Download biomeasurement 2202

BIOMEASUREMENT 2202
Sensing - hearing
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Biological acoustics VI - the active cochlea.
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.
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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.
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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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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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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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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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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Supplementary topics:
The inner ear - Physiological summary:
• contains the semicircular canals and the cochlea
• The semi-circular canals contribute little to hearing but detect for balance
• The cochlea contains all the mechanism for transforming pressure variations into properly
coded neural pulses.
• The cochlea is a small coiled tube of about 2.5 turns with length 35 mm.
• The hardest bone in the body makes up the wall of this tube
• In cross-section 2 membranes divide the tube near the middle, creating 3 distinct chambers:
• scala vestibuli
54 mm3
• cochlea duct
6.7 mm3
• scala tympani
37 mm3
• Because the cochlea duct is so small, we can approximate the cochlea as a tube divided
lengthwise by a single membrane.
• The scala vestibuli and tympani contain the same fluid. The fluid in the cochlea duct differs.
• 2 thin membranes separate the middle ear from the fluids of the cochlea:
• oval window - attached to stapes and seals the scala vestibuli
• round window - seals the scala tympani
• The cochlea duct is separated by Reissner's membrane from the scala vestibuli
• The entire blood supply comes from a single artery, the internal auditory artery, but 2 supply
roots may be identified within the cochlea:
• from vessels below the basilar membrane, outside the cochlea duct
• from a dense capillary network lying along the lateral wall of this duct
- stria vascularis . - both of these are quite far away from the hair cells
• Opposite the wall on which the stria vascularis is located is a bony promontory that
protrudes into the cochlea tube and to which the basilar membrane connects.
• Through the bony promontory run the nerve fibres which connect to the receptor elements the hair cells.
• Nerves enter the cochlea through a small opening in the bone, the cell bodies of the nerve
fibres are located in the spiral ganglia just inside the bony wall of the cochlea itself. Axons
of ganglion cells are twisted together to form part of the 8th cranial nerve.
• Within the cochlea duct and extending from above the plane of the basilar membrane is
another membrane called the tectorial membrane.
• Moving across the basilar membrane from the bony promontory to the opposite wall, you
meet in this order:
1) inner hair cells
2) the Arch of Corti
3) outer hair cells (in most mammals)
• The hair cells are oriented in a very regular and orderly fashion,
• there are 1 inner hair cell and 3 outer hair cells in each cross-section
• each inner hair cell is topped by about 50 stereocilia
• each outer hair cell is topped by about 80-100 stereocilia
• a single row of inner hair cells contains about 7000 cells
• 24,000 outer hair cells occur in several rows
• Outer hair cells appear to be more sensitive to low level sounds
4) supporting cells both below and next to the outer hair cells
5) finally a carpet of low cells as the basilar membrane stretches to the opposite wall of the
cochlea duct
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Supplementary topics:
• Sensory reception and transmission
In both vision and audition there are significant physical structures that bear strongly on the sensory
outcome. In both cases there is also a postreceptor network where signals from individual cells are
reduced to a smaller number of nerve fibre output channels. The postreceptor network performs
several functions:
• combining cell responses to improve sensitivity or recognise certain complex inputs
• feedback onto the reception process
• reduction in information to send on to the brain
The brain can interpret the environment by having impulses from known sources and by interpreting
the frequency with which these impulses arrive. In nerve impulse conduction there is an "all-ornothing" law which means that below the threshold level there are no impulses, above there are
impulses. We measure the intensity of the stimulus not by the size of the impulse, for they are all
alike, but by the frequency of the arrival of the impulses. As information from an environment is
filtered and amplified in a receptor, the coded information is conducted along afferent neurons to a
central nervous system where integration (summation) of different information takes place. Once an
action potential is produced in an afferent neuron associated with a receptor, it is propagated
without change along the length of the neuron until it reaches the end of that nerve cell axon, which
is closely associated with another neuron.
