Download biomeasurement 2202

Appendices:
BIOMEASUREMENT 2202
Sensing - hearing
S2
Biological acoustics V - cochlea mechanics.
Fluid surface waves and the cochlear travelling wave
How does the cochlea achieve its frequency discrimination? The basilar membrane supports a
different form of wave motion quite analogous with an ocean surface wave. For this reason we
study the physics of surface waves in liquids. In particular - the relationship between speed and
wavelength.
Assumptions:
• Ignore particle nature of water and assume a continuum.
• Law of conservation of mass - equation of continuity
essentially, no mass created or destroyed within the flow
• Newton's second law of motion:
rate of change of momentum vector associated with a particle of fixed mass is equal to the
sum
of the forces acting upon it.
• Nonviscous flow (ignore friction within the fluid)
• fluid is incompressible
(liquids only, not gases)
Define the velocity potential  such that the velocity field is the gradient of , ie:
 = grad() = v
(local velocity vector)
(1)
and the above assumptions may be shown to lead to Laplace's equation for potential flows:


2 = 0
(2)
• Surface waves:
Consider now the case of irrotational flow in an ideal fluid (incompressible and inviscid) subject to
certain boundary conditions. The velocity potential must satisfy Laplace's equation, which we write
here in Cartesian coordinates (using only two dimensions for simplicity)
d2
d2
+
2
dx
dy2 = 0
(3)
We can solve this equation by the method of 'separation of variables', viz.,
 = X(x).Y(y).T(t)
(4)
Notice that the Laplace equation contributes no knowledge of the time function T(t).
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Appendices:
The solution separates into the product of three terms, one dependent only on x, one dependent only
on y, and one dependent only on t:
 = (A eikx + B e-ikx ) . (C eky + D e-ky ) T(t)
(5)
Solutions exists for any value of wave number k, but the x and y dependent terms interact through k.
A, B, C and D are constants determined by the initial conditions.
This is the solution for the velocity potential in an infinite medium. Precise solutions are determined
by the boundary conditions, at the top and the bottom of the fluid channel.
• Boundary conditions:
At the bottom (y = 0) the y - component of velocity is 0, so
d
vy = dy = 0
(6)
implying C = D, or
Y(y) = C cosh (ky)
(7)
The i in the exponent of X(x) implies a periodic component, so set B arbitrarily to 0. Then
 = A eikx cosh (ky) T(t)
where the constant C has been absorbed into A.
The solution for T is determined by the boundary condition at the surface, given by Bernoulli's
equation (see Streeter, 1961),
ps
d
+
+
dt


ps
F
1
2
   - F = 0
(8)
is the density of the liquid
is the pressure at the surface
is due to any external force (such as gravity).
which is basically a consequence of the conservation of energy at the surface. It relates to the
various forms of energy 'possessed' by an element of fluid in any part of the motion. Two essential
nonlinearities:
1
(i) the term in 2    is of second degree, and hence depends on vy2, and
(ii) the implicit assumption that it is applied at the surface, the position of which is not
known a priori but is to be determined.
2
Appendices:
Both may be ignored if we assume vanishingly small displacements of the surface - the small
amplitude approximation.
3
Appendices:
If gravity is the only surface force acting, and with the small amplitude approximation,
p
d
+ g = - dt (9)

