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Hearing
In terms of performance, hearing is the most remarkable of our senses
The sense of hearing
Frequency analysis:
responds selectively to frequencies in range
20!20,000 Hz
Tom DUKE
Cavendish Laboratory, University of Cambridge
Sensitivity:
faintest audible sounds impart no more energy
than thermal noise: 4 zJ per cycle
Dynamic range:
responds and adapts over 12 orders of magnitude
of energy: 0!120 dB
PHYSBIO 2007 St Etienne de Tinée
Dynamic range
20µPa
0dB
20Pa
30dB
whisper
The ear
60dB
conversation
90dB
motorcycle
auditory nerve
120dB
pneumatic drill
rifle
displacement of air molecule at threshold of hearing ~ 0.1nm
external
middle
inner
Middle ear
Cochlea
.
Impedance matches the air-filled
ear canal with the fluid-filled cochlea
Provides an acoustic gain owing to
different areas of eardrum and stapes
footplate
basilar membrane
Cochlea
Organ of Corti
Theories of hearing
Theories of hearing
Helmholtz 1857
von Békésy 1923-47
Detection apparatus consists of a set of strings of differing length which
vibrate in resonance with the incoming sound
Careful experiments on cadavers showed that a wave travels along the
basilar membrane, and reaches a peak amplitude at a position that depends
on frequency
Notion of
Theories of hearing
Gold 1948
Argued that damping by the fluid in the cochlea would not permit resonant oscillation
and proposed that the ear is powered
Gold’s regeneration hypothesis
place code
Gold’s regeneration hypothesis
Regeneration hypothesis:
Otoacoustic emissions
The ear in not just a sound receiver; it also
spontaneously emits sounds at a very low level
An energy supply provides a feedback
proportional to velocity, and in phase with it
Critical point
near critical point, sharply tuned response at frequency
‘!The magnitude of the feedback we require is so large as to come
precariously close to cancelling the resistive losses. Some sort of
self-regulating device would have to exist!’
Hair cells in
the bullfrog
sacculus
Kemp ‘79; Manley & Köppl ‘98
Detection apparatus
Hair bundle is composed of ~50 ‘stereocilia’, which lean
against each other. Each sterocilium is a bundle of actin
filaments, surrounded by the cell membrane, which tapers
at the base. Adjacent stereocilia are connected by a fine
filament — the ‘tip link’
stereocilia
transduction
channel
tip link
+
K
kinocilium
(absent in mammals)
tip links
source: Hudspeth
Detection apparatus
Hair bundle is composed of ~50 ‘stereocilia’, which lean
against each other. Each sterocilium is a bundle of actin
filaments, surrounded by the cell membrane, which tapers
at the base. Adjacent stereocilia are connected by a fine
filament — the ‘tip link’
Mechano-chemo-electrical transduction
When bundle is pushed in direction
of tallest stereocilium,
increased tension in tip links
pulls open transduction channels
& admits K+
which depolarizes the membrane
& opens voltage-gated channels
at the base of the cell
to nerve synapse
tip link
Rapidity of transduction process
preserves information about
timing of the signal
Transduction current
Active detetion by hair bundles
Active bundle movement
Spontaneous hair-bundle oscillations
Crawford & Fettiplace ‘86; Howard & Hudspeth ‘87; Benser, Marquis & Hudspeth ‘96
In the correct physiological conditions, hair bundles actively oscillate
spontaneous oscillation ?
twitch
Self-tuned critical oscillators
Building a critical oscillator: inertial system
Camalet, Duke, Jülicher & Prost ‘00
Gold ‘48
Vibration sensor is a nonlinear mechanical system which can generate
self-sustained oscillations at a characteristic frequency
internal active force
amplitude
A feedback control mechanism maintains it on the verge of oscillating
Hopf bifurcation
critical point:
characteristic frequency:
control parameter
remarkable response properties at critical point
Building a critical oscillator: non-inertial system
An internal active process can cause a heavily damped mechanical system
to oscillate
Hopf resonance
force:
displacement:
active force with its own dynamics,
coupled to displacement
control parameter: C
bifurcation point:
Suppose effective elasticity can be made negative by changing control parameter C
•
stimulus at characteristic frequency:
Critical point at
gain diverges
for weak stimuli
Characteristic frequency
Critical Hopf resonance: single tone response
Hopf resonance
force:
Gain and active bandwidth depend on level of stimulus
displacement:
gain
control parameter: C
•
0 db
bifurcation point:
! fa
stimulus at different frequency:
20 db
if
active bandwidth
fc
frequency
Critical Hopf resonance: response in presence of noise
Canonical equation
Eguiluz et al. ‘00
Camalet, Duke, Jülicher & Prost ‘00
phase-locking
Response at ! = !c
for different f
r
! /!c
Self-adjustment to critical point
Self-tuned critical oscillations
Slow dynamics of control parameter, coupled to displacement, provides
negative feedback which automatically adjusts system to the critical point
x
"
C
Cc
C
Eg. Ca2+ concentration is control parameter
stimulus
Spontaneous fluctuations
Hair bundle response
Martin, Hudspeth & Julicher ‘01
Auto-correlation function
Martin, Julicher & Hudspeth ‘01
Response of a frog hair bundle forced by a microneedle
Linear response
Test of fluctuation-dissipation relation
Martin, Hudspeth & Julicher ‘01
Response function # defined by:
Martin, Hudspeth & Julicher ‘01
At thermal equilibrium, the
linear response is related to
the fluctuations
Hair bundle response
Two adaptation mechanisms
Martin & Hudspeth ‘01
-2/3
fast process
slow process
Ca2+ binding to
transduction channel
movement of
myosin-1C motors
~ 1 ms
~ 100 ms
Negative elasticity
Channel gating compliance
Martin & Hudspeth ’00
Instantaneous force response to an
applied displacement
Suppose channel incorporates a lever arm
opening of channel can substantially reduce the tension in the tip link
•
negative elasticity if
Channel gating compliance
Influence of motors
Physical basis of oscillations
channels closed
channels open
Ca2+ released from motors;
causes them to produce more force
Ca2+ binds to motors;
causes them to produce less force
Self-tuning mechanism ?
