TEL AVIV UNIVERSITY

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

THE PREDICTON OF TEOAE IN A ONE DIMENTIONAL

HUMAN EAR MODEL

A thesis submitted toward the degree of

Master of Science in Biomedical Engineering�

by

Yaniv Halmut

March 2006

ii

TEL AVIV UNIVERSITY

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

THE PREDICTON OF TEOAE IN A ONE DIMENTIONAL

HUMAN EAR MODEL

A thesis submitted toward the degree of

Master of Science in Biomedical Engineering

by

Yaniv Halmut

This research was carried out in the Department of Biomedical Engineering

under the supervision of Prof. Miriam Furst-Yust

March 2006

iii

Acknowledgements

Thanks and gratitude to Prof. Miriam Furst-Yust for her devoted and patient guidance through

the entire research. Her knowledge, thoroughness and patience helped me immensely in this

work.

I would like to thank my fellow researchers at the Auditory Signal Processing Laboratory:

Udi Shtalrid, Tomer Goshen, Ram Krips and especially Naom Elbaum for his insights and

long talks we shared.

I can't be grateful enough to my wife for all her support during the passing year.

Yaniv Halmut

March 2006

iv

Abstract

In the last 30 years hearing research has advanced our understanding of the auditory system

by many folds. Although many new phenomena have been discovered and many new facts

unraveled we are still not able to fully explain the data we collect from the living ear.

Since Kemp discovered the otoacoustic emissions (OAEs) in 1978 their importance to the

field of hearing diagnostics has grown significantly. Today all newborn babies undergo

auditory tests that check their response to clicks. There are several types of OAEs, each

having its' own unique properties. This phenomenon is clinically measurable but has not yet

been reproduced convincingly in models.

A one-dimensional cochlear model with embedded outer hair cells (OHC) was recently

developed by Cohen and Furst (2004). This model incorporates a basilar membrane (BM)

model with an outer hair cell model, which control each other through cochlear partition

movement and pressure.

In this work we added a middle ear model to the cochlear model in order to simulate the

generation of transient evoked otoacoustic emissions (TEOAE). The outer and middle ears are

mimicked by a simple mechanical model.

Only when nonuniformity was introduced in one of the mechanical parameters, TEOAE were

produced by the model. According to physiological data, nonuniformity in the OHC gain

seems most reasonable. Nonuniformity can account for the variability in the OHCs population

density along the cochlear partition. Increasing the OHC nonuniformity produced an increase

in TEOAE level which grew to infinity. Introducing nonlinearity in the basilar membrane

resistance limited the TEOAE level growth. The combination of nonlinearity and

nonuniformity allowed the generation of amplitude stabilized standing waves that formed

waves resembling spontaneous otoacoustic emissions (SOAEs) in the ear canal.

The simulated TEOAEs resemble measured TEOAEs in human subjects. The average level of

TEOAE was determined by the mean gain of the OHC. Thus a damaged ear that was

simulated with low OHC gain did not produce TEOAEs. The average power spectrum of the

simulated TEOAE was band limited between 2 and 5 kHz, where the OHC are most effective.

v

Our results indicate that TEOAEs are generated due to the combination of nonuniformity and

nonlinearity in the cochlea.

vi

Table of contents

Acknowledgements ...................................................................................................................iii Abstract......................................................................................................................................iv Table of contents .......................................................................................................................vi List of symbols ........................................................................................................................viii List of Figures............................................................................................................................ix 1. Introduction to ear anatomy and otoacoustic emissions.....................................................1

Introduction ............................................................................................................................1 Ear anatomy............................................................................................................................1

The outer ear.......................................................................................................................2 The middle ear ....................................................................................................................2 The inner ear.......................................................................................................................3 The traveling wave (TW) ...................................................................................................5

OtoAcoustic Emissions (OAEs) .............................................................................................6 Types of OAEs ...................................................................................................................6 TEOAE (Transient Evoked OtoAcoustic Emission) ..........................................................8 DPOAEs (Distortion Product OtoAcoustic Emissions) .....................................................9 SFOAE (Stimulus Frequency Otoacoustic Emissions) ....................................................11 SOAEs (Spontaneous OtoAcoustic Emissions) and SSOAEs (Synchronized SOAEs)...11 OAE fine structure............................................................................................................11 OAE Screening Tests .......................................................................................................12 OAE screening protocols..................................................................................................13

2. Theory of OtoAcoustic Emissions....................................................................................14 The local oscillator theory ....................................................................................................14 Coherent Reflection Filtering ...............................................................................................15 The global standing wave theory (SOAEs) ..........................................................................17 DPOAE generation theory....................................................................................................20

DPOAE specific models...................................................................................................22 OAE generation summary ....................................................................................................23

Different representation of the same physics ...................................................................23 OAE properties .................................................................................................................24

3. Motivation for the present study.......................................................................................28 4. Model details ....................................................................................................................30

Middle Ear Model.................................................................................................................30 The Cochlear Model .............................................................................................................32 The Outer Hair Cell Model...................................................................................................34 Boundary and Initial conditions ...........................................................................................36 Numerical Solution...............................................................................................................38 Fourth order Runge-Kutta ....................................................................................................40 Adaptive Step Size ...............................................................................................................41 Time versus Frequency results comparison .........................................................................42 Spatial artifacts .....................................................................................................................44

5. Results ..............................................................................................................................46 The Model Output: Basilar Membrane velocity along the cochlear partition......................46 Simulated Audiograms .........................................................................................................47 Simulated TEOAEs ..............................................................................................................49 Simulated click response ......................................................................................................50 Nonuniformity ......................................................................................................................51 �(x) “roughness” ...................................................................................................................52

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Simulated Otoacoustic emissions .........................................................................................54 Simulated TEOAE spectrum ................................................................................................56 Nonuniform cochlea Audiograms ........................................................................................56 "Linear" versus "Nonlinear" response processing................................................................58 Tone burst responses ............................................................................................................60 Localized � "roughness" .......................................................................................................61 � mean...................................................................................................................................63 Nonuniformity and Energy Explosion..................................................................................64 Introducing Nonlinearity ......................................................................................................65 Stimulus magnitude influence on nonlinear model responses .............................................67

6. Discussion.........................................................................................................................70 Impact on the field of OAE model research.........................................................................76 Future research possibilities .................................................................................................77

References ................................................................................................................................78 Appendix A – Auditory research and model history................................................................84

Auditory research history .....................................................................................................84 OHC research progress.........................................................................................................85 Middle ear research ..............................................................................................................87 Modeling history of the Cochlea ..........................................................................................89 1D vs. 2D, 3D.......................................................................................................................90 Enhanced one dimensional cochlear models ........................................................................91

viii

List of symbols

BM basilar membrane

CA cochlear amplifier

CEOAE click evoked otoacoustic emissions

CF characteristic frequency

DPOAE distortion product OAEs

ME middle Ear

OAE otoacoustic emission

OHC outer hair cell

OW oval Window

SFOAE stimulus frequency OAEs

SOAE spontaneous otoacoustic emissions

TEOAE transient evoked otoacoustic emissions

TW traveling wave

ix

List of Figures

Figure �1-1 - A diagram of the auditory system consisting of the outer, middle and inner ears .1 Figure �1-2 - The middle ear........................................................................................................2 Figure �1-3 - The mammalian cochlea as an uncoiled cochlea....................................................3 Figure �1-4 - The basic structure of the cochlear partition ..........................................................4 Figure �1-5 - An expanded view of the organ of Corti ................................................................5 Figure �1-6 - Traveling wave example.........................................................................................6 Figure �1-7 - A TEOAE response from a newborn baby.............................................................9 Figure �2-1 - Multiple internal reflections within the cochlea...................................................17 Figure �2-2 - DPOAE generation visualized..............................................................................21 Figure �4-1 - A simple middle ear mechanical model ...............................................................30 Figure �4-2 - Cochlear Model Geometry ...................................................................................32 Figure �4-3 - An equivalent electrical circuit model of the outer hair cell ................................35 Figure �4-4 - The Fourth order Runge-Kutta method. ...............................................................41 Figure �4-5 – Time versus Frequency algorithm comparison....................................................43 Figure �4-6 – Spatial artifacts ....................................................................................................44 Figure �4-7 – Spatial resolution .................................................................................................45 Figure �5-1 - Model response to a click and a sine wave...........................................................46 Figure �5-2 - Filtered stimulus response ....................................................................................47 Figure �5-3 - Audiograms simulated by the linear model..........................................................49 Figure �5-4 – Example of a simulated TEOAE in a uniform cochlea. ......................................50 Figure �5-5 – Model response to a click simulated in a “smooth” cochlea ...............................51 Figure �5-6 – Responses with nonuniform R or S .....................................................................52 Figure �5-7 – Otoacoustic emission (OAE) simulation .............................................................53 Figure �5-8 - Examples of simulated TEOAE outputs...............................................................54 Figure �5-9 – Energy increase due to larger “roughness”..........................................................54 Figure �5-10 – Emission energy histograms ..............................................................................55 Figure �5-11 – Simulated TEOAE spectrum .............................................................................56 Figure �5-12 – Audiograms of nonuniform �(x) selections .......................................................57 Figure �5-13 – Response to a click by normal and partially impaired ears ...............................57 Figure �5-14 - Linear vs. Nonlinear click responses..................................................................59 Figure �5-15 – Linear vs. Nonlinear tone burst response spectrum...........................................60 Figure �5-16 – “Nonlinear” TEOAEs from the literature ..........................................................61 Figure �5-17 – Localized � “roughness” ....................................................................................62 Figure �5-18 – Nonlinear spectral response vs. E(�) and their corresponding audiograms.......63 Figure �5-19 – Total cochlear energy vs. different � s. .............................................................64 Figure �5-20 – Energy explosion that occurs when a large � is used. .......................................64 Figure �5-21 - Large � simulation combined with constricting nonlinear damping..................66 Figure �5-22 - � vs. Total cochlear energy ................................................................................67 Figure �5-23 – Different stimulus magnitudes in the nonlinear model......................................68 Figure �5-24 – Signal in noise....................................................................................................69 Figure �6-1 - The profile of the Cochlear Amplifier by Nobili et al..........................................72 Figure 7-1 – The effect of the middle ear on OAEs .................................................................88

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1. Introduction to ear anatomy and otoacoustic emissions

Introduction

A good understanding of the underlying anatomy of a system enables researchers to develop

better models that better mimic the inner working of the system. The auditory system is no

exception and in the first part of this chapter we will give an introduction to help us better

understand its anatomy. The second part will be dedicated to the characteristics of

OtoAcoustic Emissions (OAEs). The next chapter is dedicated to current theories in the field

of auditory models.

In the passing century researchers have unveiled many aspects of the auditory system through

observing and modeling. This past decade has seen many new discoveries and many new

theories. The auditory system still exhibits phenomena that are not yet fully understood and

there are still several "rivaling" theories that haven't been proven yet. These are very

interesting years for researchers in the field of OAE analysis.

Ear anatomy

The mammalian ear is traditionally divided into three regions, the outer, middle and inner ear

regions (Figure �1-1). The task of the outer and middle ears is to translate the movement of air

molecules in the environment to the movement of fluid inside our body. The inner ear is the

part of the auditory sensor that transforms fluid movement to nerve excitation.

Figure �1-1 - A diagram of the auditory system consisting of the outer, middle and inner ears

2

The outer ear

The outer ear consists of the pinna and the external ear canal. The pinna functions as a sound

gathering horn. It intercepts sound waves from free space and funnels them via the external

ear canal to the eardrum. When sound waves hit the eardrum they impart kinetic energy in the

form of mechanical vibrations.

The eardrum (called the tympanic membrane) forms the boundary between the outer and

middle ears.

The middle ear

Connected to the eardrum is a chain of three tiny bones (the ossicle chain) that bridge the

space between the outer and inner ears. The three ossicles (malleus, incus and stapes) are

located in the air filled middle ear. They are the smallest bones in the human body.

The first major challenge the auditory system has to overcome is the task of transmitting

sound from the air medium to the fluid medium of the inner ear. The ratio of the acoustic

impedances of water and air is 3880:1.3, meaning that 99.9% of the sound would be lost if the

ear was a simple air to water interface.

As we see in Figure �1-2 the bone chain, in the air-filled middle ear, serves to “match” the

impedances of air (of the outer ear) with the impedance of fluid (of the inner ear).

Figure �1-2 - The middle ear is an impedance “matcher”, amplifying the incoming sound waves

so that they may enter the fluid-filled inner ear without significant loss of acoustic power

3

The middle ear is a mechanical energy transformer. The three ossicles work like a lever

system that increases the force transmitted from the eardrum to the stapes by decreasing the

ratio of their oscillation amplitudes. The footplate of the stapes acts like a small piston on the

cochlear fluid through a membranous connection that seals the oval window of the cochlea.