with
p a constant (atmospheric pressure) everywhere,
 is the density of water
 is the displacement of the surface from its resting position
The derivative of the velocity potential evaluated at the surface then becomes
d
1 d2
vy = dy = - g dt2
(y = h)
Putting (x, h, t) = A eikx cosh (kh) T(t)
(10)
d2T
leads to dt2 + T g k tanh (kh) = 0
which has as a solution:
T = a eit + b e-it
where  is the time-frequency given by
2 = g k tanh (kh)
(11)
which is known as the dispersion relation for gravity driven surface waves.
100
6
4
2
10
Dispersion Relation
h = 10 m
h=1m
h = 1 cm
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
4
2
1
6
4
2
0.1
1
10
100
k
This equation cannot be inverted analytically, but for investigation may be calculated as if k and not
 were the variable which is freely manipulated.
The full solution for (x, y, t) is then:
(x, h, t) = A cosh (kh) ei ( kx - t )
(12)
which has the standard form for travelling waves, f(kx - t) but is not a solution to the general
wave equation since k and  are not linearly related, as the wave equation requires.
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Appendices:
• Fluid motion
The particle motion is elliptical with semi-axes a, b such that
cosh(ky)
a = h sinh(kh)
(15)
sinh(ky)
b = h sinh(kh)
(16)
kh << 1
kh >> 1
• Wave phase velocity
c =
g.tanh(kh)
k
(17)
Note: Shallow water - velocity dependent on
depth, independent of frequency
Dispersion in Gravity-Driven Water Waves:
Plot of Phase Velocity versus Frequency
Deep water - velocity independent of
depth, dependent on frequency
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Appendices:
• Mass-loaded surface wave
A similar analysis, but the surface is now defined to have the property of mass, thinly distributed
over the surface, and instead of gravity we define an elastic forcing term proportional to the surface
displacement from rest. The surface boundary condition, eq.(9) now becomes:
d2y
d
h + m dt2 = -  dt
(y = h)
(18)
where m is the mass per unit area and  = g . Again, we differentiate with respect to time and
replace d/dt with d/dy. In this way we find a new dispersion relation:
2 =
Notice that as k  ∞
k
m k +  coth (kh)
(19)
the wavelength becomes vanishingly small and since kh >> 1
2 
k
mk + 
(20)
i.e., there is a maximum frequency c above which the velocity becomes undefined. This is called
the cut-off frequency. Above this frequency no wave-motion propagates and disturbances die out
exponentially with distance.
c =

m
(21)
For sufficiently high frequencies, the phase velocity c may be approximated as
c =
 - m 2

(kh >> 1)
(22)
which  0 as   m 2 , which is the condition for resonance at the surface.
i.e., the wave propagates up to the frequency
at which the surface mass resonates with its
elasticity. Above that frequency no wave
propagation is possible.
Dispersion in mass-loaded-surface water
waves:
Plot of phase velocity versus frequency
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Appendices:
Travelling waves in the cochlea
Uncoiling the cochlea we have schematically:
The basilar membrane of the cochlea is a tapered ribbon-like membrane with the properties of
elasticity and mass. Hence, it will propagate a wave only up to a certain frequency, which depends
upon the mass and elasticity. Since the membrane elasticity is tapered (it is stiffest near the stapes
(basal) end and floppiest near the far (apical) end, that cut-off frequency will vary according to
position along the cochlea.
Analytical solutions are not possible for the response of the basilar membrane. Numerical solutions
are now readily available. Assume an elastic membrane with mass/unit area m, and an
exponentially-tapered elasticity, stiffest at the basal end, floppiest at the apical end.
Experiments indicate that different frequencies produce peak responses at different positions on the
BM.
Figure: position measured from stapes of peak response in basilar membrane vs frequency.
Approximately:
dpeak = 51.4 - 10.7 log(fpeak)
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for d in mm and f in Hz.
Appendices:
In order to calculate a meaningful solution,
we must include friction, or damping, in the
membrane. A typical solution is at right: The
top graph shows the amplitude of the
vibration as a function of distance along the
membrane, for a fixed stapes amplitude and at
three different frequencies. The bottom shows
the phase angle, relative to stapes, again for
three different frequencies.
Computer-modelled responses for
a guinea pig cochlea at three
different frequencies. Top panel is
amplitude of vibration, bottom
panel is phase. Passive cochlea
(no cochlear amplifier). Stimulus
SPL would be around the
threshold of hearing.
Look first at the phase angle. Phase increases with distance at an increasing rate, until it reaches the
point at which the membrane is locally resonant. It then stops abruptly and becomes approximately
constant, i.e., the wave changes from travelling to evanescent.
Evanescent waves are common in physics: in total internal reflection in
optics, tunnelling in quantum mechanics, in microwave waveguides. In each
case it refers to an exponentially-decaying 'leakage' of wave motion into a
region or frequency range where it cannot propagate normally.
The rate of phase accumulation increases along the membrane, i.e., the wavelength becomes shorter
as the elasticity decreases. Now look at the amplitude. The amplitude increases with distance along
the membrane, in order to maintain the rate of energy flow. (If the wave is slowing down, its
amplitude must increase so that the energy travelling along with the wave is the same everywhere,
i.e., energy does not increase at any point.) When it approaches the place at which the membrane is
resonant for the particular frequency, friction starts to attenuate the wave and the amplitude falls
rapidly. At the foot of the falling part of the curve, the amplitude switches abruptly to an
exponentially decaying wave at precisely the place at which the phase breaks to a constant value.
sinusoidal
evanescent
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Appendices:
Computer-modelled responses for a guinea
pig cochlea at three different frequencies. Top
panel is amplitude of vibration, bottom panel
is phase. Active cochlea (with cochlear
amplifier). Stimulus SPL would be around
threshold of hearing.
Damping is too great to permit sharply-tuned responses, but reducing the model's friction simply
leads to standing waves. Something else must be going on.
The cochlear amplifier
Evidence developed in the period 1978-1980 for a source of energy within the cochlea. The figure
below shows 'echoes' recorded in the external ear canal in response to click stimuli. Very low
amplitude, not present in a dummy cavity, saturate at higher intensities.
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Appendices:
About 30% of the population has spontaneous tones emitting from their ears. Known as
spontaneous oto-acoustic emissions.
Recent evidence shows the outer hair cells are responsible. They detect the vibrations of the basilar
membrane and 'kick back' in the precise phase to add to the motion. A form of positive feedback.
Assume the simple circuit below. The incoming voltage Vin, summed with the feedback voltage
after passing through amplifier of gain G. Simple analysis gives the output signal as:
Vout = G (Vin +  Vout)
Vout =
G
V
1 -  G in
Where  is the fraction of the output fed back to the input. So we can define the overall gain with
feedback as
=
G
1-G
voltage gain
Vout
Vin
Positiv e f eedback gain
G/(1 - G)
G=1
100
10
1
0.0
0.2
0.4

