TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of Engineering
The Zandman-Slaner School of Graduate Studies
THE PREDICTON OF TEOAE IN A ONE DIMENTIONAL
HUMAN EAR MODEL
A thesis submitted toward the degree of
Master of Science in Biomedical Engineering�
by
Yaniv Halmut
March 2006
ii
TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of Engineering
The Zandman-Slaner School of Graduate Studies
THE PREDICTON OF TEOAE IN A ONE DIMENTIONAL
HUMAN EAR MODEL
A thesis submitted toward the degree of
Master of Science in Biomedical Engineering
by
Yaniv Halmut
This research was carried out in the Department of Biomedical Engineering
under the supervision of Prof. Miriam Furst-Yust
March 2006
iii
Acknowledgements
Thanks and gratitude to Prof. Miriam Furst-Yust for her devoted and patient guidance through
the entire research. Her knowledge, thoroughness and patience helped me immensely in this
work.
I would like to thank my fellow researchers at the Auditory Signal Processing Laboratory:
Udi Shtalrid, Tomer Goshen, Ram Krips and especially Naom Elbaum for his insights and
long talks we shared.
I can't be grateful enough to my wife for all her support during the passing year.
Yaniv Halmut
March 2006
iv
Abstract
In the last 30 years hearing research has advanced our understanding of the auditory system
by many folds. Although many new phenomena have been discovered and many new facts
unraveled we are still not able to fully explain the data we collect from the living ear.
Since Kemp discovered the otoacoustic emissions (OAEs) in 1978 their importance to the
field of hearing diagnostics has grown significantly. Today all newborn babies undergo
auditory tests that check their response to clicks. There are several types of OAEs, each
having its' own unique properties. This phenomenon is clinically measurable but has not yet
been reproduced convincingly in models.
A one-dimensional cochlear model with embedded outer hair cells (OHC) was recently
developed by Cohen and Furst (2004). This model incorporates a basilar membrane (BM)
model with an outer hair cell model, which control each other through cochlear partition
movement and pressure.
In this work we added a middle ear model to the cochlear model in order to simulate the
generation of transient evoked otoacoustic emissions (TEOAE). The outer and middle ears are
mimicked by a simple mechanical model.
Only when nonuniformity was introduced in one of the mechanical parameters, TEOAE were
produced by the model. According to physiological data, nonuniformity in the OHC gain
seems most reasonable. Nonuniformity can account for the variability in the OHCs population
density along the cochlear partition. Increasing the OHC nonuniformity produced an increase
in TEOAE level which grew to infinity. Introducing nonlinearity in the basilar membrane
resistance limited the TEOAE level growth. The combination of nonlinearity and
nonuniformity allowed the generation of amplitude stabilized standing waves that formed
waves resembling spontaneous otoacoustic emissions (SOAEs) in the ear canal.
The simulated TEOAEs resemble measured TEOAEs in human subjects. The average level of
TEOAE was determined by the mean gain of the OHC. Thus a damaged ear that was
simulated with low OHC gain did not produce TEOAEs. The average power spectrum of the
simulated TEOAE was band limited between 2 and 5 kHz, where the OHC are most effective.
v
Our results indicate that TEOAEs are generated due to the combination of nonuniformity and
nonlinearity in the cochlea.
vi
Table of contents
Acknowledgements ...................................................................................................................iii Abstract......................................................................................................................................iv Table of contents .......................................................................................................................vi List of symbols ........................................................................................................................viii List of Figures............................................................................................................................ix 1. Introduction to ear anatomy and otoacoustic emissions.....................................................1
Introduction ............................................................................................................................1 Ear anatomy............................................................................................................................1
The outer ear.......................................................................................................................2 The middle ear ....................................................................................................................2 The inner ear.......................................................................................................................3 The traveling wave (TW) ...................................................................................................5
OtoAcoustic Emissions (OAEs) .............................................................................................6 Types of OAEs ...................................................................................................................6 TEOAE (Transient Evoked OtoAcoustic Emission) ..........................................................8 DPOAEs (Distortion Product OtoAcoustic Emissions) .....................................................9 SFOAE (Stimulus Frequency Otoacoustic Emissions) ....................................................11 SOAEs (Spontaneous OtoAcoustic Emissions) and SSOAEs (Synchronized SOAEs)...11 OAE fine structure............................................................................................................11 OAE Screening Tests .......................................................................................................12 OAE screening protocols..................................................................................................13
2. Theory of OtoAcoustic Emissions....................................................................................14 The local oscillator theory ....................................................................................................14 Coherent Reflection Filtering ...............................................................................................15 The global standing wave theory (SOAEs) ..........................................................................17 DPOAE generation theory....................................................................................................20
DPOAE specific models...................................................................................................22 OAE generation summary ....................................................................................................23
Different representation of the same physics ...................................................................23 OAE properties .................................................................................................................24
3. Motivation for the present study.......................................................................................28 4. Model details ....................................................................................................................30
Middle Ear Model.................................................................................................................30 The Cochlear Model .............................................................................................................32 The Outer Hair Cell Model...................................................................................................34 Boundary and Initial conditions ...........................................................................................36 Numerical Solution...............................................................................................................38 Fourth order Runge-Kutta ....................................................................................................40 Adaptive Step Size ...............................................................................................................41 Time versus Frequency results comparison .........................................................................42 Spatial artifacts .....................................................................................................................44
5. Results ..............................................................................................................................46 The Model Output: Basilar Membrane velocity along the cochlear partition......................46 Simulated Audiograms .........................................................................................................47 Simulated TEOAEs ..............................................................................................................49 Simulated click response ......................................................................................................50 Nonuniformity ......................................................................................................................51 �(x) “roughness” ...................................................................................................................52
vii
Simulated Otoacoustic emissions .........................................................................................54 Simulated TEOAE spectrum ................................................................................................56 Nonuniform cochlea Audiograms ........................................................................................56 "Linear" versus "Nonlinear" response processing................................................................58 Tone burst responses ............................................................................................................60 Localized � "roughness" .......................................................................................................61 � mean...................................................................................................................................63 Nonuniformity and Energy Explosion..................................................................................64 Introducing Nonlinearity ......................................................................................................65 Stimulus magnitude influence on nonlinear model responses .............................................67
6. Discussion.........................................................................................................................70 Impact on the field of OAE model research.........................................................................76 Future research possibilities .................................................................................................77
References ................................................................................................................................78 Appendix A – Auditory research and model history................................................................84
Auditory research history .....................................................................................................84 OHC research progress.........................................................................................................85 Middle ear research ..............................................................................................................87 Modeling history of the Cochlea ..........................................................................................89 1D vs. 2D, 3D.......................................................................................................................90 Enhanced one dimensional cochlear models ........................................................................91
viii
List of symbols
BM basilar membrane
CA cochlear amplifier
CEOAE click evoked otoacoustic emissions
CF characteristic frequency
DPOAE distortion product OAEs
ME middle Ear
OAE otoacoustic emission
OHC outer hair cell
OW oval Window
SFOAE stimulus frequency OAEs
SOAE spontaneous otoacoustic emissions
TEOAE transient evoked otoacoustic emissions
TW traveling wave
ix
List of Figures
Figure �1-1 - A diagram of the auditory system consisting of the outer, middle and inner ears .1 Figure �1-2 - The middle ear........................................................................................................2 Figure �1-3 - The mammalian cochlea as an uncoiled cochlea....................................................3 Figure �1-4 - The basic structure of the cochlear partition ..........................................................4 Figure �1-5 - An expanded view of the organ of Corti ................................................................5 Figure �1-6 - Traveling wave example.........................................................................................6 Figure �1-7 - A TEOAE response from a newborn baby.............................................................9 Figure �2-1 - Multiple internal reflections within the cochlea...................................................17 Figure �2-2 - DPOAE generation visualized..............................................................................21 Figure �4-1 - A simple middle ear mechanical model ...............................................................30 Figure �4-2 - Cochlear Model Geometry ...................................................................................32 Figure �4-3 - An equivalent electrical circuit model of the outer hair cell ................................35 Figure �4-4 - The Fourth order Runge-Kutta method. ...............................................................41 Figure �4-5 – Time versus Frequency algorithm comparison....................................................43 Figure �4-6 – Spatial artifacts ....................................................................................................44 Figure �4-7 – Spatial resolution .................................................................................................45 Figure �5-1 - Model response to a click and a sine wave...........................................................46 Figure �5-2 - Filtered stimulus response ....................................................................................47 Figure �5-3 - Audiograms simulated by the linear model..........................................................49 Figure �5-4 – Example of a simulated TEOAE in a uniform cochlea. ......................................50 Figure �5-5 – Model response to a click simulated in a “smooth” cochlea ...............................51 Figure �5-6 – Responses with nonuniform R or S .....................................................................52 Figure �5-7 – Otoacoustic emission (OAE) simulation .............................................................53 Figure �5-8 - Examples of simulated TEOAE outputs...............................................................54 Figure �5-9 – Energy increase due to larger “roughness”..........................................................54 Figure �5-10 – Emission energy histograms ..............................................................................55 Figure �5-11 – Simulated TEOAE spectrum .............................................................................56 Figure �5-12 – Audiograms of nonuniform �(x) selections .......................................................57 Figure �5-13 – Response to a click by normal and partially impaired ears ...............................57 Figure �5-14 - Linear vs. Nonlinear click responses..................................................................59 Figure �5-15 – Linear vs. Nonlinear tone burst response spectrum...........................................60 Figure �5-16 – “Nonlinear” TEOAEs from the literature ..........................................................61 Figure �5-17 – Localized � “roughness” ....................................................................................62 Figure �5-18 – Nonlinear spectral response vs. E(�) and their corresponding audiograms.......63 Figure �5-19 – Total cochlear energy vs. different � s. .............................................................64 Figure �5-20 – Energy explosion that occurs when a large � is used. .......................................64 Figure �5-21 - Large � simulation combined with constricting nonlinear damping..................66 Figure �5-22 - � vs. Total cochlear energy ................................................................................67 Figure �5-23 – Different stimulus magnitudes in the nonlinear model......................................68 Figure �5-24 – Signal in noise....................................................................................................69 Figure �6-1 - The profile of the Cochlear Amplifier by Nobili et al..........................................72 Figure 7-1 – The effect of the middle ear on OAEs .................................................................88
1
1. Introduction to ear anatomy and otoacoustic emissions
Introduction
A good understanding of the underlying anatomy of a system enables researchers to develop
better models that better mimic the inner working of the system. The auditory system is no
exception and in the first part of this chapter we will give an introduction to help us better
understand its anatomy. The second part will be dedicated to the characteristics of
OtoAcoustic Emissions (OAEs). The next chapter is dedicated to current theories in the field
of auditory models.
In the passing century researchers have unveiled many aspects of the auditory system through
observing and modeling. This past decade has seen many new discoveries and many new
theories. The auditory system still exhibits phenomena that are not yet fully understood and
there are still several "rivaling" theories that haven't been proven yet. These are very
interesting years for researchers in the field of OAE analysis.
Ear anatomy
The mammalian ear is traditionally divided into three regions, the outer, middle and inner ear
regions (Figure �1-1). The task of the outer and middle ears is to translate the movement of air
molecules in the environment to the movement of fluid inside our body. The inner ear is the
part of the auditory sensor that transforms fluid movement to nerve excitation.
Figure �1-1 - A diagram of the auditory system consisting of the outer, middle and inner ears
2
The outer ear
The outer ear consists of the pinna and the external ear canal. The pinna functions as a sound
gathering horn. It intercepts sound waves from free space and funnels them via the external
ear canal to the eardrum. When sound waves hit the eardrum they impart kinetic energy in the
form of mechanical vibrations.
The eardrum (called the tympanic membrane) forms the boundary between the outer and
middle ears.
The middle ear
Connected to the eardrum is a chain of three tiny bones (the ossicle chain) that bridge the
space between the outer and inner ears. The three ossicles (malleus, incus and stapes) are
located in the air filled middle ear. They are the smallest bones in the human body.
The first major challenge the auditory system has to overcome is the task of transmitting
sound from the air medium to the fluid medium of the inner ear. The ratio of the acoustic
impedances of water and air is 3880:1.3, meaning that 99.9% of the sound would be lost if the
ear was a simple air to water interface.
As we see in Figure �1-2 the bone chain, in the air-filled middle ear, serves to “match” the
impedances of air (of the outer ear) with the impedance of fluid (of the inner ear).
Figure �1-2 - The middle ear is an impedance “matcher”, amplifying the incoming sound waves
so that they may enter the fluid-filled inner ear without significant loss of acoustic power
3
The middle ear is a mechanical energy transformer. The three ossicles work like a lever
system that increases the force transmitted from the eardrum to the stapes by decreasing the
ratio of their oscillation amplitudes. The footplate of the stapes acts like a small piston on the
cochlear fluid through a membranous connection that seals the oval window of the cochlea.
The buckling motion of the tympanic membrane decreases the velocity two-fold and increases
the force two-fold, changing the impedance ratio four-fold. Thanks to the large surface ratio
between tympanic membrane and oval window (~35), and the ossicle system lever gain
(~1.32), the forward impedance gain is about 30 dB. In this way the middle ear mechanism
couples the low acoustic impedance of air to the high mechanical impedance of the cochlea.
The oval window forms the boundary between the middle and inner ears. The vibrations of
the stapes cause the oval window to vibrate, resulting in fluid displacement inside the cochlea.
Filtering effects due to resonances of the middle ear cavity and mechanical parameters of the
ossicle system produce a peak between 1 and 2 kHz. The transmission of sound energy
through the middle ear, in humans, is most efficient at frequencies between 0.5 to 4 kHz.
The inner ear
The inner ear, also called the cochlea, consists of a fluid-filled duct coiled as a snail shell or
corkscrew. The propagation of sound waves in the cochlea is almost exactly as it would be in
a straight cochlea or an “uncoiled” one (Figure �1-3).
Figure �1-3 - The mammalian cochlea as an uncoiled cochlea having longitudinal, vertical and radial
dimensions
The perilymphatic space has the shape of an elongated U, the top arm of which is called scala
vestibuli and the bottom arm which is called scala tympani. The space between the two arms
of the mammalian perilymphatic space is the endolymphatic space, labeled scala media. An
4
extremely thin Reissner's membrane separates the scala media from the scala vestibuli. The
cochlear partition, a flexible structure that contains the sensory hair cells, separates the scala
media and the scala tympani. At the apical end is the helicotrema, a short duct connecting the
two perilymphatic scalae. Thus, when the stapes pushes the oval window inward, the U-
shaped column of perilymph is free to slide through its casing and push the round window
outward.
The round window plays an important role in releasing cochlear fluid pressure caused by
stapes displacement thereby greatly reducing the cochlear input impedance. Such movements
result in pressure differences between both sides of the basilar membrane causing the flexible
cochlear partition to vibrate.
The region of the cochlea adjacent to the oval window is called the base and the region
farthest away from the stapes is appropriately named the apex.
Figure �1-4 - The basic structure of the cochlear partition
Forming the basic platform of the cochlear partition (Figure �1-4) is the basilar membrane,
which is attached on one side to the bony spiral lamina and on the other side to the spiral
ligament. The basilar membrane is narrower and thicker in the base than it is in the apex.
These longitudinal differences in the structure of the basilar membrane are presumed to
account in large part for the different resonant measured at different points along the cochlear
partition. Resting on the basilar membrane is a small but complicated superstructure, known
as the organ of Corti (Figure �1-5), which contains the sound-sensing cells. The tectorial
5
membrane extends from the lip of the spiral limbus to overlie the apical surface of the organ
of Corti.
Figure �1-5 - An expanded view of the organ of Corti
The sound sensing cells are called hair cells because they appear to have tufts of hairs, called
stereocilia, protruding from their top. The hair cells are divided into inner and outer hair cells.
The inner hair cells (IHC) form a single row, running from base to apex, whereas the outer
hair cells (OHC) form three to five rows. In humans, there are about 3,500 inner hair cells,
each with about 40 stereocilia and 15,000 outer hair cells, each with around 140 stereocilia
protruding from them. When the basilar membrane moves up and down, a shearing motion is
created. Thus, the tectorial membrane moves to the side, relative to the tips of the hair cells.
As a result, the stereocilia of the hair cells move and rotate. The movement of the stereocilia
leads to the flow of an electrical current through the hair cells, which leads to the generation
of action potentials. These potentials give rise to nerve spikes in the neurons of the auditory
nerve. The inner hair cells act as transducers and translate the mechanical movement into
neural activity. The outer hair cells change their length and size due to these potentials, and
thus affect the physical properties of the basilar membrane.
The traveling wave (TW)
Fluid motion in the basal regions of the scala vestibuli and tympani displaces the basilar
membrane (BM). Initially only the basal part of the BM moves, but induced transverse
oscillations of the BM begin to propagate apically. Oscillatory exchanges occur between fluid
motion energy and the energy held in elastic BM displacement. Adjacent BM sections are
excited resulting in a traveling wave.
6
The traveling wave (TW) conveys stimulus energy towards the apex at less than 1/100th of
the speed of sound in air. Wave amplitude increases with distance along the BM, and reaches
its maximum at the place where the force of inertia equals and cancels the elastic restoring
force (place of resonance). Because inertial forces increase with frequency, the place along
the BM at which this peak in TW amplitude occurs is progressively nearer to the base for
higher frequency stimuli. The overall result is an asymmetric peak of excitation for each
frequency component (Figure �1-6). The TW envelope represents the excitation intensity
applied to the organ of Corti as a function of distance along the length of the cochlea. The
organ of Corti mechanism then converts BM motion to fluid motion across the IHC
stereocilia, leading to neural excitation. More about the history of auditory research is
described in Appendix A.
Figure �1-6 - Traveling wave example
OtoAcoustic Emissions (OAEs)
Types of OAEs
Otoacoustic emissions (OAEs) are sounds which can be recorded by a microphone fitted into
the ear canal. OAEs are essentially divided according to their way of stimulation. Otoacoustic
emissions that occur without any stimulus are called Spontaneous OtoAcoustic Emissions
(SOAEs). OAEs that are evoked by some sort of stimulus presented to the ear are named
Evoked Otoacoustic Emissions (EOAEs) and are divided into three groups. Transient Evoked
OtoAcoustic Emissions (TEOAEs) are created using stimuli with transients; i.e. clicks and
tone bursts. Distortion Product OtoAcoustic Emissions (DPOAEs) are generated by the use of
a stimulus containing more than one frequency component. Stimulus Frequency OtoAcoustic
Emissions (SFOAEs) are generated using continuous pure tones.