The buckling motion of the tympanic membrane decreases the velocity two-fold and increases

the force two-fold, changing the impedance ratio four-fold. Thanks to the large surface ratio

between tympanic membrane and oval window (~35), and the ossicle system lever gain

(~1.32), the forward impedance gain is about 30 dB. In this way the middle ear mechanism

couples the low acoustic impedance of air to the high mechanical impedance of the cochlea.

The oval window forms the boundary between the middle and inner ears. The vibrations of

the stapes cause the oval window to vibrate, resulting in fluid displacement inside the cochlea.

Filtering effects due to resonances of the middle ear cavity and mechanical parameters of the

ossicle system produce a peak between 1 and 2 kHz. The transmission of sound energy

through the middle ear, in humans, is most efficient at frequencies between 0.5 to 4 kHz.

The inner ear

The inner ear, also called the cochlea, consists of a fluid-filled duct coiled as a snail shell or

corkscrew. The propagation of sound waves in the cochlea is almost exactly as it would be in

a straight cochlea or an “uncoiled” one (Figure �1-3).

Figure �1-3 - The mammalian cochlea as an uncoiled cochlea having longitudinal, vertical and radial

dimensions

The perilymphatic space has the shape of an elongated U, the top arm of which is called scala

vestibuli and the bottom arm which is called scala tympani. The space between the two arms

of the mammalian perilymphatic space is the endolymphatic space, labeled scala media. An

4

extremely thin Reissner's membrane separates the scala media from the scala vestibuli. The

cochlear partition, a flexible structure that contains the sensory hair cells, separates the scala

media and the scala tympani. At the apical end is the helicotrema, a short duct connecting the

two perilymphatic scalae. Thus, when the stapes pushes the oval window inward, the U-

shaped column of perilymph is free to slide through its casing and push the round window

outward.

The round window plays an important role in releasing cochlear fluid pressure caused by

stapes displacement thereby greatly reducing the cochlear input impedance. Such movements

result in pressure differences between both sides of the basilar membrane causing the flexible

cochlear partition to vibrate.

The region of the cochlea adjacent to the oval window is called the base and the region

farthest away from the stapes is appropriately named the apex.

Figure �1-4 - The basic structure of the cochlear partition

Forming the basic platform of the cochlear partition (Figure �1-4) is the basilar membrane,

which is attached on one side to the bony spiral lamina and on the other side to the spiral

ligament. The basilar membrane is narrower and thicker in the base than it is in the apex.

These longitudinal differences in the structure of the basilar membrane are presumed to

account in large part for the different resonant measured at different points along the cochlear

partition. Resting on the basilar membrane is a small but complicated superstructure, known

as the organ of Corti (Figure �1-5), which contains the sound-sensing cells. The tectorial

5

membrane extends from the lip of the spiral limbus to overlie the apical surface of the organ

of Corti.

Figure �1-5 - An expanded view of the organ of Corti

The sound sensing cells are called hair cells because they appear to have tufts of hairs, called

stereocilia, protruding from their top. The hair cells are divided into inner and outer hair cells.

The inner hair cells (IHC) form a single row, running from base to apex, whereas the outer

hair cells (OHC) form three to five rows. In humans, there are about 3,500 inner hair cells,

each with about 40 stereocilia and 15,000 outer hair cells, each with around 140 stereocilia

protruding from them. When the basilar membrane moves up and down, a shearing motion is

created. Thus, the tectorial membrane moves to the side, relative to the tips of the hair cells.

As a result, the stereocilia of the hair cells move and rotate. The movement of the stereocilia

leads to the flow of an electrical current through the hair cells, which leads to the generation

of action potentials. These potentials give rise to nerve spikes in the neurons of the auditory

nerve. The inner hair cells act as transducers and translate the mechanical movement into

neural activity. The outer hair cells change their length and size due to these potentials, and

thus affect the physical properties of the basilar membrane.

The traveling wave (TW)

Fluid motion in the basal regions of the scala vestibuli and tympani displaces the basilar

membrane (BM). Initially only the basal part of the BM moves, but induced transverse

oscillations of the BM begin to propagate apically. Oscillatory exchanges occur between fluid

motion energy and the energy held in elastic BM displacement. Adjacent BM sections are

excited resulting in a traveling wave.

6

The traveling wave (TW) conveys stimulus energy towards the apex at less than 1/100th of

the speed of sound in air. Wave amplitude increases with distance along the BM, and reaches

its maximum at the place where the force of inertia equals and cancels the elastic restoring

force (place of resonance). Because inertial forces increase with frequency, the place along

the BM at which this peak in TW amplitude occurs is progressively nearer to the base for

higher frequency stimuli. The overall result is an asymmetric peak of excitation for each

frequency component (Figure �1-6). The TW envelope represents the excitation intensity

applied to the organ of Corti as a function of distance along the length of the cochlea. The

organ of Corti mechanism then converts BM motion to fluid motion across the IHC

stereocilia, leading to neural excitation. More about the history of auditory research is

described in Appendix A.

Figure �1-6 - Traveling wave example

OtoAcoustic Emissions (OAEs)

Types of OAEs

Otoacoustic emissions (OAEs) are sounds which can be recorded by a microphone fitted into

the ear canal. OAEs are essentially divided according to their way of stimulation. Otoacoustic

emissions that occur without any stimulus are called Spontaneous OtoAcoustic Emissions

(SOAEs). OAEs that are evoked by some sort of stimulus presented to the ear are named

Evoked Otoacoustic Emissions (EOAEs) and are divided into three groups. Transient Evoked

OtoAcoustic Emissions (TEOAEs) are created using stimuli with transients; i.e. clicks and

tone bursts. Distortion Product OtoAcoustic Emissions (DPOAEs) are generated by the use of

a stimulus containing more than one frequency component. Stimulus Frequency OtoAcoustic

Emissions (SFOAEs) are generated using continuous pure tones.

7

The oscillatory sound pressure waveform seen in the outer ear corresponds to the motion of

the eardrum being pushed backwards and forwards by fluid pressure fluctuations generated

inside the cochlea. The response is long and complex because responses from different parts

of the cochlea arrive at the ear canal at different times and at different frequencies. Several

different cochlear locations may contribute to a single frequency component of an OAE and

these may fortuitously summate or interfere with each other.

Otoacoustic emissions are intimately related to the status of the cochlea and they provide the

researcher and clinician with noninvasive tools to peer into the inner ear with an acoustic

microscope. Today OAE screening is widely used in newborn hearing screening programs

and can be used to monitor the effects of treatment.

Individual healthy ears differ greatly in the level and the spectrum of the OAEs they exhibit.

Stimuli of slightly differing frequency or spectral composition can give rise to quite different

OAE patterns. Taking an ‘average’ OAE characteristic over a range of stimuli provides a

more meaningful description of cochlear status, but even so the intensity of OAEs alone is a

very imperfect index of cochlear status. The ‘frequency’ at which an emission can be evoked

is more significant. OAEs are frequency-specific responses and tend to emerge only in

frequency bands where hearing is near normal. This fact may provide a useful pointer to

normally and abnormally functioning parts of a cochlea.

Changes in cerebrospinal fluid pressure induced by posture changes affect SOAE frequency

and evoked OAE intensity, probably by their influence on cochlear fluid pressure and

stapedial position. Drugs known to depress hearing, including aspirin and quinine, also

depress OAEs, and loop diuretics known to depress the endocochlear potential also depress

OAEs. OAEs also exhibit a physical analogue of ‘masking’ where the perception of one

sound is blocked by another. This may indicate that some forms of masking originate

preneurally in the cochlea. Tracing the suppression of an OAE response to one tone by

adjusting the intensity and frequency of a second suppressor tone allows an OAE suppression-

tuning curve to be constructed. The sharpness of such curves confirms the close association

between OAEs and auditory function, and demonstrates that sharp mechanical tuning is

present at the cochlear level.

Because the emission pressure is measured by a microphone inserted into the ear canal, where

the stimulus is also present, the measured signal contains both emission and stimulus.

8

Somehow these two signals have to be separated. There are essentially two ways to do this: in

the time domain, or in the frequency domain. Separating stimulus and emission in the time

domain is done by using a stimulus of a very short duration (a few milliseconds). The signal

measured by the microphone is then divided in time into a stimulus part and an emission part.

Examples of such delayed evoked otoacoustic emissions are click evoked otoacoustic

emissions (CEOAE). An example of separation of stimulus and emission in the frequency

domain are distortion product otoacoustic emissions (DPOAE) that result when a stimulus

containing more than one frequency component is presented to the ear. The formation of

distortion product frequencies in the cochlea was known from psychophysical measurements

for a long time, but they were not measured in the ear canal until after the discovery of OAEs.

The following section describes the basic characteristics of the four different OAE,

categorized according to their eliciting stimuli.

TEOAE (Transient Evoked OtoAcoustic Emission) / CEOAE (Click

Evoked OtoAcoustic Emission)

TEOAEs are OAEs that are generated in response to clicks, i.e. impulses or tone burst. These

OAEs are also called “Kemp Echoes” after David Kemp who discovered the phenomenon in

1978.

The recorded response is split into frequency bands and analyzed. TEOAE responses are

strongest and easiest to detect in the primary speech frequency band, 1–4 kHz. In young ears,

TEOAEs extend up to 6–7 kHz, but many clinically normal adult ears give weak TEOAEs

(less than 3 dB SPL), with no substantial response above 4 kHz.

An example of a clinical recorded TEOAE is presented in Figure �1-7. In the upper right

corner the stimulus response is plotted. We can see that the stimulus response has a much

larger magnitude than that of the OAE signal. The main window depicts the much smaller

emission signal with the stimulus artifact discarded (the first 3ms post-stimulus). The y-axis is

scaled so as to create a clear image of the response. In this example the OAE signal lasts for

about 12ms post-stimulus.

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Figure �1-7 - A TEOAE response from a newborn baby.

The time delay between the stimulus and the response allows the examiner to isolate the

response. TEOAEs detected from normal ears mirror the spectral properties of the stimulus

(Glattke & Robinette, 2002). Although clicks are ‘wide-band’ stimuli, exciting the whole of

the cochlea, TEOAE responses can give a frequency specific indication of cochlear status.

When employing a click stimulus the TEOAE spectrum will be that of the click, or broad

band. If a tone burst is used to elicit a TEOAE, then the response will mirror the frequency

composition of the tone.

The TEOAE stimulus typically is presented at an overall level of 80 dB peak SPL. Because

the energy in the stimulus is spread over a broad frequency range, the energy at any individual

frequency is about 35 dB below the overall level. That is to say, the spectrum level of the

energy in the transient stimulus is about 35 dB below the overall level. The stimulus,

therefore, is at about 40-45 dB SPL for any specific frequency.

Because TEOAEs are highly sensitive to cochlear pathology and dysfunction, TEOAEs have

found wide-spread application in newborn hearing screening programs. Although no universal

standard exists, the measures involved in the determination of whether or not a TEOAE is

present are reproducibility and signal to noise ratio.

DPOAEs (Distortion Product OtoAcoustic Emissions)

The healthy ear produces OAEs not only in response to clicks, but to any sound applied to the

ear. A common way to record DPOAEs is to present two continuous signals, called primary

10

tones, and analyze the spectrum of the sound detected in the external ear canal. DPOAEs are

relatively easy to extract because they appear at frequencies that can be exactly predicted

from the frequencies of the primary tones.

Nonlinear intermodulation between two tones is a purely mechanical process and distortion

products satisfy the frequency relationship 1 2 1( )dpf f n f f� � � where n is any positive or

negative whole number. The distortion components can be separated from the stimuli by

signal frequency analysis. The strongest component 1 2 1 1 2( ( 1)( ) 2 )dpf f f f f f� � � � � � is

used as an indicator of cochlear status. The most robust distortion product (DP) occurs

where 2 11.2f f� � . For example, the ear's response to primary tones at 2000 and 2400 Hz will

produce a robust distortion product at 1600 Hz.

DPOAEs offer a wide frequency range of observation (up to 10 kHz) in adults. More

powerful excitation with continuous tones allows DPOAEs to be recorded with moderate

losses when no TEOAE can be detected. However, DPOAE recordings provide no greater

frequency specificity than TEOAEs despite the use of pure tones. Even when proper controls

are in place DPOAE techniques do not produce results that have more frequency specificity

than TEOAE methods and they are not more robust than TEOAE and thus can not be obtained

under more challenging recording conditions.

