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TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of EngineeringThe Zandman-Slaner School of Graduate Studies
A GENERALIZED ONE-DIMENSIONAL HUMAN EAR MODEL
A thesis submitted toward the degree ofMaster of Science in Biomedical Engineering
by
Dan Mackrants
September 2008
TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of EngineeringThe Zandman-Slaner School of Graduate Studies
A GENERALIZED ONE-DIMENSIONAL HUMAN EAR MODEL
A thesis submitted toward the degree ofMaster of Science in Biomedical Engineering
by
Dan Mackrants
This research was carried out at the Department of Biomedical Engineering
under the supervision of Prof. Miriam Furst-Yust
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September 2008
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Abstract
Many aspects of understanding the auditory system have been evolved due to advanced research done in the last few decades. This on going research revealed many phenomena and facts; nevertheless, data collected from human ear can not be fully explained yet.
In order to explain much of the known phenomena and in particularly the difference between normal and abnormal cochlear performances, a relatively simple model was developed. The model is based on Cohen and Furst (2004) one-dimensional cochlear model with embedded outer hair cells (OHC), in which a model for the middle ear was incorporated (Halmut and Furst, 2005) and two types of nonlinearities were inserted in the inner ear model. Non-linearity was incorporated at the resistance of the cochlear partition and at the OHC length change.
The model was able to predict the compressive behavior of the hearing system and frequency selectivity of the auditory system, along with generation of distortion product otoacoustic emission and combination tones. It particularly demonstrated the loss of tuning, reduction in dynamic range and reduction in otoacoustic emissions in ears with outer hair cell loss. The model can be used as a quantitative description of many types of damaged ears.
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Table of Content1. Introduction.................................................................................9
Ear Anatomy....................................................................................9Behavior Characteristics of the Hearing System........................15Cochlear Models Overview...........................................................24Motivation of the present study....................................................28
2. The Generalized Model...........................................................29Model description..........................................................................29Boundary and initial conditions...................................................38Solving the non-linear model........................................................41Model Parameters..........................................................................45
3. Determination of non-linear parameters..............................46Estimating Loudness.....................................................................46BM resistance non-linearity..........................................................47Introducing OHC non-linearity...................................................49Combining of OHC and R non-linearity.....................................51
4. Model Simulation for simple tones........................................55Pure tone stimuli - Frequency selectivity estimation..................55Pure tone stimuli - loudness estimation.......................................57Two-tone stimuli............................................................................57
5. Results......................................................................................61Pure tone stimuli............................................................................61Two-tone stimuli............................................................................73
6. Conclusions and further research..........................................80
Appendix A: Using CF frequencies.....................................................85Appendix B: Experimental Data about DPOAE...............................87References.............................................................................................89
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List of figuresFigure 1-1 - The mammalian ear.............................................................................................................9Figure 1.2: Outer ear..........................................................................................................................10Figure 1.3: Ossicle chain....................................................................................................................10Figure 1.4: Middle ear.......................................................................................................................11Figure 1.5: The cochlea......................................................................................................................12Figure 1.6: Left – wave propagation; right – "uncoiled" cochlea.................................................12Figure 1-7: Cross-section of the cochlea...........................................................................................14Figure 1-8: organ of corti...................................................................................................................14Figure 1-9: BM Velocity I/O functions............................................................................................16Figure 1-10 - compression of a chinchilla cochlea............................................................................17Figure 1-11 : Tuning curves...............................................................................................................18Figure 1-12: BM velocity of 2f2-f1.....................................................................................................19Figure 1-13: TEOAE used in hearing test........................................................................................22Figure 1-14: Response to two tone stimuli........................................................................................23Figure 2-1: The Cochlea Model.........................................................................................................29Figure 2-2: Sigmoid function for different sets of parameters.......................................................33Figure 2-3: OHC Model......................................................................................................................35Figure 2-4: The middle ear (and ear canal) model..........................................................................39Figure 3-1: Calibrated loudness, with BM resistance non-linearity..............................................49Figure 3-2: Loudness for different values of ..............................................................................50Figure 3-3 (a): Loudness. =0.02....................................................................................................52Figure 3-3 (b): Loudness. =1.........................................................................................................52Figure 3-3 (c): Loudness. =500......................................................................................................52Figure 3-4: Summary of parameters.................................................................................................53Figure 3-5: I/O function for non-linearity in both R and OHC......................................................53Figure 4-1: CT examples. 100% active OHC (Gamma=0.5)...........................................................58Figure 4-2: Typical emission signal...................................................................................................59Figure 4-3: Typical emission FFT signal...........................................................................................60Figure 5-1 (a): Time domain maps. =0.5.........................................................................................61Figure 5-1 (b): Time domain maps. =0.25......................................................................................62Figure 5-1 (c): Time domain maps. =0............................................................................................62Figure 5-2 (a): Frequency domain maps. =0.5...............................................................................63Figure 5-2 (b): Frequency domain maps. =0.25.............................................................................64Figure 5-2 (c): Frequency domain maps. =0...................................................................................64Figure 5-1: Loudness. Effect of value............................................................................................65Figure 5-2: Loudness@=0.5.............................................................................................................66Figure 5-3: Loudness@=0.25...........................................................................................................67Figure 5-4: Loudness@=0................................................................................................................67Figure 5-5: Gain. =0.5.......................................................................................................................69Figure 5-7: Gain. =0..........................................................................................................................71Figure 5-8: Iso-Loudness....................................................................................................................72Figure 5-9: CT examples. =0.5.........................................................................................................73Figure 5-10: CT examples. =0.25.....................................................................................................74Figure 5-11: CT examples. Wider scale............................................................................................75Figure 5-12: DPOAE @ -60dB...........................................................................................................76Figure 5-13: DPOAE with noise pattern...........................................................................................77Figure 5-14: Zoom on noise pattern for two-tone stimuli...............................................................78Figure 5-15: Noise pattern for two-tone, single-tone and click stimulus.......................................78Figure A-1: FFT Frequencies closest to CFs....................................................................................86Figure B-1: Threshold difference histogram.........................................................................................87
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List of symbolsBM basilar membraneCA cochlear amplifierCF characteristic frequencyDPOAE distortion product otoacoustic emissionsME middle earOAE Otoacoustic emissionOHC outer hair cellOW oval windowTW traveling wave
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Preface A basic, linear one-dimensional cochlear model was developed by Azi Cohen and Miriam Furst (2004). In their work, an outer hair cell (OHC) model was incorporated into a one-dimensional basilar membrane model. The two models controlled each other through cochlear partition movement and pressure. The model output was used to predict normal and hearing impairment audiograms. The model predicted high frequency loss when OHC gain was relatively small at the basal part of the cochlea. The model also predicted phonal trauma when the OHC gain was random along the cochlea.
For predicting otoacoustic emissions, Yaniv Halmut (2005) introduced a modification to the model based on the work carried out by Tallmadge at 1998. The modified model incorporates Tallmadge's' middle ear model into the basic Cohen and Furst model. The modified model was able to predict generation of transient evoked otoacoustic emissions, but suffered from energy problems when a loud signal was introduced. Therefore, applying some kind of non-linearity to the model that would cause saturation behavior seemed inevitable.
Noam Elbaum and Miriam Furst (2005) subsequently modified Furst's basic model by introducing different types of non-linear functions at various places along the cochlea. In the course of each simulation, only one type of non-linearity was tested at any particular place along the cochlea. Although introducing such non-linearity predicted the typical compression in the ear response as well as the generation of combination tones (CT), compliance with physiological results could not be found. Their research indicated that,
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The goal of the present study was to produce a general model that would include both middle ear and non-linear functions (based on the work of Yaniv Halmut and Noam Elbaum) that would be able predict such non-linear phenomena as combination tones and otoacoustic emissions, while being stable and immune to energy explosion problems. To achieve that goal, the modifications relating to the two models (developed by Halmut and Furst (2005) and Noam and Furst (2005) mentioned above were integrated together into the basic model. . The new model which we developed incorporates Tallmadge's middle ear model and introduces non-linearity at the basilar membrane as well as at the OHCs. The previous model (Elbaum (2005)) displayed only one type of non-linearity at any single place along the cochlea.
This thesis first reviews ear anatomy as well as the non-linear phenomenon of otoacoustic emissions (Chapter 1). The modified ear model is then described (Chapter 2) and an analysis of the model's parameter is presented (Chapter 3).Simulation method are presented in chapter 4 Finally simulation results (Chapter 5) and conclusions regarding the current work (Chapter 6 are presented.
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1. Introduction In order to develop a model that simulates accurately the functionality of the hearing system, we must first understand its anatomy. This chapter is divided into three sections. The first section presents an overview of the ear's anatomy while in the second part, we introduce the model that we used in our research. The last part of the chapter presents an evaluation of the model's results.
Ear Anatomy The mammalian ear is usually regarded as consisting of three basic parts: the outer ear, the middle ear and the inner ear. (Figure 1-1). The outer and middle ears transform the movement of air molecules in the environment to the movement of fluid inside our body The inner ear then transforms the fluid movement of the middle ear to nerve excitations.
Figure 1-1 - The mammalian ear
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The Outer EarThe outer ear consists of three major parts: the pinna, the external ear canal and the ear
drum. (See figure 1-2). The pinna gathers sound waves from the environment and
transfers them through the external ear canal to the eardrum. When a sound wave strikes
the eardrum, kinetic energy creates mechanical vibrations. The eardrum, also known as
the tympanic membrane, is the boundary between the outer and middle ears.
The outer ear's depth, curves and firm walls protect the ear drum. This structure also
amplifies resonances at a basic frequency of 3.43KHz.
.
Figure 1.2: Outer ear
The Middle EarThe middle ear consist of a chain of three tiny bones, referred to as the ossicle chain
(Figure 1-3). This chain connects, at the oval window (figure 1-4), the outer ear on one
side of the tympanic membrane and the inner ear on the other side, The three ossicles --
the malleus, incus and stapes -- are the smallest bones in the human body.
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Figure 1.3: Ossicle chain
The middle ear perform two important tasks. The first task is to transmit sound from the
air medium to the fluid medium of the inner ear. Since the ratio of the acoustic
impedances of water and air is 3880:1.3, if the ear was a just a simple interface between
air and water, 99.9% of the sound would be lost. The middle ear thus serves as an
impedance matcher between the air outside the ear and the fluid that lies inside the ear.
Figure 1.4: Middle earThe second task of the middle ear is to serve as an amplifier for the transmitted energy.
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The amplification occurs due to the mechanical structure of the ossicle chain. The stapes
footplate area is 0.032 cm2 while the effective eardrum area is 0.594 cm2 providing an .
area ratio of 18:6 between the eardrum and the stapes. This ratio leads to an amplication,
by a factor of 18:6, of the pressure transferred from the eardrum to the oval window .
Moreover, the handle of the mallus is 1.3 longer than that of the incus. Effectively this is
like a lever system that amplifies the pressure by an additional factor of 1.3. Multiplying
these two factors, we get an amplification factor of 24.18 or 28 dB. In humans,
transmission of sound through the middle ear is most efficient at frequencies between 0.5
and 4 kHz. Resonances of the middle ear cavity and filtering effects caused by the
mechanics of the ossicle chain produce a sesitivity peak between 1 and 2 kHz..
The Inner Ear
The inner ear, is basically a coiled duct somewhat like a snail shell, filled with fluid. The
inner ear also known as the cochlea. (Figure 1.5)
Figure 1.5: The cochlea
Although the structure of the cochlea is curved, the sound waves move along it as if in a
a straight line. Therefore the cochlear is often described as uncoiled .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. (Figure 1.6)
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Figure 1.6: Left – wave propagation; right – "uncoiled" cochlea
The cochlea is divided into three sections: the scala vestibuli, scala media and scala
tympany. The scala vestibuli is seperated from the scala media by Reissner's membrane
while the basilar membrane separates the scala media from the scala tympani. (Figure
1.7).
As a result of the oval window's vibrations, a pressure difference between both sides of
the basilar membrane is created. This results in a corresponding movement of the basilar
membrane.
The helicotrema is a small hole at the apical end in which the scala vestibuli and tympani
are joined.
The basilar membrane is attached to the spiral lamina on its inner edge, and to the spiral
ligament on its outer edge. The width of the cochlear duct and the spiral lamina decreases
from base to apex, while the width of the basilar membrane increases from base to apex.