7
The oscillatory sound pressure waveform seen in the outer ear corresponds to the motion of
the eardrum being pushed backwards and forwards by fluid pressure fluctuations generated
inside the cochlea. The response is long and complex because responses from different parts
of the cochlea arrive at the ear canal at different times and at different frequencies. Several
different cochlear locations may contribute to a single frequency component of an OAE and
these may fortuitously summate or interfere with each other.
Otoacoustic emissions are intimately related to the status of the cochlea and they provide the
researcher and clinician with noninvasive tools to peer into the inner ear with an acoustic
microscope. Today OAE screening is widely used in newborn hearing screening programs
and can be used to monitor the effects of treatment.
Individual healthy ears differ greatly in the level and the spectrum of the OAEs they exhibit.
Stimuli of slightly differing frequency or spectral composition can give rise to quite different
OAE patterns. Taking an ‘average’ OAE characteristic over a range of stimuli provides a
more meaningful description of cochlear status, but even so the intensity of OAEs alone is a
very imperfect index of cochlear status. The ‘frequency’ at which an emission can be evoked
is more significant. OAEs are frequency-specific responses and tend to emerge only in
frequency bands where hearing is near normal. This fact may provide a useful pointer to
normally and abnormally functioning parts of a cochlea.
Changes in cerebrospinal fluid pressure induced by posture changes affect SOAE frequency
and evoked OAE intensity, probably by their influence on cochlear fluid pressure and
stapedial position. Drugs known to depress hearing, including aspirin and quinine, also
depress OAEs, and loop diuretics known to depress the endocochlear potential also depress
OAEs. OAEs also exhibit a physical analogue of ‘masking’ where the perception of one
sound is blocked by another. This may indicate that some forms of masking originate
preneurally in the cochlea. Tracing the suppression of an OAE response to one tone by
adjusting the intensity and frequency of a second suppressor tone allows an OAE suppression-
tuning curve to be constructed. The sharpness of such curves confirms the close association
between OAEs and auditory function, and demonstrates that sharp mechanical tuning is
present at the cochlear level.
Because the emission pressure is measured by a microphone inserted into the ear canal, where
the stimulus is also present, the measured signal contains both emission and stimulus.
8
Somehow these two signals have to be separated. There are essentially two ways to do this: in
the time domain, or in the frequency domain. Separating stimulus and emission in the time
domain is done by using a stimulus of a very short duration (a few milliseconds). The signal
measured by the microphone is then divided in time into a stimulus part and an emission part.
Examples of such delayed evoked otoacoustic emissions are click evoked otoacoustic
emissions (CEOAE). An example of separation of stimulus and emission in the frequency
domain are distortion product otoacoustic emissions (DPOAE) that result when a stimulus
containing more than one frequency component is presented to the ear. The formation of
distortion product frequencies in the cochlea was known from psychophysical measurements
for a long time, but they were not measured in the ear canal until after the discovery of OAEs.
The following section describes the basic characteristics of the four different OAE,
categorized according to their eliciting stimuli.
TEOAE (Transient Evoked OtoAcoustic Emission) / CEOAE (Click
Evoked OtoAcoustic Emission)
TEOAEs are OAEs that are generated in response to clicks, i.e. impulses or tone burst. These
OAEs are also called “Kemp Echoes” after David Kemp who discovered the phenomenon in
1978.
The recorded response is split into frequency bands and analyzed. TEOAE responses are
strongest and easiest to detect in the primary speech frequency band, 1–4 kHz. In young ears,
TEOAEs extend up to 6–7 kHz, but many clinically normal adult ears give weak TEOAEs
(less than 3 dB SPL), with no substantial response above 4 kHz.
An example of a clinical recorded TEOAE is presented in Figure �1-7. In the upper right
corner the stimulus response is plotted. We can see that the stimulus response has a much
larger magnitude than that of the OAE signal. The main window depicts the much smaller
emission signal with the stimulus artifact discarded (the first 3ms post-stimulus). The y-axis is
scaled so as to create a clear image of the response. In this example the OAE signal lasts for
about 12ms post-stimulus.
9
Figure �1-7 - A TEOAE response from a newborn baby.
The time delay between the stimulus and the response allows the examiner to isolate the
response. TEOAEs detected from normal ears mirror the spectral properties of the stimulus
(Glattke & Robinette, 2002). Although clicks are ‘wide-band’ stimuli, exciting the whole of
the cochlea, TEOAE responses can give a frequency specific indication of cochlear status.
When employing a click stimulus the TEOAE spectrum will be that of the click, or broad
band. If a tone burst is used to elicit a TEOAE, then the response will mirror the frequency
composition of the tone.
The TEOAE stimulus typically is presented at an overall level of 80 dB peak SPL. Because
the energy in the stimulus is spread over a broad frequency range, the energy at any individual
frequency is about 35 dB below the overall level. That is to say, the spectrum level of the
energy in the transient stimulus is about 35 dB below the overall level. The stimulus,
therefore, is at about 40-45 dB SPL for any specific frequency.
Because TEOAEs are highly sensitive to cochlear pathology and dysfunction, TEOAEs have
found wide-spread application in newborn hearing screening programs. Although no universal
standard exists, the measures involved in the determination of whether or not a TEOAE is
present are reproducibility and signal to noise ratio.
DPOAEs (Distortion Product OtoAcoustic Emissions)
The healthy ear produces OAEs not only in response to clicks, but to any sound applied to the
ear. A common way to record DPOAEs is to present two continuous signals, called primary
10
tones, and analyze the spectrum of the sound detected in the external ear canal. DPOAEs are
relatively easy to extract because they appear at frequencies that can be exactly predicted
from the frequencies of the primary tones.
Nonlinear intermodulation between two tones is a purely mechanical process and distortion
products satisfy the frequency relationship 1 2 1( )dpf f n f f� � � where n is any positive or
negative whole number. The distortion components can be separated from the stimuli by
signal frequency analysis. The strongest component 1 2 1 1 2( ( 1)( ) 2 )dpf f f f f f� � � � � � is
used as an indicator of cochlear status. The most robust distortion product (DP) occurs
where 2 11.2f f� � . For example, the ear's response to primary tones at 2000 and 2400 Hz will
produce a robust distortion product at 1600 Hz.
DPOAEs offer a wide frequency range of observation (up to 10 kHz) in adults. More
powerful excitation with continuous tones allows DPOAEs to be recorded with moderate
losses when no TEOAE can be detected. However, DPOAE recordings provide no greater
frequency specificity than TEOAEs despite the use of pure tones. Even when proper controls
are in place DPOAE techniques do not produce results that have more frequency specificity
than TEOAE methods and they are not more robust than TEOAE and thus can not be obtained
under more challenging recording conditions.
Primary tones used to elicit DPOAE are presented at stimulus levels of 50 to 70 dB SPL,
depending on the interest of the examiner. Healthy ear canal distortion levels can be above 20
dB SPL. DPOAE generation is much reduced and usually absent if there is significant sensory
hearing loss.
The clinical significance of DPOAEs has not been fully evaluated. Clinical DPOAE
measurements are generally made with both stimulus intensity and frequency ratios optimized
for maximum DPOAE 2f1–f2 intensity. Many different DPOAEs co-exist and their
generation is intimately linked to the operating characteristics of the outer hair cells. It is
possible that one day we will be able to reconstruct OHC operating characteristics from
DPOAE data.
11
SFOAE (Stimulus Frequency Otoacoustic Emissions)
SFOAEs are generated with continuous pure tones. During stimulation the energy in the ear
canal includes both the incident stimulus, sound reflected from the tympanic membrane and
sound that mirrors energy leaking back from the cochlea. The contribution of the cochlea is a
version of the stimulus that appears with a time (phase) delay.
SFOAEs are difficult to extract from the signals that are present in the ear canal and as a
result have not found their way into routine clinical practice.
SOAEs (Spontaneous OtoAcoustic Emissions) and SSOAEs
(Synchronized SOAEs)
SOAEs are narrow-band signals that occur in the absence of any known stimulus. SOAEs are
typically highly stable pure tones. Their level ranges from the noise floor of the equipment to
approximately 30 dB SPL. They are found in 30–40% of healthy young ears. SOAEs are
absent in frequency regions associated with hearing loss greater than about 30 dB HL. SOAEs
are normally detected by executing a spectral analysis on the sound recorded from the ear
canal of a participant seated in a quiet test environment. Because of their intrinsic stability and
critical dependence on cochlear status, SOAEs are, when present, particularly sensitive
indicators of metabolic and physiological changes in the cochlea.
SOAEs can be synchronized to external stimuli and extracted as Synchronized Spontaneous
Otoacoustic Emissions (SSOAEs) using time-averaging procedures commonly employed to
detect small physiological signals in the presence of background noise. Because they are not
detected universally in normal hearing individuals and because they are peculiar in terms of
frequency distribution and amplitudes, SSOAEs have not found widespread use in the clinic.
When SSOAEs are present one can conclude that the ear is functioning normally across the
outer hair cells that respond to the frequencies revealed by the SSOAEs. When they are
absent, no conclusion can be made about the hearing status (Kemp 2002, Glattke 2002).
OAE fine structure
There are consistent patterns of amplitude maxima and minima in the frequency dependence
of OAEs. The variations with frequency are collectively referred to as otoacoustic emission
fine structure (or otoacoustic emission microstructure).
12
The frequency separations of threshold microstructure and cochlear emission fine structure
appear to be about 2/5 the average critical band (frequency resolution) estimates for human
subjects, which correspond to a 0.4-mm tonotopic displacement along the basilar membrane
with respect to Greenwood’s estimate of the cochlear map (Talmadge et al. 1998).
The patterns of cochlear fine structures have been found to move down in frequency (with
little change in frequency spacing) when the overall stimulus level is increased.
OAE Screening Tests
For more than 10 years TEOAEs have been employed in large-scale newborn hearing
screening programs. The OAE recordings are made via a probe which is inserted deep into the
ear canal. This way the ear is sealed off and the recorded OAE sound pressure (below 3 kHz)
is increased. Without sealing of the ear canal, ear drum vibrations would simply move air in
and out and the emission pressure would be lost.
Click stimuli of around 84 dB SPL (peak equivalent level) normally evoke a robust TEOAE
response only if hearing threshold is 20 dB HL or better. Frequencies at which hearing
thresholds exceed 20–30 dB HL are typically absent in the TEOAE response. Middle ear
status affects OAEs and can prevent their detection (Glattke and Robinette, 2002).
Healthy infant ears typically produce strong OAE levels of more than 15 dB SPL. Little signal
processing is required to extract these strong responses from the noise. Fully validated
frequency-specific measurements can often be made in just a few seconds.
In practice, sensory hearing impairment in the newborn population appears to be mainly of the
sensory transmissive type which is easily detectable by measurement of OAEs. This fact and
the favorable ergonomic and economic factors mean that OAEs are reliable and cost effective
for newborn screening programs.
Intensity is a primary factor in OAE detectability but, it is the presence of a detectable OAE
response to a particular stimulus that is clinically important and not its strength.
13
Although OAE screening has been used clinically for over a decade, the protocols used are
not yet fixed. Prieve (2002) indicates in her review that criteria for “pass” and “refer” vary
considerably between investigators: reproducibility, signal-to-noise ratio criteria for
individual frequency bands, overall amplitude of the TEOAE and combinations of amplitude
and reproducibility measurements. Relatively few investigations have focused on the use of
DPOAEs to screen newborn infants for hearing loss and here also the “pass” versus “refer”
criteria is not uniform among investigators.
OAE screening protocols
As neonatal screening has become widespread the test protocol had to be agreed upon. A
good recommendation was set forth by Stevens et al. 2002.
1) With TEOAEs, to minimize stimulus artifacts from contaminating the waveform
the analysis window for data collection should start 2.5-4 milliseconds after
delivery of the stimulus. The proposed start time for this protocol is set at 4ms.
2) The proposed end of the data collection and analysis window is between 10 and
12.5 ms.
3) The results should be analyzed in half octave bands centered at 1, 1.5, 2, 3 and 4
kHz. A response should be reported as present within a particular half octave band
if the signal to noise is >=6dB.
The successful stimulation and detection of OAEs indicates a high degree of normality in the
functioning of the middle ear and inner ear, in particular the environment of the inner ear is
shown to be healthy. This is a necessary but not in itself a sufficient condition for normal
hearing.
Some clinical devices allow the click stimulus waveform to be viewed. The ideal is a clean,
clear, positive and negative deflection lasting no longer than 1ms and followed by a straight
line indicating no or very limited ‘ringing’, or oscillation of the waveform. This condition is
much easier to obtain in a newborn ear than in an adult.
14
2. Theory of OtoAcoustic Emissions
The field of OAE research has grown enormously since their discovery by David Kemp, 28
years ago. Their discovery has led to new insights into the mechanisms and function of the
cochlea and to a new understanding of the nature of sensory hearing impairment. As a
research tool OAEs provide a noninvasive window on intra-cochlear processes.
Cochlear transmission line models were originally developed by Zwislocki (1950) and further
refined by Hall (1974) with the introduction of non-linearites. These models are able to
analyze some of the nonlinear auditory phenomena taking place at the level of the cochlea
(like two-tone suppression and two-tone distortion products).
The discovery of cochlear emissions raised the question if those phenomena can be explained
by a transmission line model. Furst and Lapid (1988) showed that the properties of acoustic
distortion products can be predicted by the nonlinear transmission line model, including the
discrepancy between animal and human data. Resistance mismatches between adjacent points
along the cochlea evoked TEOAEs. The nonlinear transmission line model was not adequate
in order to predict the other two types of emissions (SOAE and click evoked otoacoustic
emissions). These emissions were predicted by Furst and Lapid by introducing a
noncontinuous resistance along the length of the cochlear. They proposed that such
discontinuity can occur if the connection between the basilar membrane and tectorial
membrane, via the cilia of the OHC, is not uniform along the cochlea length. Thus, resistance
mismatches between adjacent points along the cochlea evoke emissions. The click evoked
responses produced by the model did not resemble CEOAE clinical data because the
responses did not have higher frequencies than the helicotrema's CF (Furst and Lapid, 1988).
The local oscillator theory
The first model for SOAE generation, which had wide acceptance, was the local-oscillator
model. In this model the cochlea is modeled as an oscillator chain. When some of the
oscillators vibrate they create waves that travel to the base of the cochlea and out into the ear
canal. In the ear canal the waves resemble SOAEs.
Properties of SOAEs such as their interactions with one another and with external tones have
been successfully described by representing individual SOAEs using a nonlinear, limit cycle
15
oscillator such as the Van der Pol. These phenomenological, limit cycle oscillator models
were not developed to describe the “oscillating elements” within the cochlea; rather, their aim
was to approximate the behavior of a complex system of equations by a single effective
oscillator, thereby providing simple, analytically tractable representations of SOAEs as they
appear in the ear canal.
Sisto and Moleti used the time evolution of the spectral lines associated to SOAEs after
presenting a click stimulus in order to determine the correct functional form of the nonlinear
oscillator equation describing the cochlear resonances. Their model is a very simplified model
in respect to full cochlear models. The model is capable of describing, with very few
parameters, both the saturation phase and the slow decaying phase that are experimentally
observed in the time evolution of OAEs after an impulsive stimulus. The model predicts the
observed exponential decay of the lines of frequency corresponding to measurable SOAEs
after excitation by a click stimulus, which is not compatible with a Van der Pol oscillator
model (Sisto and Moleti, 1999).
Coherent Reflection Filtering
A different theory regarding SOAE generation, which was first suggested by Kemp in 1979,
predicts that mammalian SOAEs arise not via autonomous cellular oscillations but as cochlear
standing-wave resonances. In this theory, SOAEs result from multiple internal reflections of
traveling-wave energy initiated either by sounds from the environment or by physiological
noise. Kemp’s original standing-wave model postulated that the backward-traveling wave
originates from a point reflection. Since the original standing-wave model did not include the
effects of traveling-wave propagation gains and losses, the model needed to associate large
reflection coefficients with many points along the basilar membrane in order to generate
sizable standing waves. Kemps theory was subsequently elaborated in models of evoked
otoacoustic emissions and formed the basis for the coherent reflection filtering theory.
In 1995 Zweig and Shera wrote down their theory on coherent reflection filtering. According
to the theory of coherent reflection filtering, reflection-source OAEs arise by reflecting off
densely and irregularly distributed cochlear impedance perturbations. At all frequencies the
net backward-traveling wave is dominated by wavelets reflected within the region about the
peak of the traveling wave, where the wave amplitude is much larger than it is elsewhere.
16
These perturbations presumably include both those clearly visible in the anatomy, such as
spatial variations in OHC number and geometry, as well as morphologically less conspicuous
perturbations, such as variations in OHC forces due to random, cell-to-cell variations in hair-
bundle stiffness or the number of somatic motor proteins.
Intrinsic variations in emission amplitude and phase are predicted by the theory. The theory
indicates that SFOAEs are analogous to “band pass filtered noise”. In this analogy, the
“noise” is the irregular spatial arrangement and strength of the impedance perturbations that
scatter the wave and the “band pass filter” results from interference among the multiple
wavelets originating from the scattering region. Unlike distortion-source emissions, whose
amplitudes and phases typically vary relatively slowly with frequency, reflection-source
emissions often vary considerably with frequency.
For example, SFOAE amplitude spectra are often punctuated by relatively sharp notches.
According to the model, such notches result from random spatial fluctuations in the
irregularities that scatter the wave. At some frequencies, wavelets scattered from different
locations within the scattering region combine nearly out of phase, resulting in near
cancellation of the net reflected wave.
One of the key assumptions of the model by Zweig and Shera is that the spatial activity
pattern of the traveling wave is both “tall and broad”. The traveling wave has to be tall
enough to produce significant reflection from a very small level of cochlear inhomogeneities,
and the activity pattern peak region has to be broad enough to contain 1 to 2 wavelengths of
the traveling wave, a requirement for coherency of the cochlear reflections. Tall and broad
activity patterns were obtained by the introduction of time delayed stiffness.
Kim et al. observed in 1980 that adding distributed negative resistance to the BM (as an
energy source to generate tall activity patterns) and assuming that the amplifier resided in a
region basal to the activity pattern peak was sufficient in order to obtain tall and broad activity
patterns. It has since been argued by de Boer using increasingly general and more
sophisticated models that placing the cochlear amplifier basal to the activity pattern peak is a
necessary condition for obtaining tall and broad activity patterns.