Primary tones used to elicit DPOAE are presented at stimulus levels of 50 to 70 dB SPL,

depending on the interest of the examiner. Healthy ear canal distortion levels can be above 20

dB SPL. DPOAE generation is much reduced and usually absent if there is significant sensory

hearing loss.

The clinical significance of DPOAEs has not been fully evaluated. Clinical DPOAE

measurements are generally made with both stimulus intensity and frequency ratios optimized

for maximum DPOAE 2f1–f2 intensity. Many different DPOAEs co-exist and their

generation is intimately linked to the operating characteristics of the outer hair cells. It is

possible that one day we will be able to reconstruct OHC operating characteristics from

DPOAE data.

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SFOAE (Stimulus Frequency Otoacoustic Emissions)

SFOAEs are generated with continuous pure tones. During stimulation the energy in the ear

canal includes both the incident stimulus, sound reflected from the tympanic membrane and

sound that mirrors energy leaking back from the cochlea. The contribution of the cochlea is a

version of the stimulus that appears with a time (phase) delay.

SFOAEs are difficult to extract from the signals that are present in the ear canal and as a

result have not found their way into routine clinical practice.

SOAEs (Spontaneous OtoAcoustic Emissions) and SSOAEs

(Synchronized SOAEs)

SOAEs are narrow-band signals that occur in the absence of any known stimulus. SOAEs are

typically highly stable pure tones. Their level ranges from the noise floor of the equipment to

approximately 30 dB SPL. They are found in 30–40% of healthy young ears. SOAEs are

absent in frequency regions associated with hearing loss greater than about 30 dB HL. SOAEs

are normally detected by executing a spectral analysis on the sound recorded from the ear

canal of a participant seated in a quiet test environment. Because of their intrinsic stability and

critical dependence on cochlear status, SOAEs are, when present, particularly sensitive

indicators of metabolic and physiological changes in the cochlea.

SOAEs can be synchronized to external stimuli and extracted as Synchronized Spontaneous

Otoacoustic Emissions (SSOAEs) using time-averaging procedures commonly employed to

detect small physiological signals in the presence of background noise. Because they are not

detected universally in normal hearing individuals and because they are peculiar in terms of

frequency distribution and amplitudes, SSOAEs have not found widespread use in the clinic.

When SSOAEs are present one can conclude that the ear is functioning normally across the

outer hair cells that respond to the frequencies revealed by the SSOAEs. When they are

absent, no conclusion can be made about the hearing status (Kemp 2002, Glattke 2002).

OAE fine structure

There are consistent patterns of amplitude maxima and minima in the frequency dependence

of OAEs. The variations with frequency are collectively referred to as otoacoustic emission

fine structure (or otoacoustic emission microstructure).

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The frequency separations of threshold microstructure and cochlear emission fine structure

appear to be about 2/5 the average critical band (frequency resolution) estimates for human

subjects, which correspond to a 0.4-mm tonotopic displacement along the basilar membrane

with respect to Greenwood’s estimate of the cochlear map (Talmadge et al. 1998).

The patterns of cochlear fine structures have been found to move down in frequency (with

little change in frequency spacing) when the overall stimulus level is increased.

OAE Screening Tests

For more than 10 years TEOAEs have been employed in large-scale newborn hearing

screening programs. The OAE recordings are made via a probe which is inserted deep into the

ear canal. This way the ear is sealed off and the recorded OAE sound pressure (below 3 kHz)

is increased. Without sealing of the ear canal, ear drum vibrations would simply move air in

and out and the emission pressure would be lost.

Click stimuli of around 84 dB SPL (peak equivalent level) normally evoke a robust TEOAE

response only if hearing threshold is 20 dB HL or better. Frequencies at which hearing

thresholds exceed 20–30 dB HL are typically absent in the TEOAE response. Middle ear

status affects OAEs and can prevent their detection (Glattke and Robinette, 2002).

Healthy infant ears typically produce strong OAE levels of more than 15 dB SPL. Little signal

processing is required to extract these strong responses from the noise. Fully validated

frequency-specific measurements can often be made in just a few seconds.

In practice, sensory hearing impairment in the newborn population appears to be mainly of the

sensory transmissive type which is easily detectable by measurement of OAEs. This fact and

the favorable ergonomic and economic factors mean that OAEs are reliable and cost effective

for newborn screening programs.

Intensity is a primary factor in OAE detectability but, it is the presence of a detectable OAE

response to a particular stimulus that is clinically important and not its strength.

13

Although OAE screening has been used clinically for over a decade, the protocols used are

not yet fixed. Prieve (2002) indicates in her review that criteria for “pass” and “refer” vary

considerably between investigators: reproducibility, signal-to-noise ratio criteria for

individual frequency bands, overall amplitude of the TEOAE and combinations of amplitude

and reproducibility measurements. Relatively few investigations have focused on the use of

DPOAEs to screen newborn infants for hearing loss and here also the “pass” versus “refer”

criteria is not uniform among investigators.

OAE screening protocols

As neonatal screening has become widespread the test protocol had to be agreed upon. A

good recommendation was set forth by Stevens et al. 2002.

1) With TEOAEs, to minimize stimulus artifacts from contaminating the waveform

the analysis window for data collection should start 2.5-4 milliseconds after

delivery of the stimulus. The proposed start time for this protocol is set at 4ms.

2) The proposed end of the data collection and analysis window is between 10 and

12.5 ms.

3) The results should be analyzed in half octave bands centered at 1, 1.5, 2, 3 and 4

kHz. A response should be reported as present within a particular half octave band

if the signal to noise is >=6dB.

The successful stimulation and detection of OAEs indicates a high degree of normality in the

functioning of the middle ear and inner ear, in particular the environment of the inner ear is

shown to be healthy. This is a necessary but not in itself a sufficient condition for normal

hearing.

Some clinical devices allow the click stimulus waveform to be viewed. The ideal is a clean,

clear, positive and negative deflection lasting no longer than 1ms and followed by a straight

line indicating no or very limited ‘ringing’, or oscillation of the waveform. This condition is

much easier to obtain in a newborn ear than in an adult.

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2. Theory of OtoAcoustic Emissions

The field of OAE research has grown enormously since their discovery by David Kemp, 28

years ago. Their discovery has led to new insights into the mechanisms and function of the

cochlea and to a new understanding of the nature of sensory hearing impairment. As a

research tool OAEs provide a noninvasive window on intra-cochlear processes.

Cochlear transmission line models were originally developed by Zwislocki (1950) and further

refined by Hall (1974) with the introduction of non-linearites. These models are able to

analyze some of the nonlinear auditory phenomena taking place at the level of the cochlea

(like two-tone suppression and two-tone distortion products).

The discovery of cochlear emissions raised the question if those phenomena can be explained

by a transmission line model. Furst and Lapid (1988) showed that the properties of acoustic

distortion products can be predicted by the nonlinear transmission line model, including the

discrepancy between animal and human data. Resistance mismatches between adjacent points

along the cochlea evoked TEOAEs. The nonlinear transmission line model was not adequate

in order to predict the other two types of emissions (SOAE and click evoked otoacoustic

emissions). These emissions were predicted by Furst and Lapid by introducing a

noncontinuous resistance along the length of the cochlear. They proposed that such

discontinuity can occur if the connection between the basilar membrane and tectorial

membrane, via the cilia of the OHC, is not uniform along the cochlea length. Thus, resistance

mismatches between adjacent points along the cochlea evoke emissions. The click evoked

responses produced by the model did not resemble CEOAE clinical data because the

responses did not have higher frequencies than the helicotrema's CF (Furst and Lapid, 1988).

The local oscillator theory

The first model for SOAE generation, which had wide acceptance, was the local-oscillator

model. In this model the cochlea is modeled as an oscillator chain. When some of the

oscillators vibrate they create waves that travel to the base of the cochlea and out into the ear

canal. In the ear canal the waves resemble SOAEs.

Properties of SOAEs such as their interactions with one another and with external tones have

been successfully described by representing individual SOAEs using a nonlinear, limit cycle

15

oscillator such as the Van der Pol. These phenomenological, limit cycle oscillator models

were not developed to describe the “oscillating elements” within the cochlea; rather, their aim

was to approximate the behavior of a complex system of equations by a single effective

oscillator, thereby providing simple, analytically tractable representations of SOAEs as they

appear in the ear canal.

Sisto and Moleti used the time evolution of the spectral lines associated to SOAEs after

presenting a click stimulus in order to determine the correct functional form of the nonlinear

oscillator equation describing the cochlear resonances. Their model is a very simplified model

in respect to full cochlear models. The model is capable of describing, with very few

parameters, both the saturation phase and the slow decaying phase that are experimentally

observed in the time evolution of OAEs after an impulsive stimulus. The model predicts the

observed exponential decay of the lines of frequency corresponding to measurable SOAEs

after excitation by a click stimulus, which is not compatible with a Van der Pol oscillator

model (Sisto and Moleti, 1999).

Coherent Reflection Filtering

A different theory regarding SOAE generation, which was first suggested by Kemp in 1979,

predicts that mammalian SOAEs arise not via autonomous cellular oscillations but as cochlear

standing-wave resonances. In this theory, SOAEs result from multiple internal reflections of

traveling-wave energy initiated either by sounds from the environment or by physiological

noise. Kemp’s original standing-wave model postulated that the backward-traveling wave

originates from a point reflection. Since the original standing-wave model did not include the

effects of traveling-wave propagation gains and losses, the model needed to associate large

reflection coefficients with many points along the basilar membrane in order to generate

sizable standing waves. Kemps theory was subsequently elaborated in models of evoked

otoacoustic emissions and formed the basis for the coherent reflection filtering theory.

In 1995 Zweig and Shera wrote down their theory on coherent reflection filtering. According

to the theory of coherent reflection filtering, reflection-source OAEs arise by reflecting off

densely and irregularly distributed cochlear impedance perturbations. At all frequencies the

net backward-traveling wave is dominated by wavelets reflected within the region about the

peak of the traveling wave, where the wave amplitude is much larger than it is elsewhere.

16

These perturbations presumably include both those clearly visible in the anatomy, such as

spatial variations in OHC number and geometry, as well as morphologically less conspicuous

perturbations, such as variations in OHC forces due to random, cell-to-cell variations in hair-

bundle stiffness or the number of somatic motor proteins.

Intrinsic variations in emission amplitude and phase are predicted by the theory. The theory

indicates that SFOAEs are analogous to “band pass filtered noise”. In this analogy, the

“noise” is the irregular spatial arrangement and strength of the impedance perturbations that

scatter the wave and the “band pass filter” results from interference among the multiple

wavelets originating from the scattering region. Unlike distortion-source emissions, whose

amplitudes and phases typically vary relatively slowly with frequency, reflection-source

emissions often vary considerably with frequency.

For example, SFOAE amplitude spectra are often punctuated by relatively sharp notches.

According to the model, such notches result from random spatial fluctuations in the

irregularities that scatter the wave. At some frequencies, wavelets scattered from different

locations within the scattering region combine nearly out of phase, resulting in near

cancellation of the net reflected wave.

One of the key assumptions of the model by Zweig and Shera is that the spatial activity

pattern of the traveling wave is both “tall and broad”. The traveling wave has to be tall

enough to produce significant reflection from a very small level of cochlear inhomogeneities,

and the activity pattern peak region has to be broad enough to contain 1 to 2 wavelengths of

the traveling wave, a requirement for coherency of the cochlear reflections. Tall and broad

activity patterns were obtained by the introduction of time delayed stiffness.

Kim et al. observed in 1980 that adding distributed negative resistance to the BM (as an

energy source to generate tall activity patterns) and assuming that the amplifier resided in a

region basal to the activity pattern peak was sufficient in order to obtain tall and broad activity

patterns. It has since been argued by de Boer using increasingly general and more

sophisticated models that placing the cochlear amplifier basal to the activity pattern peak is a

necessary condition for obtaining tall and broad activity patterns.

In 2002 Shera and Guinan tested the key predictions of the theory of coherent reflection

filtering for the generation of reflection-source OAEs determined by the group delay of the

17

BM transfer function at its peak. The prediction is tested in cats and guinea pigs using

measurements of SFOAE group delay. A comparison with group delays calculated from

published measurements of BM mechanical transfer functions supports the theory only at the

basal-most 60% of the cochlea. At the apical end of the cochlea the measurements disagree

with neural and mechanical group delays. This disagreement suggests that there are important

differences in cochlear mechanics and/or mechanisms of emission generation between the

base and apex of the cochlea (Shera and Guinan, 2002).