Tinévez & Martin ‘06
Tinévez & Martin ‘06
Maybe Ca2+ affects rigidity of tip-links, or size of channel movement
force/displacement
bundle position
motor position
transduction current
Alternative physical basis of self-tuned oscillations
Alternative physical basis of self-tuned oscillations
Vilfan & Duke ‘02
•
Oscillations generated by
interaction of Ca2+ with
transduction channels
Vilfan & Duke ‘02
•
Oscillations generated by
interaction of Ca2+ with
transduction channels
motors
Ca2+
frequency
depends on bundle geometry
Ca
Ca
•
An active role for the kinocilium ?
• kinocilium is motile
(Rüsch & Thurm ‘90)
+
K
kinocilium
tip link
• axoneme can vibrate at ~1000 Hz
(Kamimura & Kamiya ‘89)
• but spontaneous oscillations of
frog hairbundles still occur when the
bundle is knocked down (Martin)
detection
apparatus
Hearing in fruit flies
The antenna of mosquitoes sense both odorants and sounds
stereocilia
transduction
channel
Self-tuning accomplished
by movement
of molecular motors,
regulated by Ca2+
force
generator
Spontaneous oscillations and nonlinear response of antennae
level (dB)
Göpfert ‘03
Köppl & Manley ‘93
Spontaneous
otoacoustic
emissions in
reptiles
frequency (Hz)
Reptilian inner ear
Coupled critical oscillators
Köppl & Manley ‘93
.
Vilfan & Duke
bobtail lizard
mi-1
mi
mi+1
sallet: massive structure
!i-1
!i
!i+1
real for dissipative coupling
Spontaneous oscillations of a coupled chain
oscillation frequency
Cochlear mechanics
Ermentrout & Koppel ‘84
!i
Human cochlea
Békésy’s travelling wave
.
Incoming sound excites a slow travelling wave on the basilar membrane
cochlea
tectoral membrane
basilar membrane
inner hair cells
outer hair cells
•
location of peak depends on frequency
Place code
Cochlear travelling wave
Travelling wave: one-dimensional model
Zwislocki ‘48
membrane displacement h
oval window
pressure difference
difference in flows
round window
p = P1 - P2
j = J1 - J2
helicotrema
•
•
fluid flow
•
incompressibility
•
membrane response
sound sets fluid into motion
•
variation in flow rate is accommodated by movement of membrane
•
membrane acceleration is caused by difference in fluid pressure
wave velocity
Travelling wave: damped dispersive waves
decreases by two orders of magnitude from base to apex
as wave propagates, it slows down
& its amplitude increases
Wave peaks when damping becomes
significant
Location of peak depends on frequency
4.6 kHz
Sound level required to see
an observable response at a
given place
level (dB)
Stiffness
Tuning curve
1.3 kHz
370 Hz
frequency (kHz)
Tuning curve
Energy flow in the cochlea
Lighthill ‘81
In order to obtain a very localized peak, require:
•
only waves with frequency lower than
!c(x) carry energy past point x
•
as wave approaches characteristic place, velocity of energy transport falls
to zero
•
velocity drops so fast that there is time to dissipate all energy before wave
reaches characteristic place
energy piles up at characteristic place
Outer hair cells
Active cochlear partition
Kachar et al. ‘86; Ashmore ‘87
Critical oscillators are ranged along basilar membrane, and are positioned so
that they can drive its motion
•
characteristic frequency
diminishes from base to apex,
spanning the audible range
Partition is an excitable medium with a nonlinear active response
inner hair cells
outer hair cells
•
active oscillators negate friction and damping at the characteristic place
where the oscillator frequency matches the sound frequency
Active travelling wave: waveforms
Basilar membrane motion
Rhode ‘71; Ruggero et al. ‘97; Russell & Nilsen ‘97
BM response
is nonlinear
4.6 kHz
1.3 kHz
370 Hz
Basilar membrane motion
Energy flow in the cochlea
Rhode ‘71; Ruggero et al. ‘97; Russell & Nilsen ‘97
Lighthill ‘81
Cochlea is an unusual type of waveguide:
BM response
is nonlinear
1/3
1
•
only waves with frequency lower than
!c(x) carry energy past point x
•
as wave approaches characteristic place, velocity of energy transport falls
to zero
•
velocity drops so fast that there is time to dissipate all energy before wave
reaches characteristic place
energy piles up at characteristic place
Nonlinearities due to active amplification
A critical oscillator is ideal for detecting a single tone …
… but it causes tones of different frequency to interfere
Psychoacoustic effects
& their physical origin
Response to multiple tones:
Two-tone suppression
Two-tone suppression
f1
Psychoacoustics & auditory illusions
Range of phenomena associated with multiple tones: f1, f2, ...