The tectorial membrane lies above the basilar membrane and also runs along the length of
the cochlea. Between the basilar membrane and the tectorial membrane lie the hair cells
-- part of the organ of Corti. (Figure 1-8). These cells have on their upper end hairs
referred to as stereocilia. The hair cells are divided into two groups by the tunnel of Corti.
The outer hair cells (OHCs) are arranged in several rows (maximum of five) while the
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inner hair cells (IHCs) are arranged in a single row. The human ear contains about
15,000 outer hair cells and 3,500 inner hair cells.
Figure 1-7: Cross-section of the cochlea The OHCs make contact with the tectorial membrane which is hinged at one side. When
the basilar membrane vibrates, a shearing motion is created which causes the tectorial
membrane to move to the side relative to the tops of the hair cells. As a result, the OHCs
move sideways. The movement of the OHCs creates an electrical current through the hair
cell which eventually generates action potentials. These potentials cause nerve spikes in
the neurones of the auditory nerve. The IHCs thus transform the mechanical movement of
the OHCs into neural activity.
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Figure 1-8: organ of corti
Behavior Characteristics of the Hearing System
The main motivation behind the research presented in this thesis was to develop a model
that would predict the behavior of some well known phenomena that characterize hearing
systems. This chapter introduces these phenomena.
Compression
In order to process sound waves with a dynamic range of 100 dB of more, the
mammalian hearing system implements compressive transformation at several stages of
processing (Cooper, 2004). Rhode (1971) revealed that a significant degree of
compression is already achieved in the cochlea at the mechanical vibration stage. Basilar
membrane vibration in response to tones grows at compressive rates that can be as low as
0.2 dB/dB, while the dynamic range over which compression is significant can be as
large as 80 dB (Robles and Ruggero, 2001).
One significant aspect of the auditory system is its ability to process sound over a wide
range of levels, about 120 dB . To understand the compressive capability of the basilar
membrane, it is necessary to consider the increase in the magnitude of the basilar
membrane vibrations at a given point along the membrane as a function of stimulus level.
Figure 1-9 presents an example of this process where the velocity of the basilar
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membrane movement is plotted as a function of stimulus level. These results are taken
from the basal region of a chinchilla cochlea, a region that responds best to high
frequencies. The characteristic frequency (CF) of this particular recording site was 10
KHz. (Data from Ruggero et al. 1997).
Figure 1-9: BM Velocity I/O functions
Figure 1-9 basically shows basilar-membrane response to tones with a frequency equal to
or higher than characteristic frequency of 10KHz. The straight dashed line at the right has
a linear slope (1 dB/dB). For 10KHz, the input-output (I/O) function is compressive. The
magnitude of the response generally increases with an increasing stimulus level, but the
growth is quite compressive. At moderate to high stimulus levels, the I/O function has a
slope of about 0.2 dB/dB, which corresponds to a compression ratio of about 5:1. As can
be clearly seen , the I/O function is compressive only for stimulus frequencies near the
CF of the recording site. The input-output function is linear for stimulus frequencies well
below or well above the CF. Due to compression, any point along the basilar membrane
is able to respond to a large range of stimulus levels.
As noted above, basilar membrane mechanics has a compressive behavior. Nevertheless,
the basilar membrane is assumed to be linear at low stimulus level. Hence, there must be
another source of compressive behavior.
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Figure 1-10 presents basilar membrane I/O functions from the basal region of a chinchilla
cochlea in response to a tone at the CF of the recording site. The data were obtained when
the cochlea was healthy (solid line) and after death, when the OHCs was not active and
the cochlea was not functioning normally (dashed line). The data is taken from Ruggero
(Ruggero, 1997)
Figure 1-10 - compression of a chinchilla cochlea.As expected, for low and moderate stimulus levels, the response of the living cochlea is
significantly stronger and more compressive than the response of the dead cochlea.
Similar results are obtained if the drug quinine, which is thought to directly affect the
outer hair cells, in the cochlea is injected Into the ear .(Zheng et al. 2001). Quinine
reduces the magnitude of the basilar membrane response at the lower stimulus levels, but
not at the higher levels. This data suggests that there are at least two sources for
nonlinearitis. One is active at low levels and the other at high levels. In other words, a
drug that affects the outer hair cells can cause a smaller mechanical response at the
basilar membrane. This suggests that injury or damage to the outer hair cells (such as in
the case of quinine) results in a more linear input-output function, resulting in a loss of
amplification or gain espacially at low and moderate stimulus levels ) This is can also be
seen in Figure 1-10.
Tuning Curves
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Figure 1-11 shows direct cochlear mechanical measurements representing the cochlear
sensitivity, or gain (displacement divided by stimulus pressure), of BM responses to
tones, as a function of frequency and intensity. The lower CF (10 kHz) data were
recorded at the 3.5 mm site of a chinchilla cochlea. The higher CF (17 kHz) data are from
a basal site of a guinea pig cochlea.(Robles and Ruggero, 2001).
Figure 1-11 : Tuning curves.
Combination Tones
Combination tones were discovered by the Italian violinist and composer Giuseppe
Tartini, who described the perception of a tone not present in the stimulus. These tones
are perceived at frequencies which are combinations of the primary tones f1 and f2 (f2 >
f1). (is this clear to your readers? Perhaps you should indicate where the primary tones
derive from?) Among these combination tones the most dominant and distinctive is the
cubic combination tone (CT) at a frequency of 2f1–f2.
Goldstein (Goldstein, 1970 and Goldstein et al., 1978) and others (e.g. Smoorenburg
1972, Shannon and Houtgast, 1980) have investigated many aspects of the
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psychophysical perception of combination tones with emphasis on 2f1– f2. These
researchers found that :
the relative amplitude of CT decreases sharply with f2/f1 ratio.
the relative level of 2f1–f2 is much greater than f2–f1 and greater than a higher
order CT.
the relative amplitude of the combination tone (2f1–f2) grows linearly with
primary tone amplitude. And in the case of f2/f1=1.2 with equal primary
frequency levels, the CT's relative amplitude is lower by about 20dB from
primary tone levels.
CT phases decrease linearly with primary tone amplitude.
In a physiological study of CTs a Robles et al. (1997) investigated the basal turn of a
chinchilla’s cochlea. They tested the relationship between such parameters as CT level
behavior with primary level increases and the relationship of f2/f1 and different primary
amplitudes. They found that in an equal level of primaries, the magnitude of CT grows at
the primary level with linear or faster rates at low stimulus levels, but saturates at higher
levels as seen in Figure 1-12. Robles et al. found that the phase in cases of f2/f1=1.2
decreases linearly at a primary level until about 70dB and then increases rapidly.
Figure 1-12: BM velocity of 2f2-f1
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Otoacoustic Emissions (OAEs)
Otoacoustic emissions were discovered in 1978 by David Kemp. Their discovery
contributed to the understanding of the cochlear functionality and mechanism, and led to
some new insights about hearing impairment. Otoacoustic emissions (OAEs) are sounds
which can be recorded by a microphone placed in the ear canal. Our work studies the
phenomena of distortion product otoacoustic emission. (DPOAE) which is one type of
otoacoustic emission (OAE).
Types of OAEs
The types of OAEs are separated by the way they are created.
Spontaneous otoacoustic emissions (SOAEs) occur without any stimulus
Evoked Otoacoustic Emissions (EOAEs) are evoked after presenting some kind of
stimulus to the ear and are divided into three groups:
1. Transient Evoked Otoacoustic Emissions (TEOAEs) are created using stimuli
with transients such as clicks and tone bursts.
2. Distortion Product Otoacoustic Emissions (DPOAEs) are generated by a stimulus
containing two different frequency components.
3. Stimulus Frequency OtoAcoustic Emissions (SFOAEs) are generated by
continuous pure tones.
The fluid pressure fluctuations,generated in the cochlea are responible for creating the
sound signal in the outer ear,by pushing the eardrum back and forth.Signals from
different parts of the cochlea arrive at the ear canal at different times and at different
frequencies and combined to give the actual signal in the outer ear.
The intensity and spectrum of OAEs may be different from one ear to another and
patterns will seem different as the spectrum of the stimuli change. Since OAEs are
evoked mainly when hearing is normal or near normal, OAEs are, in some respects, a
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mirror to cochlear functionality status, with the frequency in which the OAEs occur
providing a more meaningful criteria for testing hearing functionality than its amplitude.
Measuring emissions is done by inserting a microphone into the ear canal. Since the
stimulus is also present, the measured signal contains the emission (if it exists) as well as
the stimulus. Obviously these two signals have to be separated either in the time domain
or in the frequency domain. Separating these two signals in the time domain is carried out
by using a very short stimulus (a few milliseconds). The signal measured by the
microphone is divided in time into the stimulus section and the emission section.
Seperation in the frequency domain is implemented, for example, when one is trying to
measure the OAE distortion product (DPOAE). DPOAE can be measured when using a
stimulus containing two frequencies (f1 and f2 , the primary frequencies).The emission
signal is evoked at the CT frequency 2f1-f2, and is naturally seperated in frequency from
the primary frequencies..
Transient Evoked Otoacoustic Emissions (TEOAEs)
Transient Evoked Otoacoustic Emissions (TEOAEs) are OAEs that are produced when
the stimulus is an impulse or tone burst. The existence of TEOAEs was discovered by
David Kemp in 1978. When introducing a tone burst or an impulse, the TEOAE response
occurs with a few milliseconds time delay which makes it possible to isolate the
response. TEOAEs are very sensitive to cochlear damage. In healthy ears, TEOAE basic
spectral characteristics are similar to those of the stimulus. Thus, an impulse response
will be a wide-band signal and the response to tone will have the spectral properties of
the stimulus.
TEOAE penomena are most dominant in the frequency range of 1- 4KHz, although
TEOAE can be detected at frequencies above 4KHz (6-7KHz) in young ears. TEOAEs
are measured when testing the hearing of infants. Figure 1-13 presents the results of a
TEOAE test.
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Figure 1-13: TEOAE used in hearing test
In the upper-left corner is the stimulus. Below the stimulus, in the main window, the
response is presented. The upper right graph shows the spectrum of the response and
noise. Below this graph we see the stimulus spectrum. Both the response and stimulus are
wide-band signals as expected.
Stimulus frequency otoacoustic emissions (SFOAEs)
SFOAEs are evoked when the stimulus is a continiuous pure tone. During stimulation,
the signal in the ear canal consists of the incident stimulus, sound reflected from the
tympanic membrane and a signal which is caused by an energy leakage from the cochlea.
This signal is basically a time delay version of the stimulus. Since one cannot isolate a
SFOAE from the total signal in the ear canal, it isn't currently used for clinical purposes.
Stimulus Frequency OtoAcoustic Emissions (SFOAEs)
SFOAEs are narrow-band signals that occur when no stimulus is introduced to the ear.
One can measure SOAEs by analyzing the spectrum of the sound in the ear canal. SOAEs
are found in 30-40% of healthy young ears, usually, as pure tones with amplitude
between the measurement noise and approximately 30 dB SPL. SOAEs are sensitive to
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physiological changes in the cochlea. When hearing is impaired in some of the frequency
regions and the hearing loss is greater than 30dB SPL, SOAEs do not occur.
Distortion Product Otoacoustic Emissions (DPOAEs)
DPOAEs are signals that evoked when two continuous signals, called primary tones, are
introduced into the ear. When introducing to the ear a two-tone stimuli,consist of two
tones, f1 and f2, called primary tones, in addition to the response in f1 and f2 tonotopic
sites, a third tone can be measured at a third siteall three tones can be observed by
analysing the spectrom of the response (see figure 2-2). The third tone frequency mis
symboled as fDP.
DPOAE is most dominant when f2=1.22f1. Thus, for f1=4000Hz, a strong DPOAE will
occure if f2=4880Hz. In this case, fDP=3120Hz
In adults, DPOAEs occur at frequencies up to 10 kHz. Primary tone levels used to creat
DPOAE are between 50 to 70 dB SPL. The DPOAE level in a healthy ear can exceed 20
dB SPL. When hearing is impaired, DPOAE level is reduced. When hearing impairment
is significant DPOAE may be absent.
Generation of DPOAEs is related to the functionality of the outer hair cells (OHC).