In 2002 Shera and Guinan tested the key predictions of the theory of coherent reflection
filtering for the generation of reflection-source OAEs determined by the group delay of the
17
BM transfer function at its peak. The prediction is tested in cats and guinea pigs using
measurements of SFOAE group delay. A comparison with group delays calculated from
published measurements of BM mechanical transfer functions supports the theory only at the
basal-most 60% of the cochlea. At the apical end of the cochlea the measurements disagree
with neural and mechanical group delays. This disagreement suggests that there are important
differences in cochlear mechanics and/or mechanisms of emission generation between the
base and apex of the cochlea (Shera and Guinan, 2002).
The global standing wave theory (SOAEs)
The theory of reflection-source emissions predicts that backward-traveling cochlear waves are
generated by the coherent scattering of forward-traveling waves off densely and randomly
distributed perturbations in the mechanics of the cochlea (Shera 2003). Because wavelets
scattered near the peak of a forward-traveling wave have much larger amplitudes than those
reflected elsewhere, the net reflected wave is dominated by scattering that occurs in the region
about the response maximum.
The resulting backward-traveling waves are then reflected by the impedance mismatch at the
cochlear boundary with the middle ear, generating additional forward-traveling waves that
subsequently undergo another round of coherent reflection near their characteristic places
(Figure �2-1). This process continues for each backward-traveling wave in the cochlea,
partially being reflected into a forward-traveling wave. At frequencies for which the total
phase change due to round-trip wave travel is an integral number of cycles, standing waves
can build up within the cochlea, which is then acting, in effect, as a tuned resonant cavity.
Figure �2-1 - Multiple internal reflections within the cochlea (Shera 2003 poster)
18
The process of multiple reflection continues, each subsequent stapes reflection and cochlear
re-emission contributing an additional backward-traveling wave whose amplitude at the
stapes differs by a factor of R*Rstapes from the one before. Shera (Shera 2003) showed that
adding up all the backward traveling waves yields the factor /(1 )stapesR RR� for the total
outgoing wave at the stapes.
Whenever the product R*Rstapes is positive real all the high order forward traveling waves
combine in phase with the primary traveling wave at the stapes. The multiple internal
reflections then reinforce one another, creating a significant standing wave component in the
cochlear response whose amplitude depends on the product of cochlear and stapes reflection
factors, R*Rstapes. The theory predicts that the standing wave grows without bound as
R*Rstapes approaches 1. In the real cochlea, of course, unconstrained growth is prevented by
compressive nonlinearities that limit the energy produced.
Cochlear sites corresponding to SOAE frequencies need manifest no special distinguishing
features. In the global standing wave model, SOAE frequencies are determined by R*Rstapes,
and SOAEs therefore trace their origin to aspects of the mechanics as subtle, and as non-local
to the site in question, as the magnitude and angle of the impedance mismatch at the cochlear
boundary with the middle ear, the spatial frequency content of the cochlear impedance
perturbations that scatter the wave, and the total round trip traveling wave gain and phase shift
experienced en route.
In the global standing wave model, SOAE frequencies are determined in part by the
impedance mismatch at the cochlear boundary with the middle ear. Manipulations that modify
this basal boundary condition can therefore modulate both SOAE amplitude and SOAE
frequency. In accord with these predictions, middle ear impedance changes (caused by
postural changes) have been found to alter SOAE characteristics, including frequency.
The modern standing wave model predicts that most SOAEs result from normal mechanical
variability rather than from pathologically large impedance discontinuities. Cochlear standing
waves can become self-sustaining and thus appear in the ear canal as SOAEs when the total
round-trip power gain matches the energy losses experienced en route.
The standing-wave model differs fundamentally from the local-oscillator scenario. Rather
than supposing that the “oscillating elements” generating SOAEs are localized to particular
19
cells or subcellular structures within the organ of Corti, the standing-wave model identifies
SOAEs as a global collective phenomenon necessarily involving the mechanics,
hydrodynamics, and cellular physiology of the entire cochlea, as well as the mechanical and
acoustical loads presented to it by the middle and external ears.
In the local-oscillator model these macromechanical structures and processes play no
fundamental role, they serve merely to connect the autonomous oscillating element with the
external environment, providing a conduit for the acoustic energy it produces to escape from
the inner ear.
The coherent reflection model predicts that the SFOAE evoked by a tone comprises a sum of
wavelets scattered by perturbations located throughout the peak of the traveling wave. The
SFOAE therefore arises from a distributed region, roughly equal in extent to the width of the
traveling wave envelope. In the 1–2 kHz region of the human cochlea, this distance spans on
the order of 100 rows of outer hair cells at sound levels near threshold (Zweig and Shera,
1995). This is in contrast to Kemp’s original standing wave model of point reflection. In the
global standing wave model, by contrast, the oscillating element comprises the entire cochlea,
and the collective response of the hearing organ as a whole contributes essentially to creating,
maintaining, and determining the characteristics of the emission.
The evident success of the global standing-wave model contradicts the notion, often implicit
in the local-oscillator framework, that SOAEs measured in the ear canal provide direct access
to the local elementary cellular oscillators within the organ of Corti (Shera 2003).
The global standing wave model resolves the paradox noted by Geisler (1998) in his
discussion of the van der Pol oscillator as a local oscillator model for SOAEs:
“Why doesn’t every section of the cochlea act as a limit-cycle oscillator and the cochlea
therefore produce emissions at all frequencies? It follows that there must be something
different about those cochlear sites that generate the relatively few emissions observed.
Unfortunately, the search for such differences has not been successful”.
Shera tested the two alternative models for mammalian SOAEs generation. His work focused
on key predictions of the global standing-wave model that distinguish it from the local-
oscillator alternative. He showed that although some of the model predictions could perhaps
be obtained by artful adjustment of local-oscillator models, they all arise quite naturally
20
within the standing-wave framework. His tests provide strong support for the idea that human
SOAEs arise via global standing-wave resonances (Shera, 2003).
Sheras' quantitative tests provide strong support for the global standing wave model and its
prediction that SOAE frequencies are determined by R*Rstapes. The results demonstrate that
in addition to predicting the existence of multiple emissions with characteristic minimum
frequency spacing, the global standing wave model also accurately predicts the mean value of
this spacing, its standard deviation, and its power law dependence on SOAE frequency.
Furthermore, the statistics of SOAE time waveforms demonstrate that SOAEs are coherent,
amplitude stabilized signals.
DPOAE generation theory
DPOAEs are generated in the cochlea via the nonlinear interaction of the excitations produced
by two primary tones of frequencies f1 and f2. Their initial production is in the region of
strong overlap of the f1 and f2 activity patterns, which is around the f2 tonotopic site. From
the generation region, DPOAE components propagate both basally (backward) and apically
(forward) (seen in Figure �2-2C).
The DP cochlear wave reaching the DP tonotopic place will be partially reflected by small
irregularities in the cochlear properties. A tall and broad DP activity peak will allow coherent
scattering from many reflectors to give a large basal-ward reflection of the cochlear traveling
wave.
A portion of the basally traveling distortion product component will be transmitted through
the middle ear to the ear canal and detected as a DPOAE. The remaining signal will be
reflected back into the cochlea.
In general these two components will not arrive in phase. The relative phase difference
between them will mainly depend on the DP frequency and the resulting interference will give
an observable fine structure as a function of the DP frequency.
This phase is related to the round trip group delay for waves traveling from the generation
region to the DP place, and will thus depend on the frequency separation between the two
21
primaries, especially for narrower frequency ratios of the two primaries (Talmadge et al.
1998).
When the ratio f2/f1 is nearly one (e.g. 1.05), f1 and f2 TW velocities are very similar at all
points. The phase distribution of DP elements then necessarily forms a forward (apical) TW
with little DP sent backwards to form a DPOAE. Even so, some DPOAE signal escapes via
the SOAE route.
For large f2/f1 (e.g. 1.5), the densely packed phase changes within the f2 envelope generate
an undulating DP phase distribution that will be largely self-canceling and little DP waves
will propagate from that region. However, because there is a minus sign in ‘2f1–f2’, for f2 >
f1, the spatial phase gradients of TW ‘f1’ and TW ‘f2’ counteract each other in 2f1–f2 DP
production. Consequently, at some optimum f2/f1 ratio (around 1.2), the relative velocities of
TW ‘f1’ and TW ‘f2’ are such that the spatial distribution of DP elements actually becomes
that of a backward traveling wave over a considerable length of OHCs. Interestingly, there is
no optimum frequency ratio for the ‘alternative’ DPOAE 2f2–f1 (see Figure �2-2C), which
emanates from a place basal to both f1 and f2 peaks over a wide range of f2/f1 ratios.
Figure �2-2 - DPOAE generation visualized from (Kemp 2002). (A) A ‘dead’ cochlea where natural
damping absorbs most of the stimulus energy before any clear separate excitation peaks for f1 and f2 can develop. (B) A linear ‘live’ cochlea, where linear OHC amplification cancels the damping and sharp
‘images’ of stimuli f1 and f2 can be seen. (C) A “real” non-linear ‘live’ cochlea, where OHC motility is non-linear and this results in intermodulation distortion products being created under the entire f2
envelope (including 2f1–f2 and 2f2–f1).
22
DPOAE specific models
A good example for a “simple” model that by introducing nonlinearities manages to create
DPOAE can be seen in the paper by Jaffer et al. (Jaffer et al, 2002). Jaffer et al. presented a
model of the cochlear partition in which a weak elastic longitudinal coupling has been
included between the resonant viscoelastic plates of the cochlea. The connective tissue
between plates was modeled by springs oriented in the longitudinal direction, perpendicular to
the transverse motion of the cochlear partition. The model includes the linear cochlear
partition mechanics and linear cochlear hydrodynamics but no active mechanics are included.
The addition of longitudinal elastic dynamics produce cubic distortion product otoacoustic
emissions where the 2f1-f2 intermodulation component is largest (in response to two primary
frequencies 2f1-2f2, 2f2-2f1, 2f2-1f1, and 2f1-1f2 were produced).
Dhar et al. evaluated the relative contributions of two sources to the DPOAE, the distortion
and reflection components (Dhar et al., 2005). The nonlinear interaction between the stimulus
tones around the tonotopic region of the higher frequency stimulus tone (f2) generates the
distortion or generation component. The reflection component is generated due to linear
coherent reflections from a randomly distributed roughness about the tonotopic region of the
DPOAE. This reflected energy signal contribution is the product of the initial apical moving
DP component with the apical reflectance.
By separating the ear canal DPOAE signal into its two major components, Dhar et al. have
been able not only to investigate the relative levels of the two components, but also to better
understand the properties of each component. While the generator component when plotted as
a function of the stimulus-frequency ratio showed a distinct band-pass shape, the pattern for
the reflection component was more variable with the stimulus level and across ears. One of
the most striking characteristics of the results is the stability of the generator component and
the variability of the reflection component across ears. Also, the influence of stimulus level on
absolute and relative component levels was found to be significant.
It should be emphasized that the apical reflectance (Ra), according to the coherent reflection
model, depends mainly on the sharpness of tuning of the BM around the DP CF region and
the strength of cochlear inhomogeneities around that region. If both of these factors are
present the apical reflectance will generally be significant. The apical reflectance is expected
to fluctuate significantly across healthy ears due to differences in the degree of cochlear
roughness across such ears.
23
Dhar et als’ results are consistent with other reports in that the reflection component is shown
to be dominant at low stimulus levels only. Furthermore, they have shown the dominance of
the reflectance component to be consistent across the entire range of stimulus-frequency ratios
tested. Great variability in the reflectance component across subjects is also reported in their
study.
Dhar et al. conclude that most of the characteristic features of the generation component and
the reflection component that they find are at least qualitatively described by cochlear models
that incorporate a nonlinear generation process around the f2 CF region combined with the
coherent reflection of the initial apical moving DP component.
OAE generation summary
Different representation of the same physics
An interesting example of the dynamics of OAE modeling research can be seen in the Nobili
et al. paper from 2003. Nobili et al. proposed a “new and different” interpretation of OAEs
based on the instantaneous fluid coupling between the stapes footplate and the BM, and
among the BM oscillating elements themselves. Nobili et al. claimed that this interpretation
differs from modeling the cochlea as a transmission line. Their OAE time-domain simulations
were based on a hydrodynamic model adapted so as to fit physical and geometrical
characteristics of the human inner ear. The model was completed with the inclusion of
forward and reverse middle ear transfer functions. In their analysis of the results Nobili et al.
tried to show differences between their results and the results obtained by transmission line
models. They related their results to the hydrodynamic character of cochlear dynamics, in
particular, the instantaneous character of fluid coupling between BM and stapes.
In response to the paper by Nobili et al. Shera, Tubis and Talmadge wrote two papers (2004a,
2004b). In the first paper, Shera, Tubis and Talmadge (2004a) demonstrated that Nobilis'
model fails to reproduce basic empirical properties of actual evoked OAEs. By circumventing
uncertainties about the numerical accuracy of Nobili et al.'s published simulations, they
demonstrated that the middle ear filtering mechanism proposed by Nobili fails to reproduce
basic empirical properties of actual evoked OAEs.
24
The second paper (Shera, Tubis and Talmadge, 2004b) was aimed against the critic claimed
by Nobili et al. that the transmission line models fundamentally misrepresent the
hydrodynamics of the cochlea. Nobili et al. argued that although the concepts of wave
propagation and reflection may apply in idealized hypothetical situations, transmission line
models cannot describe anything resembling the physics of an actual ear.
Shera, Tubis and Talmadge resolve and synthesize the two different approaches, arguing that
the wave-equation and hydrodynamic formulations of cochlear mechanics are different
mathematical representations of the same underlying physics.
Long range fluid coupling underlying Nobili et al.’s integrodifferential equation is shown to
be identical to that in a one-dimensional, tapered transmission line model. They are shown to
be two different mathematical representations of a single model based on Newton’s laws.
Since they both represent the same physics, both ultimately yield the same solutions.
Although both formulations provide valid representations of the physics of the cochlea, the
two approaches are attended by strikingly different conceptual and computational
frameworks. Although no less physically appropriate than the Green’s function alternative,
the wave-equation formulation often provides considerably more aid to the intuition and is
easier to visualize. Shera, Tubis and Talmadge argue that the wave-equation formulation
provides compelling advantages, at least in the context of modeling OAEs.
OAE properties
Otoacoustic emissions have been explained as arising from a combination of two cochlear
mechanisms: coherent, linear reflection and nonlinear distortion. Reflections are described as
scattering from multiple randomly spaced discontinuities along the basilar membrane. Only
those reflections that sum constructively with the incident traveling wave and that arise for an
incident traveling wave that is both broad and tall will have sufficient amplitude to contribute
to the recorded OAE. This type of traveling wave response involves nonlinear mechanical
amplification supplied by the cochlear outer hair cells that has its highest gain at the lowest
stimulus levels. The reflection process itself is thought to be linear. In contrast, the nonlinear
distortion mechanism for OAE generation is described as a byproduct of intermodulation
distortion in the basilar membrane traveling-wave response. Both coherent linear reflection
25
and nonlinear distortion are thought to contribute to all evoked OAEs, with the level of the
stimulus partially determining which mechanism is dominant (Konrad and Keefe, 2003).
SFOAEs and TEOAEs at low to moderate levels are thought to be predominantly generated
by linear reflection near the tonotopic region associated with the spectral content of the
stimulus. The SFOAE may have a quasi-regular fluctuation in amplitude and phase with small
changes in the stimulus frequency. This spectral fine structure has a local frequency
separation between maxima that is inversely related to the round-trip time delay between the
ear-canal microphone and the apical CF reflection site in the cochlea. Interference from
multiple reflections between the oval window and the apical reflection site may produce a
temporal fine structure in the response envelope, and may contribute to the spectral fine
structure.
In contrast to SFOAEs and TEOAEs, DPOAEs are thought to be initiated by nonlinear
distortion in the basilar membrane’s response to the two-tone stimulus (f1 and f2). For the
cubic distortion product (2f1-f2) and other DPs tuned more apically than the f2 place a
secondary component arises as a coherent linear reflection near the place associated with the
DP frequency.
DPOAE fine structure has been explained as originating from interference between these two
“sources” and from multiple reflections of these two components between the oval window
and the DP tonotopic place (Dhar et al., 2002). Another proposed source of the spectral fine
structure in evoked OAEs is related to spatial variations in the magnitude of the reflected
wave, which may be due to variations in the effective reflectance with position along the
basilar membrane (Shera and Guinan, 1999).
SOAEs are narrow-band emissions measurable in the ear canal in the absence of acoustic
stimulation, which are generated within the cochlea by stable limit-cycle oscillations (Tubis
and Talmadge, 1998).
Konrad and Keefe (2003) studied the influence of SSOAEs on the rest of the OAEs. They
showed that in ears with significant SSOAEs, multiple SSOAE sources and the presence of
multiple internal reflections influence the fine structure of TEOAEs elicited by low to
moderate level stimuli.
26
Konrad and Keefe used the TFR (time-frequency analyses) method in order to better
understand the OAE frequency components and their time of appearance. In their paper they
showed that they can align SSOAE peeks with their counterparts in SFOAE and DPOAE
recordings. Thus the “hot spots” (strong sites of cochlear reflection) contribute to the two
phenomena. Also shown is the “hot spots” contribution to DPOAEs via stimulus and SSOAE
intermodulations. This suggests the importance of two-tone suppression processes to the
interpretation of OAE responses elicited by any stimuli more complicated than a single sine
tone. The stimulus spectrum is also seen mirrored in the OAE spectrum (its' main lobe at
least).
This suggests that the coherent reflection theory of OAE generation is still incomplete for
describing responses at levels (at approximately 20 dB SPL and higher) for which the basilar
membrane response becomes compressive due to the saturation effects of outer hair cell
functioning.
They also showed that the relative contributions by SSOAEs and multiple internal reflections
to the total OAE response increased with decreasing stimulus level (the relative SOAE
amplitude will increase). This means that the evoked OAE spectrum, as measured by a
nonlinear residual technique, is less likely to resemble that of the eliciting stimulus as the
stimulus level decreases.