Combination tones:
presence of a second tone diminishes
the neural response to the first tone
Tartini 1714
frequencies not present in the stimulus can be heard,
most prominently 2f1 - f2
f2
Residue pitch:
Schouten 1938
frequencies close to, but not necessarily equal to
"f = f2 - f1 can be heard
f1 & f2
suppression is greater if frequencies
are close
Consonance:
Pythagoras
two musical notes whose frequencies are in a
simple integer ratio sound pleasant
basis of diatonic scale
Two-tone suppression
Distortion products
Presence of second tone can extinguish the nonlinear amplification
=0
Nonlinearities create a characteristic spectrum of distortion products
! 0
Active travelling wave: two-tone response
Active travelling
wave:
two-tone
response
Andor
!2
!1
2 ! 1- !2
2!1-!2
!1
!2
distance along cochlea
frequency
component
location vibrating almost exclusively at 2!1-!2
distortion product can be heard
Neural representation of pitch and level
Auditory nerve: spontaneous activity
Liberman ‘77
Two competing theories:
pitch represented by the place in the cochlea where
nerves fire most frequently
Time code
pitch represented by time interval between spikes
threshold (dB)
level represented by firing rate at that place
number of units
Place code
level represented by number of nerves firing at an elevated
rate
spontaneous rate (spikes/sec)
Inference of pitch and level
First passage time to traversal of threshold
Jülicher, Andor & Duke ‘01
Model
•
a spike is elicited whenever the hair
bundle deflection traverses a threshold
•
there are hair cells with a wide range of
thresholds for each characteristic frequency
Algorithm
•
•
construct histogram of inter-spike intervals T,
summing over all hair cells in the cochlea
•
perceived tones correspond to peaks in histogram
and are assigned pitch 1/T
perceived level corresponds to height of peak
spontaneous rate (spikes/sec)
Temporal coding permits detection of stimuli
20 dB below rate-threshold
First passage time to traversal of threshold
Temporal coding permits detection of stimuli
20 dB below rate-threshold
Distortion products: spectra & waveforms
spontaneous
high level
stimulus
Spike timings: two tones
Two-tone stochastic resonance
Chialvo et al. ‘02
.
firing prob. density
high threshold neuron
interval / s
0.009s ~ 110 Hz
0.045s ~ 22 Hz
residue pitch
Dissonance: psychophysical experiments
Nature of dissonance
Kaestner 1909
Helmholtz ascribed dissonance to close
mis-matches in frequency between harmonics
5:4
Integer ratios of frequencies are preferred
when notes are played on musical instruments
4:3
For pure tones, there is no such preference,
but only a dislike of close frequencies
1:1
8:5
6:5 4:3
3:2 5:3
5:4
0
0.2
2:1
1
pleasantness
0.8
" 2 :1
0.6
0.4
0.2
pure tones
+ harmonics
0
Two tones can be clearly distinguished
if they differ in frequency by more than 15%
Close frequency mis-matches generate
complicated hair bundle responses
f2 / f1
loudness
1.03
0.8
time
1.10
The difficulty of inferring frequency
components from partial information
about a complex waveform results
in an indeterminacy of pitch
1.3
rough
inferred pitch
Two tones are interpreted ambiguously
if they differ in frequency by 5 - 15%
0.6
Dissonance: a physical interpretation
Pitch discrimination
Two tones are heard as a single tone
if they differ in frequency by less than 5%
0.4
!f / f
log(frequency)
1.2
1.1
1
1.30
1
1.1
1.2
f2 / f1
0.8
1
1.2
1.4
inferred pitch T -1/ f1
‘What distinguishes dissonances from consonances is not a greater or lesser
degree of beauty, but a greater or lesser degree of comprehensibility’
Arnold Schoenberg
1.3
1