The global standing wave theory (SOAEs)

The theory of reflection-source emissions predicts that backward-traveling cochlear waves are

generated by the coherent scattering of forward-traveling waves off densely and randomly

distributed perturbations in the mechanics of the cochlea (Shera 2003). Because wavelets

scattered near the peak of a forward-traveling wave have much larger amplitudes than those

reflected elsewhere, the net reflected wave is dominated by scattering that occurs in the region

about the response maximum.

The resulting backward-traveling waves are then reflected by the impedance mismatch at the

cochlear boundary with the middle ear, generating additional forward-traveling waves that

subsequently undergo another round of coherent reflection near their characteristic places

(Figure �2-1). This process continues for each backward-traveling wave in the cochlea,

partially being reflected into a forward-traveling wave. At frequencies for which the total

phase change due to round-trip wave travel is an integral number of cycles, standing waves

can build up within the cochlea, which is then acting, in effect, as a tuned resonant cavity.

Figure �2-1 - Multiple internal reflections within the cochlea (Shera 2003 poster)

18

The process of multiple reflection continues, each subsequent stapes reflection and cochlear

re-emission contributing an additional backward-traveling wave whose amplitude at the

stapes differs by a factor of R*Rstapes from the one before. Shera (Shera 2003) showed that

adding up all the backward traveling waves yields the factor /(1 )stapesR RR� for the total

outgoing wave at the stapes.

Whenever the product R*Rstapes is positive real all the high order forward traveling waves

combine in phase with the primary traveling wave at the stapes. The multiple internal

reflections then reinforce one another, creating a significant standing wave component in the

cochlear response whose amplitude depends on the product of cochlear and stapes reflection

factors, R*Rstapes. The theory predicts that the standing wave grows without bound as

R*Rstapes approaches 1. In the real cochlea, of course, unconstrained growth is prevented by

compressive nonlinearities that limit the energy produced.

Cochlear sites corresponding to SOAE frequencies need manifest no special distinguishing

features. In the global standing wave model, SOAE frequencies are determined by R*Rstapes,

and SOAEs therefore trace their origin to aspects of the mechanics as subtle, and as non-local

to the site in question, as the magnitude and angle of the impedance mismatch at the cochlear

boundary with the middle ear, the spatial frequency content of the cochlear impedance

perturbations that scatter the wave, and the total round trip traveling wave gain and phase shift

experienced en route.

In the global standing wave model, SOAE frequencies are determined in part by the

impedance mismatch at the cochlear boundary with the middle ear. Manipulations that modify

this basal boundary condition can therefore modulate both SOAE amplitude and SOAE

frequency. In accord with these predictions, middle ear impedance changes (caused by

postural changes) have been found to alter SOAE characteristics, including frequency.

The modern standing wave model predicts that most SOAEs result from normal mechanical

variability rather than from pathologically large impedance discontinuities. Cochlear standing

waves can become self-sustaining and thus appear in the ear canal as SOAEs when the total

round-trip power gain matches the energy losses experienced en route.

The standing-wave model differs fundamentally from the local-oscillator scenario. Rather

than supposing that the “oscillating elements” generating SOAEs are localized to particular

19

cells or subcellular structures within the organ of Corti, the standing-wave model identifies

SOAEs as a global collective phenomenon necessarily involving the mechanics,

hydrodynamics, and cellular physiology of the entire cochlea, as well as the mechanical and

acoustical loads presented to it by the middle and external ears.

In the local-oscillator model these macromechanical structures and processes play no

fundamental role, they serve merely to connect the autonomous oscillating element with the

external environment, providing a conduit for the acoustic energy it produces to escape from

the inner ear.

The coherent reflection model predicts that the SFOAE evoked by a tone comprises a sum of

wavelets scattered by perturbations located throughout the peak of the traveling wave. The

SFOAE therefore arises from a distributed region, roughly equal in extent to the width of the

traveling wave envelope. In the 1–2 kHz region of the human cochlea, this distance spans on

the order of 100 rows of outer hair cells at sound levels near threshold (Zweig and Shera,

1995). This is in contrast to Kemp’s original standing wave model of point reflection. In the

global standing wave model, by contrast, the oscillating element comprises the entire cochlea,

and the collective response of the hearing organ as a whole contributes essentially to creating,

maintaining, and determining the characteristics of the emission.

The evident success of the global standing-wave model contradicts the notion, often implicit

in the local-oscillator framework, that SOAEs measured in the ear canal provide direct access

to the local elementary cellular oscillators within the organ of Corti (Shera 2003).

The global standing wave model resolves the paradox noted by Geisler (1998) in his

discussion of the van der Pol oscillator as a local oscillator model for SOAEs:

“Why doesn’t every section of the cochlea act as a limit-cycle oscillator and the cochlea

therefore produce emissions at all frequencies? It follows that there must be something

different about those cochlear sites that generate the relatively few emissions observed.

Unfortunately, the search for such differences has not been successful”.

Shera tested the two alternative models for mammalian SOAEs generation. His work focused

on key predictions of the global standing-wave model that distinguish it from the local-

oscillator alternative. He showed that although some of the model predictions could perhaps

be obtained by artful adjustment of local-oscillator models, they all arise quite naturally

20

within the standing-wave framework. His tests provide strong support for the idea that human

SOAEs arise via global standing-wave resonances (Shera, 2003).

Sheras' quantitative tests provide strong support for the global standing wave model and its

prediction that SOAE frequencies are determined by R*Rstapes. The results demonstrate that

in addition to predicting the existence of multiple emissions with characteristic minimum

frequency spacing, the global standing wave model also accurately predicts the mean value of

this spacing, its standard deviation, and its power law dependence on SOAE frequency.

Furthermore, the statistics of SOAE time waveforms demonstrate that SOAEs are coherent,

amplitude stabilized signals.

DPOAE generation theory

DPOAEs are generated in the cochlea via the nonlinear interaction of the excitations produced

by two primary tones of frequencies f1 and f2. Their initial production is in the region of

strong overlap of the f1 and f2 activity patterns, which is around the f2 tonotopic site. From

the generation region, DPOAE components propagate both basally (backward) and apically

(forward) (seen in Figure �2-2C).

The DP cochlear wave reaching the DP tonotopic place will be partially reflected by small

irregularities in the cochlear properties. A tall and broad DP activity peak will allow coherent

scattering from many reflectors to give a large basal-ward reflection of the cochlear traveling

wave.

A portion of the basally traveling distortion product component will be transmitted through

the middle ear to the ear canal and detected as a DPOAE. The remaining signal will be

reflected back into the cochlea.

In general these two components will not arrive in phase. The relative phase difference

between them will mainly depend on the DP frequency and the resulting interference will give

an observable fine structure as a function of the DP frequency.

This phase is related to the round trip group delay for waves traveling from the generation

region to the DP place, and will thus depend on the frequency separation between the two

21

primaries, especially for narrower frequency ratios of the two primaries (Talmadge et al.

1998).

When the ratio f2/f1 is nearly one (e.g. 1.05), f1 and f2 TW velocities are very similar at all

points. The phase distribution of DP elements then necessarily forms a forward (apical) TW

with little DP sent backwards to form a DPOAE. Even so, some DPOAE signal escapes via

the SOAE route.

For large f2/f1 (e.g. 1.5), the densely packed phase changes within the f2 envelope generate

an undulating DP phase distribution that will be largely self-canceling and little DP waves

will propagate from that region. However, because there is a minus sign in ‘2f1–f2’, for f2 >

f1, the spatial phase gradients of TW ‘f1’ and TW ‘f2’ counteract each other in 2f1–f2 DP

production. Consequently, at some optimum f2/f1 ratio (around 1.2), the relative velocities of

TW ‘f1’ and TW ‘f2’ are such that the spatial distribution of DP elements actually becomes

that of a backward traveling wave over a considerable length of OHCs. Interestingly, there is

no optimum frequency ratio for the ‘alternative’ DPOAE 2f2–f1 (see Figure �2-2C), which

emanates from a place basal to both f1 and f2 peaks over a wide range of f2/f1 ratios.

Figure �2-2 - DPOAE generation visualized from (Kemp 2002). (A) A ‘dead’ cochlea where natural

damping absorbs most of the stimulus energy before any clear separate excitation peaks for f1 and f2 can develop. (B) A linear ‘live’ cochlea, where linear OHC amplification cancels the damping and sharp

‘images’ of stimuli f1 and f2 can be seen. (C) A “real” non-linear ‘live’ cochlea, where OHC motility is non-linear and this results in intermodulation distortion products being created under the entire f2

envelope (including 2f1–f2 and 2f2–f1).

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DPOAE specific models

A good example for a “simple” model that by introducing nonlinearities manages to create

DPOAE can be seen in the paper by Jaffer et al. (Jaffer et al, 2002). Jaffer et al. presented a

model of the cochlear partition in which a weak elastic longitudinal coupling has been

included between the resonant viscoelastic plates of the cochlea. The connective tissue

between plates was modeled by springs oriented in the longitudinal direction, perpendicular to

the transverse motion of the cochlear partition. The model includes the linear cochlear

partition mechanics and linear cochlear hydrodynamics but no active mechanics are included.

The addition of longitudinal elastic dynamics produce cubic distortion product otoacoustic

emissions where the 2f1-f2 intermodulation component is largest (in response to two primary

frequencies 2f1-2f2, 2f2-2f1, 2f2-1f1, and 2f1-1f2 were produced).

Dhar et al. evaluated the relative contributions of two sources to the DPOAE, the distortion

and reflection components (Dhar et al., 2005). The nonlinear interaction between the stimulus

tones around the tonotopic region of the higher frequency stimulus tone (f2) generates the

distortion or generation component. The reflection component is generated due to linear

coherent reflections from a randomly distributed roughness about the tonotopic region of the

DPOAE. This reflected energy signal contribution is the product of the initial apical moving

DP component with the apical reflectance.

By separating the ear canal DPOAE signal into its two major components, Dhar et al. have

been able not only to investigate the relative levels of the two components, but also to better

understand the properties of each component. While the generator component when plotted as

a function of the stimulus-frequency ratio showed a distinct band-pass shape, the pattern for

the reflection component was more variable with the stimulus level and across ears. One of

the most striking characteristics of the results is the stability of the generator component and

the variability of the reflection component across ears. Also, the influence of stimulus level on

absolute and relative component levels was found to be significant.

It should be emphasized that the apical reflectance (Ra), according to the coherent reflection

model, depends mainly on the sharpness of tuning of the BM around the DP CF region and

the strength of cochlear inhomogeneities around that region. If both of these factors are

present the apical reflectance will generally be significant. The apical reflectance is expected

to fluctuate significantly across healthy ears due to differences in the degree of cochlear

roughness across such ears.

23

Dhar et als’ results are consistent with other reports in that the reflection component is shown

to be dominant at low stimulus levels only. Furthermore, they have shown the dominance of

the reflectance component to be consistent across the entire range of stimulus-frequency ratios

tested. Great variability in the reflectance component across subjects is also reported in their

study.

Dhar et al. conclude that most of the characteristic features of the generation component and

the reflection component that they find are at least qualitatively described by cochlear models

that incorporate a nonlinear generation process around the f2 CF region combined with the

coherent reflection of the initial apical moving DP component.

OAE generation summary

Different representation of the same physics

An interesting example of the dynamics of OAE modeling research can be seen in the Nobili

et al. paper from 2003. Nobili et al. proposed a “new and different” interpretation of OAEs

based on the instantaneous fluid coupling between the stapes footplate and the BM, and

among the BM oscillating elements themselves. Nobili et al. claimed that this interpretation

differs from modeling the cochlea as a transmission line. Their OAE time-domain simulations

were based on a hydrodynamic model adapted so as to fit physical and geometrical

characteristics of the human inner ear. The model was completed with the inclusion of

forward and reverse middle ear transfer functions. In their analysis of the results Nobili et al.

tried to show differences between their results and the results obtained by transmission line

models. They related their results to the hydrodynamic character of cochlear dynamics, in

particular, the instantaneous character of fluid coupling between BM and stapes.

In response to the paper by Nobili et al. Shera, Tubis and Talmadge wrote two papers (2004a,

2004b). In the first paper, Shera, Tubis and Talmadge (2004a) demonstrated that Nobilis'

model fails to reproduce basic empirical properties of actual evoked OAEs. By circumventing

uncertainties about the numerical accuracy of Nobili et al.'s published simulations, they

demonstrated that the middle ear filtering mechanism proposed by Nobili fails to reproduce

basic empirical properties of actual evoked OAEs.