Models incorporating cochlear reflectance predict that an OAE response would resemble the
eliciting stimulus. For example, the model predicts that the SFOAE spectral energy should lie
within the pass-band of the stimulus energy, and that the SFOAE spectrum, aside from fine
structure, should be similar to the stimulus spectrum within the pass-band. The results of this
study show that other factors are involved. In reality, for tone-pip-evoked SFOAEs, the OAE
spectrum is narrow compared to the eliciting stimulus, and response components correspond
primarily to the higher-frequency portion of the stimulus pass-band.
The round-trip magnitude of the cochlear reflectance was measured and found to decrease
with increasing effective input level, which means that realistic models of OAE generation
should include nonlinearity in the apical cochlear reflectance. A recursive formulation of the
coherent reflection theory was presented that may be useful in time-domain simulations.
Predictions of DPOAE and SFOAE latencies by cochlear models were in general accord
although the onset of the SFOAE and the lack of level dependence in the simplified cochlear
27
models present difficulties. An additional effect at high stimulus levels in some ears is the
observed dynamical linking between SSOAE and stimulus-evoked OAE components, which
produces intermodulation at frequencies not present in the original SSOAE or in the pass-
band of the stimulus (Konrad and Keefe, 2003).
Sisto and Moleti found a difference in latency, in the “nonlinear” OAE response, between
normal and impaired ears. They analyzed the “nonlinear” OAE response to TEOAEs using
the wavelet analysis. They also showed that the latency-frequency relationship predicted by
scale-invariant full cochlear models does not agree with experimental measurements of the
TEOAE latency as a function of frequency (Sisto and Moleti, 2002).
The field of OAE research has not yet come to a full understanding of the ways OAEs are
generated inside the cochlea. There have been many breakthroughs in the past 20 years in the
field, and more are expected to come in the next 5-10 years. Until a model can be created that
can simulate all OAEs with high accuracy the field of OAE research and modeling will
continue to be an active one.
28
3. Motivation for the present study
Although there are many models that try to explain the existence of OAEs, our purpose is to
predict the existence of OAEs in a complete human ear model. Recently a cochlear model was
developed in our lab (Cohen and Furst, 2004). This model incorporates cochlear fluid
dynamics with an OHC model. The model is confined to the inner ear alone (i.e. the cochlea)
and cannot on its' own simulate OAEs. The model was successful in predicting both normal
and abnormal audiograms and tuning curves.
In this study we include a middle ear model and test the constraints that enable us to predict
the existence of OAEs in normal and abnormal ears. The model will serve as a tool in the
testing of OAE phenomena. Our hope is to better understand the different effects leading to
OAE formation through the use of the model. A computational model of the entire
presynaptic system can help validate hypotheses and help interpret data collected. If emissions
can be properly simulated by a cochlear model and details about them described in terms of
structures and mechanisms in the model, the correspondence between model structures and
physical structures in the real cochlea can be used to extract information about the interior of
the cochlea without actually having to physically “look inside”. There has not yet been a
model created with the capability of truly simulating all the OAEs generated by the biological
ear.
The work described in this thesis describes computations performed with a one-dimensional
cochlear model. This model has the advantage that it is not too complex to understand and
leads to a numerical code that does not require extensive computer time and memory. The fact
that the model computes the behavior of the cochlea in the time domain makes it especially
useful for the study of nonuniform and nonlinear effects, which play an important role in
otoacoustic emissions.
This is a theoretical research focused on studying the OAE sources in a mathematical model.
The research goal is to shed some light on the sources of TEOAEs and SOAEs. In general,
OAE responses carry a large amount of information about the status, activity and environment
of OHCs, which we are currently unable to interpret. OAEs tend to be dominated by
microscopic details of little relevance to hearing. Nevertheless, OAEs provide the only
detailed noninvasive window to the cochlea and by their very presence confirm normal
29
presynaptic cochlear function. Thus far, only a small portion of the OAE potential has been
tapped, primarily by efforts to use OAEs to screen for hearing loss. Although useful today, if
we can learn how to extract definitive data on OHC status from OAE data, then their clinical
importance will be greatly enhanced.
The next chapter gives a detailed description of the mathematical model used throughout this
thesis. Chapter 5 demonstrates the different phenomena simulated by our complete human ear
model.
30
4. Model details
The model presented in this thesis is comprised of a 1-D cochlear model. The cochlear model,
which includes an embedded outer hair cell model, was developed by Azi Cohen (Cohen
2004, Cohen and Furst 2004). The cochlear models’ boundary condition was the movement of
the stapes. In order for the model to predict OAEs a middle and outer ear model were added,
and the models boundary condition replaced. Because the embedded model cannot be treated
as time-invariant when large magnitude random variances are applied, our implementation
will focus on the time domain solution to the model equations. For clarity reasons the entire
model is described below with emphasis on the original part of this work, the middle ear
model.
Middle Ear Model
The mechanics of the middle ear and ear canal are based on a simple mechanical model
(Talmadge et al., 1998). In this model the tympanic membrane is treated as a single piston that
has a fixed incudostapedial joint (Figure �4-1).
Figure �4-1 - A simple middle ear mechanical model
The ear canal is assumed to be sealed off by a stimulus delivering microphone assembly and
the length of the ear canal (from microphone to tympanic membrane) is assumed to be small
relative to the sound wavelength. Thus, the pressure ( )eP t in the ear canal may be considered
to be uniform. It is also assumed that all air pressure changes occur without loss or gain of
heat, so that the mechanical model gives rise to a single oscillator equation of the form:
31
� �2 1( ) * ( ) * ( ) * ( , ) * ( )OW OW OW ow OW me e
OW
t t t P o t G P t� � ��
� � � ��� � (4.1)
where (0, )P t is the pressure difference between the scala tympani and scala vestibuli near the
stapes, OW� is the effective areal density of the oval window (effective mass of oval window
+ ossicles / area of oval window), OW is the middle ear damping constant, OW is the middle
ear frequency and meG is the mechanical gain of the ossicle chain. All middle ear parameters
are phenomenological constants and are defined in Table I. OW was chosen so as to be as
close as possible to the middle ear transfer function by Puria (2003).
TABLE I. Table of middle ear parameters. The parameters are taken from Talmadge et al.
(1998).
Parameter Value Definition
OW� 1.85 g/cm^2 areal density of oval window
OW 500 1/s middle ear damping constant
OW 1500 Hz*2� middle ear frequency
meG 21.4 mechanical gain of ossicles
meC 2 60.059
(2 1340 ) * 60.49*1.4
x HZ e� � coupling of oval window displacement to ear
canal pressure
owC 0.0322.909
0.011� coupling of oval window to basilar membrane
Eq. 4.1 relates the displacement of the oval window ( )OW t� to the fixed pressure ( )eP t in the
ear canal and the pressure (0, )P t near the stapes. For most experimental setups, however, the
pressure ( )eP t is an observable rather than a fixed experimental input. Instead, the
experimental input is the “calibrated ear canal pressure”, ( )inP t , which is the pressure (created
by a microphone) in the ear canal in the case of a rigid ear drum. ( )eP t is the “total” pressure
in the ear canal influenced by pressure created by the microphone ( )inP t and by pressure from
the displacement of the tympanic membrane. With the assumption of adiabatic
compression/expansion, the relation between ( )eP t and ( )inP t is
( ) ( ) * ( )e in me owP t P t C t�� � (4.2)
32
where meC is the coupling of the oval window displacement to the tympanic membrane
displacement contribution to the ear canal pressure (defined in Table I).
The emission pressure ( eP ) can be easily determined by calculating the ear canal pressure,
resulting from a tone injected into the ear canal ( inP ), and the cochlear response ( ( )OW t� ).
The Cochlear Model
In the simple one-dimensional model the cochlea is considered as an uncoiled structure with
two fluid-filled rigid-walled compartments separated by an elastic partition. The basic
equations are obtained by applying fundamental physical principles such as conservation of
mass and the dynamics of deformable bodies. In the model the elastic partition is responsible
for the mechano-neural transduction of sound.
Cohen and Furst integrated an OHC model into the one dimensional cochlear model. The two
models control each other through cochlear partition movement and cochlear partition cross
pressure variables. If we assume that the cochlea is uncoiled and approximated by two fluid-
filled rigid-walled compartments separated by an elastic partition, then it may be represented
by a one-dimensional model as shown in Figure �4-2.
Figure �4-2 - Cochlear Model Geometry
Let x be the longitudinal coordinate such that at the basal end 0x � and at the apical end
x L� , where L is the uncoiled cochlea length. Let t be the time variable. Let ( , )vP x t be the
pressure through the scala vestibuli and ( , )tP x t the pressure through the scala tympani.
33
The intermediate channel between the scala vestibuli and the scala tympani is called the scala
media and is represented by the elastic partition. The vertical displacement of the partition
along the x dimension is denoted by ( , )bm x t� . The fluid velocity along the x dimension is
( , )vU x t and ( , )tU x t for the scala vestibuli and the scala tympani, respectively.
The principle of conservation of mass yields the following equations:
0v bmUA
x t�
� �
� �� �
(4.3)
0t bmUA
x t�
� �
� �� �
(4.4)
where ( )x is the basilar membrane width and ( )A x is the scalae cross section area. Since
both scalae tympani and vestibuli contain perilymph, which we can assume is an almost
incompressible fluid, the equation of motion for each scala can be written as:
0v vP Ux t
�� �
� �� �
(4.5)
0t tP Ux t
�� �
� �� �
(4.6)
where � is the perilymph density.
This set of equations is completed by the equation of motion of the cochlear partition. The
partition is, mechanically, a flexible structure embedded in a rigid framework. It is assumed
that the flexible part, the basilar membrane, and the structure above it have point wise
mechanical properties. This means that the partition velocity at any point is related to the
pressure difference across the partition at that point only and not at neighboring points.
We define the pressure difference across the cochlear partition as:
t vP P P� � (4.7)
The cochlear partition is regarded as a flexible boundary between the scala tympani and the
scala vestibuli, whose mechanical properties are describable in terms of point-wise mass
34
density, stiffness and damping. Thus, at every point along the cochlear duct, the partition’s
velocity is driven by the pressure difference P across the partition. From the principle of
conservation of mass we can derive the relationship between the fluid velocity and the basilar
membrane displacement bm� .
Combining equations, Eq 4.3 - Eq 4.7, yields the differential equation for P :
22
2 2
2 ( )0bmP x
x A t�� ��
� �� �
(4.8)
The pressure difference across the partition ( P ) is the combined result of the pressure
generated by the basilar membrane model and the pressure generated by the OHC model. Eq.
4.9 depicts the combined pressure from the contributions of the two models:
bm ohcP P P� � (4.9)
where the basilar membrane is imitated as an electrical transmission line.
2
2( , ) ( ) ( , ) ( )bm bmbm bmP x t m x r x t s x
t t� �
�� �
� � �� �
(4.10)
where ( ), ( , )m x r x t and ( )s x represent the basilar membrane mass, resistance, and stiffness
per unit area, respectively.
The Outer Hair Cell Model
The outer hair cell (OHC) membrane is divided into two regions, the apical part facing the
scala media and the basolateral part embedded in the organ of Corti. The basic OHC model
represents these two cell membrane segments as two parallel resistance and capacitance
circuits. Figure �4-3 represents an equivalent electrical circuit model for the OHC.
35
Figure �4-3 - An equivalent electrical circuit model of the outer hair cell
Changes in the OHC length are controlled by the voltage change across the OHC basolateral
membrane� . Solving the electrical circuit in Figure �4-3 yields a differential equation for� :
0( )aohc a ohc
dCdG
dt dt�
� � �� � � � (4.11)
where aC and aG are the capacitance and conductance of the apical part, respectively.
ohc and � are defined as . 2 1000a b bohc
a b b
G G Gconst
C C C �
�� � � � �
� and .sm sm
b a b
V Vconst
C C C� � � �
�.
The capacitance aC and conductance aG of the apical part are affected by the stereocilia
movement. The OHC stereocilia are shallowly but firmly embedded in the under-surface of
the tectorial membrane. Since the tectorial membrane is attached on one side to the basilar
membrane, a sheer motion arises between the tectorial membrane and the organ of Corti as
the basilar membrane moves up and down. The model assumes that aG and aC are functions
of bm� (the basilar membrane vertical displacement).
The voltage variation across the basolateral part of the OHC causes a length change ( OHCl� ) in
the OHC. Thus, the force OHCF that an OHC exhibits due to voltage change is derived by:
( ( ) )ohc ohc ohc bmF K l � �� �� (4.12)
The pressure that the OHCs contribute to the basilar membrane pressure is derived from
36
( )ohc ohcP x F� (4.13)
where ( )x is the relative density of healthy OHCs per unit area along the cochlear duct.
( )x is referred to as the OHC gain, whose value ranges from 0 to 0.5. When ( )x is larger
than 0.5 the model represents a nonrealistic cochlea whose motion approaches infinity and
thus, will not be used.
When linear dependencies ( ( ),A bmG �� ( ),A bmC �� ( )OHCl ��� ) are assumed and substituted
into equations 4.11, 4.12 and 4.13 we derive the differential equation for ohcP :
2 1( )ohc bmohc ohc bm
dP dP x
dt dt�
� � �� �� � �� �� � (4.14)
where the values of 1( )x� and 2 ( )x� are:
1
( ) ( )( )
( )r x s x
xm x
� � � (4.15)
2 ( ) ( ) ohcx r x� �
Boundary and Initial conditions
The boundary condition between the middle ear and the cochlea relate the cochlear fluid
velocity to the velocity of the oval window. Thus, the model boundary conditions are:
(0, ) (0, ) ( )v t ow owU t U t C t�� � � � � (4.16)
( , ) 0P L t �
where ( )ow t�� is the oval window (OW) velocity, owC is the ratio between the area of the oval
window and the cross-section of the cochlear scalae (Table I) and L is the cochlear length.
From Eq. 4.5, 4.6 and 4.7 we obtain
37
t v v tP P U UPx x x t t
� �� � � �� � � � �� � � �� � � �� � � � �� � � �
(4.17)
substituting Eq. 4.16 into Eq. 4.17 yields:
0
( , ) ( , )2 ( )v t
ow owx
U o t U o tPC
x t t t� � �
�
� �� �� � � �� � � �� � � �� � � �� �� �� (4.18)
Thus our boundary conditions are:
0t� �
(0, )
2 ( )ow OW
P tC t
x� �
�� � �
��� (4.19)
( , ) 0P x t � x L�
where the pressure difference derivative near the stapes is related to the oval window
acceleration.
( )ow t��� can be derived from Eq.4.19, therefore substituting Eq 4.1 in Eq 4.19 yields:
2(0, )(0, ) * ( ) * * * * * ow
me in OW OW OW OW OW OWOW
CP tP t G P t
x � � � �
�� � �� � � �� ��
� (4.20)
As we can see the boundary condition influences the pressure difference equation only in the
first section (the boundary between the cochlea and the middle ear). We stimulate the model
through ( )inP t which represents the ear canal pressure generated by the ear canal microphone.
The initial value conditions [0, ]x l� � were defined as:
( ,0) 0bm x� � (4.21)
( ,0) 0bm xdt�
�
( ,0) 0OHCP x �
38
TABLE II. Table of cochlear parameters. The parameters are taken from Cohen and Furst
(2004).
Parameter Value Definition
L 3.5 cm length of uncoiled cochlea � 1 g/cm^3 Perilymph density 0.003 cm Basilar membrane width
A 0.5 cm^2 Scalae cross section area
( )m x 6 1.5 21.286 10 /xx e g cm� � �� Basilar membrane mass per unit area
( )r x 4 1.5 2 21.282 10 /xx e g cm s�� �� Basilar membrane damping per unit area
( )s x 0.06 20.25 /xe g cm s�� �� Basilar membrane stiffness per unit area
ohc 1000 Hz*2� OHC cutoff frequency
Numerical Solution
The time domain solution is performed in two sequential steps (Cohen and Furst, 2004). In the
first step, the boundary value problem is solved by the finite differences method while the
time is held as a parameter. In the second step, the initial value condition problem is solved by
the fourth order Runge-Kutta method. The first step is run in the spatial domain, and the
second step in the time domain.
We use the finite difference method to solve the second degree differential equation (Eq. 4.8).
In order to solve the boundary value problem we rewrite the equation by substituting Eq. 4.9
and Eq. 4.10 into Eq. 4.8, which yields:
2
2 ( ) ( ) ( )P
P Q x G x Q xx
�� � � �
� (4.22)
where 2
( )( )
Q xA m x
� �
� and ( ) ( ) ( )BM
BM OHCG x r x s x Pt
��
�� �� � � �� ��� �.
Eq. 4.19 is used as the boundary condition. The natural three-point approximation to the
second derivative of x is:
39
2
2 2
( , ) ( , ) 2 ( , ) ( , )P x t P x x t P x t P x x tx x
� � � � � � ��
� � (4.23)
where L
xN
� � and N is the number of spatial sections of the cochlea. In this way we define
a uniform grid of 1N � points in the interval [0,L], so that lx l x� � and ( , )l lP P x t� , where
0,1,...,l N� .
The initial value differential equations to be solved are Eq. 4.9, 4.10 and 4.14 with the initial
conditions:
( ,0) 0, ( ,0) 0, ( ,0) 0bmbm OHCx x P x
dt�
� � ����� � ����� � (4.24)
The boundary value problem (Eq. 4.22) can be expressed as a set of linear equations:
P Y� � � where
� �0 1 1, ,..., ,T
N NP P P P P��
2 2 20 1 1 2 2 1 1, , ,..., ,0
T
N NY Y G Q x G Q x G Q x� �� �� � � �� � (4.25)
2 20 0 0
1* ( ) * * * * *
2ow
me in ow ow ow ow ow owow
CY G Q x G P t h � � � �
�� �� �� � � �� �
�
and
!
!
2
0
21
21
1 * 0 0 0 02
1 2 1 0 0
0 0 1 2 1
0 0 0 0 1
ow
ow
N
CxQ x
x Q
x Q
�
�
� �" #�� � � �� �$ %& '� �
� �� � �� �� � � �� �� �� � �� �� �� �
� � � � �
All parameters used in the simulations are listed in Tables I and II.
40
Assuming that , ,bm bm OHCP� �� are known (for t T� and for every ix ) and the boundary
condition variables ( ,ow ow� �� ) are also known, an approximation of the pressure difference P
can be obtained for every nodal point ix , using the finite difference method.