Figure 1-14: Response to two tone stimuli.
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Cochlear Models Overview
Over the years, a variety of models were developed.for elucidating various element of
the hearing system. Helmhotz (1862) developed the first cochlea model which linked the
cochlea to a bank of resonators, each representing a specific locaton on the basilar
membrane. His model, however, failed to account for the cochlear fluid which, in fact,
couples the resonators. Wegel and Lane (1924) proposed a model consisting of a cascade
of capacitors, resistors and inductors. Their model was, theoretically, able to predict the
traveling wave, but they couldn't solve the model's equations. In 1928, von Bekesey
performed some experiments that revealed the nature of the basilar membrane's traveling
waves. He suggested that the cochlea can be compared to a dispersive transmision line in
which different frequency elements move at different velocities and therefore they are
located at different locations along the basilar membrane In 1948, Zwisloci presented the
basic equations for the one-dimensional transmission line model. In 1976 George Zweig
and colleagues found an aproximate but accurate solution for the one-dimensional
transmission line model. His result were similar to Rhode's tuning curves.
Rank (1950) was the first to formulate a two-dimensional model The motivation for the
two-dimensional model was that near the membrane's maximum amplitude site, the
perpendicular velocity vector could not be neglected relative to the longitudal vector, as
was assumed in one-dimensional, long wave models. Ranke's model wasn't accurate for
areas far basal from the maximum amplitude site. For these areas, Ranke suggested the
use of a long wave model. Two- dimensional models were considered as theoretically
more natural as opposed to one- dimensional models, but they were also more difficult to
understand and to numerically solve.
Three-dimensional models have also been presented. Many of the early works ignored
three dimensional variations in the physical properties of the cochlear partition.
(Viergever 1980). Most of the three-dimensional models involve complicated
mathematics and demand simplifying assumptions for solution, which ignores the
cochlea physiology. Lien and Cox (1974) simplified a three-dimensional model into a
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one-dimensional model by introducing some simplifing assumptions. The interpretation
that a one-dimensional model can mimic a three dimensional model, justified the use of
one-dimensional models.
The discovery of non-linear phenomena, such as compression and otoacoustic emissions,
that couldn't be explained by linear models, brought up the need for more extended
models. These models, incorporating non-linear elements. are solved by computer
modeling. Naturaly, one-dimensional models demand less memory than two- or three-
dimensional models.
Hubberd (1993) used non-linear damping that increased with the basilar membrane
velocity. The main disadvantage of this model was that the frequency filters' Q factor had
to be enlarged in order to achieve the non-linear effect. This caused some tips at the
response (basilar mebrane velocity) that should occur at much higher input levels. Some
researchers showed, that only using damping non-linearity does not predict accurate
reults. Furst and Goldstein (1980) showed that combining damping with stiff non-
linearity can predict results that explain the amplutide and phase of the distortion product.
Hubberd and Mountain (1998) established a non-linear, traveling wave amplifier model,
bases on the work of Hubberd (1993). Their model was based on two transmission lines
coupled by a feedback. The coupling was non-linear. Hubberd modified the model done
by incorporating an electric charactaristic of the scala media and the OHC. The non-
linearity was incorporated into the OHC's conductivity. Zwicker (1986) also presented a
non-linear transmission line OHC model, in which the OHC is described as a non-linear
amplifier which feedbacks the basilar membrane movements. The OHC is modelled as an
amplifier followed by a non-linear sigmoid compressor. Nobili (2000) suggested a model
for the OHC and the tactorial membrane. His model was composed of an array of non-
linear oscillators, each of which was coupled instantly to all the others through
hydrodynamic forces transmitted by the fluid that filled its interior. The input to the
cochlea through the ossicle chain of the middle ear was also transmitted
hydrodynamically to the oscillator array. Due to the different physical parameters of the
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oscillators (mass, stiffness and damping constants), the response of the array to a
frequency tone peaks at a frequency dependent location within the cochlea. A typical
response was a traveling wave, characterized by a phase delay that increases
monotonically along the oscillator array, accumulating a few cycles up to the location of
the response peak. The nonlinearity was incorporated at the basilar membrane velocity
using a sigmoid function. Nobili showed (2000) that for low stimulus levels, frequency
selectivity and hearing system sensitivity were better than at high levels. In the work of
Nobili, Mammano and Ashmore,(1998) a shearing force was added, representing OHC
control of the mutual displacement of the tactorial to the basilar membrane. The idea was
that there must be another component that enhances
location-specific frequencies, other than the transmission line filters. Talmadge, Long and
Tubis (2000) established a model which predicted spontaneous otoacoustic emission by
incorporating damping non-linearity. The non-linearity was incorporated in terms of a
Van Der Pol oscillator in the BM motion equation.
Reviewing the development of models over the years and the ability of each one to
predict the compressive behavior of the auditory system, along with such non-linear
phenomenon as OAEs, leads to the conclusion that a non-linear transmission line model,
acounting for the OHC contribution to the BM movement and incorporating non-linearity
both in the basialr membrane and OHC, should be able to predict the audiotory system
behavior. In the case of OHC non-linearity, the sigmoid function should be used as
mentioned above. Needless to say, since the model should be non-linear, a one-
dimensional model is preferable, since it is less complicated to solve numerically.
Cohen and Furst (2004) suggested a one-dimensional transmission line model, in which a
hair cell model was incorporated in a complete, time-domain, one-dimensional cochlear
model. The two models controlled each other through cochlear partition movement and
pressure. The model simulations revealed typical normal and abnormal excitation
patterns. The model output was used to estimate normal and hearing-impairment
audiograms. Since the Cohen and Furst model (2004) is a one-dimensional transmission
27
line model, which accounts for the contribution of both the OHC and the cochlea to
hearing, it should be a good basis for developing a complete one-dimentioanl model.
28
Motivation of the present studyThe main goal of this thesus is to develop a general cochlear model that will include the
middle ear, active OHC and non-linearity and which will predict such properties and
phenomena as compression, combination tones and OAEs. The model that is used in the
current work is based on the research of Cohen and Furst (2004). Their model is a one-
dimensional, transmission line model, which incorporates the OHC into the cochlear
model. The model allows, in a rather simple way, integration of a middle ear model and
the incorporation of non-linear functions at the OHC and basilar membrane
characteristics. Since the model is one-dimensional, numerical solution is applicable.
The next chapter presents a thorough explanation of the model beginning with a
description of the cochlea model. This is followed by a description of the OHC model,
which is incorporated in the cochlea model. Tthe middle ear model, which also lays the
boundary condition for the cochlea model, is also elucidated. All model parameters are
defined at the end of the chapter.
29
2. The Generalized Model
Model descriptionThe cochlea model is a simple one-dimensional model that regards the cochlea as an
uncoiled structure with an elastic partition that separates two rigid-walled sections filled
with fluid. A representation of the model is shown in Figure 2-1. The elastic partition is
the scalla media,which is the intermediate channel between the scala vestibuli and the
scala tympani. This partition play the roll of transforming mechanical vibrations to neural
activity.
Figure 2-1: The Cochlea Model
For the sake of clarity, we now define a few variables:
- L : the length of the cochlea.
- x : the longitudinal coordinate. At the base x=0, at the apex x=L.
- t : the time variable.
- P(x,t): the pressure difference between in the scala vestibuli and the scala tympani.
- bm(x,t) : the vertical displacement of the partition along the x dimension.
- : the basilar membrane width
- A(x) : the scalae cross-section area.
30
bm
L
- : the perilymph density.
By applying conversation of mass and fluid dynamic laws and by assuming that the
perilymph (which is the fluid both scalae tympani and vestibuli contain ) is an almost
incompressible fluid and that the cochlea has the mechanical properties of a point mass
(i.e. its velocity at any point is related to the pressure difference across it at that point
only and not at neighboring points), Cohen and Furst (2004) derived the following
equation:
The basilar membrane pressure (Pbm) results from the combination of P, which is the total
pressure difference between the scala tympani and the scala vestibuli, and the pressure
generated by the OHC length change (Pohc), which yields:
In the transmission line model, Pbm is obtained as follows:
Where m(x) is the basilar membrane mass per unit area, s(x) is the basilar membrane
stiffness per unit area and r(x,t) is the the basilar membrane resistance per unit area.
In the current model, non-linearity was introduced into the BM resistance r(x,t) by
applying a dependency on the basilar membrane velocity, .
Elbaum and Furst (2005) have tested a list of non-linear functions and showed that using
a cubic function at the BM resistance generates combination tones in cases involving
two-tone stimuli.
Basing our model on this research, we chose to model BM resistance as a cubic function
with one parameter, , whose value has to be determined (Eq. 2-4).
31
)2-1(
)2-2(
)3-3(
)3-3(
)2-3(
where r0(x) is the linear basilar membrane resistance per unit area.
Substituting Eq. 2-2 and Eq. 2-4 in Eq. 2-3 yields:
Substituting Eq. 2-5 in Eq. 2-1 yields:
POHC in Eq. 2-6 refers to the pressure that is added by the OHCs.
The OHC model in use is based on the work of Cohen and Furst (2004). In their work,
Cohen and Furst suggested a linear model. Since the present study's basic assumtion is
that the system is non-linear, especially OHC behavior, the model must be modified.
We assume that the pressure depends on the OHC force for unit area multiplied by the
number of OHCs in unit area, thus
Where represents the density of active OHCs along the cochlear partition, it is
referred to as the OHC gain, and its value ranges from 0 to 0.5. A value of 0.5 represents
a healthy cochlea (Cohen and Furst 2004) .A value greater then 0.5 represent a
nonrealistic cochlea.
We further assume that the OHC force ( ) is due to the elasticity of the OHC, which
acts as a spring.
Thus,
32
)2-4(
)2-5(
)2-6(
)2-7(
)2-8(
Where:
- FOHC is the force
- KOHC is the OHC's spring constant
- bm is the vertical displacement of the basialr membrane
- lOHC is the length change.
Thus,
For the linear case
In thus work ΔlOHC was chosen to be a non-linear, sigmoid function of ψ the voltage
difference across
the basolateral part.
The sigmoid function is one of the most common non-linear functions used for
physiological modeling. Robles (1997) described sigmoid characteristics in OHC
physiological measurements while Zwicker (1986) also presented a non-linear OHC
model, in which the OHC was modelled as an amplifier followed by a non-linear
sigmoid compressor.
The sigmoid function depends on two parameters: , which determines the point of
transition between linear to non-linear behavior and ,which determines the slope of the
linear portion of the function, as can be seen in Figure 2-2.
33
)2-9(
)2-10(
)2-14(
Figure 2-2: Sigmoid function for different sets of parameters
From Eq. (2-9) , Eq. (2-14) we get the OHC pressure equation:
Let us define
34
)2-15(
-10 -8 -6 -4 -2 0 2 4 6 8 10-1
0
1
Cell voltage (psi) [V]
Leng
th C
hang
e [c
m]
Sigmoid:B2=-1
B1=0.25
B1=0.5
B1=1
B1=2
-10 -8 -6 -4 -2 0 2 4 6 8 10-1
0
1
Cell voltage (psi) [V]
Leng
th C
hang
e [c
m]
Sigmoid:B2=1
B1=0.25
B1=0.5
B1=1
B1=2
-10 -8 -6 -4 -2 0 2 4 6 8 10-1
0
1
Cell voltage (psi) [V]
Leng
th C
hang
e [c
m]
Sigmoid:B2=4
B1=0.25
B1=0.5
B1=1
B1=2
Substituting Eq. 2-16 in Eq. 2-6 yields the boundary values' differential equation for the
pressure difference between the scala tympany and scala vestibuli.
The basic model represents the OHC as an electric circuit, which includes few elements,
such as adjustable capacitors and resistors (the resistance is represented by conductivity),
and two voltage sources, as shown in Figure 2-3.
The OHC model divides the OHC into two parts. One part is the apical which is closer
to the scala media while the second is the basolateral, which is, in fact, part of the organ
of Corti.
Figure 2-3: OHC Model
Cohen and Furst (2004) posited the following assumptions:
35
)2-17(
)2-16(
1.
2.
3.
4.