24

The second paper (Shera, Tubis and Talmadge, 2004b) was aimed against the critic claimed

by Nobili et al. that the transmission line models fundamentally misrepresent the

hydrodynamics of the cochlea. Nobili et al. argued that although the concepts of wave

propagation and reflection may apply in idealized hypothetical situations, transmission line

models cannot describe anything resembling the physics of an actual ear.

Shera, Tubis and Talmadge resolve and synthesize the two different approaches, arguing that

the wave-equation and hydrodynamic formulations of cochlear mechanics are different

mathematical representations of the same underlying physics.

Long range fluid coupling underlying Nobili et al.’s integrodifferential equation is shown to

be identical to that in a one-dimensional, tapered transmission line model. They are shown to

be two different mathematical representations of a single model based on Newton’s laws.

Since they both represent the same physics, both ultimately yield the same solutions.

Although both formulations provide valid representations of the physics of the cochlea, the

two approaches are attended by strikingly different conceptual and computational

frameworks. Although no less physically appropriate than the Green’s function alternative,

the wave-equation formulation often provides considerably more aid to the intuition and is

easier to visualize. Shera, Tubis and Talmadge argue that the wave-equation formulation

provides compelling advantages, at least in the context of modeling OAEs.

OAE properties

Otoacoustic emissions have been explained as arising from a combination of two cochlear

mechanisms: coherent, linear reflection and nonlinear distortion. Reflections are described as

scattering from multiple randomly spaced discontinuities along the basilar membrane. Only

those reflections that sum constructively with the incident traveling wave and that arise for an

incident traveling wave that is both broad and tall will have sufficient amplitude to contribute

to the recorded OAE. This type of traveling wave response involves nonlinear mechanical

amplification supplied by the cochlear outer hair cells that has its highest gain at the lowest

stimulus levels. The reflection process itself is thought to be linear. In contrast, the nonlinear

distortion mechanism for OAE generation is described as a byproduct of intermodulation

distortion in the basilar membrane traveling-wave response. Both coherent linear reflection

25

and nonlinear distortion are thought to contribute to all evoked OAEs, with the level of the

stimulus partially determining which mechanism is dominant (Konrad and Keefe, 2003).

SFOAEs and TEOAEs at low to moderate levels are thought to be predominantly generated

by linear reflection near the tonotopic region associated with the spectral content of the

stimulus. The SFOAE may have a quasi-regular fluctuation in amplitude and phase with small

changes in the stimulus frequency. This spectral fine structure has a local frequency

separation between maxima that is inversely related to the round-trip time delay between the

ear-canal microphone and the apical CF reflection site in the cochlea. Interference from

multiple reflections between the oval window and the apical reflection site may produce a

temporal fine structure in the response envelope, and may contribute to the spectral fine

structure.

In contrast to SFOAEs and TEOAEs, DPOAEs are thought to be initiated by nonlinear

distortion in the basilar membrane’s response to the two-tone stimulus (f1 and f2). For the

cubic distortion product (2f1-f2) and other DPs tuned more apically than the f2 place a

secondary component arises as a coherent linear reflection near the place associated with the

DP frequency.

DPOAE fine structure has been explained as originating from interference between these two

“sources” and from multiple reflections of these two components between the oval window

and the DP tonotopic place (Dhar et al., 2002). Another proposed source of the spectral fine

structure in evoked OAEs is related to spatial variations in the magnitude of the reflected

wave, which may be due to variations in the effective reflectance with position along the

basilar membrane (Shera and Guinan, 1999).

SOAEs are narrow-band emissions measurable in the ear canal in the absence of acoustic

stimulation, which are generated within the cochlea by stable limit-cycle oscillations (Tubis

and Talmadge, 1998).

Konrad and Keefe (2003) studied the influence of SSOAEs on the rest of the OAEs. They

showed that in ears with significant SSOAEs, multiple SSOAE sources and the presence of

multiple internal reflections influence the fine structure of TEOAEs elicited by low to

moderate level stimuli.

26

Konrad and Keefe used the TFR (time-frequency analyses) method in order to better

understand the OAE frequency components and their time of appearance. In their paper they

showed that they can align SSOAE peeks with their counterparts in SFOAE and DPOAE

recordings. Thus the “hot spots” (strong sites of cochlear reflection) contribute to the two

phenomena. Also shown is the “hot spots” contribution to DPOAEs via stimulus and SSOAE

intermodulations. This suggests the importance of two-tone suppression processes to the

interpretation of OAE responses elicited by any stimuli more complicated than a single sine

tone. The stimulus spectrum is also seen mirrored in the OAE spectrum (its' main lobe at

least).

This suggests that the coherent reflection theory of OAE generation is still incomplete for

describing responses at levels (at approximately 20 dB SPL and higher) for which the basilar

membrane response becomes compressive due to the saturation effects of outer hair cell

functioning.

They also showed that the relative contributions by SSOAEs and multiple internal reflections

to the total OAE response increased with decreasing stimulus level (the relative SOAE

amplitude will increase). This means that the evoked OAE spectrum, as measured by a

nonlinear residual technique, is less likely to resemble that of the eliciting stimulus as the

stimulus level decreases.

Models incorporating cochlear reflectance predict that an OAE response would resemble the

eliciting stimulus. For example, the model predicts that the SFOAE spectral energy should lie

within the pass-band of the stimulus energy, and that the SFOAE spectrum, aside from fine

structure, should be similar to the stimulus spectrum within the pass-band. The results of this

study show that other factors are involved. In reality, for tone-pip-evoked SFOAEs, the OAE

spectrum is narrow compared to the eliciting stimulus, and response components correspond

primarily to the higher-frequency portion of the stimulus pass-band.

The round-trip magnitude of the cochlear reflectance was measured and found to decrease

with increasing effective input level, which means that realistic models of OAE generation

should include nonlinearity in the apical cochlear reflectance. A recursive formulation of the

coherent reflection theory was presented that may be useful in time-domain simulations.

Predictions of DPOAE and SFOAE latencies by cochlear models were in general accord

although the onset of the SFOAE and the lack of level dependence in the simplified cochlear

27

models present difficulties. An additional effect at high stimulus levels in some ears is the

observed dynamical linking between SSOAE and stimulus-evoked OAE components, which

produces intermodulation at frequencies not present in the original SSOAE or in the pass-

band of the stimulus (Konrad and Keefe, 2003).

Sisto and Moleti found a difference in latency, in the “nonlinear” OAE response, between

normal and impaired ears. They analyzed the “nonlinear” OAE response to TEOAEs using

the wavelet analysis. They also showed that the latency-frequency relationship predicted by

scale-invariant full cochlear models does not agree with experimental measurements of the

TEOAE latency as a function of frequency (Sisto and Moleti, 2002).

The field of OAE research has not yet come to a full understanding of the ways OAEs are

generated inside the cochlea. There have been many breakthroughs in the past 20 years in the

field, and more are expected to come in the next 5-10 years. Until a model can be created that

can simulate all OAEs with high accuracy the field of OAE research and modeling will

continue to be an active one.

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3. Motivation for the present study

Although there are many models that try to explain the existence of OAEs, our purpose is to

predict the existence of OAEs in a complete human ear model. Recently a cochlear model was

developed in our lab (Cohen and Furst, 2004). This model incorporates cochlear fluid

dynamics with an OHC model. The model is confined to the inner ear alone (i.e. the cochlea)

and cannot on its' own simulate OAEs. The model was successful in predicting both normal

and abnormal audiograms and tuning curves.

In this study we include a middle ear model and test the constraints that enable us to predict

the existence of OAEs in normal and abnormal ears. The model will serve as a tool in the

testing of OAE phenomena. Our hope is to better understand the different effects leading to

OAE formation through the use of the model. A computational model of the entire

presynaptic system can help validate hypotheses and help interpret data collected. If emissions

can be properly simulated by a cochlear model and details about them described in terms of

structures and mechanisms in the model, the correspondence between model structures and

physical structures in the real cochlea can be used to extract information about the interior of

the cochlea without actually having to physically “look inside”. There has not yet been a

model created with the capability of truly simulating all the OAEs generated by the biological

ear.

The work described in this thesis describes computations performed with a one-dimensional

cochlear model. This model has the advantage that it is not too complex to understand and

leads to a numerical code that does not require extensive computer time and memory. The fact

that the model computes the behavior of the cochlea in the time domain makes it especially

useful for the study of nonuniform and nonlinear effects, which play an important role in

otoacoustic emissions.

This is a theoretical research focused on studying the OAE sources in a mathematical model.

The research goal is to shed some light on the sources of TEOAEs and SOAEs. In general,

OAE responses carry a large amount of information about the status, activity and environment

of OHCs, which we are currently unable to interpret. OAEs tend to be dominated by

microscopic details of little relevance to hearing. Nevertheless, OAEs provide the only

detailed noninvasive window to the cochlea and by their very presence confirm normal

29

presynaptic cochlear function. Thus far, only a small portion of the OAE potential has been

tapped, primarily by efforts to use OAEs to screen for hearing loss. Although useful today, if

we can learn how to extract definitive data on OHC status from OAE data, then their clinical

importance will be greatly enhanced.

The next chapter gives a detailed description of the mathematical model used throughout this

thesis. Chapter 5 demonstrates the different phenomena simulated by our complete human ear

model.

30

4. Model details

The model presented in this thesis is comprised of a 1-D cochlear model. The cochlear model,

which includes an embedded outer hair cell model, was developed by Azi Cohen (Cohen

2004, Cohen and Furst 2004). The cochlear models’ boundary condition was the movement of

the stapes. In order for the model to predict OAEs a middle and outer ear model were added,

and the models boundary condition replaced. Because the embedded model cannot be treated

as time-invariant when large magnitude random variances are applied, our implementation

will focus on the time domain solution to the model equations. For clarity reasons the entire

model is described below with emphasis on the original part of this work, the middle ear

model.

Middle Ear Model

The mechanics of the middle ear and ear canal are based on a simple mechanical model

(Talmadge et al., 1998). In this model the tympanic membrane is treated as a single piston that

has a fixed incudostapedial joint (Figure �4-1).

Figure �4-1 - A simple middle ear mechanical model

The ear canal is assumed to be sealed off by a stimulus delivering microphone assembly and

the length of the ear canal (from microphone to tympanic membrane) is assumed to be small

relative to the sound wavelength. Thus, the pressure ( )eP t in the ear canal may be considered

to be uniform. It is also assumed that all air pressure changes occur without loss or gain of

heat, so that the mechanical model gives rise to a single oscillator equation of the form:

31

� �2 1( ) * ( ) * ( ) * ( , ) * ( )OW OW OW ow OW me e

OW

t t t P o t G P t� � ��

� � � ��� � (4.1)

where (0, )P t is the pressure difference between the scala tympani and scala vestibuli near the

stapes, OW� is the effective areal density of the oval window (effective mass of oval window

+ ossicles / area of oval window), OW is the middle ear damping constant, OW is the middle

ear frequency and meG is the mechanical gain of the ossicle chain. All middle ear parameters

are phenomenological constants and are defined in Table I. OW was chosen so as to be as

close as possible to the middle ear transfer function by Puria (2003).

TABLE I. Table of middle ear parameters. The parameters are taken from Talmadge et al.

(1998).

Parameter Value Definition

OW� 1.85 g/cm^2 areal density of oval window

OW 500 1/s middle ear damping constant

OW 1500 Hz*2� middle ear frequency

meG 21.4 mechanical gain of ossicles

meC 2 60.059

(2 1340 ) * 60.49*1.4

x HZ e� � coupling of oval window displacement to ear

canal pressure

owC 0.0322.909

0.011� coupling of oval window to basilar membrane

Eq. 4.1 relates the displacement of the oval window ( )OW t� to the fixed pressure ( )eP t in the

ear canal and the pressure (0, )P t near the stapes. For most experimental setups, however, the

pressure ( )eP t is an observable rather than a fixed experimental input. Instead, the

experimental input is the “calibrated ear canal pressure”, ( )inP t , which is the pressure (created

by a microphone) in the ear canal in the case of a rigid ear drum. ( )eP t is the “total” pressure

in the ear canal influenced by pressure created by the microphone ( )inP t and by pressure from

the displacement of the tympanic membrane. With the assumption of adiabatic

compression/expansion, the relation between ( )eP t and ( )inP t is

( ) ( ) * ( )e in me owP t P t C t�� � (4.2)

32

where meC is the coupling of the oval window displacement to the tympanic membrane

displacement contribution to the ear canal pressure (defined in Table I).

The emission pressure ( eP ) can be easily determined by calculating the ear canal pressure,

resulting from a tone injected into the ear canal ( inP ), and the cochlear response ( ( )OW t� ).