The time domain model equations are solved numerically. We assign a variable ( stept ) to be
the time variable step size. The spatial step size is LN
x �� and each point along the cochlear
partition is denoted by ix .
Once the pressure difference ( P ) for every location along the partition is known (for t T� ),
then , ,bm ow ohcP� ��� �� � can be calculated. From Eq. 4.9 and Eq. 4.10 we calculate bm��� , Eq. 4.14
enables us to calculate ohcP� and Eq. 4.1 combined with Eq. 4.2 are used for ow��� .
After the spatial domain step, the pressure difference along the cochlear partition is known,
allowing us to calculate , ,OHC bm bmP � �� and P along the cochlear partition (and the boundary
condition variables ,ow ow� �� ), at all time points that satisfy t T( . Now an approximation (of
the following variables: , , , ,bm bm OHC ow owP� � � �� � ) at time stept T t� � is achievable by an initial
value numerical method.
In this research work, we have chosen the multi-step Fourth order Runge-Kutta method as the
numerical methods to approximate the above differential system solution. Although the
Runge-Kutta method is more “computation consuming” than the simple modified Euler
method it is more stable and accurate (the model by Cohen and Furst used the modified Euler
method).
Fourth order Runge-Kutta
The fourth order Runge-Kutta algorithm is similar to the Euler and improved Euler methods.
Rather than approximating the area of a rectangle, as the Euler method does, or by the area of
a trapezoid, as the improved Euler method does, it approximates by the area under a parabola.
In order to do so it uses Simpson's rule:
41
6 2 2( , ( )) ( , ( )) 4 ( , ( )) ( , ( ))n
n
t hh h h
n n n n n nt
f t t dt f t t f t t f t h t h) ) ) )�
� �� � � � � � �� �* (4.26)
where 2( ), ( )hn nt t) ) � and ( )nt h) � are unknown and need to be approximated. The fourth
order Runge-Kutta algorithm which incorporates all the approximations is:
,1
1,2 ,12 2
1,3 ,22 2
,4 ,3
1 ,1 ,2 ,3 ,46
( , )
( , )
( , )
( , )
2 2
n n n
hn n n n
hn n n n
n n n n
hn n n n n n
k f t y
k f t h y k
k f t h y k
k f t h y hk
y y k k k k�
�
� � �
� � �
� � �
� �� � � � �� �
(4.27)
The fourth order Runge-Kutta method does four function evaluations per step (depicted as the
hollow circles and ny in Figure �4-4) in order to give a method with fourth order accuracy. In
each step the derivative is evaluated four times: once at the initial point (1), twice at trial
midpoints (2, 3), and once at a trial endpoint (4). From these derivatives the final function
value (shown as 1ny � in Figure �4-4) is calculated.
Figure �4-4 - The Fourth order Runge-Kutta method.
Diependaal et al. showed that the variable step size fourth order Runge-Kutta scheme is both
more stable and much more efficient than other published numerical solution techniques
(Diependaal et al. 1987, Gear 1971).
Adaptive Step Size
To ensure that the Runge-Kutta iterations converge, we have to be sure that the time step size
is adequate. In order to do so we use the ‘step doubling’ technique.
42
This technique uses two separate estimations for each step. The first approximation uses the
time step to calculate the parameters at stept T t� � . The second approximation is done using
two small time steps which are half the size of the previous time step. The first step estimates
the parameters at 2
tstept T� � and the second step uses them as a basis in order to estimate the
parameters at stept T t� � .
Once the two different estimations are computed we compare them and calculate the error. A
relative error threshold is used in order to decide if the step size used is too big and the
numeric solution does not converge. If the error threshold is passed the step size is halved and
the process is repeated with a half the initial step size. The time step will continue to decrease
as long as the calculated error is larger than the error threshold (and the solution does not
converge).
We double the time step when 100 steps are calculated without a single one of then crossing
the error threshold. No time step changes, in 100 cycles, reveals that the method is stable and
has enough “margin” so we can try and increase the step size, thus continuing faster through
the time domain.
In order to keep the computation error from growing uncontrolled, the time step size should
be bound. If the time step is too small the computation is not efficient and the error due to
rounding increases.
Time versus Frequency results comparison
The time domain model algorithm was used throughout this study because of its ability to
simulate complex inputs. The model algorithm implementation was verified by comparing the
results with the results obtained by solving the model equations in the frequency domain
(according to Cohen and Furst, 2004). Steady state inputs were used to excite both models.
The models were stimulated with sine waves at frequencies: 250Hz, 500Hz, 1000Hz, 2000Hz,
3000Hz, 4000Hz, 6000Hz and 8000Hz.
A comparison between the outputs of the time and frequency domain algorithms is
demonstrated in Figure �4-5. Both algorithms are based on the same model parameters. Basilar
43
membrane energy curves corresponding to an ideal cochlear model (�=0.5) are plotted in
Figure �4-5a. All eight responses, from both the time and frequency solutions, are plotted in
the same figure. Continuous lines representing the time domain solutions and dashed lines the
frequency domain solutions. Figure �4-5b has the same eight stimuli, but a dysfunctional OHC
gain model (�=0.0) was used. In these simulations a constant � was implemented along all the
cochlear partition.
(a)
(b)
Figure �4-5 – Time versus Frequency algorithm comparison. (a) Basilar membrane energy curves from an
ideal cochlea (�=0.5). (b) Basilar membrane energy curves for a totally dysfunctional OHC cochlea (�=0.0). Continuous lines represent the time domain solution and dashed lines the frequency domain
solution.
It is obvious from Figure �4-5 that with the increase of � the location of resonance for each
input frequency moves towards the helicotrema, and the peak of the resonance becomes more
significant.
All time domain simulations were done with 512 sections and a constant time step. In order to
compare “steady state” responses the initial 30ms of data from the time domain solution was
discarded (in order to minimize the stimulus artifact contributing to the BME curve). The
stimulus artifact is not totally cancelled out by this action and low frequency energy still
resides in the BME curves.
Although the frequency domain simulations are more accurate (for steady state stimuli the
frequency domain simulations do not contain any artifacts resulting from initial conditions or
signal transients) the fit between the two sets of curves is so close that in most places one can
hardly tell that two lines are plotted rather than one. Near the low frequency part of each
curve the time domain solution parts from the “ideal” frequency solution. There are several
reasons for this: the limited dynamic range of the time domain computations, the accumulated
rounding errors due to long simulation times and the low energy stimulus artifact (due to the
44
zero velocity and zero displacement initial conditions). The frequency domain solution is
superior for linear “steady state” responses, but for complex input signals (especially ones
containing transients) and nonlinearities the only option is the time domain algorithm.
Spatial artifacts
Since the model algorithm is solved numerically, non-convergence of the solution is possible
when too little spatial sections are used. By not using enough sections in the calculations
spatial artifacts may be triggered. The same also applies for performing time steps that are too
big for the numeric convergence of the solution.
In his 1-D model Hengel (1996) used a constant number of sections (400). We implemented
400 sections into our model and used Hengels cochlear parameters. The resulting BM velocity
matrix contained spatial artifacts, as can be seen in Figure �4-6a as the second vertical pulse at
around 25ms. Once the section count was increased to a reasonable amount the artifacts
completely disappeared (Figure �4-6c), thus proving that Hengels 1-D OAE results are nothing
more than an artifact.
(a)
(b)
(c)
Figure �4-6 – Spatial artifacts. (a) The figure is contaminated with spatial artifacts when only 400 sections were used. (b) The spatial artifact can still be seen when 512 sections are used. (c) Here we see a clear
result after increasing the section count to 1024. All three simulations use the Hengel cochlear parameter configuration.
In order to eliminate the same problem in our simulations, all the results in this work were
checked for spatial artifacts by running important findings multiple times with different
section counts. The model was run with different spatial resolutions and the output monitored
45
for significant changes. Also, all the simulations in this work were done with a minimum of
512 sections.
(a)
(b)
Figure �4-7 – Spatial resolution. (a) The results obtained with 512 sections. (b) The same results obtained
with 2048 sections. Both simulations were done with the same �(x) distribution.
If the model solution is numerically stable then the output from different simulations with
different section counts should be the same. Figure �4-7 is an example of a simulation that is
not contaminated by spatial artifacts. The model was run twice, with two different section
counts, resulting in the same output.
46
5. Results
In the previous chapter we introduced the auditory model design and algorithm. In this
chapter we present the different results that were obtained using that model. The first section
of this chapter presents the model solution for a linear-uniform human ear model. The second
part of the chapter will describe a linear-nonuniform model and its results. In the last part the
simulated results for a nonlinear-nonuniform model will be demonstrated.
The Model Output: Basilar Membrane velocity along the cochlear
partition
Several different types of input signals were used as stimuli in this research. Clicks and tone
bursts were used primarily in order to study the models OAE response to transients, while
pure sine waves were used to calculate basilar membrane energy curves and audiograms.
Figure �5-1 depicts the basilar membrane velocity response to a click and a sine wave.
The model simulation results are described as BM velocity in a time-place matrix. In this
representation it is possible to visualize the stimulus energy dissipating along the cochlear
partition, from base to apex. The two-dimensional matrix as shown in Figure �5-1 represents
the basilar membrane partition velocities’ magnitude ( bm�� ) for every section along the BM
partition versus time as a response to a click (a) and a sine wave (f=250Hz) (b). The matrix
rows represent the longitudinal axis along the cochlear partition and the matrix columns
represent time. The top section (near the base of the cochlea) has the highest characteristic
frequency (CF) and the bottom section (near the apex) has the lowest CF.
(a) (b)
Figure �5-1 - Model response to a click and a sine wave. (a) BM velocity response to a click. (b) BM
velocity response to a 250 Hz sine wave.
47
Figure �5-2 represents the BM response when band pass stimuli are fed into the model. The
BM response is place-bound in correlation to the frequency band of the stimulus. Only a
partial BM movement is initiated by band passed stimuli. Low frequency specific sections do
not move in response to stimuli which was high passed.
(a) (b)
Figure �5-2 - Filtered stimulus response. (a) The response to a band pass (500Hz - 5000Hz) filtered click
stimulus. (b) The response to a 2 KHz tone burst stimulus (2
0( )sin(2 )* t tKhz e� � ).
Simulated Audiograms
One of the formal measurements of human hearing is the pure tone audiogram. In pure tone
audiometry, hearing is measured at frequencies varying from low pitches (250 Hz) to high
pitches (8000 Hz). Calibrated tones are provided to a person via earphones, allowing that
person to increase the level until the tone can just be heard. Audiograms compare hearing to
the “normal” threshold of hearing, which is an average threshold calculated from many
individuals with intact hearing. The threshold of hearing varies with frequency and the
audiogram is normalized so that a straight horizontal line (at 0 dB) represents a normal
hearing individual.
The hearing level is quantified relative to “normal” hearing in decibels (dB), i.e. normal being
0 dB and higher numbers indicating decreased thresholds. An adult with a hearing level of
less than 25 dB is said to have normal hearing, while in children the threshold is a bit stricter
with a hearing level of 15 dB defining normal hearing. The dB score is not hearing percent
loss, but a 100 dB hearing loss is nearly equivalent to complete deafness for that particular
frequency. It is possible to have scores less than 0, which indicate better than average hearing.
48
We can use the model simulation to compute "loudness", which will be used to calculate
estimated audiograms (Figure �5-3) for a particular simulated ear. Loudness corresponds to the
subjective impression of the magnitude of a sound. For the purpose of estimating the model
outcomes in terms of this perceptual concept, we will use the following definition: Loudness
in terms of the model is the energy acquired by the whole cochlea due to the basilar
membrane velocity (Furst et al. 1992):
2
0 0
1( ( , ))
T l
d bmL x t dxdtT
+� * * � (5.1)
where T is the stimulus duration and l is the cochlear length.
Since the audiogram is a relative measurement, Cohen and Furst (2004) defined an ideal
cochlea as a cochlea with OHCs that are optimally activated. One parameter was used to
describe the OHC activity, the OHC gain factor (�). � was defined as 0 1( ( where �=0
represents a cochlea with no active OHC and �=1 represents a non-realistic cochlea whose
BM motion reaches infinity. �=0.5 was chosen to be the optimal cochlea. This choice reflects
the best match to physiological tuning curves and gain (Cohen, 2004).
The model used for the audiogram calculations is a linear model with low level stimuli. It is
accepted that inner ear responses are linear for low level stimuli magnitudes and hearing
threshold measurements are usually done in magnitude ranges in which the cochlea is
regarded as linear.
Sine waves were used as stimuli for the calculation of each point in the audiogram. A full
simulation of a single sine wave was done and the initial 25ms of the output matrix discarded
(during this time the output is manifested with stimulus artifacts). The “steady state” part of
the response was summed for the loudness calculation. Our “ideal” cochlea (�=0.5) was used
to define a reference threshold to which all other loudness calculations were compared:
( ) ( 0.5)d dThres L L � � � (5.2)
49
Figure �5-3 - Audiograms simulated by the linear model. Red represents a normal hearing cochlea, green
represents a cochlea with partially functioning OHCs and blue represents a dysfunctional OHC cochlea.
For input frequencies below 1000 Hz the difference in the estimated threshold for different
values of � is less than 30 dB. However, there is a significant difference in the estimated
threshold for higher frequencies. For � < 0.2, each of the estimated audiograms has a
maximum threshold at a frequency between 4 and 6 kHz. These types of audiograms resemble
typical phonal trauma audiograms.
Simulated TEOAEs
OAE pressure is calculated using equation 4.2, where eP represents the emission pressure in
the ear canal. An example of a simulated TEOAE is demonstrated in Figure �5-4. Equation 4.2
links the output pressure to the stimulus pressure and the tympanic membrane pressure
(influenced by the pressure emitted by the cochlea). This linkage causes the stimulus pressure
to always be mirrored in the output at the time of stimulus onset. After the stimulus ends
(when clicks and tone bursts are used) we see a short period of stimulus artifacts. Thus, the
first 2.5 ms after stimulus are blocked and thrown out from the rest of the computations.
These initial responses have high amplitude low-frequency components and interfere with the
OAE signal which is several magnitudes weaker.
The emission pressure calculated by the linear-uniform model is depicted in Figure �5-4. The
TEOAE in this example has no similarity to real TEOAEs recorded in the clinic. The model
generated pressure contains only the stimulus artifact. From now on when we refer to
TEOAEs we refer to them in post-stimulus time, where the first 2.5 ms of the recorded time
(stimulus artifact) have been thrown away.
50
Figure �5-4 – Example of a simulated TEOAE in a uniform cochlea.
Simulated click response
Figure �5-5 demonstrates a simulation of BM velocity as a response to a click. In the upper
part of Figure �5-5 the corresponding OAE pressure vs. time is plotted. As can be seen, no
emissions are generated in this particular simulation. On the right, basilar membrane energy
(BME) curves are plotted (also called excitation patterns). The BME curves are obtained by
computing the sum of the squared BM velocity over time. This gives us an energy-per-section
plot for sine wave stimuli. As can be seen from the BME curves the model amplifies the
higher frequencies (the top sections) more than the lower frequencies (bottom sections). This
accounts for the difference in the colors of the energy lines in the two-dimensional matrix
representation.
Since clicks are wideband stimuli, we see that almost all the different sections of the basilar
membrane start to move in response to the click stimulus. We see the different click
frequency components traveling to their different resonant places along the cochlear partition,
low frequency energy taking the longest time to reach its specific CF (located near the apex).
As can be seen from Figure �5-5, the click (or any other stimulus with low frequency
components) response takes a long time to diminish. This is in agreement with experimental
data which states that it normally takes click responses 20-25 ms to decay (Konrad and Keefe,
2003).
51
Figure �5-5 – Model response to a click simulated in a “smooth” cochlea. In the upper left the calculated
outer ear pressure shows no OAE. On the right we see the Basilar Membrane Energy curves as a response
to frequencies of: 250Hz, 500Hz, 1000Hz, 2000Hz, 3000Hz, 4000Hz, 6000Hz and 8000Hz. The time-place
representation demonstrates what happens inside the cochlea as time progresses.
Nonuniformity
The generation of OAEs requires that part of the energy returns from within the cochlea, back
to the ear canal. As said before, the energy enters the cochlea near the stapes and dissipates
along the BM partition. When all model parameters are uniform/smooth, the stimulus energy
is spent on damping forces of the BM partition. In such a case no OAEs are generated.
Previous models (Furst and Lapid 1988, Zweig and Shera 1995, Talmadge et. al. 1998) have
shown that nonuniformity (random spatial variations) must be assumed in order for the model
to generate some kind of OAE.
In our work we verified the assumption that when small impedance mismatches
(nonuniformities or inhomogeneities or “roughness”) are spread throughout the BM partition
parameters, small energy scatterings occur. These energy scatterings form backward-traveling
52
waves that propagate toward the base of the cochlea, thus, causing OAEs to be generated in
our simulations.
Small impedance mismatches were tested in the following parameters: R (BM partition
damping), S (BM partition stiffness) and � (OHC gain). Inserting the randomness into the R or
S parameters of the BM partition caused the creation of OAEs (Figure �5-6).
Figure �5-6a represents the response to a click of a cochlea which has “roughness” in the BM
partition damping. The small impedance mismatches along the path of the traveling stimulus
energy cause reflections that travel back to the base of the cochlea. If we compare Figure �5-6a
with Figure �5-5 we can clearly see a “ringing” of the maximal velocity area of the cochlea.
Energy reflecting off that part of the cochlea returns to the outer ear canal and is recorded as
OAEs. The exact same phenomena is seen when the “roughness” is inserted into the BM
partition stiffness (Figure �5-6b).
(a) (b)
Figure �5-6 – Responses with nonuniform R or S. (a) The cochlear response to “roughness” in the damping
parameter of the BM partition. (b) The same test with “roughness” in the BM stiffness parameter.
�(x) “roughness”
We decided to incorporate the "roughness" into the OHC gain �(x) parameter in our
simulations. It is well known that OHC damage causes hearing loss and that impaired ears
lack intact OAEs. Thus, it seams that the most "educated guess" to where the randomness
should be is in �(x), controlling OHC functionality. OHC gain was interpreted by Cohen and
Furst (2004) as the relative density of the functioning OHCs along the cochlear partition. It is
53
reasonable to assume that this density is not fixed along the cochlea, but varies randomly. We
assume that �(x) is a Gaussian random variable with a mean of 0.5 (for healthy ears) and a
standard deviation of �.
