Egbert de Boer

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Waves in Cochlear Fluids
(Part I Introduction)
Egbert de Boer
Dept. of Experimental Audiology, Academic Medical
Center, Amsterdam, The Netherlands
Oregon Hearing Research Center, Oregon Health
Science University, Portland, OR, USA
2007
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• Since my retirement (in 1993) I have
• participated in several projects in hearing
• research in the USA. I have been both
• an experimenter and a theoretician.
• The data I am reporting here are from
• experiments that have been done in
• close collaboration with
• Dr. Alfred (Fred) L. Nuttall,
• (Oregon Health Science University, Portland,
  OR, USA)
• and his coworkers.
• The theoretical background of that work has
• mostly been developed at my home,
• in Amsterdam, The Netherlands
• (but not without help from many friends and
  colleagues!).

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Waves in cochlear fluids (Part I
Introduction) The title has been modeled
after Waves in Fluids Sir James
Lighthill Cambridge University Press, Cambridge,
UK, 1980
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• I need some structure in this talk.
• For that purpose I have chosen

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La cucina Italiana (the Italian cuisine)
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Gli antipasti, start of the meal process
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Waves .
Compression wave, shear wave, dilatation
wave, Raleigh wave, bending wave.
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A demo of compression waves will be run after
this session, or later.
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• Surface waves
• In the cochlea
• long waves,
• short waves,
• evanescent waves.

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Surface waves and compression waves have to be
combined in order to explain the paradoxical
direction of wave propagation in BONE
CONDUCTION. For some time I will restrict myself
now to surface waves.
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short wave
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long wave
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HIGH
MIDDLE
LOW
Békésys findings frequency is translated into
place
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In later years, starting with Rhode (1971,
1978), mechanical tuning of the cochlea has been
found to be much sharper.
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High frequencies (above 3 kHz)
1) From Rhodes measurements (1971-1978), tuning
of mechanical and neural responses of the cochlea
has turned out to be qualitatively similar
(except for a notch in neural tuning and
deviations at low frequencies). 2) Mechanical
tuning is extremely susceptible to
(physiological) damage. 3) There is a large
difference in mechanical response between the
live and the dead cochlea.
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High frequencies (above 3 kHz)

4) The cochlea is highly nonlinear.
Large-amplitude signals are
compressed. 5) Mechanical nonlinearity is
frequency dependent. 6) The nonlinearity is
susceptible to damage.
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Low frequencies (below 0.7 kHz)
1) At low frequencies the sharpness of tuning is
much less than at high frequencies. 2)
Compressive nonlinearity is small and of a
different type. (Occasionally, an expansive
nonlinearity has been reported.)
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Il primo piatto (la pasta), revealing the basic
facts
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Sebastian Well, I am standing
water. Antonio Ill teach you how to
flow. The Tempest, William Shakespeare, II, 1.
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Composite-spectrum files BM velocity versus
stapes velocity
Amplitude
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low level
Phase
high level
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• In constructing a model of the mechanics of the
  cochlea
• we can separate two aspects
• Hydrodynamics (fluid), and
• mechanics-dynamics of the cochlear partition.