The Cochlear Model

In the simple one-dimensional model the cochlea is considered as an uncoiled structure with

two fluid-filled rigid-walled compartments separated by an elastic partition. The basic

equations are obtained by applying fundamental physical principles such as conservation of

mass and the dynamics of deformable bodies. In the model the elastic partition is responsible

for the mechano-neural transduction of sound.

Cohen and Furst integrated an OHC model into the one dimensional cochlear model. The two

models control each other through cochlear partition movement and cochlear partition cross

pressure variables. If we assume that the cochlea is uncoiled and approximated by two fluid-

filled rigid-walled compartments separated by an elastic partition, then it may be represented

by a one-dimensional model as shown in Figure �4-2.

Figure �4-2 - Cochlear Model Geometry

Let x be the longitudinal coordinate such that at the basal end 0x � and at the apical end

x L� , where L is the uncoiled cochlea length. Let t be the time variable. Let ( , )vP x t be the

pressure through the scala vestibuli and ( , )tP x t the pressure through the scala tympani.

33

The intermediate channel between the scala vestibuli and the scala tympani is called the scala

media and is represented by the elastic partition. The vertical displacement of the partition

along the x dimension is denoted by ( , )bm x t� . The fluid velocity along the x dimension is

( , )vU x t and ( , )tU x t for the scala vestibuli and the scala tympani, respectively.

The principle of conservation of mass yields the following equations:

0v bmUA

x t�

� �

� �� �

(4.3)

0t bmUA

x t�

� �

� �� �

(4.4)

where ( )x is the basilar membrane width and ( )A x is the scalae cross section area. Since

both scalae tympani and vestibuli contain perilymph, which we can assume is an almost

incompressible fluid, the equation of motion for each scala can be written as:

0v vP Ux t

�� �

� �� �

(4.5)

0t tP Ux t

�� �

� �� �

(4.6)

where � is the perilymph density.

This set of equations is completed by the equation of motion of the cochlear partition. The

partition is, mechanically, a flexible structure embedded in a rigid framework. It is assumed

that the flexible part, the basilar membrane, and the structure above it have point wise

mechanical properties. This means that the partition velocity at any point is related to the

pressure difference across the partition at that point only and not at neighboring points.

We define the pressure difference across the cochlear partition as:

t vP P P� � (4.7)

The cochlear partition is regarded as a flexible boundary between the scala tympani and the

scala vestibuli, whose mechanical properties are describable in terms of point-wise mass

34

density, stiffness and damping. Thus, at every point along the cochlear duct, the partition’s

velocity is driven by the pressure difference P across the partition. From the principle of

conservation of mass we can derive the relationship between the fluid velocity and the basilar

membrane displacement bm� .

Combining equations, Eq 4.3 - Eq 4.7, yields the differential equation for P :

22

2 2

2 ( )0bmP x

x A t�� ��

� �� �

(4.8)

The pressure difference across the partition ( P ) is the combined result of the pressure

generated by the basilar membrane model and the pressure generated by the OHC model. Eq.

4.9 depicts the combined pressure from the contributions of the two models:

bm ohcP P P� � (4.9)

where the basilar membrane is imitated as an electrical transmission line.

2

2( , ) ( ) ( , ) ( )bm bmbm bmP x t m x r x t s x

t t� �

�� �

� � �� �

(4.10)

where ( ), ( , )m x r x t and ( )s x represent the basilar membrane mass, resistance, and stiffness

per unit area, respectively.

The Outer Hair Cell Model

The outer hair cell (OHC) membrane is divided into two regions, the apical part facing the

scala media and the basolateral part embedded in the organ of Corti. The basic OHC model

represents these two cell membrane segments as two parallel resistance and capacitance

circuits. Figure �4-3 represents an equivalent electrical circuit model for the OHC.

35

Figure �4-3 - An equivalent electrical circuit model of the outer hair cell

Changes in the OHC length are controlled by the voltage change across the OHC basolateral

membrane� . Solving the electrical circuit in Figure �4-3 yields a differential equation for� :

0( )aohc a ohc

dCdG

dt dt�

� � �� � � � (4.11)

where aC and aG are the capacitance and conductance of the apical part, respectively.

ohc and � are defined as . 2 1000a b bohc

a b b

G G Gconst

C C C �

�� � � � �

� and .sm sm

b a b

V Vconst

C C C� � � �

�.

The capacitance aC and conductance aG of the apical part are affected by the stereocilia

movement. The OHC stereocilia are shallowly but firmly embedded in the under-surface of

the tectorial membrane. Since the tectorial membrane is attached on one side to the basilar

membrane, a sheer motion arises between the tectorial membrane and the organ of Corti as

the basilar membrane moves up and down. The model assumes that aG and aC are functions

of bm� (the basilar membrane vertical displacement).

The voltage variation across the basolateral part of the OHC causes a length change ( OHCl� ) in

the OHC. Thus, the force OHCF that an OHC exhibits due to voltage change is derived by:

( ( ) )ohc ohc ohc bmF K l � �� �� (4.12)

The pressure that the OHCs contribute to the basilar membrane pressure is derived from

36

( )ohc ohcP x F� (4.13)

where ( )x is the relative density of healthy OHCs per unit area along the cochlear duct.

( )x is referred to as the OHC gain, whose value ranges from 0 to 0.5. When ( )x is larger

than 0.5 the model represents a nonrealistic cochlea whose motion approaches infinity and

thus, will not be used.

When linear dependencies ( ( ),A bmG �� ( ),A bmC �� ( )OHCl ��� ) are assumed and substituted

into equations 4.11, 4.12 and 4.13 we derive the differential equation for ohcP :

2 1( )ohc bmohc ohc bm

dP dP x

dt dt�

� � �� �� � �� �� � (4.14)

where the values of 1( )x� and 2 ( )x� are:

1

( ) ( )( )

( )r x s x

xm x

� � � (4.15)

2 ( ) ( ) ohcx r x� �

Boundary and Initial conditions

The boundary condition between the middle ear and the cochlea relate the cochlear fluid

velocity to the velocity of the oval window. Thus, the model boundary conditions are:

(0, ) (0, ) ( )v t ow owU t U t C t�� � � � � (4.16)

( , ) 0P L t �

where ( )ow t�� is the oval window (OW) velocity, owC is the ratio between the area of the oval

window and the cross-section of the cochlear scalae (Table I) and L is the cochlear length.

From Eq. 4.5, 4.6 and 4.7 we obtain

37

t v v tP P U UPx x x t t

� �� � � �� � � � �� � � �� � � �� � � � �� � � �

(4.17)

substituting Eq. 4.16 into Eq. 4.17 yields:

0

( , ) ( , )2 ( )v t

ow owx

U o t U o tPC

x t t t� � �

�

� �� �� � � �� � � �� � � �� � � �� �� �� (4.18)

Thus our boundary conditions are:

0t� �

(0, )

2 ( )ow OW

P tC t

x� �

�� � �

��� (4.19)

( , ) 0P x t � x L�

where the pressure difference derivative near the stapes is related to the oval window

acceleration.

( )ow t��� can be derived from Eq.4.19, therefore substituting Eq 4.1 in Eq 4.19 yields:

2(0, )(0, ) * ( ) * * * * * ow

me in OW OW OW OW OW OWOW

CP tP t G P t

x � � � �

�� � �� � � �� ��

� (4.20)

As we can see the boundary condition influences the pressure difference equation only in the

first section (the boundary between the cochlea and the middle ear). We stimulate the model

through ( )inP t which represents the ear canal pressure generated by the ear canal microphone.

The initial value conditions [0, ]x l� � were defined as:

( ,0) 0bm x� � (4.21)

( ,0) 0bm xdt�

�

( ,0) 0OHCP x �

38

TABLE II. Table of cochlear parameters. The parameters are taken from Cohen and Furst

(2004).

Parameter Value Definition

L 3.5 cm length of uncoiled cochlea � 1 g/cm^3 Perilymph density 0.003 cm Basilar membrane width

A 0.5 cm^2 Scalae cross section area

( )m x 6 1.5 21.286 10 /xx e g cm� � �� Basilar membrane mass per unit area

( )r x 4 1.5 2 21.282 10 /xx e g cm s�� �� Basilar membrane damping per unit area

( )s x 0.06 20.25 /xe g cm s�� �� Basilar membrane stiffness per unit area

ohc 1000 Hz*2� OHC cutoff frequency

Numerical Solution

The time domain solution is performed in two sequential steps (Cohen and Furst, 2004). In the

first step, the boundary value problem is solved by the finite differences method while the

time is held as a parameter. In the second step, the initial value condition problem is solved by

the fourth order Runge-Kutta method. The first step is run in the spatial domain, and the

second step in the time domain.

We use the finite difference method to solve the second degree differential equation (Eq. 4.8).

In order to solve the boundary value problem we rewrite the equation by substituting Eq. 4.9

and Eq. 4.10 into Eq. 4.8, which yields:

2

2 ( ) ( ) ( )P

P Q x G x Q xx

�� � � �

� (4.22)

where 2

( )( )

Q xA m x

� �

� and ( ) ( ) ( )BM

BM OHCG x r x s x Pt

��

�� �� � � �� ��� �.

Eq. 4.19 is used as the boundary condition. The natural three-point approximation to the

second derivative of x is:

39

2

2 2

( , ) ( , ) 2 ( , ) ( , )P x t P x x t P x t P x x tx x

� � � � � � ��

� � (4.23)

where L

xN

� � and N is the number of spatial sections of the cochlea. In this way we define

a uniform grid of 1N � points in the interval [0,L], so that lx l x� � and ( , )l lP P x t� , where

0,1,...,l N� .

The initial value differential equations to be solved are Eq. 4.9, 4.10 and 4.14 with the initial

conditions:

( ,0) 0, ( ,0) 0, ( ,0) 0bmbm OHCx x P x

dt�

� � ����� � ����� � (4.24)

The boundary value problem (Eq. 4.22) can be expressed as a set of linear equations:

P Y� � � where

� �0 1 1, ,..., ,T

N NP P P P P��

2 2 20 1 1 2 2 1 1, , ,..., ,0

T

N NY Y G Q x G Q x G Q x� �� �� � � �� � (4.25)

2 20 0 0

1* ( ) * * * * *

2ow

me in ow ow ow ow ow owow

CY G Q x G P t h � � � �

�� �� �� � � �� �

�

and

!

!

2

0

21

21

1 * 0 0 0 02

1 2 1 0 0

0 0 1 2 1

0 0 0 0 1

ow

ow

N

CxQ x

x Q

x Q

�

�

� �" #�� � � �� �$ %& '� �

� �� � �� �� � � �� �� �� � �� �� �� �

� � � � �

All parameters used in the simulations are listed in Tables I and II.

40

Assuming that , ,bm bm OHCP� �� are known (for t T� and for every ix ) and the boundary

condition variables ( ,ow ow� �� ) are also known, an approximation of the pressure difference P

can be obtained for every nodal point ix , using the finite difference method.

The time domain model equations are solved numerically. We assign a variable ( stept ) to be

the time variable step size. The spatial step size is LN

x �� and each point along the cochlear

partition is denoted by ix .

Once the pressure difference ( P ) for every location along the partition is known (for t T� ),

then , ,bm ow ohcP� ��� �� � can be calculated. From Eq. 4.9 and Eq. 4.10 we calculate bm��� , Eq. 4.14

enables us to calculate ohcP� and Eq. 4.1 combined with Eq. 4.2 are used for ow��� .

After the spatial domain step, the pressure difference along the cochlear partition is known,

allowing us to calculate , ,OHC bm bmP � �� and P along the cochlear partition (and the boundary

condition variables ,ow ow� �� ), at all time points that satisfy t T( . Now an approximation (of

the following variables: , , , ,bm bm OHC ow owP� � � �� � ) at time stept T t� � is achievable by an initial

value numerical method.

In this research work, we have chosen the multi-step Fourth order Runge-Kutta method as the

numerical methods to approximate the above differential system solution. Although the

Runge-Kutta method is more “computation consuming” than the simple modified Euler

method it is more stable and accurate (the model by Cohen and Furst used the modified Euler

method).

Fourth order Runge-Kutta

The fourth order Runge-Kutta algorithm is similar to the Euler and improved Euler methods.

Rather than approximating the area of a rectangle, as the Euler method does, or by the area of

a trapezoid, as the improved Euler method does, it approximates by the area under a parabola.