0.6
0.8
1.0
Note that as  increases, the gain increases rapidly and is very sensitive to small changes in p. This
is an example of positive feedback.
Adding positive feedback to the cochlea numerical model increases the amplitude only near the
cut-off point (why?). A sharp peak is produced with much better frequency selectivity. The phase is
relatively unaffected. Note the presence still of the cut-off point at membrane resonance.
• Active feedback, cochlear amplifier.
Movement of the basilar membrane
stimulates outer hair cells in the organ of
Corti. Stimulation of OHCs causes
motion of the cells, in an as-yet
undetermined way, and feeds mechanical
energy back into the cochlea. The inner
hair cells appear to be passive signallers
on the basilar membrane motion.
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Appendices:
Characteristics of positive feedback
• Sensitivity to parameters. If  changed by a small amount, amplitude changes by a much
larger amount. (Typically, a change of 10% in  might produce a fall in amplitude from
10,000 to 1,000.) This is the major cause of hearing impairment acquired during life. The
outer hair cells are damaged by loud sounds, mechanical trauma, drugs, disease, etc, by even a
relatively small percentage, causing a substantial loss in hearing sensitivity,
• Compressive nonlinearity. Because outer hair cell transduction saturates (Boltzmann
function), so does the positive feedback at higher intensities. The feedback fraction  is
reduced at higher intensities, so gain depends upon signal amplitude.
It is this compressive nonlinearity in the basilar membrane which explains the extraordinary
dynamic range of mammalian hearing. At low sound intensities the basilar membrane motion is
linear, that is, a doubling of stimulus pressure results in a doubling of basilar membrane vibration
amplitude. Then, at around 2W0 dB SPL (depending on species, sound frequency, condition of the
animal/human, etc) the vibration suddenly enters a compressive mode, where increases in SPL are
not matched by corresponding increases in basilar membrane vibration. In fact, the BM vibration
grows something like a 1/3rd to 1/8th power of stimulus pressure. So a change of stimulus pressure
of, say, 10 times might only double the basilar membrane vibration amplitude. So something like 80
dB of input range might correspond to an increase of only 3 - 10 times in the amplitude of vibration.
This is one major factor in the huge dynamic range of hearing.
The feedback pathway for hearing includes the outer hair cells (OHCs) which have a Boltzmann
relationship between displacement of their stereocilia and the current they pass. i.e., the current
saturates at higher BM vibration amplitudes. Saturation is obvious at larger displacement levels, but
even at small amplitudes of vibration there is a small deviation from linearity, towards saturation.
BM displacement enters its saturating region when OHC current saturates of the order of a part in a
few hundred.
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Appendices:
This nonlinearity of BM motion is responsible
for the characteristic shape of so-called
Fletcher-Munsen curves which show the
relative apparent loudness of sounds of
various intensities.
Loss of a few hair cells due to aging, exposure to noise, some oto-toxic drugs, etc, reduces the
amplification disproportionately (see  above). Hence, the sensitivity to soft sounds decreases
dramatically while the apparent loudness of louder tones is not greatly affected (recruitment).
People with some hearing loss have a reduced dynamic range of hearing, i.e., the range between the
softest sound they can hear and the loudest they can tolerate is reduced considerably.