5. (same order of magnitude)
which yields the differential equation for potential difference across the basolateral part
ψ:
where ωOHC is the OHC cutoff frequency (~1kHz - Dallos and Evans (1995)), 0 is
the resting OHC potential ( ~-70mV - Mountain and Hubbard (1995)) and
λ=Vsm/Cb=Const. (Vsm~80mV)
Cohen and Furst assumed that Ga and Ca are linear functions of . In the present
study, Ga and Ca are still linear function as described in Equation 2-19.
Under these assumptions Eq. 2-18 becomes:
For small values of , the sigmoid function should coincide with Cohen and Furst's
linear aproximation of , which is given in Equation 2-10.
36
)2-18(
)2-19(
)2-20(
Assuming a small , the Taylor approximation for the sigmoid function yields:
Hence,
Comparing Eq. 2-21 with Eq. 2-10 derives the following relations:
Since there is no restraint on the value of , it is subjected to our choice.
Since BM displacement is caused by the pressure changes and these changes are caused
by OHC length changes, which, according to the assumption do not cause a significant
change in the cell characteristics, Ga and Ca, are assumed to be linear.
In accordance with Cohen and Furst's assumptions and in order to estimate Ga and Ca,
we assume that the BM r, s and m and are linear and depend on x. One of our
working assumptions is that OHC are active, mainly at low stimulus levels. At such
levels, a linear approximation can be also used for . Thus, to estimate Ga and Ca,
37
)2-22(
)2-21(
we can follow Cohen and Furst's assumptions. Substituting Eq. (2-10) and Eq. (2-18) in
Eq.(2-19) and performing frequency region manipulations (see Cohen and Furst [2004])
leads to Eq. 2-23:
Manipulating the relationship above yields:
Finally, we can substitute Eq. 2-24 in Eq. 2-20 :
For the sake of convenience, Eq. 2-25 can be written as follows:
Where
Boundary and initial conditions
38
)2-23(
)2-24(
)2-25(
)2-26(
)2-27(
The boundary condition for the model are determined by the middle ear. The middle ear
model is based on the work on Talamadge (1998) and it is a rather simple mechanical
model. According to the model (Figure 2-4), the tympanic membrane is described as a
single piston that has a fixed incudostapedial joint.
Figure 2-4: The middle ear (and ear canal) model
Three assumption are taken into consideration. One considers the ear canal to be sealed
by the microphone which produces the stimulus.The second assumption is that the
length of the ear canal is small relative to the wavelength. These two assumption allow us
to consider the pressure in the ear canal as uniform. The third assumption is that since all
air pressure changes occur without loss or gain of heat, we can use a single oscillator
equation to describe the mechanical model.
Where:
- is the displacement of the oval window
- P(0,t) is the pressure difference between scala tympany and scala vestibuli at the
stapes
39
ow
Ear canal Middle ear
Pin
Oval windowTympanic membraneIncudostapedial joint enohporciM
)2-28(
- Pe(t) is the pressure in the ear canal
- ow is the effective aerial density of the oval window and is calculated as follows:
effective mass of oval window+ ossicles / area of oval window
- ow is the middle ear damping constant
- ow is the middle ear frequency
- Gme is the mechanical gain of the ossicle chain.
Equation 2-28 indicates the relation between the displacement of the oval window, the
pressure in the ear canal and the pressure near the stapes. Thus, the pressure in the ear
canal (Pe(t)) is influenced by pressure created by the microphone (P in(t) ) and by pressure
caused by the displacement of the tympanic membrane.
where Cme is coupling of oval window displacement to ear canal pressure.
The relation between the the oval window acceleration and the pressure difference across
the cochlear partition is given in Eq. 3-23:
where is the oval window acceleration, a function of and (from Eq. 2-28
and 2-29) . Cow is the ratio between the area of the oval window and the cross-section of
the cochlear scalae and L is the cochlear length. Since for any given time, the pressure
difference, P, at the helicoterma is equal to zero, we can write the following boundery
condition:
The initial conditions are as follows:
40
)2-29(
)2-30(
)2-31(
Solving the non-linear modelThe simulation method used in this work is based on the time-domain solution method
used by Cohen and Furst (2004). Here we add of the middle ear by integrating the oval
window equations and POHC calculation through ψ calculation.
The model equations are solved iteratively. Each iteration consist of two steps.
• solving differential equations for the boundary value of the pressure difference between
the scala tympani and scala vestibuli when the time is regarded as a parameter.
• Solving differential equations for the initial value time-dependent of the derivation of
basilar membrane displacement ξBM and the potential difference across the
basolateral part of the OHC ψ .
The model equations
The model equations comprises the boundary value problem equation and the initial
value time dependent problem equations. The boundery problem equation for the
pressure difference between scala tympani and scala vestibuli is given in Eq. (i)
i.
The boundery conditions are given in equation 2-30 and 2-31:
By using Eq. 2-28 ,the second boundary condition can be written as:
41
)2-32(
owmemeowowowowowinmeC
xxtxP CGtPGtP
ow
ow )()(),0( 22
0),(
The initial value time dependent probelm is given in equations (ii) to (iv). Eq. (ii) is
derived from Eq. 2-5 and 2-15. Eq. (iii) is derived from Eq. 2-28 and 2-29, and Eq. (iv) is
simply Eq. 2-26.
ii.
iii.
iv.
The initial conditions are given as follows:
Soluation procedure
The pressure equation (Eq. i) can be approximated by a three-point aproximation.
Where and N is the number of section in the cochlea. Thus, a uniform grid is
defined, such that for every i=0..N , and .The pressure equation can now be presented as a set of linear equations:
42
Where:
Since the elements in are independent on time (t), can only be calculated once, at
t=0. The value of (t=0) will also be applicable for the calculation of P at t>0. Since at
every iteration (time step), the values of basilar membrane displacement and velocity as
well as OHC pressure are known, the value of P can be approximated using the method
shown above.
The set of equations is solved iteratively. Each iteration represents a time step and the
pressure vector is derived directly from Eq. (i). Based on P value, the basilar membrane
acceleration is calculated using Eq. (ii). The acceleraton is used to estimate basilar
membrane displacement and velocity for the next time step. In the same manner, the oval
window acceleration is calculated using Eq. (iii). The acceleration is used to estimate
oval window displacement and velocity for the next time step. The potential change rate
is calculated using Eq. (iv). This rate is then used to estimate the potential for the next
time step.
43
)2-33(
)2-35(
)2-34(
Below we present an illustration for the solution procedure:
Model ParametersAll model parameters are constants and are defined in the table below
44
Calculate (x) (Eq. 2-35)
t=0
Set initial conditions:
) ii .qE( dna etaluclaC
)iii .qE( dnaetaluclaC
) vi .qE( etaluclaC
(t,x)Y etaluclaC (43-2 .qE) .qE)
=(t,x)P (t,x)Y(33-2 .qE)
+t=t(,)
(),()
(,),(,)
tx
tt
t
txt
tx
wowo
mbmb
Parameter Description Value basilar membrane width 0.003 cm
A(x) scalae cross-section area. 0.5 cm2
perilymph density. 1 g/cm3
m(x) Basilar membrane mass density per unit area 1.286e-6*e1.5x g/cm2
r(x) Basilar membrane mass resistance per unit area 0.25*e-0.06x g/cm2s
s(x) Basilar membrane mass stiffness per unit area 1.282e4*e-1.5x g/cm2s2
(x) OHCs relative density (0.5=healthy) 0.5 1/cm2
KOHC Normalizing constant 1e-2
WOHC OHC cutoff frequency 1Khz
l For the linear case -1
ow Oval window aerial density 1.85 gm/cm2
ow Middle ear damping constant 500 1/s
ow Middle ear frequency 1500*2 Hz
Gme Mechanical gain of the ossicles 21.4
CmeCoupling of oval window displacement
to ear canal pressure6e6
Cow Coupling of oval window to basilar membrane 2.909
Table 1: Model Parameters
45
3. Determination of non-linear parametersIn this work, we chose to introduce non-linearity in r(x) and OHC length change, as
follows (Eqs. 2-4 and 2-14):
where =4
In this section, we we present the process of combining both types of non-linearity
(r(x) and OHC length change, using the modified 1D model.
Estimating LoudnessLoudness is a useful parameter measure for testing the effect of the different non-linear.
Based on the work of Cohen and First (2004), loudness is defined as follows.
xdtdLL T
tTdMB
0 0
21
where L is the cochlea length and T is the stimuli duration.
We chose loudness as a test parameter since it is smoother and less noisy than the BM
velocity. Nevertheless, the behavior of both is very similar; hence we can use Ruggero's
velocity measurements as a reference as to how our results should look (Fig. 1-9).
.
46
)3-3(
)3-1(
)3-2(
BM resistance non-linearity In the current work we chose to use a cubic non-linear function in the basilar membrane
resistance (Eq. 3-1).Our purpose is to find that will best fit experimented data.
After introducing this function to the BM motion equation and simulating the solution of
the non-linear set of equations (see Chapter 3), we can get I/O function for loudness as a
function of the stimuli level as seen in figure 4-1.
Simulations were performed for input levels of -200dB to 0dB; for four stimuli
frequencies (0.25Khz, 1Khz, 3Khz, 6Khz); and for different value of the parameter
(0.01,1,10). As can be easily seen from Fig. 4-1, the I/O function is linearfor low and
medium stimuli levels. When the input level reaches a certain value which is different for
each frequency, I/O function is no longer linear but exhibits characteristics of
compression and, in the end, saturation.
Input level Calibration
The effect of the cubic function parameter a on the loudness I/O function is shown in
Fig: 4-1 for different input frequncies. As a increases, the I/O function becomes non-
linear at lower input values. One other observation that can be made is that for low
frequencies -- 250Hz and 1000Hz -- the I/O function is slowly compressing. For the
higher frequencies that were simulated -- 3000Hz, and 6000Hz -- full saturation is
observed.
As Fig. 4-1 clearly indicates, loudness behavior at low stimuli level is always linear up to
a certain input stimuli level depending on stimuli frequency.
In the simulation, we considered a simple sine wave:
Ain represents amplitude level (in volts) where .
47
)3-4(
Input signal values should be presented as a function of reference pressure.
Fig. 4-1 also shows that since changes in the non-linear cubic function parameter cause
a shift of the compression point, relative to input stimuli level and since the input value is
an arbitrary value, we can "shift" the input level axis to fit the dynamic range of the
auditory system, such that 0dB will be defined as the hearing threshold. Accordingly, all
input levels should be shifted upward 140dB. This shift actually means that we choose a
reference pressure value P0 such that
Hence, P0=10-14.
Output level Calibration
In order to calibrate output (loudness) level, a threshold value should be defined. The
threshold was defined as the maximum loudness level at which the loudness I/O function
is still linear. This value was found to be -58dB. Loudness results were normalized to this
value such that the loudness level of -58dB was represented as 0dB which became the
loudness threshold.
Input threshold
For each frequency, the input threshold was defined as the input level at which loudness
reaches its threshold. Figure 3-1 shows the calibrated results.
48
)3-5(
0 05 001 0510
05
001
051
.zH052=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE
10.0=a1=a
01=a
0 05 001 0510
05
001
051
.zH0001=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE
0 05 001 0510
05
001
051
.zH0003=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE 0 05 001 051
0
05
001
051
.zH0006=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE
Figure 3-1: Calibrated loudness, with BM resistance non-linearity.
Introducing OHC non-linearity As noted in Section 3, we incorporated non-linearity in the OHC's legnth change. The
non-linearity function that was chosen was the sigmoid function with two parameters that
had to be found. On of the parameters, , was determined by comparingthe Taylor
approximation to the linear case, which reduced the problem to the determination of a
single parameter, .
Figure 3-2 shows the results for a few selections of .
Each of the sub-figures shows the loudness I/O function for three values of and the
result of a simulation in which the OHC contribution to the pressure is zero (POHC=0). The
simulation was performed at input stimuli levels of -60dB to 150dB and at four input
frequencies: 250Hz,1000Hz, 3000Hz., and 6000Hz.