In order to do so it uses Simpson's rule:

41

6 2 2( , ( )) ( , ( )) 4 ( , ( )) ( , ( ))n

n

t hh h h

n n n n n nt

f t t dt f t t f t t f t h t h) ) ) )�

� �� � � � � � �� �* (4.26)

where 2( ), ( )hn nt t) ) � and ( )nt h) � are unknown and need to be approximated. The fourth

order Runge-Kutta algorithm which incorporates all the approximations is:

,1

1,2 ,12 2

1,3 ,22 2

,4 ,3

1 ,1 ,2 ,3 ,46

( , )

( , )

( , )

( , )

2 2

n n n

hn n n n

hn n n n

n n n n

hn n n n n n

k f t y

k f t h y k

k f t h y k

k f t h y hk

y y k k k k�

�

� � �

� � �

� � �

� �� � � � �� �

(4.27)

The fourth order Runge-Kutta method does four function evaluations per step (depicted as the

hollow circles and ny in Figure �4-4) in order to give a method with fourth order accuracy. In

each step the derivative is evaluated four times: once at the initial point (1), twice at trial

midpoints (2, 3), and once at a trial endpoint (4). From these derivatives the final function

value (shown as 1ny � in Figure �4-4) is calculated.

Figure �4-4 - The Fourth order Runge-Kutta method.

Diependaal et al. showed that the variable step size fourth order Runge-Kutta scheme is both

more stable and much more efficient than other published numerical solution techniques

(Diependaal et al. 1987, Gear 1971).

Adaptive Step Size

To ensure that the Runge-Kutta iterations converge, we have to be sure that the time step size

is adequate. In order to do so we use the ‘step doubling’ technique.

42

This technique uses two separate estimations for each step. The first approximation uses the

time step to calculate the parameters at stept T t� � . The second approximation is done using

two small time steps which are half the size of the previous time step. The first step estimates

the parameters at 2

tstept T� � and the second step uses them as a basis in order to estimate the

parameters at stept T t� � .

Once the two different estimations are computed we compare them and calculate the error. A

relative error threshold is used in order to decide if the step size used is too big and the

numeric solution does not converge. If the error threshold is passed the step size is halved and

the process is repeated with a half the initial step size. The time step will continue to decrease

as long as the calculated error is larger than the error threshold (and the solution does not

converge).

We double the time step when 100 steps are calculated without a single one of then crossing

the error threshold. No time step changes, in 100 cycles, reveals that the method is stable and

has enough “margin” so we can try and increase the step size, thus continuing faster through

the time domain.

In order to keep the computation error from growing uncontrolled, the time step size should

be bound. If the time step is too small the computation is not efficient and the error due to

rounding increases.

Time versus Frequency results comparison

The time domain model algorithm was used throughout this study because of its ability to

simulate complex inputs. The model algorithm implementation was verified by comparing the

results with the results obtained by solving the model equations in the frequency domain

(according to Cohen and Furst, 2004). Steady state inputs were used to excite both models.

The models were stimulated with sine waves at frequencies: 250Hz, 500Hz, 1000Hz, 2000Hz,

3000Hz, 4000Hz, 6000Hz and 8000Hz.

A comparison between the outputs of the time and frequency domain algorithms is

demonstrated in Figure �4-5. Both algorithms are based on the same model parameters. Basilar

43

membrane energy curves corresponding to an ideal cochlear model (�=0.5) are plotted in

Figure �4-5a. All eight responses, from both the time and frequency solutions, are plotted in

the same figure. Continuous lines representing the time domain solutions and dashed lines the

frequency domain solutions. Figure �4-5b has the same eight stimuli, but a dysfunctional OHC

gain model (�=0.0) was used. In these simulations a constant � was implemented along all the

cochlear partition.

(a)

(b)

Figure �4-5 – Time versus Frequency algorithm comparison. (a) Basilar membrane energy curves from an

ideal cochlea (�=0.5). (b) Basilar membrane energy curves for a totally dysfunctional OHC cochlea (�=0.0). Continuous lines represent the time domain solution and dashed lines the frequency domain

solution.

It is obvious from Figure �4-5 that with the increase of � the location of resonance for each

input frequency moves towards the helicotrema, and the peak of the resonance becomes more

significant.

All time domain simulations were done with 512 sections and a constant time step. In order to

compare “steady state” responses the initial 30ms of data from the time domain solution was

discarded (in order to minimize the stimulus artifact contributing to the BME curve). The

stimulus artifact is not totally cancelled out by this action and low frequency energy still

resides in the BME curves.

Although the frequency domain simulations are more accurate (for steady state stimuli the

frequency domain simulations do not contain any artifacts resulting from initial conditions or

signal transients) the fit between the two sets of curves is so close that in most places one can

hardly tell that two lines are plotted rather than one. Near the low frequency part of each

curve the time domain solution parts from the “ideal” frequency solution. There are several

reasons for this: the limited dynamic range of the time domain computations, the accumulated

rounding errors due to long simulation times and the low energy stimulus artifact (due to the

44

zero velocity and zero displacement initial conditions). The frequency domain solution is

superior for linear “steady state” responses, but for complex input signals (especially ones

containing transients) and nonlinearities the only option is the time domain algorithm.

Spatial artifacts

Since the model algorithm is solved numerically, non-convergence of the solution is possible

when too little spatial sections are used. By not using enough sections in the calculations

spatial artifacts may be triggered. The same also applies for performing time steps that are too

big for the numeric convergence of the solution.

In his 1-D model Hengel (1996) used a constant number of sections (400). We implemented

400 sections into our model and used Hengels cochlear parameters. The resulting BM velocity

matrix contained spatial artifacts, as can be seen in Figure �4-6a as the second vertical pulse at

around 25ms. Once the section count was increased to a reasonable amount the artifacts

completely disappeared (Figure �4-6c), thus proving that Hengels 1-D OAE results are nothing

more than an artifact.

(a)

(b)

(c)

Figure �4-6 – Spatial artifacts. (a) The figure is contaminated with spatial artifacts when only 400 sections were used. (b) The spatial artifact can still be seen when 512 sections are used. (c) Here we see a clear

result after increasing the section count to 1024. All three simulations use the Hengel cochlear parameter configuration.

In order to eliminate the same problem in our simulations, all the results in this work were

checked for spatial artifacts by running important findings multiple times with different

section counts. The model was run with different spatial resolutions and the output monitored

45

for significant changes. Also, all the simulations in this work were done with a minimum of

512 sections.

(a)

(b)

Figure �4-7 – Spatial resolution. (a) The results obtained with 512 sections. (b) The same results obtained

with 2048 sections. Both simulations were done with the same �(x) distribution.

If the model solution is numerically stable then the output from different simulations with

different section counts should be the same. Figure �4-7 is an example of a simulation that is

not contaminated by spatial artifacts. The model was run twice, with two different section

counts, resulting in the same output.

46

5. Results

In the previous chapter we introduced the auditory model design and algorithm. In this

chapter we present the different results that were obtained using that model. The first section

of this chapter presents the model solution for a linear-uniform human ear model. The second

part of the chapter will describe a linear-nonuniform model and its results. In the last part the

simulated results for a nonlinear-nonuniform model will be demonstrated.

The Model Output: Basilar Membrane velocity along the cochlear

partition

Several different types of input signals were used as stimuli in this research. Clicks and tone

bursts were used primarily in order to study the models OAE response to transients, while

pure sine waves were used to calculate basilar membrane energy curves and audiograms.

Figure �5-1 depicts the basilar membrane velocity response to a click and a sine wave.

The model simulation results are described as BM velocity in a time-place matrix. In this

representation it is possible to visualize the stimulus energy dissipating along the cochlear

partition, from base to apex. The two-dimensional matrix as shown in Figure �5-1 represents

the basilar membrane partition velocities’ magnitude ( bm�� ) for every section along the BM

partition versus time as a response to a click (a) and a sine wave (f=250Hz) (b). The matrix

rows represent the longitudinal axis along the cochlear partition and the matrix columns

represent time. The top section (near the base of the cochlea) has the highest characteristic

frequency (CF) and the bottom section (near the apex) has the lowest CF.

(a) (b)

Figure �5-1 - Model response to a click and a sine wave. (a) BM velocity response to a click. (b) BM

velocity response to a 250 Hz sine wave.

47

Figure �5-2 represents the BM response when band pass stimuli are fed into the model. The

BM response is place-bound in correlation to the frequency band of the stimulus. Only a

partial BM movement is initiated by band passed stimuli. Low frequency specific sections do

not move in response to stimuli which was high passed.

(a) (b)

Figure �5-2 - Filtered stimulus response. (a) The response to a band pass (500Hz - 5000Hz) filtered click

stimulus. (b) The response to a 2 KHz tone burst stimulus (2

0( )sin(2 )* t tKhz e� � ).

Simulated Audiograms

One of the formal measurements of human hearing is the pure tone audiogram. In pure tone

audiometry, hearing is measured at frequencies varying from low pitches (250 Hz) to high

pitches (8000 Hz). Calibrated tones are provided to a person via earphones, allowing that

person to increase the level until the tone can just be heard. Audiograms compare hearing to

the “normal” threshold of hearing, which is an average threshold calculated from many

individuals with intact hearing. The threshold of hearing varies with frequency and the

audiogram is normalized so that a straight horizontal line (at 0 dB) represents a normal

hearing individual.

The hearing level is quantified relative to “normal” hearing in decibels (dB), i.e. normal being

0 dB and higher numbers indicating decreased thresholds. An adult with a hearing level of

less than 25 dB is said to have normal hearing, while in children the threshold is a bit stricter

with a hearing level of 15 dB defining normal hearing. The dB score is not hearing percent

loss, but a 100 dB hearing loss is nearly equivalent to complete deafness for that particular

frequency. It is possible to have scores less than 0, which indicate better than average hearing.

48

We can use the model simulation to compute "loudness", which will be used to calculate

estimated audiograms (Figure �5-3) for a particular simulated ear. Loudness corresponds to the

subjective impression of the magnitude of a sound. For the purpose of estimating the model

outcomes in terms of this perceptual concept, we will use the following definition: Loudness

in terms of the model is the energy acquired by the whole cochlea due to the basilar

membrane velocity (Furst et al. 1992):

2

0 0

1( ( , ))

T l

d bmL x t dxdtT

+� * * � (5.1)

where T is the stimulus duration and l is the cochlear length.

Since the audiogram is a relative measurement, Cohen and Furst (2004) defined an ideal

cochlea as a cochlea with OHCs that are optimally activated. One parameter was used to

describe the OHC activity, the OHC gain factor (�). � was defined as 0 1( ( where �=0

represents a cochlea with no active OHC and �=1 represents a non-realistic cochlea whose

BM motion reaches infinity. �=0.5 was chosen to be the optimal cochlea. This choice reflects

the best match to physiological tuning curves and gain (Cohen, 2004).

The model used for the audiogram calculations is a linear model with low level stimuli. It is

accepted that inner ear responses are linear for low level stimuli magnitudes and hearing

threshold measurements are usually done in magnitude ranges in which the cochlea is

regarded as linear.

Sine waves were used as stimuli for the calculation of each point in the audiogram. A full

simulation of a single sine wave was done and the initial 25ms of the output matrix discarded

(during this time the output is manifested with stimulus artifacts). The “steady state” part of

the response was summed for the loudness calculation. Our “ideal” cochlea (�=0.5) was used

to define a reference threshold to which all other loudness calculations were compared:

( ) ( 0.5)d dThres L L � � � (5.2)

49

Figure �5-3 - Audiograms simulated by the linear model. Red represents a normal hearing cochlea, green

represents a cochlea with partially functioning OHCs and blue represents a dysfunctional OHC cochlea.

For input frequencies below 1000 Hz the difference in the estimated threshold for different

values of � is less than 30 dB. However, there is a significant difference in the estimated

threshold for higher frequencies. For � < 0.2, each of the estimated audiograms has a

maximum threshold at a frequency between 4 and 6 kHz. These types of audiograms resemble

typical phonal trauma audiograms.

Simulated TEOAEs

OAE pressure is calculated using equation 4.2, where eP represents the emission pressure in

the ear canal. An example of a simulated TEOAE is demonstrated in Figure �5-4. Equation 4.2

links the output pressure to the stimulus pressure and the tympanic membrane pressure

(influenced by the pressure emitted by the cochlea). This linkage causes the stimulus pressure

to always be mirrored in the output at the time of stimulus onset. After the stimulus ends

(when clicks and tone bursts are used) we see a short period of stimulus artifacts. Thus, the

first 2.5 ms after stimulus are blocked and thrown out from the rest of the computations.

These initial responses have high amplitude low-frequency components and interfere with the

OAE signal which is several magnitudes weaker.