Saturation of active feedback alters the loop gain and reduces amplification in the cochlea. The
result is an expanded dynamic range.
• Summary
In these lectures we have reviewed the essential physical acoustics necessary to understand and
quantify the various processes from sound source to cochlea. We considered both linear and
nonlinear wave motion and saw how the cochlea mechanical design separates the frequencies in a
complex signal. Ultimately the brain receives the message via the mechanosensitive ion channels in
the stereocilia of the organ of Corti.
References:
Kemp, D.T. (1978). Stimulated acoustic emissions from within the human auditory system.
J.Acoust.Soc.Am. 64: 1386-1391.
Pickles, J.O. (1982). An Introduction to the Physiology of Hearing. Academic Press London: (book)
Pickles, J.O. (1993). A model for the mechanics of the stereociliar bundle on acousticolateral hair
cells. Hear.Res. 68: 159-172.
Yates, G.K. (1992). The ear as an acoustical transducer. Acoustics Australia 21, 3, 1 - 5.
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Appendices:
Review questions:
• Show that /k is equal to f. Under what conditions is this true?
• What wavenumber k corresponds to a wavelength of 1 m? What wavelength corresponds to a
wavenumber of 1? If k increases 10-fold what happens to ?
• In the graph below equation (11) showing the dispersion relation for depths 1cm, 1m and 10m,
how would the relation c = f appear? If c were adjusted to best overlap the plotted curves what
values of c would be required?
• Shallow water approximation: For small x we can approximate tanh x ≈ x
In equation (11) this leads to
2 = g.h.k2 , and wave speed c = g.h .
(a) In a bath h ≈ 0.5 m. What wavelengths may be treated in the shallow water approximation?
(b) In Geographe Bay near Dunsborough, the water depth is sometimes less than a metre out to
several hundred meters offshore. What wavelengths may be treated by our approximation?
(c) When a series of waves spaced 10 m apart arrive at the edge of Geographe Bay what
happens subsequently as they travel towards the shore? What of the parameters f, c and ?
• In a linear medium such as air we have c = f where c is assumed constant. f plotted against 
produces a hyperbola. For gravity driven water waves find the corresponding expression
involving f,  and constants. What might the plot of f vs  be like?
• Consider a suburban swimming pool covered by a plastic cover (in contact with the water
surface) 2 mm thick. Estimate the maximum frequency that could be propagated across the
plastic membrane. Assume 0.9 kg m-3 for the density of the plastic.
• Consider a layer of castor oil on top of water left to settle creating a planar interface. What is the
speed of a wave travelling along the interface? See appendix 2).
[castor oil:  = 969 kg m-3, c = 1477 m s-1. water:  = 998 kg m-3, c = 1497 m s-1]
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Appendices:
1) Hyperbolic trigonometry
• Euler's formula allows us to write:
eia + e-ia
cos a =
and
2
sin a =
cos2a + sin2a = 1
So
for all a
• Hyperbolic sines and cosines can be defined as:
ea + e-a
cosh a =
and
2
sinh a =
cosh2a - sinh2a = 1
Then
eia - e-ia
2i
ea - e-a
2
for all a
4
Hyperbolic functions of x
3
2
sinh
cosh
1
tanh
0
0.0
• Series expansions:
Trigonometric
0.5
1.0
1.5
2.0
x
x3
x5
x7
sin x = x - 3! + 5! - 7! + ...
x2
x4
x6
cos x = 1 - 2! + 4! - 6! + ...
1
2
17
tan x = x + 3 x3 + 15 x5 + 315 x7 + ...
Hyperbolic
x3
x5
x7
sinh x = x + 3! + 5! + 7! + ...
x2
x4
x6
cosh x = 1 + 2! + 4! + 6! + ...
1
2
17
tanh x = x - 3 x3 + 15 x5 - 315 x7 + ...
tanh x = 1 - 2 e-2x - 2 e-4x - 2 e-6x - ...
14
good for small x
good for large x
Appendices:
2) Surface waves - more detailed treatment
Propagation of displacement of the free surface. Fourier components:
 = A cos(kx - t)
g