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stylized 3-dim. model
cochlear partition
basilar membrane
Coordinates x longitudinal, y radial, z
vertical
OW
x-axis
RW
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The hydrodynamics of the model is simplified by
considering the fluid as ideal. The
hydrodynamical properties of the fluid depend
upon the shape of the walls and membranes that
enclose the fluid.
The cochlear partition is modeled as a plate, the
most important part of it is formed by the
basilar membrane (BM).
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If the basilar-membrane-organ-of-Corti assembly
is linear, its mechanics can be described by the
basilar-membrane mechanical impedance, abbreviated
BM impedance.
I am aware of the fact that all these steps are
simplifications or approximations.
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The basilar-membrane impedance has two
components the real part ----- has to do
with power dissipation and creation, and the
imaginary part ---- has to do
with stiffness and mass, and with wave
propagation
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The fundamental model equation (for the
hydrodynamics) reads
where
is the pressure near the BM (a vector), is the BM
velocity (a vector), is the stapes velocity (a
vector)
is the hydrodynamics matrix, is the stapes
propagation vector.
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A well-known approximation is that of a
long-wave or one-dimensional model the model
equation reads
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In the more accurate approach, the full
dimensionality of the enclosure is included in
part of the model, in the peak region, there will
occur short waves.
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The essence of the (forward) solution
When
is substituted,
can be computed.
This requires a matrix inversion.
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With modern-day computers this can be done to any
degree of accuracy, in a short time.
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On the basis of Rhodes high-frequency results,
Kim et al. (1980) posed that the cochlea is
locally active.
What does that mean?
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In this region the cochlear wave is
amplified, this is assumed to be accomplished by
the Outer Hair Cells (OHCs) in the organ of
Corti.
The cochlea is locally active.
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In more technical terms The real part of the
basilar-membrane impedance is negative over a
part of the x-axis (the longitudinal axis of the
cochlea)
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How do we find this basilar-membrane impedance?
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The inverse solution. Mathematical procedures to
find the impedance from a given response
function have been published by (amongst
others) Zweig (1991), de Boer (1995), Chadwick
et al. (1996), de Boer and Nuttall (1999), de
Boer et al. (2007).
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The fundamental model equation reads
where
is the pressure near the BM (a vector), is the BM
velocity (a vector), is the stapes velocity (a
vector)
is the hydrodynamics matrix, is the stapes
propagation vector.
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The essence of the inverse solution
When and are given,
When and are given,
can be computed.
With
can be found.
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A typical result
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alive
20 dB
dead
dead
20 dB
alive
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How to visualize activity?
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OHCs
Outer Hair Cell
Basilar Membrane (BM)
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x
stapes
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With the proper BM impedance function inserted,
the (three-dimensional) model can be made to
simulate a measured response. (We have called
that procedure resynthesis.) This is true for
a viable model (the impedance has to be locally
active) as well as for the response of the dead
cochlea (the impedance is passive).
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We have eaten (enjoyed, I hope) the first course
of the meal the basic facts.
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Il secondo piatto, here come the refinements
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What about nonlinearity?
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When stimulus level is varied, the derived BM
impedance varies, too
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20 dB
20 dB
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50 dB
50 dB
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70 dB
70 dB
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80 dB
80 dB
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100 dB
100 dB
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post-mortem
100 dB
100 dB
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Explanation
For stronger signals the OHCs tend to saturate
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A problem
Formally, the concept of impedance cannot be
used in a nonlinear system.
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The solution (for a system with 'weak
nonlinearity)
By using noise signals as stimuli the concept of
impedance can be extended and retained. The
impedance is to be considered as a kind of
average.
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Outer hair cells are assumed to be nonlinear
transducers.
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LINEAR
Outer Hair Cell
LINEAR
NONLINEAR
Basilar Membrane (BM)
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Nonlinear transduction function of a
stereociliary gate
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for very large signals (average)
Partial saturation in an outer hair cell
(output)
for very small signals
(input)
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This mechanism explains (even quantitatively) the
compression (or self-suppression) of strong
stimuli.

This mechanism also explains the generation of
Distortion Products (DPs).
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When two tones with frequencies and
are given, with , the lower
Distortion Products (DPs) have frequencies
, ,
, etc. etc., all below . The upper
DPs have frequencies, ,
, , etc. etc.,
all above . The principal DP has the
frequency .
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Present two tones to the cochlea, and measure the
spectrum of the primary tones and the DPs, at a
specific location of the BM there results an
organ front.
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Using the concept of activity, and including
the nonlinearity of OHCs, it is possible to
predict the amplitude of the DPs (not always
successfully!).
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x
location x
stapes
time
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nonlinear transformation
x
location x
stapes
time
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nonlinear transformation
x
location x
time
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The pressure components
with DP frequencies
generate a set of waves with the same frequencies.
A problem How do these waves propagate?
Each of these components will be suppressed
by other components, and may be
undergoing self-suppression.
In the computation this requires iteration.
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circles computed responses for DPs, at the
location of the hole.
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Prospero ... Their understanding Begins to
swell and the approaching tide Will shortly
fill the reasonable shore. The Tempest, William
Shakespeare, V, 1.
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We may feel satisfied But we have skipped many
problems. Two years ago I compiled a
list (perhaps not complete), and today I feel
that most of these problems have not been
solved. And I can add many more. Hopefully, we
will hear more about solutions in this Symposium!
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?
?
?
?
?
?
How are OHCs anchored at their apex?
?
How is the Reticular Lamina hinged?
?
?
How is the low-pass filtering of OHCs overcome?
How large is the damping in the sub-tectorial
space?
?
?
Can stereocilia produce large enough forces for
activity?
?
?
How does reciprocal operation of transducer
channels work?
?
?
How are the stereocilia coupled to the TM?
?
What part is played by stiffness variations of
OHCs?
?
What are the essential differences between low
and high frequencies?
?
?
?
?
?
?
?
?
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Are we nearing the end of the meal?
No, there is more to come.
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La Gorgonzola, cochlear waves and DPs (do we
smell something here?)
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We will next consider A) coherent reflection
(smooth or spicy?) and B) the propagation
direction of DP waves (spicy, or perhaps smelly?).
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Both subjects will be deferred to my second talk.
I repeat, many important subjects have not even
been touched upon, let alone, treated. I have
only tried to follow a straight line..
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Egbert de Boer
Waves in Cochlear Fluids
(Part I Introduction)
The End
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1) Academisch Medisch Centrum Amsterdam, NL
The End
OHSU
2) Oregon Hearing Research Center, Oregon Health
Science University, Portland,
Oregon, USA
EdB
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