From this point on the model equations include the nonuniformity described, in their OHC
gain �(x) parameter.
(a) (b)
Figure �5-7 – Otoacoustic emission (OAE) simulation. (a) No OAEs are generated in a cochlea with a
constant �(x)=0.5. (b) The generation of OAEs by a cochlea with E(�)=0.5 and �=1e-6.
In Figure �5-7 OAE generation is demonstrated in the cochlear model. A model with uniform
parameters is depicted generating no OAEs (a) while a model with "roughness" inserted into
the �(x) parameter (b) demonstrates the generation of OAE that travel backward toward the
stapes. In the upper part of each figure we can see the energy that dissipated (via the middle
ear) into the ear canal and was recorded as OAEs.
By closely examining Figure �5-7a it is noticeable that the energy reflections take part around
the maximum velocity areas. This is in conjunction with the notion that the backward-
traveling wave is dominated by wave reflections within the region about the peak of the
traveling energy wave. Around the peak the wave amplitude is much larger than it is
elsewhere, thus leading to larger backward-traveling wave contributions.
54
Simulated Otoacoustic emissions
Two examples of simulated TEOAEs by a nonuniform model are demonstrated in Figure �5-8.
(a) (b)
Figure �5-8 - Examples of simulated TEOAE outputs.
In black we see the stimulus artifact (that is mirrored in the TEOAE) generated by a linear
model. The blue line represents the simulated TEOAE response created by the nonuniformity.
If we compare the figure to the uniform TEOAE (Figure �5-4) we notice that the uniform
models' contribution is the stimulus artifact and the nonuniform contribution is the TEOAE
drawn in blue. Thus, by including a nonuniformity or "roughness" into the cochlear
parameters we are able to create energy reflections that are seen in the ear canal as TEOAEs.
Figure �5-9 demonstrates the correlation between increasing � and the energy amounting
inside the cochlea. All simulations were run with E(�)=0.5. The simulations were calculated
in the time-domain on a linear cochlear model.
Figure �5-9 – Energy increase due to larger “roughness”.
55
We can see an uncontrolled energy increase as the magnitude of � rises above a “roughness”
threshold ( 610� ). This phenomenon will be explained in detail later in this chapter. We should
remember that these graphs were simulated in a linear model and that the energy explosion is
not realistic.
Figure �5-10 depicts the correlation between increasing � and the TEOAE energy emitted by
the cochlea. The figure demonstrates energy histograms calculated by 100 simulations with
different random selections of �(x) with constant � . Figure �5-10a was run with a � of 610� ,
Figure �5-10b with 710� �� , Figure �5-10c with 810� �� and Figure �5-10d with 1010� �� .
The mean value of TEOAE energy for Figure �5-10c and Figure �5-10d is around 52.15 dB
which represents the noise floor for our simulations. From the mean values of Figure �5-10a
and Figure �5-10b we can calculate that the TEOAE average energy changes by around 18dB
for each magnitude of the OHC gain factor ( � ) (this fact was observed in many different �
histograms with different E(�) values).
(a)
(b)
(c)
(d)
Figure �5-10 – Emission energy histograms. (a) �=1e-6 (b) �=1e-7 (c) �=1e-8 (d) �=1e-10
all simulations run with E(�)=0.5
56
Simulated TEOAE spectrum
The OAEs generated by the "rough" model have unique frequency properties. The average
frequency spectrum of 20 TEOAE responses is demonstrated in Figure �5-11. In contrast to
measurements done in the human ear, there are no “low frequencies” below 1500 Hz in the
spectrum. All of our simulated TEOAE frequencies are between 2 and 6 KHz. If we look at a
particular click response simulation we see that the OAE spectrum varies most in the
frequencies of 2 – 6 KHz from the average spectrum. These frequencies match the sections
that are in the area of maximum velocity (in the click response). The low frequencies in the
spectrum (up to 1 KHz) belong to stimulus artifact components that are present more than 3
ms after stimulus onset.
Figure �5-11 – Simulated TEOAE spectrum. The spectrum obtained from 20 runs of the model in the time
domain, where E(�)=0.5 and �=1e-6.
Nonuniform cochlea Audiograms
We have seen that the model generated OAEs change with each random selections of �(x).
Surprisingly, the effect of different random selections on the calculated audiogram is
negligible. The colored lines in Figure �5-12 were obtained for random selections of �(x)
where 610� �� . The colored lines represent the simulation of a healthy cochlea, i.e. E(�)=0.5,
while the black line depicts a cochlea with partially functioning OHCs, i.e. E(�)=0.2. There
are 30 different random selections of healthy cochleae and 30 different random selections of
partially functioning cochleae plotted in the figure.
57
Figure �5-12 demonstrates that the random selection of �(x) has a “negligible” effect on the
audiogram drawn (the change in threshold level is less than +/-5dB) when E(�) is fixed.
Changing E(�) (from 0.5 to 0.2) causes a substantial reduction in threshold (of about 40dB).
Figure �5-12 – Audiograms of nonuniform �(x) selections. The colored lines represent random selections of
�, in a healthy cochlea, where E(�)=0.5 and �=1e-6. The black line represents the simulated audiogram of
a cochlea with partially functioning OHCs.
The corresponding time-place representations are shown in Figure �5-13. Figure �5-13a
demonstrates a typical result of E(�)=0.5 with 610� �� and Figure �5-13b represents a typical
results of E(�)=0.2 also with 610� �� . From the simulations it is clear that a normal hearing
cochlea with "roughness" will create OAEs while a partially functioning OHC cochlea with
the same "roughness" will not.
(a)
(b)
Figure �5-13 – Response to a click by normal and partially impaired ears. (a) Functioning OHCs, i.e.
E(�)=0.5
(b) Partially functioning OHCs, i.e. E(�)=0.2. Both simulations have the same � (=1e-6).
58
It is well known that OHC damage causes hearing loss and that impaired ears lack intact
OAEs. Our simulations demonstrate the connection between the lowering of � (the
functionality of the OHCs) and the lack of generated OAEs.
"Linear" versus "Nonlinear" response processing
In typical experiments the TEOAE recorded are analyzed by two different techniques:
1) The “linear average” technique is the average of a train of several identical stimuli
responses (these are also called Linear clicks). Four click responses are summed together, thus
increasing the SNR of the signal. The implementation of this technique on our model results
is demonstrated in the left column of Figure �5-14.
2) The “nonlinear average” technique is performed on the responses of a train of four stimuli
in which three identical stimuli of a given polarity are followed by one of the opposite
polarity and triple the amplitude. All four auditory responses are summed together, thus
canceling-out the "linear" component of the signal. This way the summed output is less
sensitive to the stimulus artifact (composed mainly of linear components) in the response,
such as the ringing phenomenon in the first few milliseconds of recording. This “nonlinear
average” is referred to as the Derived Nonlinear Technique (DNT). The implementation of
this technique is demonstrated in the right column of Figure �5-14.
In order to verify our assumptions, we implemented the "nonlinear average" technique
combined with a completely linear cochlear model. The linear cochlear model output is four
identical click responses. Their sum completely eliminates the signals, outputting a zero
matrix from that simulation (not shown in the figures).
Figure �5-14 demonstrates response processing by both methods. The top plots (a, e) reveal the
stimulus train, plots (b) and (f) the time-place representation of the BM velocity, plots (c) and
(g) demonstrate the OAE output and plots (d) and (h) the OAE spectrum. The left column
represents the linear technique and the right column the nonlinear technique. Although both
OAE spectrums seem alike, they differ in their low frequency content. The low frequencies
are the result of processing the uncensored click response (containing the linear components).
The OAE "linear" response has its peak magnitude near the stimulus peak and then the
response decays while the OAE "nonlinear" response gradually increases in magnitude to a
59
delayed peak (due only to the nonlinear components in the response). Both models used were
nonlinear models, as explained later in this chapter.
(a) (e)
(b) (f)
(c) (g)
Figure �5-14 - Linear vs. Nonlinear click responses. The left column represents responses to a linear click
train and the right column a nonlinear click train. (a) and (e) are the click train stimuli. (b) and (f) are the
response inside the cochlea. (c) and (g) are the time plot of the calculated OAE. (d) and (h) are the OAE
spectrum.
60
Tone burst responses
Tone burst (2
0( )sin( )* t te � � ) trains were simulated in the nonlinear model. Their response was
processed with the “Derived Nonlinear Technique”. Three identical positive polarity tone
bursts were followed by a negative polarity tone burst which had triple their magnitude. All
four responses were summed together.
In Figure �5-15 the resulting TEOAE spectral response to tone bursts (at frequencies: 1KHz,
2KHz, 2.5KHz, 3KHz, 3.5KHz, 4KHz, 4.5KHz, 5KHz and 6KHz) is demonstrated.
(a)
(b)
Figure �5-15 – Linear vs. Nonlinear tone burst response spectrum. (a) The OAE spectrum of a cochlea with
normal OHCs, i.e. E(�)=0.5. (b) The OAE spectrum from a cochlea with dysfunctional OHCs ,i.e.
E(�)=0.2. The tone burst frequencies are: 1 KHz, 2 KHz, 2.5 KHz, 3 KHz, 3.5 KHz, 4 KHz, 4.5 KHz, 5
KHz and 6 KHz.
Figure �5-15a demonstrates a cochlea with fully active OHCs (E(�)=0.5) while Figure �5-15b is
taken from simulations of a cochlea with partially activated OHCs (E(�)=0.2). In Figure �5-15a
we can see the stimulus spectrum mirrored in the normal ear response. This is in conjunction
with research stating that TEOAEs detected in normal ears mirror the spectral properties of
the stimulus (Glattke & Robinette, 2002 - Figure �5-16b). Our simulations show that impaired
ears do not mirror the stimulus spectrum in their TEOAEs and that their TEOAEs are filled
only by noise.
By comparing our simulated TEOAE spectrum to “nonlinear” TEOAE responses taken from
the literature (Figure �5-16a) we can see several similarities. The picture is taken from the ILO
software and is a TEOAE tested in a healthy normal ear. The bumps in the spectrum correlate
with the spectrum of TEOAEs from differential tone bursts. The low frequency noise (seen in
red) is in correlation with the 1-2 KHz “noise” in our simulations.
61
(a)
(b)
Figure �5-16 – “Nonlinear” TEOAEs from the literature. (a) The ILO spectral screen for a normal hearing
ear. (b) The stimulus spectral properties are demonstrated mirrored in the TEOAE.
Localized � "roughness"
We have already described the simulated click responses concerning a cochlea with “smooth”
�(x) (OHC gain). We have also gone over the influence of � magnitude on the response when
�(x) has a random distribution. Here localized “roughness” is injected into an otherwise
smooth �(x). This alters the models behavior and the results are quite different, instead of a
short multi-frequency response we see a narrow band long-term effect. The stimulus used was
a click, the E(�) was set at 0.5 and the localized “roughness” used was 210� �� .
By localizing the “roughness” the OAEs generated have a dominant semi-sine wave
frequency. The location of the injected “roughness” controlled the frequency emitted by the
cochlea.
The correlation between the OAE frequency emitted and the section into which the
“roughness” was injected is demonstrated in Figure �5-17. Four different “roughness”
locations are depicted generating four different OAE frequencies. Each figure shows the OAE
generated (top left), the OAE spectrum (top right) and the time-place representation from
within the cochlea (bottom plot). In each of the simulations in Figure �5-17 the OAE emission
is clearly seen resonating as a “standing-wave” inside the cochlea. It seems that the localized
“roughness” transforms the cochlea into a tuned resonant cavity. The interaction between the
reflected energy (by the “roughness” impedance mismatch) and the OHC gain generates a
“long lasting” OAE frequency.
62
Figure �5-17a has roughness along sections 150-200 and generated OAEs with a mean
frequency of 2250 Hz, Figure �5-17b has roughness at sections 200-250 and a frequency of
1250Hz, Figure �5-17c has roughness at sections 250-300 and a frequency of 950Hz and
Figure �5-17d has roughness at sections 300-350 and a low frequency that resides under the
noise floor of the simulation environment and can only be seen in the time-place
representation.
(a)
(b)
(c)
(d)
Figure �5-17 – Localized � “roughness”. The figures demonstrate the simulation output when localized
“roughness” was added along different sections of the BM partition. The top left plot in each figure shows
the OAE created while the top right plot shows the OAE spectrum. The bottom time-place representation
clearly depicts the resonating energy within the cochlear.
Although it would seem that the "long lasting" resonance should change the simulated
audiogram created by their unique �(x) distribution, this is not so. Because audiograms are
measured with sine waves over a long period of time, the resonating wave dies down and does
not influence the simulated audiogram by more that 3 [dB] (results not shown).
63
� mean
In order to study the influence of E(�) on the simulated TEOAE spectral properties we ran
multiple model simulations with different mean values. The results are shown in Figure �5-18.
In Figure �5-18a we see the simulated spectral properties versus E(�) and in Figure �5-18b the
corresponding audiograms are plotted.
(a) (b)
Figure �5-18 – Nonlinear spectral response vs. E(�) and their corresponding audiograms. (a) The figure
demonstrates the spectral click response vs. the decrease of E(�). (b) The audiograms generated by the
same E(�) show no significant deviation from constant �(x) audiograms.
Each spectral line was obtained from 20 different runs of the model, while �(x) was randomly
selected having a constant E(�) and 610� �� . The average spectrum is plotted for each
constant E(�) revealing a frequency “migration” with the decrease of E(�). The OAE
magnitude is lowered by more than 100 dB while the amount of different frequencies in each
OAE is reduced drastically. The maximal frequency component changes from 8 KHz in
E(�)=0.5 to around 1.5K Hz at E(�)=0. Thus, it is clear that low values of E(�) yield OAEs
without high frequency components. This estimate is in conjunction with the traveling wave
peak moving toward the stapes with the decrease of E(�).
In clinical tests OAEs are separated into bands of frequencies and each band is checked for
the presence of the OAE energy. Our results suggest that with the onset of hearing loss
(degradation of the OHC gain factor) the high frequency bands are the first OAE components
that are lost.
As already seen, in the beginning of this chapter, changes in E(�) cause a considerable change
to the simulated audiogram (Figure �5-18b). By lowering E(�) the threshold level decreases
64
substantially reaching a minimum at frequencies around 5 KHz. A healthy ear (E(�)=0.5) sets
the baseline for all simulated audiograms, generating a straight line across the plot. When
dysfunctional OHC ears (E(�)=0) are estimated the damage amounts to around 60dB in
frequencies associated with human speech (4 - 6 KHz).
Nonuniformity and Energy Explosion
In Figure �5-19 the total energy inside the cochlea is plotted versus different distributions of
�(x). When � is increased above a threshold an energy explosion occurs inside the cochlea.
Figure �5-19 – Total cochlear energy vs. different � s.
The energy explosion is demonstrated as seen inside the cochlea in Figure �5-20. Figure �5-20
demonstrates a simulation run in the linear time domain model where E(�)=0.5 and 410� �� .
Figure �5-20 – Energy explosion that occurs when a large � is used.
65
The energy explosion represents the speed of the BM partition increasing indefinitely.
Without anything to restrain the energy (in the linear model) the BM velocity increases to
infinity. This phenomenon is not realistic because in human ears a one time high intensity
stimulus does not trigger indefinite ringing.
The energy explosion can be explained if backward-traveling energy, reaching the base of the
cochlea, is partially reflected back into the cochlea by the middle ear boundary condition. The
reflected energy becomes a forward-traveling wave and generates new backward-traveling
waves that are also reflected back into the cochlea by the middle ear. At frequencies for which
the total phase change due to round-trip wave travel is an integral number of cycles, standing
waves can build up within the cochlea, which is then acting, in effect, as a tuned resonant
cavity.
Up to this point the model was treated as a Linear Time-Invariant (LTI) system. In such a
system for an input that includes sinusoids the output can be described as a sum of the same
sinusoids with phase and amplitude changes. Up to the inclusion of nonuniformities in the
cochlear equations the system acted as a LTI system, all equations were linear and the system
was time-invariant. After we include a large enough nonuniformity into the model, the model
stops to act as a LTI system. It is not time-invariant anymore. This fact is obvious when we
pay attention to the system stimulus as simulation time progresses. At the beginning of the
simulation only our stimulus excites the cochlear model. As time progresses energy returning
off impedance mismatches reaches the cochlear base and is reflected by the middle ear
boundary condition back into the cochlea. This energy is added to the cochlear stimulus,
which means that the stimulus to the cochlear model is the outside stimulus contribution plus
the contribution of the twice reflected energy from within the cochlea itself. This fact breaks
the time-invariant property and prevents us from treating the complete system as an LTI
system.
Introducing Nonlinearity
Because no active damping is present in the linear model, the energy buildup leads to an
energy explosion inside the cochlea. This energy buildup starts locally and draws all the BM
sections into the process. In order to constrict the energy explosion we added nonlinearity into
the model.
66
There are many known nonlinear cochlear phenomena. For example combination tones and
emitted distortion products (ADP). It is clear that every realistic cochlear model must include
nonlinear terms. In the study by Elbaum and Furst (2005) different nonlinear functions were
introduced into the BM-OHC cochlear model. The nonlinearities were inserted in three
different locations: the BM damping, the BM stiffness and the OHC electromotility
components. The research purpose was to analyze different functions integrated in a variety of
model components and to check the generation of Combination Tones (CTs) and their
characteristics. Their results show the insensitiveness of the model to the location of the
nonlinearity. They concluded that it is reasonable to assume that there are a number of sources
for nonlinearity inside the cochlea, one nonlinear source causing linear CT amplitude growth
and an additional nonlinear source causing saturation effects.
Thus we included the nonlinearity in the BM damping factor by assuming:
20 1
( , )( , ) ( )*(1 *[ ] )bm x t
r x t r xdt
��
�� � (5.3)
Eq. 5.3 is substituted in Eq. 4.10.
By implementing nonlinearity (in the damping factor) we constrict the energy buildup which
still occurs, but now reaches an upper limit (Figure �5-21). With the nonlinearity in place the
OAE resembles stabilized amplitude standing waves. From this point on the model described
includes nonlinearity in the BM damping factor plus the nonuniformity in the OHC gain �(x)
parameter.
Figure �5-21 - Large � simulation combined with constricting nonlinear damping. The nonlinear damping
in the BM partition limits the energy generated inside the cochlea.