with k and  both positive. The wave speed is
c = k =
(wavelength << depth)
k .
Thus waves of longer wavelength travel faster. The propagation is therefore dispersive.
d
The group velocity is defined
cg = dk .
Frequency is  =
gk ,
so group velocity is
1
cg = 2 c
Fluid interface - between fluid with densities 1 and 2:
g  1 - 2 
c2 = k 

1 + 2
Surface waves (ie air above) correspond to the case 2 ≈ 0. Waves at an internal fluid interface
where 1 ≈ 2 travel much slower than at the surface.
• surface tension effects - capillary waves
Using the small amplitude leads to a dispersion relation:
2
 k3
= gk +
,

g
k
c2 = k +

Surface tension effects are important when  k2 >> g
Shorter wavelengths travel faster - in direct contrast to the case of gravity waves (above).
15
Appendices:
Example - 2 cm of water at 20˚C.
 = 0.073 N.m, g = 9.8 m s-2, = 1000 kg m-3, h = 0.02 m
f (Hz)
0
100
150
capillary
waves
0.4
c (m/s)
50
200
gravity
waves
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
(m)
Effect of depth of fluid:
f (Hz)
0
c (m/s)
0.8
50
100
150
200
20 cm
finite depth
gravity waves
10 cm
0.6
5 cm
0.4
2 cm
1 cm
0.2
0.1
0.2
0.3
(m)
16
0.4
0.5
Appendices:
3) Recent experiments (Hudspeth et al, 2000) found a surprising phenomenon while studying signal
transduction by the inner ear. A hair cell contains bundle of stiff fibres (stereocilia) that project from
the cell. The fibres sway when the surrounding inner ear fluid moves. They measured the forcedisplacement relation for the bundle by using a tiny glass fibre to poke it. A feedback circuit
maintained a fixed displacement for the bundle's tip and reported back the force needed to maintain
this displacement. The surprise is the complex curve shown in (c) in the figure. A simple spring has
a stiffness k = df/dx that is constant (independent of x). The diagram shows that the bundle of
stereocilia behaves like a simple spring at large deflections but in the middle it has a region of
negative stiffness! To explain their results they used a simple model with two linear springs and a
"trap door" mechanism.
a) Scanning electron micrograph of a bundle of stereocilia projecting from an auditory hair cell.
b) Model - pushing the bundle to the right causes a relative motion between two neighbouring
stereocilia in the bundle, stretching the tip link, a thin filament joining them. At large enough
displacement, the tension in the tip link can open a "trap door".
c) Force exerted by the hair bundle in response to imposed displacements. Positive f values
correspond to forces directed to the left in (b); positive x values represent displacements to the
right.
d) Mechanical model for stereocilia. The left spring represents the tip link. The spring on the right
represents the stiffness of the attachment point where the stereocilium joins the main body of
the hair cell. The two springs exert a combined force f. The model envisages N of these units in
parallel.
Ref: Nelson P, (2004). "Biological physics", W.H. Freeman
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