49
=0.01
=1
=10
05- 0 05 001001-
05-
0
05
001
051.zH052=f .ssenduoL
[Bd] edutilpmA ilumitS
[Bd]
sse
nduo
L de
tam
itsE 05- 0 05 001
001-
05-
0
05
001
051.zH0001=f .ssenduoL
[Bd] edutilpmA ilumitS
[Bd]
sse
nduo
L de
tam
itsE
05- 0 05 001001-
05-
0
05
001
051.zH0003=f .ssenduoL
[Bd] edutilpmA ilumitS
[Bd]
sse
nduo
L de
tam
itsE 05- 0 05 001
001-
05-
0
05
001
051.zH0006=f .ssenduoL
[Bd] edutilpmA ilumitS
[Bd]
sse
nduo
L de
tam
itsE
1
20.0=
11=
1005=
CHO oN
Figure 3-2: Loudness for different values of
As can clearly be seen from Figure 3-2, OHC functionality is dominant only for low and
moderate stimulus levels. At low stimuli levels, the OHC gain has a clear contribution to
the ear response, which is still linear. This contribution depends on input frequency. As
stimuli levels grow, the sigmoid function exhibits compression; eventually, for each
value of at some medium stimuli levels, the response coincides with cases in which
there are no active OHC. This implies that there is no OHC contribution to the cochlea
response.
The value of the parameter directly affects that point where the contribution of OHC
to hearing response becomes weaker and the response is no longer linear but exhibits
compression. As increases, this phenomena occurs at higher input levels.
50
Combining of OHC and R non-linearity After incorporating non-linearity in r(x) and in the OHCs in the model and testing its
parameters, we combined OHC non-linearity with R non-linearity. This was simply done
by operating both OHC and R non-linearity in the model with two constraints. The first
constaint was that OHC non-linearity should take place at low stimuli levels and R non-
linearity at high stimuli levels. This is to say, OHC non-linearity should occur at input
levels where R non-linearity is still not dominant . The second constraint was that once
OHC non-linearity occurs, the response will no longer be linear, even at stimuli levels at
which OHC activity vanishes. That means that R non-linearity should occur at input
levels in which the OHC non-linearity vanishes. By choosing the proper non-linear
parameters, we can arrive the desired result as Figure 3-4 indicates.
Since the choice of both and only shifts the response relative to the input pressure
level, we first chose the value of such that compression will occur at a pressure level of
about 80dB. Next, we determined the value of . Figures 3-3 (a) – (c) represent three
alternatives for the choise of ((a) =0.02 (b) =1 (c) =500). Each figure shows an
estimated loudness I/O function for three configurations of the model – non-linearity only
at the BM resistance (red); non-linearity at OHC length change (blue); and the linear
model (black).
As noted earlier, our goal is to combine both non-linearities such that at input pressure
levels (slightly different for each frequency) at which OHC non-linearity vanishes, BM
resistance non-linearity occurs. As can be seen in Figures 3-3 (a) – (c), for the choice
=0.02, OHC non-linearity vanishes at input levels way below the input levels at which
BM resistance non-linearity occurs. Similarly, for =500, OHC non-linearity vanishes at
an input level somewhat above the input level at which BM resistance non-linearity
occurs. On the other hand, for =1, at the input level slightly above the level at which
OHC non-linearity vanishes, BM resistance non-linearity occurs. This means that 1 is a
proper value for .
51
A summary of the non-linear parameters that were chosen appear in the table below
(Figure 3-4) :
-50 0 50 100 150
-50
0
50
100
150
Pin [dB]
Loud
ness
[dB
]
Frequency=3000KHz. alpha=10. Beta1=0.02.
NL BM resistanceNL OHC length changeLinear
Figure 3-3 (a): Loudness. =0.02
-50 0 50 100 150
-50
0
50
100
150
Pin [dB]
Loud
ness
[dB
]
Frequency=3000KHz. alpha=10. Beta1=1.
NL BM resistanceNL OHC length changeLinear
Figure 3-3 (b): Loudness. =1
-50 0 50 100 150
-50
0
50
100
150
Pin [dB]
Loud
ness
[dB
]
Frequency=3000KHz. alpha=10. Beta1=500.
NL BM resistanceNL OHC length changeLinear
Figure 3-3 (c): Loudness. =500
52
Frequency=3000Hz. =10 , =0.02
Frequency=3000Hz. =10 , =1
Frequency=3000Hz. =10 , =500
0 50 100 1500
50
100
150
Loudness. f=250Hz.
Stimuli Amplitude [dB]
Est
imat
ed L
oudn
ess
[dB
]
0 50 100 1500
50
100
150
Loudness. f=1000Hz.
Stimuli Amplitude [dB]
Est
imat
ed L
oudn
ess
[dB
]
0 50 100 1500
50
100
150
Loudness. f=3000Hz.
Stimuli Amplitude [dB]
Est
imat
ed L
oudn
ess
[dB
]
0 50 100 1500
50
100
150
Loudness. f=6000Hz.
Stimuli Amplitude [dB]
Est
imat
ed L
oudn
ess
[dB
]
Place Non-linear type Parameters
Basilar membrane (R) Cubic =10
OHCs Sigmoid =1, =4
Figure 3-4: Summary of parameters
Figure 3-5 below presents estimated loudness results for the selection of non-linear
parameters shown in Fig. 3-4. Each one of the sub-figures presents estimated loudness for
different frequency. Stimuli level was changed from -60dB to 160dB every 10dB. Four
values of stimuli frequency (in Hz) were tested (250, 1000, 3000 and 6000).
Figure 3-5: I/O function for non-
linearity in both R and OHC
53
1 2 3 4
As may be seen from the figures above, particularly for the higher frequencies, there are
four stages in the response to pure tone (marked for 3Khz): (1) at low stimulus levels, up
to approximatly 40dB , depending on the stimuli frequency, the response is linear. This
linear region is wider for low frequency stimuli than for medium and high frequency
stimuli; (2) at the end of the first region, a compression occurs in the response and the
slope of the response decreases to nearly zero; (3) moderate, almost linear response,
followed by a (4) saturation at high input levels.
Regions (2) and (3) are direct consequences of OHC non-linearity, while the region (4)
source is R non-linearity. The distinction between the four stages is more obvious for the
higher frequencies (3000Hz and 6000Hz) than for the lower frequencies (250Hz and
1000Hz).
54
4. Model Simulation for simple tones Two types of simulations were performed. Pure tone simulation was performed in order
to ensure that the generalized model predicted the compression and saturation nature of
the hearing system along with its well known frequency selectivity characteristics. Two-
tone simulations were performed in order to ensure that the generalized model generated
of DPOAE and CT by . Both pure and two-tone simulations were performed for several
gamma values in order to understand the hearing system's dependency on ear
functionality status.
Using the simulated data
Some of the characteristics of the hearing system, such as frequency selectivity,
combination tones and DPOAEs were extracted from the frequency domain
representation of the simulated data. Transforming the data to frequency domain was
performed using a 512-point FFT routine. (512 was chosen to avoid long computation
time). The maximum stimulus frequency that was used was 8KHz. Since the sampling
rate of the simulated data was 50KHz (hence the maximum frequency allowed was
actually 25KHz), frequency resolution was 50KHz/512~97Hz. To obtain more accurate
results, we used FFT frequencies that are also close to the CFs of the cochlea (see
appendix A).
Since 512 points were taken in the frequency domain, the same number of points should
be taken in the time domain. Hence the length of the data needed is 512/50KHz~10msec.
The length of the output signal was actually 20msec; we took the last 10msec at which
the output signal was stable and transient effects were over.
Pure tone stimuli - Frequency selectivity estimationOne of the main characteristic of the hearing system is its ability to separate between
different frequencies. In order to observe the model predictions in this respect, we
performed a simulation for generating estimated frequency filters and iso-loudness data.)
55
Input signal
The input signal was a sinus. For each simulation, the amplitude and frequency were
determined. Amplitude values ranged from 0dB to 140dB every 20dB (eight amplitude
values). Twenty-eight frequencies were selected, between 100Hz and 8 KHz, as noted in
the previous section.
Gamma value
For each set of amplitude and frequency, gamma was determined to be one of three
values (0.5, 0.25, and 0) where 0.5 represented a healthy ear. Consequently, there were
672 sets of amplitude-frequency-gamma to be simulated and analyzed.
Data processing
Each simulation produces the basilar membrane velocity for each point on the cochlea at
every point in time. Based on the data, a time-place velocity map was built. FFT was
performed on this map to create a frequency-place velocity map in which each cell
corresponded to a specific place along the cochlea and to a specific frequency with a
frequency resolution of 97Hz.
Four frequencies were selected (the same frequencies mentioned in the previous section).
Each had its own characteristic place along the cochlea. For each of the four frequencies,
the frequency-place velocity map value in the cell corresponding to the specific
characteristic place, at each one of the 28 frequencies, was extracted, and normalized to
the input level, producing gain-frequency characteristics for each of the four frequencies.
This was done for each of the stimuli amplitude values.
56
Pure tone stimuli - loudness estimationInput signal
The input signal was a sinus. For each simulation, the amplitude and frequency were
determined. Amplitude values ranged from -60dB to 160dB every 10dB (21 amplitude
values). Four frequencies were selected -- approximately 500Hz, 1KHz, 3KHz, 6KHz --
such that the frequency values would coincide with the FFT frequencies while being
close enough to the CFs. The actual values were 488Hz, 1074Hz, 3027Hz, and 6055Hz.
Gamma value
For each set of amplitude and frequency, gamma was determined to be one of three
values 0.5, 0.25, or 0 , which represent a 100% active OHC; 50% active OHC; and 0%
active OHC, respectively for . Consequently, there were 252 amplitude-frequency sets for
gamma simulation and analysis.
Data processing
Each simulation produces the basilar membrane velocity for each point on the cochlea at
every point in time. Based on the data, the loudness value (Ld) is calculated. (Eq. 4-1).
Iso-loudness data was generated directly from the loudness data
Two-tone stimuliAnother important step in our work was to verify that the modified model containing the
middle ear, along with non-linear behavior in the OHC and the basilar membrane
resistance could also predict the generation of CT and DPOAE.
Input signal
The input signal consisted of two sinus waves (primary tone 1 and primary tone 2 with a
frequency ratio of f2/f1=1.2 which is a good ratio for DPOAE). This ratio also makes it
possibly to select primaries such that f1, f2 and 2f1-f2, which is a DPOAE frequency,
will be a whole multiplication of the basic FFT frequency. Twelve primary tone 1
frequencies were chosen ranging from 400Hz and 6000Hz. Eight amplitude values for
57
primary 1 were selected from between 0dB and 140dB. The amplitude of primary tone 2
was identical to primary tone 1 amplitude.
Gamma value
For each amplitude and frequency set, gamma was determined to be one of three values
(0.5, 0.25, and 0) with 0.5 representing a healthy ear.
Consequently, there were 288 amplitude-frequency-gamma sets to be simulated and
analyzed.
Data processing of CT extraction
Space-time matrixes of basilar membrane velocity were produced by the model for each
set of frequency, input level and gamma values. An FFT was performed on each one,(on
each set??) producing space-frequency matrixes of the basilar membrane velocity . Each
matrix should contain :(a) primary 1 response (b) primary 2 response and (c) a CT, each
one in its own characteristic place along the cochlea, and on its own frequency. See
Figure 4-1.
100 3.125 6.25
0
0.875
1.75
1953Hz
Frequnecy [KHz]
Dis
tanc
e [c
m]
0 3.125 6.25
0
0.875
1.75
2441Hz
Frequnecy [KHz]
Dis
tanc
e [c
m]
0 3.125 6.25
0
0.875
1.75
4394Hz
Frequnecy [KHz]
Dis
tanc
e [c
m]
0 3.125 6.25
0
0.875
1.75
5371Hz
Frequnecy [KHz]
Dis
tanc
e [c
m]
-120
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
CT P2
P1
CT
P1
P2
CT
P1
P2 CT
P1
P2Figure 4-1: CT examples. 100% active OHC (Gamma=0.5)
58
Data processing of DPOAE extraction
Each simulation produces the emission signal in the ear canal. An example of such a
signal is shown in Figure 4-2.
0 2 4 6 8 01 21 41 61 81 023-
2-
1-
0
1
2
301 x
3- noissimE[]
eutil
pmA
[cesm] emiT
5.7 8 5.8 9 5.9 01 5.01 113-
2-
1-
0
1
2
301 x
3- noissimE
[] eu
tilpm
A
[cesm] emiT
Figure 4-2: Typical emission signal
59
When using FFT on the emission signal, it was expected that three peaks would be found.