67
Once the standing wave is initiated inside the cochlea it never decreases. In simulations
conducted the process continued without losing amplitude for over 1000ms. The process
reaches a steady state and becomes the models “noise floor”. It seems that the nonuniformity
creates noise inside the cochlea that might cause a significant hearing loss.
By applying different magnitudes of � to the nonlinear model we can generate a � versus total
cochlear energy graph (Figure �5-22). The model was run with multiple �'s (having E(�)=0.5)
and different damping factors (�1 in equation 5.3). In Figure �5-22 we can see that with the
nonlinear equation in place the energy buildup reaches a saturation level. Our conclusion is
that above a threshold of 610� �� the energy explosion is initiated and the total cochlear
energy stabilizes, due to the nonlinearity in the damping factor. In Figure �5-22 two different
damping factors (�1 in Eq. 5.3) generating two different energy stabilization levels are
demonstrated. We can clearly see that the damping factor (�1) from equation 5.3 controls the
energy stabilization magnitude (the “noise floor” amplitude).
(a) (b)
Figure �5-22 - � vs. Total cochlear energy. The � magnitude was varied while E(�)=0.5. The total energy
that developed inside the cochlea was calculated in response to a click stimulus. (a) damping factor �1=10
(b) damping factor �1=1e-5.
Stimulus magnitude influence on nonlinear model responses
When working with a nonlinear cochlear model (for example where E(�)=0.5 and 610� �� )
we can divide the stimulus magnitude into three distinct levels. The magnitudes are separated
according to the kind of cochlear response they generate.
68
For small stimulation magnitudes the models' response is similar to a linear models response.
The OAE generated are well defined and the reflected energy is easily separated from the
background noise in the time-place representation (Figure �5-23a).
For larger stimulus energies, where the model is not linear anymore, the stimulation
magnitudes divide into two phenomena. Medium stimulation magnitudes generate an
emission pressure that is under the cochlear “noise floor”. The emission pressure is dominated
by the standing wave energy being emitted by the cochlea and the TEOAE we are interested
in is not seen. Inside the cochlea a large energy mass is being generated blocking the view on
the TEOAE of interest (Figure �5-23b).
For high stimulus magnitudes the emission signal we are interested in is well above the “noise
floor”. The simulated ear canal pressure signal is dominated by the TEOAE high amplitude
response and we can easily see the BM partition responding to the strong energy stimulus.
The high energy input activates the BM partition so intensely that the velocity created by the
stimulus is well above the movement created by the standing waves inside the cochlea (Figure
�5-23c).
(a)
(b)
(c)
Figure �5-23 – Different stimulus magnitudes in the nonlinear model. (a) A low level sine wave stimulus
response creating a “linear” response. (b) A medium energy sine wave response, under the “noise floor”.
(c) A high energy sine wave response, dominating the movement of the BM partition.
In the medium stimulus magnitude range the cochlear response can be partially retrieved by
averaging a large amount of cochlear responses. The “noise” created by the standing wave
cancels itself out and we are left with an almost “normal” cochlear response (Figure �5-24).
This response is similar to the low energy linear response.
69
(a) (b)
Figure �5-24 – Signal in noise. (a) A plot of 100 simulations with different random selections of �(x) (all
with the same �). (b) The average of the same 100 OAEs demonstrates the cancellation of the random
noise and reveals the “low-level-linear-response” we would get from a cochlea with a “normal” �.
Figure �5-24 demonstrates the averaging of 100 cochlear responses to a medium sized input.
As we can see in Figure �5-24b, the averaged signal of 100 responses resembles the “linear”
low level response created by the model. More than 100 responses are needed in order to
achieve the fine details of the low level response and all its properties.
70
6. Discussion
The human ear is partially still a mystery, and there are several cochlear phenomena that are
not yet fully understood. This research is the first step towards the implementation of a model
capable of simulating different emissions generated by the human ear.
The model presented in this thesis is an extension of the one-dimensional model created by
Cohen and Furst (2004). Their model was enhanced in order to create OtoAcoustic Emissions.
A simple middle ear model and an outer ear canal were integrated into the cochlear model to
form an entire human ear system. The main assumption of the model is that the middle ear is
equivalent to a simple mechanical piston translating the tympanic membrane motion to oval
window displacement. The model was tested and its ability to generate TEOAEs was verified.
In this enhanced model the stimulus is injected through changes in ear canal pressure and the
output is recorded as OAE pressure (also in the ear canal).
The model algorithm solution was implemented in the time-domain. A frequency-domain
solution was also developed for steady-state stimuli and both solutions yielded exactly the
same results. However, when nonuniformity was introduced the time-domain solution
“exploded” while the energy in the frequency-domain solution was unchanged. The
nonuniformity reveals the fact that the model is not time-invariant. Thus, a frequency-domain
solution is not justified.
The time-place representation which is obtained by the time-domain solution reveals temporal
phenomena in the cochlear responses. The time-place cochlear representation changes with
each sample of the input signal; hence, it enables discrimination between short time events.
Thus, the time-domain model enables response prediction for both click and tone burst signals
(transient signals).
The model of Cohen and Furst stands out from the rest of the cochlear models because of it’s
incorporation of an OHC model into the BM model. This special feature allows us to
distinguish between normal functioning cochleae and dysfunctional cochleae. This is done by
changing the OHC gain factor and influencing the OHC functionality along the cochlea. The
influence of OHC functionality on TEOAE generation is clearly seen in the difference
between normal and impaired ears time-place representations.
71
Other models have shown that some kind of “roughness” must exist in the model in order to
generate TEOAEs (Talmadge et al. 1998, Zweig and Shera 1995, Shera and Guinan 2003).
These models did not incorporate an OHC model and so had to insert the roughness in an
artificial way into other model parameters, i.e. Talmadge et al. (1998) incorporated the
“roughness” into their place-frequency map.
The first research goal was to verify whether the cochlear model by Cohen & Furst could be
enhanced into a model capable of emulating TEOAEs. We succeeded in generating a model
capable of simulating TEOAEs. The TEOAEs were generated after nonuniformity
(“roughness”) was introduced into the OHC gain parameter. Without “roughness” the
emission pressure waves generated resemble stimulus artifacts, with no TEOAE
characteristics. Only the addition of minute impedance mismatches caused the generation of
TEOAEs with realistic properties. It seems that a totally “smooth” human ear is not capable of
creating TEOAEs, and some kind of “roughness” must exist somewhere inside the inner ear.
We have shown that the addition of “roughness” in several of the model parameters creates
the TEOAE phenomena. There is no conclusive evidence as to where the “roughness” is
located in the real human ear. The main conclusion of this research is that a “too smooth”
cochlea is unnatural and the cochlea must have some kind of nonuniformity in order for the
cochlea to be able to emit any kind of sound.
In the present model the OHC mode of activity is characterized by one parameter �(x), which
can vary along the cochlear partition and is regarded as the OHC functionality. This research
focuses on how TEOAE responses vary due to variations in �(x). We refer to a cochlea with
E(�)=0 as a dysfunctional cochlea (passive/dead cochlea - without active OHCs). A cochlea
with E(�)=0.5 is regarded as a normal cochlea (healthy with functioning OHCs). The
dysfunctional cochlea reproduces the behavior of the basilar membrane typical of postmortem
measurements and the healthy cochlea reproduces the high tuning basilar membrane motion
of an active cochlea.
When the OHC gain factor is normal, i.e. E(�)=0.5, cochlear displacement and velocity are
enhanced in the vicinity of the characteristic frequencies. The enhancement is mostly
significant for frequencies above 1 KHz (the cutoff frequency of the OHC membrane).
Frequencies between 3 and 6 KHz receive the most amplification by the OHC models
contribution to the traveling wave. Changes to the OHC gain factor cause the decrease of the
72
basilar membrane motion peek. The most significant energy component is reflected within the
region of maximum displacement (the traveling wave peek), thus the OHC gain factor has a
large influence on the generation of TEOAEs. A low OHC gain does not enable the
generation of OAEs with enough energy to reach the ear canal.
In contrast to incorporating an OHC model in the cochlear model, other researchers have to
“hand manufacture” the gain profile in their models. Nobili et al. (2003) had to manually
create a cochlear amplifier (CA) gain profile for their model (Figure �6-1) by comparing
psychoacoustic data from subjects with normal hearing to data from patients with acquired
hearing loss of cochlear origin. In our model the gain profile is inherent in the design. The
OHC model is the one responsible for the heightened area in the 1-5 KHz frequencies.
Figure �6-1 - The profile of the Cochlear Amplifier by Nobili et al.
The fact that the cochlear model has OHC functionality incorporated into it makes the
insertion of roughness/inhomogeneity much more intuitive. Because TEOAEs are thought to
originate as a side-effect of the cochlear amplifier (Kemp 2002) which has become associated
with the OHC motility (Liberman et al. 2002, Liberman et al. 2004) we decided to locate the
“roughness” in our model in the OHC gain factor. Although there is no evidence in the
auditory physics literature that there is OHC gain nonuniformity in the human ear, it is only
natural that minute differences are present between neighboring OHCs. It is not conceivable
that the biological tissue is totally uniform throughout the cochlea and small perturbations will
be enough to cause energy reflections. Even a small bending of the OHCs in relation to one-
another, or inhomogeneities in the OHC forces due to random, cell-to-cell, variations in the
number of somatic motor proteins will suffice to create the inhomogeneities incorporated into
73
our model (changes of less than 0.0001% were incorporated into the OHC gain factor). These
small changes between adjacent sections of the BM partition are sufficient to create reflection
source OAEs.
The assumption of random distributed OHC gain along the cochlea seems realistic since in
humans the loss of sensitivity in 4 KHz was found independently of the type of noise they
were exposed to (Saunders et al 1985; Moore 1998). Cohen and Furst (2004) showed that a
random �(x) along the cochlea will generate simulated audiograms with a maximum threshold
at 4 KHz.
Hearing-impaired people who suffer from OHC loss exhibit a significant degradation in the
performance of all known nonlinear phenomena such as two-tone suppression, combination
tones, and cochlear otoacoustic emissions (Moore 1998). This research provides model
predictions of the influence of nonuniformity on the TEOAE generation in the human ear. We
have shown that decreased OHC functionality leads to degradation in cochlear reflection
source otoacoustic emissions.
Simulated audiograms were generated, on the basis of the basilar membrane velocity, for
‘normal’ ears and ears with OHC loss. Our results show that the mean OHC gain magnitude
has a substantial influence on the simulated audiogram. Even a relatively small decrease in
E(�) will create a hearing threshold change in the modeled ear. On the other hand, the
modeled "roughness" has very little influence on the audiograms obtained (in the range
tested). This enhances the notion that the goal of the human ear is to transform sound waves
to neural excitation patterns and not the creation of TEOAEs. TEOAEs are only a side-effect
(an artifact) of the normal workings of the human ear.
In contrast to audiograms, TEOAEs are very small signals that are influenced considerably by
both the “roughness” magnitude and the mean OHC gain magnitude. The mean OHC gain
controls the formation of traveling wave peeks and the “roughness” magnitude controls the
proportion of energy reflected along the cochlea. Without both, TEOAEs will not be created.
By comparing the generated TEOAE from our model with recorded TEOAEs from the
literature we see a resemblance in the time domain characteristics but the clinical recorded
74
TEOAE spectrum is different from our simulated results. The literature details that the
recorded TEOAE spectrum has frequency components between 0 and 7 KHz, with a
maximum at about 1.7 KHz. In comparison, our generated TEOAEs lack low frequency
components. We were not able to generate frequencies below 1.5 KHz in our simulated
OAEs.
Our results clearly show that the OAE frequency is dependent on the “roughness” place. In
the some time-place representations we can clearly see a particular BM section resonating and
“creating” the OAEs. This happens when we inject localized “roughness” into the BM
parameters.
The coherent reflection model (Zweig and Shera, 1995) predicts that the TEOAE evoked by a
click comprises a sum of waves scattered by perturbations located throughout the peak of the
traveling wave. In our model we can see that the TEOAE arises from a distributed region,
roughly equal in extent to the width of the traveling wave envelope.
The two techniques used in the clinic to improve the OAE recorded SNR are Averaging and
the Derived Nonlinear Technique (DNT). Both methods were simulated and the results
resembled clinical responses. When averaging was applied, the tail of the acoustical stimulus
waveform (up to 6ms after stimulus onset) was seen interfering with the early parts of the
response. When we implemented the DNT method, the stimulus linear artifact was completely
canceled out (in the linear model). When run on a model with nonlinearity, the low
frequencies (mainly stimulus artifact) in the calculated DNT were significantly smaller and
we were able to get better results for the high frequency components. In real practice the
stimulus artifact is not completely canceled, but is considerably attenuated (about 40 dB)
disclosing the early part of the OAE response which otherwise would be mixed with parts of
the stimulus artifact.
When the nonuniformity magnitude reached a threshold, the system behavior became
unstable. Large “roughness” in conjunction with medium sized stimuli created very large
reflected waves. The reflected energy created an increasing standing wave inside the cochlea
that led to an energy explosion. In order to prevent the energy explosion from taking place,
75
nonlinearity was added to the damping factor of the BM partition. The nonlinearity
constricted the amplitude of the standing wave and thus stopped the energy “explosion”.
Other researchers had to incorporate nonlinearities into their models in order to solve the
energy explosion problem as well. Talmadge et al. (1998) incorporated “stabilizing
nonlinearity” into their model in order to overcome this nonrealistic phenomenon.
We conclude that adding nonuniformity to the model (by itself) is not enough and a
constricting nonlinearity must be added as well. Without the nonlinearity, loud signals would
bring the system to an unreal energy explosion. We must remember that biological systems
are nonlinear in nature and our model was built on linear equations for simplicity. It is most
likely that several nonlinearities contribute to the real human ears’ energy constriction. In our
tests to constrict the energy explosion we tried just one type of nonlinearity and we placed it
in just one place inside our simplistic human ear model.
With the nonuniformity and nonlinearity in place, the energy that escapes basally and reaches
the middle ear boundary is partially reflected back into the cochlea. This energy forms a new
traveling wave that re-stimulates the OHC gain mechanism. Under conditions of high
amplification and endless recirculation of the traveling wave sustained oscillations inside the
cochlea are sustained. These oscillations create “spontaneous” OAEs in the ear canal. Unlike
clinical recorded SOAEs that have one or more pure tones, our simulated spontaneous OAEs
have a broad spectrum of frequencies and follow the traveling waves peek envelope
(controlled by the OHC contribution to the BM motion). These standing waves resemble the
noise floor of the system, once they are triggered (by a loud noise or by large “roughness”
along the BM partition) they never die down.
For these spontaneous emissions to occur, strong OHC amplification must coexist with at
least one distributed irregularity (“roughness”) in one of the BM partition parameters. The
energy reflected by the middle ear must also be sufficient to sustain a continuous oscillation
along a section of the BM, after re-amplification and re-emission.
It is not only spatial imperfections that can generate OAEs. If the forces exerted by OHCs on
the BM do not exactly follow the stimulus waveform (i.e. if the OHC electromotility is
76
“nonlinear”), they will add distortion signals to the forward traveling wave, which are one
cause of aural combination tones.
Research from the recent years has shown that there are several sources to DPOAEs and
TEOAEs, each contributing part of the energy (Shera 2004). The sources for OAE production
can be separated into two main categories: reflection and nonlinear. Reflection sources are
distributed inhomogeneities along the cochlea while the nonlinear OAE sources can be
distributed in several different places in the cochlea and the organ of Corti. In our study we
implemented nonlinearity in the BM partition damping factor. We have seen that our
implemented nonlinearity does not generate DPOAEs in the ear canal.
The work by Elbaum and Furst (2005) tested several types of nonlinearities and their impact
on the cochlear response. Their work concentrated on the nonlinearities responsible for the
creation of DPOAEs inside the cochlea. The nonlinearity implemented in our work did not
succeed in generating DPOAEs in the ear canal, although DPs were created locally along the
BM partition.
Impact on the field of OAE model research
1) Our study has shown that Nonuniformity alone can not explain the different phenomena
encountered in the cochlea. The minimum need is for Nonuniformity with constraining
Nonlinearity in order to achieve a more realistic outcome. Without the incorporation of
restricting nonlinearity into the model we can show that the model is too unstable and tends to
“explode” with unconstrained energy release.
2) During the course of this study several different cochlear models were investigated. By
comparing our own model to them we learned many new things about the data collected.
Several mistakes were found in the different models. Their conclusions were “corrected” into
a better understanding of the data they generate. By studying thoroughly the effects of
choosing ‘wrong parameters’ for the model, we have now a robust and easy to use model
which can form the base for future research.
3) Considering that our model is a simplified one-dimensional model of the human ear, the
created TEOAEs have a good resemblance to clinical recorded TEOAEs. By varying the
OHC gain factor we were able to predict several TEOAE phenomena of healthy and impaired
77
ears. The models’ simplicity and speed enables hypothesis testing in a relatively short time
opposed to the more elaborate 3D models (Bohnke and Arnold, 1999) which take a long time
to set up and run. In addition, this model serves as the start of development of a unified, more
realistic human ear model capable of simulating all OAE types.
Future research possibilities
This unique research has been focused on generating TEOAEs by adding a middle ear model
to the already existing cochlear model. In order to harvest better results the existing cochlear
model was rewritten from scratch. The model now is more robust and more user friendly. It
incorporates a fourth order Runge-Kutta technique and a robust variable step size algorithm.
Many different stimuli with many cochlear configurations were tried out during the course of
the study. During simulated data analysis we have observed very interesting results that could
benefit from further study.
These are several future research directions that should be considered:
1) Noam Elenbaum has been studying the effects of different nonlinearities on DPOAE
generation. His study and results could be combined into this model and its’ existing
nonuniformities and nonlinearities in order to produce ONE COMPLETE model with the
capability of generating TEOAEs, SOAEs and DPOAEs.
2) Our simulated TEOAEs do not exactly match all the properties of clinically measured
OAEs. Further study is needed in order to try and find the source of the missing low
frequencies in our simulated TEOAE spectrum. Parameter changes conducted during this
study have shown that low frequencies can not be produced by the model as is, and an
explanation is needed. Changes to the different model parameters could result in more
realistic OAEs. Changes to the middle ear parameters (or a change of the middle ear model
completely) and the M, R, S parameters should continue to be investigated.
