes (multiple fast
compressional waves that propagate throughout the substrate of the device), the back
of a SAW resonator is either roughened or waxed22. In the cochlea the top of the TM
is criss-crossed with a covering net23.
5. Towards the inner edge of the TM we find a sharp discontinuity – the
vestibular lip (Fig. 4.5) – and here reflections could occur, possibly acting to return
wave energy back into the amplifying cavity and allowing real-time convolution and
autocorrelation of the signal to take place23,24,25. The key idea here is that the IHC
would then sit as a central detector between the signal source (the OHC cavity) and
its reflected image. As explained by Fergason and Newhouse (1973), when two
surface waves pass through each other from opposite directions and interact
nonlinearly (to produce sum and difference frequencies), a centrally placed detector
integrates the sum frequency to form the convolution and the difference frequency
supplies the correlation26.
6. The speed of electromechanical ripples in a solid-state SAW resonator is
about 5 orders of magnitude lower than the speed of the electrical signal in its input
19 Smith (1981), p. 102. 20 Hardesty, I. (1908). On the nature of the tectorial membrane and its probable role in the anatomy of hearing. Am. J. Anat. 8: 109–179 (+ plates). 21 Edge, R. M., et al. (1998). Morphology of the unfixed cochlea. Hear. Res. 124: 1-16. 22 Smith (1981), p. 130. 23 Campbell 1998, Ch. 17.3. 24 Quate, C. F. and R. B. Thompson (1970). Convolution and correlation in real time with nonlinear acoustics. Appl. Phys. Lett. 16: 494-496. 25 Kino, G. S. (1976). Acoustoelectric interactions in acoustic-surface-wave devices. Proc. IEEE 64: 724-748. (p. 728 and Fig. 6) 26 Furgason, E. S. and V. L. Newhouse (1973). Convolution and correlation using nonlinear interactions of Lamb waves. IEEE Transactions on Sonics and Ultrasonics SU-20: 360-364.
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leads, a reduction that makes it possible to compactly store many cycles for signal
analysis27. Thus, it is possible to store in an inch of crystal the information that would
otherwise fill a mile-long cable28. In a similar way, the speed of the hypothetical
squirting wave is 4–5 orders of magnitude lower than the speed of a sonic pressure
wave in the surrounding cochlear fluids, some 1500 m/s. In the width of the tectorial
membrane, then, may be stored up to a dozen cycles of acoustic signal. One
experiment involving a microphone in a human ear canal29 observed a 1280-Hz tone
continually waxing and waning every 5.8 ms– that is, an envelope repeating every
7.4 waves.
4.4 Prima facie validity
The structural parallels are suggestive, but by themselves they are not
conclusive. A number of lines of evidence can be found in the literature, and a more
detailed presentation of some of them is set out in Chapter D9. However, in order to
justify the modeling that forms the core of this thesis (Chapter R6), it is worth noting
two main predictions of the SAW model and the major evidence supporting them.
A. Inverted response of different rows. A distinctive feature of SAW devices
is the alternating polarity of the interdigital electrodes. Translated to the cochlea, this
means that the response polarity of one or more rows of OHCs should be inverted
compared to the others. One key finding here is that the cochlear microphonic, an
electrical potential recordable from outside the cochlea, exhibits ‘dual’ polarities30, as
if it were composed of two potentials of opposite polarity (which could in turn derive
from two populations of OHCs of opposite polarity). Moreover, isolated OHCs can
either lengthen or shorten in a tuned manner in response to oscillating pressure31, and
27 Maines, J. D. and E. G. S. Paige (1976). Surface-acoustic-wave devices for signal processing applications. Proc. IEEE 64: 639-652. 28 Claiborne, L. T., et al. (1976). Scanning the issue. Proc. IEEE 64: 579-580. 29 Wit, H. P. and R. J. Ritsma (1980). Evoked acoustical responses from the human ear: some experimental results. Hear. Res. 2: 253-261. The wave continued for more than 280 ms. 30 Pierson, M. and A. R. Møller (1980). Some dualistic properties of the cochlear microphonic. Hear. Res. 2: 135-149. 31 Brundin, L., Flock, Å., and Canlon, B. (1989). Sound-induced motility of isolated cochlear outer hair cells is frequency-specific. Nature 342: 814–816.
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direct antiphasic activity of OHC rows has been observed, even (surprisingly) in an
explant32.
B. Radial wave motion. If the SAW model is valid, we would expect closely
spaced phase changes across the partition. Indeed, relative phase changes of up to
180º between locations only 10 µm apart radially have been observed on the basilar
membrane of a guinea pig33. Nilsen and Russell drilled a hole in the cochlea and
measured, in a profile across the basilar membrane, the motion of the membrane in
response to sound stimulation. By using a beam from a self-mixing laser diode,
focused to a 5-µm spot, they avoided having to place reflective beads on the
membrane, a routine practice that conceivably might interfere with normal cochlear
activity. The unique Nilsen and Russell results are highly suggestive, even though
several other workers have only seen smaller phase gradients and others have not
seen any (see §D 9.3/b). Rapid phase variations strongly support the SAW model,
and are otherwise difficult to explain.