Two of the peaks were at primary tone 1 and primary tone 2 frequencies (f1 and f2), and
the third was at 2f1-f2, which was the DPOAE signal as can be seen in Figure 4-2. In that
case, f1=2441Hz, and f2=2929Hz, thus 2f1-f2=1953Hz.
First, results for a healthy ear were inspected. For every input frequency, and amplitude,
the DPOAE signal at 2f1-f2 was extracted and compared to the primary tone 1 response
and to the mean value of the noise. Best results were found for the amplitude value of
-60dB, where the DPOAE value was rather high. (See Figure 4-3). Next, for each Gamma
value and for each frequency, an emission FFT signal at an amplitude value of -60dB was
also inspected to extract the DPOAE value.
0 0001 0002 0003 0004 0005 0006 0007
05-
54-
04-
53-
03-
52-
02-
51-
01-
[Bd]
noi
ssim
E
[zHK] ycneuqerF
Figure 4-3: Typical emission FFT signal
60
2f1-f2
f1 f2
5. Results
Pure tone stimuliTime domain maps
Figure 5-1 represents a representative image of the basilar membrane velocity as a
function of time and place along the cochlear partition for an input sine wave. The
vertical axis represents the distance from the stapes, and the horizontal axis represents the
time. Absolute velocity in dB is color-coded in the figure (See colorbar) . In Figure 5-1
four different frequencies are presented -- 488Hz, 1.074KHz, 3.027KHz and 6.055KHz.
A few observations can be made. First, for each input frequency, the image contains a
peak at a specific location which is constant in time. For low frequencies (488Hz) the
location is at the apical half of the cochlea. As frequency increases, the peak moves
toward the stapes – as predicted. Second, the intensity of the peak also depends on input
frequency – as the input frequency moves closer to ~3KHz, the peak intensity rises.
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
[cesm] emiT
[mc]
sep
ats
mor
f ecn
atsi
D
zH884
2 4 6 801 21 41 61 81 02
0
57.1
5.3
zHK470.1
[cesm] emiT
[mc]
sep
ats
mor
f ecn
atsi
D
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK720.3
[cesm] emiT
[mc]
dep
ats
mor
f ecm
atsi
D 2 4 6 801 21 41 61 81 02
0
57.1
5.3
zHK550.6
[cesm] emiT
[mc]
dep
ats
mor
f ecn
atsi
D
091-
081-
071-
061-
051-
041-
031-
021-
011-
001-
09-
Figure 5-1 (a): Time domain maps. =0.5
For values lower than 0.5, the absolute velocity decreases, as shown in Figures
61
5-1b and c. Figure 5-1b represents results for =0.25 and while Figure 5-1c(c) represents
results for =0. Looking at Figures 5-1 (a) to (c) one can easily find that for =0.5 there
is significant intensity difference between different frequencies. As the value decreases,
the intensities of the responses at different frequencies become similar.
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
[cesm] emiT
[mc]
sep
ats
mor
f ecn
atsi
D
zH884
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK470.1
[cesm] emiT
[mc]
sep
ats
mor
f ecn
atsi
D
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK720.3
[cesm] emiT
[mc]
dep
ats
mor
f ecm
atsi
D 2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK550.6
[cesm] emiT
[mc]
dep
ats
mor
f ecn
atsi
D
091-
081-
071-
061-
051-
041-
031-
021-
011-
001-
09-
Figure 5-1 (b): Time domain maps. =0.25
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
[cesm] emiT
[mc]
sep
ats
mor
f ecn
atsi
D
zH884
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK470.1
[cesm] emiT
[mc]
sep
ats
mor
f ecn
atsi
D
2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK720.3
[cesm] emiT
[mc]
dep
ats
mor
f ecm
atsi
D 2 4 6 8 01 21 41 61 81 02
0
57.1
5.3
zHK550.6
[cesm] emiT
[mc]
dep
ats
mor
f ecn
atsi
D
091-
081-
071-
061-
051-
041-
031-
021-
011-
001-
09-
Figure 5-1 (c): Time domain maps. =0
62
Frequency domain maps
Figure 5-2 shows the same data as in Figure 5-1 but in the frequency domain. The X axis
represents frequency. We can see that each frequency has its own unique and specific
location along the cochlea in which the response is the strongest. The intensity of the
response, as seen on Figure 5-1 , is also function of the input frequency, as described in
the last paragraph.
0 5213 0526
0
57.1
5.3
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
zH884
0 5213 0526
0
57.1
5.3
zH4701
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
0 5213 0526
0
57.1
5.3
zH7203
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D 0 5213 0526
0
57.1
5.3
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
041-
031-
021-
011-
001-
09-
08-
07-
06-
05-
04-
Figure 5-2 (a): Frequency domain maps. =0.5.
The effect of the value can also be seen in the frequency domain. For values lower
than 0.5, the absolute velocity decreases, as shown in Figures 5-2b and c. Figure 5-2b
represents results for =0.25 while Figure 5-2c represents results for =0. As was
explained for the time domain images, as the value decreases, the intensities of the
responses at different frequencies become similar. Moreover, the value also affects the
characteristic place along the cochlea at which a maximum response is achieved. As
value decreases, this place is moving toward the stapes.
63
0 5213 0526
0
57.1
5.3
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
zH884
0 5213 0526
0
57.1
5.3
zH4701
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
0 5213 0526
0
57.1
5.3
zH7203
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D 0 5213 0526
0
57.1
5.3
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D041-
031-
021-
011-
001-
09-
08-
07-
06-
05-
04-
Figure 5-2 (b): Frequency domain maps. =0.25.
0 5213 0526
0
57.1
5.3
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
zH884
0 5213 0526
0
57.1
5.3
zH4701
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
0 5213 0526
0
57.1
5.3
zH7203
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D 0 5213 0526
0
57.1
5.3
(zH0526-0) ycneuqerF
[mc]
sep
ats
mor
f ecn
atsi
D
041-
031-
021-
011-
001-
09-
08-
07-
06-
05-
04-
Figure 5-2 (c): Frequency domain maps. =0.
64
Effect of on loudness I/O function
Figure 5-1 shows estimated loudness vs. amplitude for three values of : =0.5 which
represents an normal ear with 100% active OHCs; =0.25 which represents a partially
damaged ear with only 50% functioning OHCs; and =0 which represents a damaged ear,
with no active OHCs. The representation is given separately for each frequency.
0 05 001 051
0
05
001.zH884=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE
5.0=ammaG52.0=ammaG
0=ammaG
0 05 001 051
0
05
001.zH4701=f .ssenduoL
[Bd] niP[B
d] s
send
uoL
deta
mits
E
0 05 001 051
0
05
001.zH7203=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE 0 05 001 051
0
05
001.zH5506=f .ssenduoL
[Bd] niP
[Bd]
sse
nduo
L de
tam
itsE
Figure 5-1: Loudness. Effect of value
Several conclusions can be drawn From Figure 5-9. First, regardless of the value, OHC
activity (or contribution) vanishes at stimuli amplitude values of about 100dB. Second,
and also regardless of , due to BM resistance non-linearity, loudness function exhibits
compression behavior as stimuli amplitude values exceed 120dB. Loudness dependency
on value is expressed as the measure of the extent to which OHC activity contributes
to the I/O function level. As the value decreases, the OHC contribution to the loudness
level becomes smaller, eventually disappearing at =0.
Another observation is that different frequencies respond differently to the change in .
Responses to frequencies near 3KHz are affected in a more severe way by OHC damage;
65
while as stimulus frequency moves away from the 3KHz zone, the effect is less
dominant. Thus different frequencies will respond differently to hearing impairment
whose origin is some damage in OHC activity.
Figures 5-2 to 5-4 show results for loudness I/O function for four separate frequencies for
each value of : (Fig. 5-2) 0.5 which represents an normal ear with 100% active OHCs;
(Fig. 5-3) 0.25 which represents a partially damaged ear with only 50% functioning
OHCs; and (Fig. 5-4) 0 which represents a damaged ear with no active OHCs.
Results for a normal ear (=0.5)
For a normal ear, good frequency selectivity is achieved at low and moderate input
stimuli levels. As input levels approach 40dB to 80dB, depending on frequency,
frequency selectivity is degraded. As noted in the last sectin, the I/O function can be
divided into four regions. (1) linear region (2) OHC deactivation region (3) A second and
narrow linear region and (3) saturation due to BM resistance non-linearity. For 3 KHz,
the regions and the distinction between them is obvious from looking at the I/O function.
As stimuli frequency moves away from the 3 KHz zone, the regions and the separation
between them are less distinctive.
02- 0 02 04 06 08 001 021 041 061
0
05
0015.0=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
02- 0 02 04 06 08 001 021 041 061
0
05
00152.0=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
02- 0 02 04 06 08 001 021 041 061
0
05
0010=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
zH 884zH 4701zH 7203zH 5506
zH 884zH 4701zH 7203zH 5506
zH 884zH 4701zH 7203zH 5506
Figure 5-2: Loudness@=0.5
Results for damaged ear (=0.25)
For partially damaged ears, with 50% active OHCs, frequency selectivity is degraded.
Moreover, apart from the saturation region (4) caused by BM resistance non-linearity, the
66
four regions at the I/O function are less apparent as is the separation between them. The
effect of is noticeable mainly at low and moderate stimulus levels, at which OHC
contribution to the gain and selectivity is dominant. At high stimulus levels, at which the
OHC contribution vanishes, the effect of is negligible.
02- 0 02 04 06 08 001 021 041 061
0
05
0015.0=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
02- 0 02 04 06 08 001 021 041 061
0
05
00152.0=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
02- 0 02 04 06 08 001 021 041 061
0
05
0010=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
zH 884zH 4701zH 7203zH 5506
zH 884zH 4701zH 7203zH 5506
zH 884zH 4701zH 7203zH 5506
Figure 5-3: Loudness@=0.25
Results for severely damaged ear (=0)
For cases in which there are no active OHC, frequency selectivity almost vanishes.
Regions (1) –(3) coincide to linear response. Compression and saturation (Region 4)
occur at a high input stimuli level since they are caused by non-linearity on the basilar
membrane resistance character which is not affected by OHC functionality.
02- 0 02 04 06 08 001 021 041 061
0
05
0015.0=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
02- 0 02 04 06 08 001 021 041 061
0
05
00152.0=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
02- 0 02 04 06 08 001 021 041 061
0
05
0010=ammaG
DLOHSERHT
(Bd) niP
(Bd)
sse
nduo
L de
tam
itsE
zH 884zH 4701zH 7203zH 5506
zH 884zH 4701zH 7203zH 5506
zH 884zH 4701zH 7203zH 5506
Figure 5-4: Loudness@=0
Effect of on Dynamic range and frequency selectivity
Frequency gain
67
Figures 5-5 to 5-7 show, the results of frequency analysis. The figures actually show
velocity gain as a function of frequency for several input levels. (See Chapter 4 –
Simulation method). Each figure represents the result of inspection of the gain at a
specific location along the cochlea, which corresponds to the CF values as indicated in
the figure.
Four locations along the cochlea were tested, regarding a CF of 488Hz, 1074Hz, 3027Hz
and 6055Hz. Those locations were 2.3cm, 1.8cm, 1.1cm and 0.65cm from the stapes,
respectively. Input level is plotted by a different color. As explained in the last chapter
(Input pressure p. 48), each frequency has its own pressure threshold level. Figures 5-5 to
5-7 shows results only for input pressure level above the threshold level.
Results for a normal ear (=0.5)
Figure 5-5 represents results for =0.5. The basic shape of the gain as function of
frequency is a band-stop filter with a peak at the CF frequency. The filters represent the
frequency selectivity characteristic of the hearing system. For frequencies below the
peak, the slope of the function is rather small while for frequencies above the peak the
filter is steeper. The peak level depends on the CF in a similar way to what the iso-
loudness or the loudness I/O functions reveal (fig. 5-4, fig. 5-8). The highest peak level
is received for a CF near 3000Hz. As the CF moves away from 3000Hz, the peak level is
decreased. Regardless of the specific frequency tested, as stimuli amplitude grow,
frequency selectivity is degraded; the peak is getting wider and smaller. This is in
conjunction with the claim that at low stimuli levels, the OHCs (which are also claimed
to be the source of frequency selectivity) are fully active, while as the stimuli level
increase, their activity is decreased.