With one action potential (arrows in figure below), the graded postsynaptic membrane
depolarisation may not reach threshold and will rapidly decay. The small depolarisation is called a
"miniature excitatory postsynaptic potential" (mEPSP). If several occur in a short enough period to
add up to a sufficient amount of depolarisation, threshold is reached and an action potential results.
Figure: Illustration of the short-term additive effects of miniature excitatory postsynaptic
potentials (mEPSPs) on postsynaptic membrane depolarisation. Each mEPSP represents
release of the contents of a presynaptic vesicle.
Figure: The effect of the presynaptic potential on the postsynaptic potential in the squid
stellate ganglion is most pronounced over a relatively narrow range of change in
presynaptic potential (after Katz & Miledi, 1967).
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Supplementary topics:
Figure: Measurements of membrane potentials for receptor cells show that the receptor
potential is graded; i.e., it depends on the magnitude of the stimulus energy. Note also
that the receptor potential changes with time while stimulus energy remains constant.
Figure: Measurement of the receptor
potential of a stretch receptor shows that the
magnitude of the potential is a nonlinear
function of stimulus intensity (after Katz,
1950).
Figure: Models of the consequences of
coding sensory information in different ways.
Linear coding with a high slope would
provide high sensitivity with no information
at high intensities (saturation). Linear coding
with a lower slope would provide information
at high intensities but less sensitivity to
stimulus changes. A logarithmic function
represents a compromise between these
extremes.
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Supplementary topics:
• Additional hearing mechanism - bone conduction:
• Some sounds are heard through the vibrations of the skull which reach the inner ear.
• Hearing via bone conduction plays an important role in speech and singing.
• During voice production the perceived sound has 2 components - via bone conduction, and
through the surrounding air. Hence recorded sound, lacking the bone component, always
sounds unconvincing at first audition.
• Masking:
• When the ear is exposed to two or more different tones, often one of these may mask the others.
Masking:
- the upward shift in the hearing threshold of the weaker tone
by the louder tone
• Masking depends on the frequencies of the 2 tones. Pure tones, complex sound, narrow and
broadband noise all have different masking abilities.
• Conclusions from experiment:
• Pure tones close together in frequency mask each other more than tones widely separated
• A pure tone masks tones of higher frequency rather than lower.
• The greater the intensity of the masking tone the wider the frequency range of masking
• If 2 tones are widely separated in frequency little or no masking occurs
• Masking by a narrow band of noise shows many of the same features as masking by a pure
tone and more so for higher frequencies than lower
• Masking of tones by broadband (white) noise shows an approximately linear relationship
between masking and noise level. ie: and increase in noise by 10 dB causes an increase in hearing
threshold by 10 dB. Broadband noise masks tones of all frequencies.
• Forward masking - occurs when a sound is masked by a tone that ends a short time (20-30 ms)
before it. Forward masking suggests that recently stimulated hair cells are less sensitive that fully
rested cells.
• Backward masking - occurs when a tone is masked by a sound that begins a few ms later. A tone
can be masked by a sound that begins up to 10 ms later although the amount of masking decreases
as the time interval increases
• Under certain conditions masking a tone in one ear can be caused by noise in the other ear. This is
called - central masking.
a)
b)
Figures: typical forward masking response: a) masking intensity vs f for fixed amplitude fo;
b) hearing threshold vs f in the presence of fixed masking tone.
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Supplementary topics:
Sensitivity of the ear to intensity
The connection between the intensity of sensation and that of the corresponding physical stimulus
was first investigated by Weber for weights, and his conclusion, known as Weber's law, was that the
increase in stimulus necessary to produce the minimum perceptible increase in the resulting
sensation is proportional to the pre-existing stimulus. Thus if W is a weight producing a sense of
pressure and ∆W is the increase in weight which gives a just perceptible increase ∆s of the pressure
sensation, then:
∆W
∆s = K W
where K is a constant.