78
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Appendix A – Auditory research and model history
Auditory research history
In the late 19th Century, von Helmholtz (1862-1885) conceived a sophisticated theory of
hearing that invoked the presence of highly selective resonators in the cochlea. Approximately
50 years later, von Bekesy (1928) discovered the cochlear traveling wave phenomenon and
fashioned a theory that disputed von Helmholtz' “resonance” theory. Although the traveling
wave phenomenon discovered by von Bekesy is believed to be the key element in the
cochlea's analysis of sound, the initial observations of von Bekesy revealed responses that
were neither sharply tuned nor of sufficient sensitivity to respond to threshold levels of
stimulation. The difficulty extrapolating from von Bekesy's observations to real-life listening
situations can be appreciated when one considers that his observations were confined to either
cadaver ears or to mechanical models constructed to replicate conditions encountered in
cadaver ears.
In discovering the traveling wave, von Bekesy observed that the cadaver cochlea has a very
poor ‘imaging’ quality. He found the traveling wave peak in response to a pure tone stimulus
to extend over a third or more of the entire cochlear length. On the contrary, in the healthy
living cochlea, the TW peak for low-level pure tone stimulation is much sharper. The TW
peak covers less than 1 mm and a shift in frequency of just one-third of an octave moves the
TW peak to stimulate an entirely different set of sensory cells.
The discrepancies between the contemporary traveling wave demonstration and the tuning
and sensitivity of the ear, were clearly described by Thomas Gold some 20 years after von
Bekesy's discovery. Gold suggested that parts of von Helmholtz' resonance theory were in
keeping with the results of listening studies conducted with persons who had normal hearing
(Gold, 1948). The idea of an amplifier to overcome physical limitations was first proposed by
Gold. At first his ideas were not well accepted, but after the discovery of hair cell motility by
Brownell they became a credible possibility.
By the early 1970s, sophisticated investigations of cochlear mechanics in living ears at near-
threshold stimulus intensities revealed that healthy systems were capable of better frequency
85
resolution and threshold sensitivity than were predicted on the basis of von Bekesy's
observations (Rhode, 1971).
Approximately 50 years after von Bekesy's discovery, David Kemp reported that sound
energy produced by the ear could be detected in the ear canal (Kemp, 1978). Called
otoacoustic emissions (OAEs), these sounds offer evidence that the ear contains a source of
energy and that the energy may fuel the sharp tuning and exquisite threshold sensitivity von
Bekesy was unable to see in his experiments. Soon after Kemp's discovery there appeared
reports that cochlear outer hair cells (OHC) were capable of movement in response to
electrical stimuli provided in vitro (Brownell, 1983). As the 1980s dawned, Hallowell Davis
(1983) coined the phrase ‘cochlear amplifier’ to describe the phenomenon in which the inner
ear responds to and resolves near-threshold stimuli.
Ren showed that the ear emits sound through the cochlear fluids as compression waves. He
used a scanning-laser interferometer and found forward-traveling waves but no backward-
traveling waves. He also noted that the stapes vibrates earlier than the basilar membrane.
These results show that the ear emits sound through the cochlear fluid as compression waves
rather than along the basilar membrane (as backward-traveling waves) (Ren, 2004).
OHC research progress
The organ of Corti houses two types of hair cells, the inner (IHC) and outer (OHC) hair cell.
Both cell types transduce mechanical stimuli into electrical signals by modulating a standing
cationic current in response to stereocilia displacement (forward transduction). This current
induces a receptor potential across the basolateral membrane of the cell, the depolarizing
phase of which may promote the release of neurotransmitterers (Santos-Sacchi, 2003).
Only after Brownell’s observation of the twitching of the OHC in response to electrical
stimulation did a potential mechanism for the highly selective and sensitive responses of the
mammalian auditory system to high frequency acoustic stimulation appear. Following his
discovery of reverse transduction, a re-evaluation of the classical concepts of mammalian
hearing has been underway. Current theories that are concerned with the basis of the cochlear
amplifier envision an acoustically evoked cycle by cycle feedback process between the OHCs
and the basilar membrane. The acoustically evoked electrical responses of the OHCs are
86
assumed to affect rapid length changes by these cells, which boost the mechanical input to the
IHC. The IHCs are the receptor cells that receive up to 95% of the afferent innervations.
In 2000 Dallos and colleagues identified the OHC lateral membrane motor, a protein of 744
amino acid residues that they named ‘prestin’. This gene is specifically expressed in outer hair
cells. The mechanical response of outer hair cells to voltage change is accompanied by a
'gating current', which is manifested as nonlinear capacitance. In their study they also
demonstrate this nonlinear capacitance in transfected kidney cells. They concluded that
prestin is the motor protein of the cochlear outer hair cell (Zheng et al., 2000).
Further studies showed that prestin expressing cells were electromotile with motility
magnitudes approaching 0.2 ,m. Actual force measurements that were carried out with an
atomic force microscope showed that prestin generates significant mechanical force and that
this force is independent of frequency up to at least 20 kHz (Dallos and Fakler, 2002).
A study conducted by Liberman et al. showed that targeted deletion of prestin in mice results
in loss of outer hair cell electromotility in vitro and a 40–60 dB loss of cochlear sensitivity in
vivo. These results suggest that prestin is indeed the motor protein, that there is a simple and
direct coupling between electromotility and cochlear amplification, and that there is no need
to invoke additional active processes to explain cochlear sensitivity in the mammalian ear
(Liberman et al. 2002, Cheatham et al. 2004).
Furthermore by working with prestin knockout mice Liberman et al. gathered evidence
suggesting that OHC stereocilia transduction is normal in prestin null ears. The round window
CM data shows that, in the absence of prestin, nonlinearities in OHC stereocilia transduction
are still producing a distortion component at 2f1-f2 in the OHC receptor currents. This fact
has no explanation according to the conventional theory (i.e., that nonlinearities in forward
transduction in OHC stereocilia produce a distortion-frequency component in receptor current
which is then reverse-transduced and amplified via OHC somatic motility into distortion-
frequency vibrations of the organ of Corti).
Liberman argues that because nonlinearities in prestin-based motility have been eliminated by
the targeted deletion the only remaining known nonlinearities are in OHC stereocilia. Thus, a
simple view of the persistent DPOAEs in the prestin-null mouse is that distortions in the
organ of Corti motion arise from the direct coupling of the mechanical nonlinearities of OHC
87
stereocilia bundles. This means that the mammalian stereocilia must be sufficiently well
coupled to the motion of the cochlear partition that they can drive the middle ear to produce
DPOAEs in the ear canal. In his studies DPOAE amplitudes fell within a few minutes after
death, clearly demonstrating a biological origin for the phenomena, i.e. an “active” process
depending on endocochlear potential.
Middle ear research
In the last century there have been many attempts to characterize the human middle ear
whether by measurements or by proposed mathematical models (surveyed in Puria 2003).
The four middle ear measurements carried out are: ear canal impedance, stapes displacement
to ear canal pressure ratio, vestibule pressure to ear canal pressure ratio (middle ear pressure
gain) and reverse middle ear pressure gain (from the cochlea, through the middle ear, and into
the ear canal).
The most extensive measurement of middle ear characteristics was recently done by Puria
(2003). The goal of this work was to allow a full characterization of the human middle ear and
to provide an empirical basis for understanding how the middle ear modifies OAEs generated
by the cochlea and measured in the ear canal. The forward and reverse middle ear pressure
gain measurements are used to quantify the effect the middle ear has on ear canal
measurements of otoacoustic emissions. In addition, the cochlear input impedance and the
reverse impedance are used to quantify the stapes reflection coefficient for OAEs.
Middle and inner ears from seven human cadaver temporal bones were stimulated in the
forward direction by an ear canal sound source, and in the reverse direction by an inner ear
sound source. The forward middle ear pressure gain, the cochlear input impedance, the
reverse middle ear pressure gain, and the reverse middle ear impedance were calculated from
measurements obtained for the first time from the same preparation. These measurements
were used to fully characterize the middle ear as a two-port system.
Presently, the effect of the middle ear on otoacoustic emissions (OAEs) is quantified by
calculating the product of forward and reverse middle ear pressure gain.
88
Figure 7-1 – The effect of the middle ear on OAEs, taken from Puria 2003
In the 2–6.8 kHz region, the roundtrip middle ear pressure gain decreases with a slope of -22
dB/oct, while OAEs (both click evoked and distortion products) tend to be independent of
frequency. This suggests a steep slope in vestibule pressure from 2 kHz to at least 4 kHz for
click evoked OAEs and to at least 6.8 kHz for distortion product OAEs. Contrary to common
assumptions, the measurements indicate that the emission generator mechanism is frequency
dependent.
Voss and Shera used DPOAEs in order to measure the middle ear forward and reverse middle
ear transmission in cat. They used DPOAEs to drive the middle ear “in reverse” without
opening the inner ear of the cats used (Voss and Shera, 2004). The technique allows
measurement of DPOAEs, middle ear input impedance, and forward and reverse middle ear
transfer functions in the same animal. Their results generally agree with the middle ear model
by Puria and Allen (1998). The reverse transfer function is shown to depend on the acoustic
load in the ear canal, and the measurements are used to compute the round-trip middle-ear
gain and delay.
Dynamic analysis of the ossicles shows that the isolated ossicles act as a rigid body in the
audible frequency range. For every measured ossicle the first natural frequency is far away
89
from the audible frequency range (above 30 KHz), so that the ossicles should be considered as
rigid bodies without any energy absorption due to structure bending within the audible
frequencies (Ferrazzini et al. 2002).
Modeling history of the Cochlea
The first recognized model of the cochlea was published by Helmholtz in 1862 in an appendix
of “On Sensation of Tone”. Helmholtz linked the cochlea to a bank of highly tuned
resonators, which were selective for different frequencies, much like a piano or a harp, with
each resonator representing a different place on the basilar membrane. The model he proposed
was not very satisfying since many important features were left out. The most important of
which includes the cochlear fluid which couples the mechanical resonators together. But,
given the publication date, it is an impressive contribution by this early great master of
physics and psychophysics.
The next major contribution was made by Wegel and Lane, and stands in a class of its own
even today. The paper was the first to quantitatively describe the details of the upward spread
of masking, and proposes a “modern” model of the cochlea. If Wegel and Lane had been able
to solve their model's equations, they would have predicted cochlear traveling waves.
It was the experimental observations of the Hungarian researcher G. Von Bekesy, starting in
1928 on human cadavers cochleae, which unveiled the physical nature of the basilar
membrane traveling wave. Von Bekesy, found that the cochlea is analogous to a “dispersive”
transmission line where different frequency components, which make up the input signal,
travel at different speeds along the basilar membrane, thereby isolating those various
frequency components at different places along the basilar membrane. He properly named this
dispersive wave a “traveling wave”. He observed the traveling wave using stroboscopic light
in dead human cochlea at sound levels well above the pain threshold (above 140 dB SPL).
These high sound pressure levels were required to obtain displacement levels that were
observable under his microscope. Von Bekesy's pioneering experiments were considered so
important that in 1961 he received the Nobel Prize.
Over the intervening years these experiments have been greatly improved, but Von Bekesy's
fundamental observations of the traveling wave still stand. Today, we find that the traveling
wave has a more sharply defined location on the basilar membrane for pure tone input than
90
observed by Von Bekesy. In fact, according to measurements made over the last 20 years, the
response of the basilar membrane to a pure tone can change in amplitude by more than five
orders of magnitude per millimeter of distance along the basilar membrane.
Zwislocki was the first to quantitatively analyze Wegel and Lane's cochlear model, explaining
Von Bekesy's traveling wave. Wegel and Lane's cochlear model is constructed from cascade
sections of inductors, capacitors, and resistors; which represent the mass of the fluids of the
cochlea and the basilar membrane mass, partition, resistance and stiffness, respectively. The
aspects of the vertical and width dimensions of each section were suppressed, which means
that each variable was taken as a constant inside the section.
In 1976, Zweig and colleagues noted that an approximate, but accurate, solution for the one
dimensional model could be obtained using a well known method in physics called the
Liouville Green or “WKB” approximation. The results of Zweig et al. were similar to Rhode's
contemporary neural tuning curve responses.
The most common model today is the transmission line model, also called the one
dimensional model. The one dimensional model is built from cascade sections of inductors,
capacitors and resistors, which represent the mass of the fluids of the cochlea, partition
resistance and stiffness, respectively.
1D vs. 2D, 3D
In the 70's, several two-dimensional model solutions became available. Rank was the first to
formulate and consider a two-dimensional model. The 2D model argues that the long wave
approximation is not fulfilled in the region of maximum response of the membrane.
The 2D model is considered to be theoretically more natural than the long wave theories. In
spite of the better results gained with the 2D model the long wave models has gained more
appreciation because they are easy to understand and simple to solve numerically.
In 3D models, the pressure and fluid flows can vary across the width of the cochlear partition.
This pressure variation is consistent with the notion that the arcuate and pectinate regions of
the basilar membrane have different properties.
91
Givelberg, Rajan and Bunn have constructed a comprehensive 3D computational model of the
cochlea using the immersed boundary method. Their pure tone experiments capture the most
important properties of the cochlear macro-mechanics. Even after extensive optimization and
parallelization a typical experiment with 2 milliseconds of simulated time takes approximately
18 hours on an HP Superdome computer (Givelberg et al., 2001).
Bohnke and Arnold developed a 3D finite element mechanical model of the cochlea including
the fluid structure couplings. The model allows the evaluation of the passive mechanical
behavior of the human cochlea with arbitrary input pressure at the stapes footplate including
all kinds of slow and fast waves in the lymph and the cochlear partition. The models linear
solutions fit early experiments which studied the wave propagation in the cochlea of human
cadavers (Bohnke and Arnold, 1999).
As we can see three dimensional models are very strenuous on computer power and do not
include nonlinear or nonuniformities of any sort (not yet).
Both, 2D and 3D models are more complex and involve complicated mathematics, thus harder
to solve numerically. This is reasonable since the 1D formulations have fewer components to
deal with. The 1D model simulations have gained more appreciation because they require less
memory and fewer computations than the 2D and 3D models, and yet are successful in
predicting a large number of phenomena. Moreover, in a 1D model, parameters can be easily
chosen using methods that make sense anatomically, physiologically, and mechanically.
Enhanced one dimensional cochlear models
With the discoveries of the nonlinear compressive basilar membrane, the inner hair cell
responses, the otoacoustic emission, and the outer hair cell motility the models mimicking the
cochlea became much more complex. The 2D and 3D models became too heavy to compute
and a paradigm shift was seen toward the 1D extended model. Using simplifying assumptions
we can collapse the 3D model formulations to a 1D model formulation. Thus a new branch of
1D extended models evolved which could better represent the complex systems under
observation, i.e. the scala media and the organ of Corti.
The large number of nonlinear phenomena discoveries, which began in the 70's, revealed the
necessity to incorporate nonlinear elements into the cochlear models.
92
Hubbard and Hall both used nonlinear damping that increased with the increase of cochlear
partition velocity. Furst and Goldstein tested a nonlinear damping model versus a nonlinear
damping and stiffness model.
A model that includes a representation of the electrical characteristics of the scala media and
the outer hair cells (OHCs) was represented by Hubbard et al. Their model used a standard 1D
model combined with a model of hydromechanical changes in the organ of Corti. A nonlinear
conductance that varies as a function of the basilar membrane displacement was also added.
Nobili et al suggested a model made of an array of nonlinear oscillators, each of which is
coupled instantly to all the others through hydrodynamic forces transmitted by the fluid in the
cochlea. Nonlinearity in this model is expressed by a sigmoid function operating on the
basilar membrane velocity, and feed back pressure difference on the basilar membrane. In
another work by Nobili, Mammano and Ashmore, a shearing viscosity term was added. This
term represents the viscous forces acting on one oscillator section, caused by possible
different velocities of adjacent oscillators.
The models by Talmadge, Long, and Tubis incorporate nonlinearity in terms of a Van Der Pol
oscillator added to the damping factor. They incorporate the concept of delayed operation, as
suggested by Zweig (both fast and slow feedback). Their model produces SOAEs although no
stimulus exists.
A model of outer hair cell motility that cooperates with a cochlear model was suggested by
Geisler. He used the basilar membrane and reticular lamina as two ‘free bodies’ and the outer
hair cell force frequency transform was described as an all pass filter with a constant delay.
Kolston et al. suggested that the sharpening effect of cochlear amplifiers may be due to
variable impedance. Their impedance was connected in parallel to the partition mass,
stiffness, and damping.
The work of Talmadge et al. showed that a class of nonlinear active cochlear models can
successfully describe a broad body of data on the quasi-periodic variation with frequency of
otoacoustic emission fine structure and the microstructure of the hearing threshold (Talmadge
et al., 1998).
93
Their model is based on a 1D macromechanical model using time delayed stiffness and
simplified models of the middle and outer ears. The cochlear model incorporates a frequency
map of the form suggested by Greenwood and the cochlear nonlinearity is modeled as a
quadratic (“Van der Pol” type) nonlinear damping function. Slow and fast feedback time
delayed stiffness was used in the model. Roughness (distributed randomness) was applied to
the place-frequency map. Talmadge et al.’s conclusion was that random spatial variations of
almost any of the cochlear parameter will give rise to the effects seen in their model.
The advantage of this model is that it allows one to make specific predictions regarding
various modeling assumptions (the effect of nonlinearity or distributed roughness), as well as
to directly test in a time domain cochlear model the predictions of the theoretical framework
laid out.
The key elements of the models are tall and broad cochlear traveling wave activity patterns
and cochlear wave reflections at the base of the cochlea and around the tonotopic place of the
traveling wave, with the latter being due to distributed cochlear inhomogeneities in
conjunction with the tall and broad activity pattern and distributed nonlinear cochlear
response.
Spontaneous emissions may arise and are associated with instability modes of the linear active
component of the cochlear mechanics. The cochlear nonlinearity provides the requisite
stabilization for converting the instabilities into limit cycle oscillations corresponding to
actual SOAEs.
The fine structures for SEOAEs, TEOAEs, threshold microstructure, DPOAEs and the
frequency spacing of neighboring spontaneous emissions are mainly determined by the
parameters for basal reflectance, apical reflectance around the tonotopic place, and the ratio of
the left and right basis functions and their spatial derivatives. The model predicts that
psychoacoustic and SOAE/SEOAE/TEOAE fine structure spacing should be similar, but that
the DPOAE spacing should be wider. The models also account for the band-pass character of
DPOAEs.
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