4.5 What is the wave propagation mode?
SAW devices can, in principle, operate using a wide range of wave
propagation modes: Rayleigh waves are the most common, but Lamb, Love,
Bleustein–Gulyaev–Shimizu, Stonely, Sezawa, and other wave modes are
employed34,35. Deciding which mode operates in the cochlea is difficult due to
complex structure and little-explored physical properties of the layers bounding the
OHC stereocilia – the tectorial membrane (TM) at the top (in which the tips of the
longest stereocilia are embedded) and the plateau of Corti (reticular lamina, or RL)
underneath. The TM is a fibre-reinforced viscoelastic gel covered with surface layers
(Hardesty’s membrane and covering net) and immersed in electrolytes and an
electrical field; the RL, for its part, is a very thin (∼1 µm) membrane composed of
32 Scherer, M. P. and A. W. Gummer (2004). Vibration pattern of the organ of Corti up to 50 kHz: evidence for resonant electromechanical force. Proc. Nat. Acad. Sci. 101: 17652-17657. 33 Nilsen, K. E. and I. J. Russell (1999). Timing of cochlear feedback: spatial and temporal representation of a tone across the basilar membrane. Nat. Neurosci. 2: 642-648. 34 Ballantine, et al. Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications. Chapters 2 and 3. 35 Campbell (1998), Chapter 2.
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the interlocking flattened heads of phalangeal processes that in turn originate from
Deiters cells (see §R7).
Wave modes can be categorised in two main ways: longitudinal, in which the
particles move back and forth in the direction of wave propagation; and transverse, in
which they move at right angles to this direction. In geophysics, the first are known
as P-waves whereas the second are S-waves. P-waves depend on compressional
moduli, and hence are relatively fast compared to S-waves which rely on shear
moduli and can be much slower in gel-like materials. In seeking a remarkably slow
wave in the cochlea, shear waves appear the better candidates.
Simple capillary (surface tension) waves, with their characteristically slow
speed, are also worth consideration since surface tension effects have been observed
in the cochlea36. The speed, c, of a surface tension wave37 of wavelength λ on the
surface between a liquid and a gas is:
c = (2πT/λρ)½ (5.1)
where T is the surface tension and ρ is the density of the underlying liquid. Surface
tension may have a role in the cochlea, but it is difficult to see how this parameter
can be made to vary systematically from base to apex by a factor of 104 or more in
order to tune the partition over a frequency range of at least 2 decades. Note from
Eq. 5.1 that capillary waves are dispersive, with speed varying as ω 0.5.
A shear wave of some type is a strong possibility because the shear moduli of
gels are small, much less than their compressional (Youngs) moduli. In its most basic
form, the speed, v, of a bulk shear wave is given by
v = (G/ρ)½ (5.2)
where G is the shear modulus and ρ is the density. These waves are non-dispersive
(speed independent of frequency); a Rayleigh wave, a shear wave on the surface of a
solid that is thick compared with the wavelength involved, travels at about 90% of
this speed. Acoustic measurements38,39 of soft ‘ringing gels’ (showing the peculiar
36 Olson, E. S. and D. C. Mountain (1990). In vivo measurement of basilar membrane stiffness. In: Mechanics and Biophysics of Hearing, edited by P. Dallos et al. (Springer: Berlin), 296-303. The attraction between the basilar membrane and a probe only occurred pre-mortem; it was also noted by Olson in subsequent work. 37 Lighthill, J. (1978). Waves in Fluids. (Cambridge University Press: Cambridge). 38 Sinn, C. (2000). Acoustic spectroscopy of aerogel precursors. Progr. Colloid Polym. Sci 115: 325-328.
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property of ringing like a bell when struck), give shear moduli as low as 14 kPa and a
corresponding shear-wave velocity of 4.1 m/s, which is not low enough for audio-
frequency SAW operation in the cochlea – some millimetres per second (point 2 on
page M 4 [8]).
Measurements of the shear modulus (elasticity) of the actual TM have
returned a remarkably wide range, as Table 4.1 shows. The table includes values of
the less precise ‘point stiffness’, which earlier experimenters used, a measure that
depends on the sharpness of the probe (elasticity of 1 kPa ≈ point stiffness of
0.01 N/m). Values vary from as low as 0.0004 N/m to as high as 10 N/m, but even
the lowest value (G ≈ 0.2 kPa) is, after using equation 5.2, still too high for our
purposes (giving a c of 800 mm/s).
For comparison, measurements on the elasticity of the gel covering the
otoliths in an amphibian are also given in Table 4.1. These values are very low, and
the unsupported gel would probably collapse under its own weight in air, but even
this material gives a shear wave speed no lower than 50 mm/s. Experiments with a
vibrating magnetic bead inside a cell40 give a shear modulus of the cell contents as
20–735 Pa, and one cannot hope to go lower than that. Taken together, the evidence
suggests that a shear wave does not seem to be the answer.