Fechner took this relation and integrated it to obtain:
s = K log W
This is known as the Weber-Fechner law and is more or less applicable to all sensations.
The sensation of light, like the sensation of sound, does not follow the Weber-Fechner law exactly.
A modification by Nutting giving satisfactory agreement at both high and low frequencies has the
form:
∆I
Io n
I = Pm + (1 - Pm)  I 
where Pm is the limiting contrast for large values of I, Io is the threshold intensity (at that frequency)
and n is a parameter depending on the frequency of light.
Knudsen (1923) showed that for sound one
also has:
∆I
Io n
I = F + (1 - F)  I 
where for a frequency of 1000 Hz, n = 1.05,
while at 200 Hz, n = 1.63.
Figure (left): Showing minimum perceptible
change in intensity by the human ear.
Numbers on the curve indicate the sensation
level of the test tone in decibels above
threshold.
Riesz (1928) found good agreement for sound of intensity greater than 60 dB above threshold with
differential sensitivity ∆I/I lying between 0.05 and 0.15. For I = 60 dB at 1000 Hz an increase of
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Supplementary topics:
intensity of 0.05 is just perceptible under the best conditions. this corresponds to a change in
sensation level of 0.2 dB. At this frequency about 370 distinct steps in loudness are perceptible.
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Supplementary topics:
Sensitivity of the ear to pitch.
Figure: Limits of audibility for normal ears
A: threshold of feeling; B: threshold of audibility
Human hearing ranges across ten or eleven octaves. Kö nig (1832-1901) found the lowest frequency
perceived is around 16 Hz. The upper limit is a strong function of age. He could hear to 23 kHz at
age 41, reducing to 20.48 kHz at age 57 and 18.4 kHz at 67. Measurements on the amplitude of the
tympanic membrane vibrations were performed by Wilska (1935). Calculations by Sivian and White
(1932) showed that the pressure due to thermal noise (Brownian motion) in the air is, between
1 kHz and 6 kHz, of the same order of magnitude as the pressure sensitivity of very sensitive ears.
This means that any further increase in sensitivity would be useless in these cases.
Figure: The circles show the amplitude of vibration of the eardrum at threshold, as
determined by Wilska. The curve represents the calculated amplitude of the air
molecules in a sound wave at threshold pressure. Where the ear is most sensitive, the
amplitude of vibration of the eardrum is less than the diameter of a hydrogen molecule.
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Supplementary topics:
Bé ké sy investigated thresholds at very low frequencies. Some kind of auditory sensation is
established for frequencies down to 2 or 3 Hz. Between 3 and 50 Hz there is evidence that the
process is quantal in nature. The most prominent step occurs at 18 Hz. The "tonal" character of the
sensation begins around 18 Hz and is established by 25 Hz.
Figure: The minimum audible
pressures for low frequencies. This
threshold curve shows the step-like
character which may indicate the
quantal nature of the process involved
(Bé ké sy, 1935).
In 1931 Knudsen made extensive measurements and concluded that sensitivity to changes in sound
frequency or pitch is much greater than to changes in intensity, being greatest for frequencies from
about 600 to 4000 Hz. Over this range the sensitivity ∆f/f is approximately 0.003 where ∆f is the
minimum perceptible change in the frequency f.
• Maximum sensitivity is about 0.0017 at 2000 Hz for a level of 70 dB.
• Above 500 Hz ∆f/f is fairly constant until rising beyond 8000 Hz.
• Below 500 Hz ∆f is fairly constant, causing ∆f/f to rise steadily.
NOTE: these results are for pure tones - sine waves with no harmonics. So, for example 60 Hz and
62 Hz are indistinguishable as pure tones. But with the presence of 4th partials (harmonics) the
difference is 8 Hz so can now be distinguished.
Figure: Variation of ∆f/f with frequency - using sensation level as a parameter.
These measurements also conform to the Weber-Fechner law and its modifications as seen above
for intensity response.
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Supplementary topics:
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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