68
102
103
104
-100
0
100
CF=488Hz @ 2.3cm from stapes
Frequency [Hz]
Gai
n [d
B]
102
103
104
-100
0
100
CF=1074Hz @ 1.8cm from stapes
Frequency [Hz]
Gai
n [d
B]
102
103
104
-100
0
100
CF=3027Hz @ 1.1cm from stapes
Frequency [Hz]
Gai
n [d
B]
102
103
104
-100
0
100
CF=6055Hz @ 0.65cm from stapes
Frequency [Hz]
Gai
n [d
B]
0dB
20dB
40dB
60dB
80dB
100dB
120dB
140dB
Figure 5-5: Gain. =0.5
Another effect of the input level increasing is the degradation of the overall gain for all
frequencies and a shift of the peak toward the low frequencies.
Comparing the results of four different frequencies (or locations) studied reveals that at
low stimuli levels, for the higher frequencies (3027Hz and 6055Hz) auditory system
tuning is more effective than for the lower frequencies (488Hz and 1074Hz). For high
stimuli levels, the tuning curves are similar for all frequencies that were studied. This is
one indication of the saturation effect.
Results for damaged ears (=0.25)
Figure 5-6 shows results for =0.25. As can be seen in the tuning curves, regardless of
the inspected frequency (location), frequency selectivity is degraded, the peaks are
smaller and the filters are wider. These effects are more dominant for the higher
frequencies especially at low stimuli levels, in which tuning was rather effective relative
to the tuning at low frequencies.
69
Results for severely damaged ears (=0)
Figure 5-7 shows results for =0. For severely damaged ears, frequency selectivity is
meaningfully degraded. Moreover, the filter shape is similar for most of the stimuli levels
tested.
012
013
014
001-
0
001
sepats morf mc3.2 @ zH884=FC
[zH] ycneuqerF[Bd]
nia
G
012
013
014
001-
0
001
sepats morf mc8.1 @ zH4701=FC
[zH] ycneuqerF[Bd]
nia
G
012
013
014
001-
0
001
sepats morf mc1.1 @ zH7203=FC
[zH] ycneuqerF
[Bd]
nia
G
012
013
014
001-
0
001
sepats morf mc56.0 @ zH5506=FC
[zH] ycneuqerF
[Bd]
nia
G
Bd0
Bd02
Bd04
Bd06
Bd08
Bd001
Bd021
Bd041
Figure 5-6: Gain. =0.25
Changes in frequency selectivity due to OHC-related hearing impairment are more
dominant for low input levels than moderate or high levels. This is due to the fact that at
moderate and high input levels, OHC are not fully active, or not active at all, while their
dominant contribution is at low stimuli levels.
70
012
013
014
001-
0
001
sepats morf mc3.2 @ zH884=FC
[zH] ycneuqerF
[Bd]
nia
G
012
013
014
001-
0
001
sepats morf mc8.1 @ zH4701=FC
[zH] ycneuqerF
[Bd]
nia
G
012
013
014
001-
0
001
sepats morf mc1.1 @ zH7203=FC
[zH] ycneuqerF
[Bd]
nia
G
012
013
014
001-
0
001
sepats morf mc56.0 @ zH5506=FC
[zH] ycneuqerF
[Bd]
nia
G
Bd0
Bd02
Bd04
Bd06
Bd08
Bd001
Bd021
Bd041
Figure 5-7: Gain. =0
Iso-Loudness lines
Based on the loudness data, one can present an iso-loudness matrix. In an iso-loudness
matrix, the input level for a specific value of loudness and for each input frequency is
described. Figure 5-8 consists of three sub-figures representing iso loudness lines - each
one for each value of , or in other words, the percentage of active OHCs.
The iso-loudness matrix actually represents hearing sensitivity, by telling the observer
what should be the input level for a given loudness (in a given frequency) but also
representing to the observer the dynamic range, which is the ratio between the highest
stimulus level at which the system is not yet saturated, to the weakest stimulus that can be
detected by the system, as indicated in the figure (plotted in black, the black numbers
representing the dynamic range values). The iso-loudness matrices above reveal the
following findings: first, for normal ears, sensitivity depends strongly on input frequency
and reaches its maximum around 3Khz. For frequencies below or above 3 KHz, the
sensitivity decreases. As input level increases, frequency selectivity decreases. In
71
damaged ears (<0.5) the difference between frequency selectivity for high and low input
levels decreases. For =0, the differences in sensitivity for high and low input levels
disappear. In this case, frequency selectivity for all input levels is similar to these of a
high level, which is the lowest selectivity.
102
103
-20
0
20
40
60
80
100
120
140
Frequency (kHz)
Gamma=0.5
102
103
-20
0
20
40
60
80
100
120
140
Frequency (kHz)
Gamma=0.25
102
103
-20
0
20
40
60
80
100
120
140
Frequency (kHz)
Inpu
t Stim
ulus
Lev
el (d
B)
Gamma=0
Ld=60Ld=50Ld=40Ld=30Ld=20Ld=10
Figure 5-8: Iso-Loudness
Another observation that can be made is that as value decreases (i.e., hearing
impairment is more severe), the required input level value for a given loudness level and
input frequency increases, as can also be seen in Figure 5-8 above. The last, but most
distinct observation that can be made is the degradation of the dynamic range by 40dB
when is reduced from 0.5 to 0. That means that hearing range for OHC-related
damaged ears is 40dB smaller than for normal ears. The degradation is due to loss of
sensitivity at low stimulus levels, at which the OHC are normally active (for normal ears)
Two-tone stimuli
72
60dB78dB
100dB
CT Generation
Figure 5-9 presents a few examples, for =0.5 (100% active OHCs):
0 521.3 52.6
0
578.0
57.1
zH3591
[zHK] ycneuqerF
[mc]
ecn
atsi
D
0 521.3 52.6
0
578.0
57.1
zH1442
[zHK] ycneuqerF[m
c] e
cnat
siD
0 521.3 52.6
0
578.0
57.1
zH4934
[zHK] ycneuqerF
[mc]
ecn
atsi
D
0 521.3 52.6
0
578.0
57.1
zH1735
[zHK] ycneuqerF
[mc]
ecn
atsi
D 021-
011-
001-
09-
08-
07-
06-
05-
04-
03-
02-
TC 2P
1P
TC
1P
2P
TC
1P
2P TC
1P
2P
Figure 5-9: CT examples. =0.5
For a primary 1 frequency of 1953Hz, the CT response is about 50dB weaker than
primary 1 (P1) or primary 2 responses (P2) respectively. As P1 frequency increases, the
CT is becoming stronger – only about 20dB weaker than P1 and P2 for a P1 frequency of
2441Hz. When the P1 frequency is high enough so that the CT frequency is around 3
KHz (which is the case for a P1 frequency of 4394Hz, in which the CT frequency is
3515Hz), the CT signal is a bit stronger than the P1 and P2 responses. As the CT
frequency exceeds 4 KHz, (in the case of a P1 frequency of 5371Hz, the CT frequency is
4296Hz), the CT response is somewhat weaker than the P1 and P2 responses. The reason
for this is the high gain and sensitivity of the ear at those frequencies. (See Figure 5-8:
Iso-loudness lines)
73
OHC activity sensitivity
In order to examine CT sensitivity to OHC activity, the same procedure described above
was performed with a value of 0.25
Figure 5-10 presents the results:
0 521.3 52.6
0
578.0
57.1
zH3591
[zHK] ycneuqerF
[mc]
ecn
atsi
D
0 521.3 52.6
0
578.0
57.1
zH1442
[zHK] ycneuqerF
[mc]
ecn
atsi
D
0 521.3 52.6
0
578.0
57.1
zH4934
[zHK] ycneuqerF
[mc]
ecn
atsi
D
0 521.3 52.6
0
578.0
57.1
zH1735
[zHK] ycneuqerF
[mc]
ecn
atsi
D 021-
011-
001-
09-
08-
07-
06-
05-
04-
03-
02-2P
1P
TC
1P
2P
1P 2P1P 2P
Figure 5-10: CT examples. =0.25In the case of =0.25, for some of the frequencies, the CT signal, if generated, is much
too weak to be detected on a 120dB wide scale. For example, for a P1 frequency of
2441Hz, the CT signal is about 100dB weaker than the P1 and P2 responses.
A closer look at the results, on a wider level scale, will expose CT signals for the other
frequency, although very weak ones. In Figure 5-11, an example of a P1 frequency of
1953Hz is given:
74
[zHK] ycneuqerF
[mc]
ecn
atsi
D
zH3591
521.3 52.6
0
578.0
57.1041-
021-
001-
08-
06-
04-
02-
Figure 5-11: CT examples. Wider scale.
In this example, the CT response level is -130dB, which is 90dB lower than the P1
response. In the case of a P1 frequency of 4394Hz, the CT response level was almost
120dB weaker than the P1 response, and in the case of 5371Hz, the CT response level
was about 140dB weaker than the P1 response.
DPOAE Generation
Results of the processing mentioned in Chapter 5 for an input level of -60dB and primary
input frequencies of 2.930 KHz and 3.515 KHz is shown in Figure 5-12. A DPOAE
signal is generated at a frequency of 2.441 KHz, as predicted. For 100% active OHCs
(=0.5), the DPOAE level was 16dB above noise (DPOAE SNR) and 37dB below P1
level. response, while for 50% and 0% active OHCs, the DPOAE SNR was 10dB and
0dB above noise respectively and 42dB and 62dB below primary 1 response,
respectively. Since the OHC activity effect on the primary 1 response was negligible, it
75
can be clearly said that in a normal ear, OHC activity contributes a gain of 16dB to
DPOAE.
0 8288.4 6567.9 846.41
06-
04-
02-
[zhK] ycneuqerF
[Bd]
lang
iS5.0=ammaG
0 8288.4 6567.9 846.41606-
04-
02-
52.0=ammaG
[zhK] ycneuqerF
[Bd]
lang
iS
0 8288.4 6567.9 846.41
06-
04-
02-
[zhK] ycneuqerF
[Bd]
lang
iS
0=ammaG
Bd0 :RNS EAOPD
Bd61 :RNS EAOPD
Bd01 :RNS EAOPD
Figure 5-12: DPOAE @ -60dB
OHCs Low frequency Gain
When observing results for a primary 1 frequency higher than 3 KHz, (3.9KHZ in the
example below), a low frequency noise, pattern-like gain can be seen. This phenomenon
is represented in Figure 5-13. The noise is most dominant for full OHC activity (=0.5)
but vanishes for partial OHC activity (=0.25) or total lack of OHC activity (=0).
76
0 5 01 51
06-
04-
02-
[zhK] ycneuqerF
[Bd]
lang
iS
sCHO evitcA %001
0 5 01 51
06-
04-
02-
sCHO evitcA %05
[zhK] ycneuqerF
[Bd]
lang
iS
0 5 01 51
06-
04-
02-
[zhK] ycneuqerF
[Bd]
lang
iS
sCHO evitcA %0
Bd0 :RNS EAOPD
Bd01 :RNS EAOPD
Bd6 :RNS EAOPD
Figure 5-13: DPOAE with noise pattern
Figure 5-14 provides a closer look at this phenomenon. The noise pattern starts to build
up at 500Hz. Between 1000Hz and 2000Hz the envelope level of the noise pattern is the
highest, while from approximately 2200Hz, the pattern is getting weaker. A similar
pattern (although not exactly) can be observed when performing single-tone simulation
(Figure 5-15). This means that the noise pattern probably has something to do with the
interaction between the two primary tones, but that the two-tone interaction isn't the
source of this phenomenon. A similar pattern can be observed even when performing a
click stimulus simulation, as also shown on Figure 5-15.
77
Gamma=0.5
Gamma=0.25
Gamma=0
005 0001 0051 0002 0052
57-
07-
56-
06-
55-
05-
54-
[Bd]
noi
ssim
E
[zH] ycneuqerF
Figure 5-14: Zoom on noise pattern for two-tone stimuli
0 5 01 51021-001-
08-06-04-02-
[zhK] ycneuqerF
[Bd]
lang
iS
ilumits enot-owT
0 5 01 51021-001-
08-06-04-02-
ilumits enot elgniS
[zhK] ycneuqerF
[Bd]
lang
iS
0 5 01 51021-001-
08-06-04-02-
[zhK] ycneuqerF
[Bd]
lang
iS
kcilC
Figure 5-15: Noise pattern for two-tone, single-tone and click stimulus
78
Since a thorough study of this phenomenon is beyond the scope of this work, this work
does not presume to explain it, but only to represent the findings and suggest areas of
further study.