A more likely avenue is to look at bending or flexural waves in plates, which
in general have lower velocities than waves based on shear. These waves also tend to
be highly dispersive. The basic surface wave is a Lamb wave, and they come in
symmetric and antisymmetric forms41,42. SAW devices based on Lamb waves can be
lowered further in frequency by immersing them in liquid or loading them with a
layer of high density and lower shear modulus, which retards the wave speed and
tends to trap wave energy at the surface. As a result, other interface modes such as
Stoneley, Sezawa, and Scholte waves begin to appear43. In this connection, we note
that the TM is not a homogeneous layer, but has its underside covered with a very
thin layer about 1 µm thick44 called Hardesty’s membrane.
39 Sinn, C. (2004). When jelly gets the blues: audible sound generation with gels and its origins. Journal of Non-Crystalline Solids 347: 11-17. 40 Bausch, A. R., et al. (1999). Measurement of local viscoelasticity and forces in living cells by magnetic tweezers. Biophys. J. 76: 573-579. 41 Wenzel, S. W. and R. M. White (1988). A multisensor employing an ultrasonic Lamb-wave oscillator. IEEE Transactions on Electron Devices 35: 735-743. 42 White, R. M. (1970). Surface elastic waves. Proc. IEEE 58: 1238-1277. 43 Ballantine et al. (1997), Ch. 3, pp. 89 ff. 44 Hardesty (1908).
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Table 4.1. Measurements of tectorial membrane stiffness
Experimenter Point stiffness (N/m) Elasticity (kPa)
Békésy (1960) 45 0.1–10
Zwislocki & Cefaratti (1989) 46 0.13 0.6
Hemmert et al. (2000) 47 0.1–0.6
Abnet & Freeman (2000) 48 0.07–1 7–100
Freeman et al. (2003) 49 0.06–0.34
Shoelson et al. (2003) 50 0.0004–0.003 0.7–3.9
Shoelson et al. (2004) 51 3–20
Measurements of otolith gel
Kondrachuk (1991) 52 1
Kondrachuk (2002) 53 0.01
The way in which liquid and gel layers combine to form low propagation-
speed layers prompts further investigation. Ballantine et al. (1997) discuss lossless
Scholte waves (their p. 124) and they mention that its phase velocity approaches zero
for very thin layers (which they show in their Fig. 3.46). A literature search on
Scholte waves returned a most astonishing reference – Hassan and Nagy (1997) – the
abstract of which describes the experimental finding of “a mode slower than the
slowest-order bending mode of the plates and [which] asymptotically approaches the
45 Békésy, G. v. (1960). Experiments in Hearing. (McGraw-Hill: New York). 46 Zwislocki, J. J. and L. K. Cefaratti (1989). Tectorial membrane II: stiffness measurements in vivo. Hear. Res. 42: 211-228. 47 Hemmert, W., et al. (2000). Mechanical impedance of the tectorial membrane. Midwinter Meeting, Florida, Association for Research on Otolaryngology http://www.aro.org/archives/2000/4838.html. 48 Abnet, C. C. and D. M. Freeman (2000). Deformations of the isolated mouse tectorial membrane produced by oscillatory forces. Hear. Res. 144: 29-46. 49 Freeman, D. M., et al. (2003). Dynamic material properties of the tectorial membrane: a summary. Hear. Res. 180: 1-10. 50 Shoelson, B., et al. (2003). Theoretical and experimental considerations for the study of anisotropic elastic moduli of the mammalian tectorial membrane. Mid-Winter Meeting, Florida, Association for Research in Otolaryngology 51 Shoelson, B., et al. (2004). Evidence and implications of inhomogeneity in tectorial membrane elasticity. Biophys. J. 87: 2768-2777. 52 Kondrachuk, A. V., et al. (1991). Mathematical simulation of the gravity receptor. The Physiologist 34: S212-S213. 53 Kondrachuk, A. V. (2002). Models of otolithic membrane–hair cell bundle interaction. Hear. Res. 166: 96-112.
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Stonely–Scholte mode” of a solid–liquid interface54. The mode, which involves the
motion of fluid in the gap between two symmetrically flexing plates, was first
predicted in 1965 by Lloyd and Redwood, and they called it a ‘squirting wave’ after
the way in which fluid squeezed between two closely spaced plates behaves. This
wave, which I call a symmetric Lloyd–Redwood (SLR) wave, applies well to the
anatomy of the cochlea’s sensing surface. Moreover, the predicted wave speed –
based on the dimensions and properties of the subtectorial space – gives a good
match to the speeds required to operate the SAW resonator. This prime candidate is
described in detail in the next chapter.
54 Hassan, W. and P. B. Nagy (1997). On the low-frequency oscillation of a fluid layer between two elastic plates. J. Acoust. Soc. Am. 102: 3343-3348. (p. 3343).
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