79
6. Conclusions and further researchCombining two sources of non-linearity in the model
The main challenge of this work was to decide where and what would be the non-linear functions and the unique parameters that would provide a model that could predict the behavior of the hearing system, including such phenomona as compression, frequency sensitivity and generation of combination tones and otoacoustic emission.
Two non-linear functions were applied in the model; A cubic function (x-x3) was applied at the basilar membrane resistance and a sigmoid function at the OHC length change. The behavior of each one of the non-linear functions depends on the choice of its parameters. In this work we applied two non-linear functions in the model, choosing appropriate parameters for each function, based on the assumption (which is supported by previous works) that OHC functionality is dominant at low stimuli levels, while basilar membrane characteristics are more relevant at high stimuli level since they cause the compression behavior of the hearing system. The model was able to predict the compressive nature of the hearing system, together with amplitude-frequency sensitivity (See iso-loudenss lines in Chapter 5.)
Along with predicting the compressive nature of the hearing system, the model allowed us to generate tuning curves, representing frequency selectivity. Tuning curves were tested for several stimuli levels between 0dB and 140dB (where 0dB is the hearing threshold). The results showed that at low stimuli levels the tuning curves are rather narrow and have a distinct peak. As stimuli levels increase above a certain value (depending on the stimuli frequency), frequency
80
selectivity was degraded, the peak was smaller and the curves were wider.
OHC effect on hearing system response
One of the most interesting aspects of this work is the effect that OHC-related hearing impairments might have on the hearing system response. OHC functionality status was controlled by changing the gamma value, which represents the percentage of the active OHCs.
OHC functionality affects a hearing system response in three aspects – dynamic range, gain and frequency selectivity. As to the gain, comparing a simulation of 100% active OHCs to 0% active OHCs reveals that the contribution of the 100% active OHCs lies between 20dB for low frequencies (~500Hz) to 60dB for medium-high frequencies (6 KHz).
The results show that frequency selectivity is also affected by OHC functionality.Two findings support this conclusion. First, when the active OHC percentage decreases, the frequency filters peak decrease and the filters become wider. Second, as was shown , as stimuli levels increase above a certain value, frequency selectivity is degraded. As was demonstrated previously in this work,when stimuli levels increase above a certain value, the OHC contribution to the gain is also becoming smaller, which implies that the OHCs are less active. As stimuli levels grow, OHCs become less active and frequency selectivity (which is assumed to be related to the OHCs) is degraded. Thus, the model also points to a relationship between frequency selectivity and OHC functionality.
81
Dynamic range degradation was demonstrated using an iso-loudness matrix, for which a degradation of about 40dB could be observed when was changed from 0.5 to 0. This degradation was due to loss of sensitivity at low stimuli levels.
CT generation
Combination tone products were predicted by the model using two-tone stimuli and extracting the 2D FFT data (place vs. frequency) of the basilar membrane velocity. For a primary frequency well below 3 KHz, such that CT frequency is also below 3 KHz, CT signals were about 50dB weaker than primary response signals. When CT frequency was around 3 KHz, the CT level reached and sometime exceeded the primary response level. The model simulation results show that CT generation depends on OHC functionality status. For 0% active OHC, CT was not generated; for 50% active OHC, CT was generated for some individual cases at a very low level. (~100dB below primary response).
It will be interesting to investigate more thoroughly several aspects regarding the behavior of CT characteristics (level, phase, generation site). One aspect is the effect of stimuli frequency on CT. Another aspect is the correlation between stimuli level (or the difference between primary 1 and primary 2 levels) and CT characteristics. Another important feature is how CT characteristics are changed with changes in OHC functionality status.
DPOAE generation
Distortion product oto-acoustic emission signals were also predicted by the model by using two-tone stimuli and extracting the FFT data from the emitted signal. DPOAE signals were about 50dB weaker than primary response signals. The model simulations results show that
82
DPOAE depend on OHC functionality status. For 0% active OHC, DPOAE was not generated; for 50% active OHC, DPOAE was 10dB above noise; and for 100% active OHC, DPOAE was 16dB above noise.
One interesting phenomenon that was observed was a noise pattern found primarily in the 1-2 KHz frequency range. This pattern exists also for single-tone simulation, and for click stimuli simulation, but does not exist for partial OHC activity. (For example 50% active OHC). A thorough study of this phenomenon is beyond the scope of the work nevertheless, it definitely warrants further study.
DPOAE has been studied by several researchers (See Appendix B) in order to find an optimal relation between primaries, amplitude and frequency and to generate some quantitative information regarding the effect of OHC impairment on DPOAE. The model predictions in regard to these aspects should also be tested.
Summary and future work
In this work, the Cohen and Furst (2004) model, which is a one-dimensional transmission
line model which accounts for OHC contribution to the BM movement has been
modified. The modified model that we present incorporates non-linearity both in the
basilar membrane and OHC to account for middle ear functionality. The model predicted
the compressive behavior of the auditory system along with non-linear phenomena such
as CT and DPOAE. Using a one-dimensional model made the numerical solution of the
non-linear differential equation applicable. The model was also able to predict the effect
of OHC-related damage on auditory system charactaristic, such as frequency selectivity
and dynamic range.
Nevertheless, further research is still required. A complete study of CT and DPOAE should be performed to gain some insights about these phenomena regarding phase and amplitude dependency on stimuli
83
amplitude and on the ratio between primary 1 and primary 2 frequencies (and amplitudes). Of course, the gamma effect in each case should be studied.
DPOAE analysis in this work revealed a phenomenon in which, a noise pattern can be observed at frequencies between 500Hz and 2KHz. This phenomenon should also be thoroughly studied.
In this work we used gamma which was constant along the cochlear partition. Previous work by Yaniv Halmut (2005) showed that for emission to occur, some roughness along the cochlear partition should be incorporated. A roughness effect on the model's results should be tested.
Simulations performed in the course of this work revealed that for stimuli levels above 160dB, the numerical solution does not converge. Further work should be done to gain some understanding for this and to suggest a proper solution.
Since the mechanical interaction between the tactorial membrane, the basilar membrane and the OHCs leads to the positive feedback created by the OHCs, and eventually to local amplification by the OHCs, which is the source of frequency selectivity, an additional modification to the model might incorporate a tactorial membrane model into the cochlear model.
84
Appendix A: Using CF frequencies In this thesis, the input frequency values for simulation that demanded frequency analysis
were done at specific frequencies. The experimental experience shows that by choosing
these specific frequencies, one can get better and more accurate results. These
frequencies are the characteristic frequencies (CFs). In general, CFs depend on the place
along the cochlea (each point along the cochlea has its own CF.)
Let X be the place along the cochlea, and define:
then CF(X) is given in the following formula:
Since the sampling frequency of the input signal is 50KHz, and the FFT length we use is
512, then the frequency resolution is 50,000/512=97.6563Hz. Clearly, the CFs don't
necessarily coincide with FFT frequencies. Consequently we have to choose FFT
frequencies which are the closest to the CFs. To do this, we calculate for each FFT
frequency, the nearest CF, and the error between them.
For this example we chose 17 FFT frequencies whose error was rather small as the
following figure indicates. The upper part presents the FFT frequencies as a function of
the CFs. The lower part presents the error between the chosen frequencies and the CFs.
85
)A-1(
)A-3(
)A-2(
0 1000 2000 3000 4000 5000 6000 7000 8000 90000
10
20
30
40
50
FFT Frequencies [Hz]
erro
r [H
z]
0 1000 2000 3000 4000 5000 6000 7000 80000
5000
10000
CF [Hz]
FFT
Freq
uenc
ies
[Hz]
All frequenciesSelected Frequencies
. Figure A-1: FFT Frequencies closest to CFs
86
Appendix B: Experimental Data about DPOAEThe scientific interest in DPOAE phenomena has been increasing in recent years. One
area of research deals with finding the optimal condition for DPOAEs regarding the
frequency and primary level ratio. One common set of conditions includes a frequency
ratio of f2/f1=1.22 and primary ratio in the form of L1=0.4L2+39. Johnson, Neely,
Gamer and Gorga (2005) studied the influence of primary level and primary frequency
ratios on human DPOAEs. They found the optimal ratio to be:
22 ()()gol 2
21
2e
Lfd
ff c
where:
a=0.137, b=18, c=1.22, d=9.6, e=415
Other researchers have tried to find a relation between hearing loss and DPOAE.
Schmuziger, Patscheke, and Probst (2005) found a correlation between the hearing and
DPOAE thresholds in cases of mild and moderate hearing loss. Their results are shown
in the figure below where LT-LEDPT is the difference between the pure-tone threshold and
the DPOAE estimated threshold. The measurment results are represented in the form of a
histogram of the threshold difference.
Figure B-1: Threshold difference histogram Gorga, Neely, Bergman, and Beauchaine (1992) measured DPOAEs in normal-hearing
and hearing-impaired subjects. They found that DPOAE is an adequate criteria for
predicting hearing loss, especially if measured around 4000 Hz. Thay also found that for
87
)B-1(
)B-2(
lower frequencies, the ability to predict hearing loss is degraded. According to their
study, in 95% of normal-hearing ears, a DPOAE level of at least 1dB can be measured
and the DPOAE threshold is 65dB or less.
88
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[37] Ruggero MA, Rich NC, Recio A, Narayan SS, Robles L (1997) Basilar membrane response to tones at the base of the chinchilla cochlea J. Acoust. Soc.Am. 101: 2151-2163.
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תקציר
שונים, בהיבטים האנושית השמיעה מערכת את הבנתנו האחרונים, התפתחה בעשורים כיום, ומעניינות. ואמנם, גם רבות תופעות חשפו אלו שבוצעו. מחקרים למחקרים הודות
את להסביר ניתן לא השמיעה מערכת אודות ונאסף שהתקבל המידע במלואו, האנושית.
קוכלארי תפקוד בין ההבדל את ובמיוחד ידועות מתופעות נרחב חלק להסביר מנת על)2004( ופירסט כהן של המודל על מבוסס פשוט. המודל מודל לפגום, פותח תקין התיכונה לאוזן מודל שולב זה השערה. במודל תאי את המכליל חד-מימדי מודל שהנו
גם שולבו אי-לינאריות של סוגים שני ) וכן2005( ופירסט הלמוט של עבודתם בסיס עלאי האוזן במודל כן הבזילרית הממברנה בהתנגדות שולבה הלינאריות הפנימית.
השערה. תאי התארכות ובפונקצית
לצד השמיעה מערכת של בתדר והסינון הדחיסה תכונות את מנבא המוכלל המודל).CT( המצורף האות ) ותופעתDPOAE( אוטו-אקוסטיים קוכלארים פלטים יצירת
הקוכלארים בפלטים וההפחתה הדינמי התחום הכוונון, צמצום אובדן את מדגים המודלהשערה. תאי של חסר או חלקי תפקוד של במקרים
השמיעה במערכת ליקויים של שונים סוגים לתאור כמותי כלי לשמש יכול המודלתפקודה. על והשפעתם
93
אביב- תל אוניברסיטתפליישמן ואלדר איבי ש"ע להנדסה הפקולטה
סליינר-זנדמן ש"ע מתקדמים לתארים הספר בית
למערכת כללי מימדי חד מודלהאנושית השמיעה
חשמל בהנדסת" אוניברסיטה מוסמך" התואר לקראת גמר כעבודת הוגש זה חיבורואלקטרוניקה
ידי – על
מקרנץ דן
מערכות- חשמל להנדסת במחלקה נעשתה העבודהיוסט פירסט מרים' פרופ בהנחיית
תשס"ח אלול
94
אביב- תל אוניברסיטתפליישמן ואלדר איבי ש"ע להנדסה הפקולטה
סליינר-זנדמן ש"ע מתקדמים לתארים הספר בית
למערכת כללי מימדי חד מודלהאנושית השמיעה
חשמל בהנדסת" אוניברסיטה מוסמך" התואר לקראת גמר כעבודת הוגש זה חיבורואלקטרוניקה
ידי – על
מקרנץ דן
תשס"ח אלול
95