The emission pressure calculated by the linear-uniform model is depicted in Figure �5-4. The

TEOAE in this example has no similarity to real TEOAEs recorded in the clinic. The model

generated pressure contains only the stimulus artifact. From now on when we refer to

TEOAEs we refer to them in post-stimulus time, where the first 2.5 ms of the recorded time

(stimulus artifact) have been thrown away.

50

Figure �5-4 – Example of a simulated TEOAE in a uniform cochlea.

Simulated click response

Figure �5-5 demonstrates a simulation of BM velocity as a response to a click. In the upper

part of Figure �5-5 the corresponding OAE pressure vs. time is plotted. As can be seen, no

emissions are generated in this particular simulation. On the right, basilar membrane energy

(BME) curves are plotted (also called excitation patterns). The BME curves are obtained by

computing the sum of the squared BM velocity over time. This gives us an energy-per-section

plot for sine wave stimuli. As can be seen from the BME curves the model amplifies the

higher frequencies (the top sections) more than the lower frequencies (bottom sections). This

accounts for the difference in the colors of the energy lines in the two-dimensional matrix

representation.

Since clicks are wideband stimuli, we see that almost all the different sections of the basilar

membrane start to move in response to the click stimulus. We see the different click

frequency components traveling to their different resonant places along the cochlear partition,

low frequency energy taking the longest time to reach its specific CF (located near the apex).

As can be seen from Figure �5-5, the click (or any other stimulus with low frequency

components) response takes a long time to diminish. This is in agreement with experimental

data which states that it normally takes click responses 20-25 ms to decay (Konrad and Keefe,

2003).

51

Figure �5-5 – Model response to a click simulated in a “smooth” cochlea. In the upper left the calculated

outer ear pressure shows no OAE. On the right we see the Basilar Membrane Energy curves as a response

to frequencies of: 250Hz, 500Hz, 1000Hz, 2000Hz, 3000Hz, 4000Hz, 6000Hz and 8000Hz. The time-place

representation demonstrates what happens inside the cochlea as time progresses.

Nonuniformity

The generation of OAEs requires that part of the energy returns from within the cochlea, back

to the ear canal. As said before, the energy enters the cochlea near the stapes and dissipates

along the BM partition. When all model parameters are uniform/smooth, the stimulus energy

is spent on damping forces of the BM partition. In such a case no OAEs are generated.

Previous models (Furst and Lapid 1988, Zweig and Shera 1995, Talmadge et. al. 1998) have

shown that nonuniformity (random spatial variations) must be assumed in order for the model

to generate some kind of OAE.

In our work we verified the assumption that when small impedance mismatches

(nonuniformities or inhomogeneities or “roughness”) are spread throughout the BM partition

parameters, small energy scatterings occur. These energy scatterings form backward-traveling

52

waves that propagate toward the base of the cochlea, thus, causing OAEs to be generated in

our simulations.

Small impedance mismatches were tested in the following parameters: R (BM partition

damping), S (BM partition stiffness) and � (OHC gain). Inserting the randomness into the R or

S parameters of the BM partition caused the creation of OAEs (Figure �5-6).

Figure �5-6a represents the response to a click of a cochlea which has “roughness” in the BM

partition damping. The small impedance mismatches along the path of the traveling stimulus

energy cause reflections that travel back to the base of the cochlea. If we compare Figure �5-6a

with Figure �5-5 we can clearly see a “ringing” of the maximal velocity area of the cochlea.

Energy reflecting off that part of the cochlea returns to the outer ear canal and is recorded as

OAEs. The exact same phenomena is seen when the “roughness” is inserted into the BM

partition stiffness (Figure �5-6b).

(a) (b)

Figure �5-6 – Responses with nonuniform R or S. (a) The cochlear response to “roughness” in the damping

parameter of the BM partition. (b) The same test with “roughness” in the BM stiffness parameter.

�(x) “roughness”

We decided to incorporate the "roughness" into the OHC gain �(x) parameter in our

simulations. It is well known that OHC damage causes hearing loss and that impaired ears

lack intact OAEs. Thus, it seams that the most "educated guess" to where the randomness

should be is in �(x), controlling OHC functionality. OHC gain was interpreted by Cohen and

Furst (2004) as the relative density of the functioning OHCs along the cochlear partition. It is

53

reasonable to assume that this density is not fixed along the cochlea, but varies randomly. We

assume that �(x) is a Gaussian random variable with a mean of 0.5 (for healthy ears) and a

standard deviation of �.

From this point on the model equations include the nonuniformity described, in their OHC

gain �(x) parameter.

(a) (b)

Figure �5-7 – Otoacoustic emission (OAE) simulation. (a) No OAEs are generated in a cochlea with a

constant �(x)=0.5. (b) The generation of OAEs by a cochlea with E(�)=0.5 and �=1e-6.

In Figure �5-7 OAE generation is demonstrated in the cochlear model. A model with uniform

parameters is depicted generating no OAEs (a) while a model with "roughness" inserted into

the �(x) parameter (b) demonstrates the generation of OAE that travel backward toward the

stapes. In the upper part of each figure we can see the energy that dissipated (via the middle

ear) into the ear canal and was recorded as OAEs.

By closely examining Figure �5-7a it is noticeable that the energy reflections take part around

the maximum velocity areas. This is in conjunction with the notion that the backward-

traveling wave is dominated by wave reflections within the region about the peak of the

traveling energy wave. Around the peak the wave amplitude is much larger than it is

elsewhere, thus leading to larger backward-traveling wave contributions.

54

Simulated Otoacoustic emissions

Two examples of simulated TEOAEs by a nonuniform model are demonstrated in Figure �5-8.

(a) (b)

Figure �5-8 - Examples of simulated TEOAE outputs.

In black we see the stimulus artifact (that is mirrored in the TEOAE) generated by a linear

model. The blue line represents the simulated TEOAE response created by the nonuniformity.

If we compare the figure to the uniform TEOAE (Figure �5-4) we notice that the uniform

models' contribution is the stimulus artifact and the nonuniform contribution is the TEOAE

drawn in blue. Thus, by including a nonuniformity or "roughness" into the cochlear

parameters we are able to create energy reflections that are seen in the ear canal as TEOAEs.

Figure �5-9 demonstrates the correlation between increasing � and the energy amounting

inside the cochlea. All simulations were run with E(�)=0.5. The simulations were calculated

in the time-domain on a linear cochlear model.

Figure �5-9 – Energy increase due to larger “roughness”.

55

We can see an uncontrolled energy increase as the magnitude of � rises above a “roughness”

threshold ( 610� ). This phenomenon will be explained in detail later in this chapter. We should

remember that these graphs were simulated in a linear model and that the energy explosion is

not realistic.

Figure �5-10 depicts the correlation between increasing � and the TEOAE energy emitted by

the cochlea. The figure demonstrates energy histograms calculated by 100 simulations with

different random selections of �(x) with constant � . Figure �5-10a was run with a � of 610� ,

Figure �5-10b with 710� �� , Figure �5-10c with 810� �� and Figure �5-10d with 1010� �� .

The mean value of TEOAE energy for Figure �5-10c and Figure �5-10d is around 52.15 dB

which represents the noise floor for our simulations. From the mean values of Figure �5-10a

and Figure �5-10b we can calculate that the TEOAE average energy changes by around 18dB

for each magnitude of the OHC gain factor ( � ) (this fact was observed in many different �

histograms with different E(�) values).

(a)

(b)

(c)

(d)

Figure �5-10 – Emission energy histograms. (a) �=1e-6 (b) �=1e-7 (c) �=1e-8 (d) �=1e-10

all simulations run with E(�)=0.5

56

Simulated TEOAE spectrum

The OAEs generated by the "rough" model have unique frequency properties. The average

frequency spectrum of 20 TEOAE responses is demonstrated in Figure �5-11. In contrast to

measurements done in the human ear, there are no “low frequencies” below 1500 Hz in the

spectrum. All of our simulated TEOAE frequencies are between 2 and 6 KHz. If we look at a

particular click response simulation we see that the OAE spectrum varies most in the

frequencies of 2 – 6 KHz from the average spectrum. These frequencies match the sections

that are in the area of maximum velocity (in the click response). The low frequencies in the

spectrum (up to 1 KHz) belong to stimulus artifact components that are present more than 3

ms after stimulus onset.

Figure �5-11 – Simulated TEOAE spectrum. The spectrum obtained from 20 runs of the model in the time

domain, where E(�)=0.5 and �=1e-6.

Nonuniform cochlea Audiograms

We have seen that the model generated OAEs change with each random selections of �(x).

Surprisingly, the effect of different random selections on the calculated audiogram is

negligible. The colored lines in Figure �5-12 were obtained for random selections of �(x)

where 610� �� . The colored lines represent the simulation of a healthy cochlea, i.e. E(�)=0.5,

while the black line depicts a cochlea with partially functioning OHCs, i.e. E(�)=0.2. There

are 30 different random selections of healthy cochleae and 30 different random selections of

partially functioning cochleae plotted in the figure.

57

Figure �5-12 demonstrates that the random selection of �(x) has a “negligible” effect on the

audiogram drawn (the change in threshold level is less than +/-5dB) when E(�) is fixed.

Changing E(�) (from 0.5 to 0.2) causes a substantial reduction in threshold (of about 40dB).

Figure �5-12 – Audiograms of nonuniform �(x) selections. The colored lines represent random selections of

�, in a healthy cochlea, where E(�)=0.5 and �=1e-6. The black line represents the simulated audiogram of

a cochlea with partially functioning OHCs.

The corresponding time-place representations are shown in Figure �5-13. Figure �5-13a

demonstrates a typical result of E(�)=0.5 with 610� �� and Figure �5-13b represents a typical

results of E(�)=0.2 also with 610� �� . From the simulations it is clear that a normal hearing

cochlea with "roughness" will create OAEs while a partially functioning OHC cochlea with

the same "roughness" will not.

(a)

(b)

Figure �5-13 – Response to a click by normal and partially impaired ears. (a) Functioning OHCs, i.e.

E(�)=0.5

(b) Partially functioning OHCs, i.e. E(�)=0.2. Both simulations have the same � (=1e-6).

58

It is well known that OHC damage causes hearing loss and that impaired ears lack intact

OAEs. Our simulations demonstrate the connection between the lowering of � (the

functionality of the OHCs) and the lack of generated OAEs.

"Linear" versus "Nonlinear" response processing

In typical experiments the TEOAE recorded are analyzed by two different techniques:

1) The “linear average” technique is the average of a train of several identical stimuli

responses (these are also called Linear clicks). Four click responses are summed together, thus

increasing the SNR of the signal. The implementation of this technique on our model results

is demonstrated in the left column of Figure �5-14.

2) The “nonlinear average” technique is performed on the responses of a train of four stimuli

in which three identical stimuli of a given polarity are followed by one of the opposite

polarity and triple the amplitude. All four auditory responses are summed together, thus

canceling-out the "linear" component of the signal. This way the summed output is less

sensitive to the stimulus artifact (composed mainly of linear components) in the response,

such as the ringing phenomenon in the first few milliseconds of recording. This “nonlinear

average” is referred to as the Derived Nonlinear Technique (DNT). The implementation of

this technique is demonstrated in the right column of Figure �5-14.

In order to verify our assumptions, we implemented the "nonlinear average" technique

combined with a completely linear cochlear model. The linear cochlear model output is four

identical click responses. Their sum completely eliminates the signals, outputting a zero

matrix from that simulation (not shown in the figures).

Figure �5-14 demonstrates response processing by both methods. The top plots (a, e) reveal the

stimulus train, plots (b) and (f) the time-place representation of the BM velocity, plots (c) and

(g) demonstrate the OAE output and plots (d) and (h) the OAE spectrum. The left column

represents the linear technique and the right column the nonlinear technique. Although both

OAE spectrums seem alike, they differ in their low frequency content. The low frequencies

are the result of processing the uncensored click response (containing the linear components).

The OAE "linear" response has its peak magnitude near the stimulus peak and then the

response decays while the OAE "nonlinear" response gradually increases in magnitude to a

59

delayed peak (due only to the nonlinear components in the response). Both models used were

nonlinear models, as explained later in this chapter.

(a) (e)

(b) (f)

(c) (g)

Figure �5-14 - Linear vs. Nonlinear click responses. The left column represents responses to a linear click

train and the right column a nonlinear click train. (a) and (e) are the click train stimuli. (b) and (f) are the

response inside the cochlea. (c) and (g) are the time plot of the calculated OAE. (d) and (h) are the OAE

spectrum.

60

Tone burst responses

Tone burst (2

0( )sin( )* t te � � ) trains were simulated in the nonlinear model. Their response was

processed with the “Derived Nonlinear Technique”. Three identical positive polarity tone

bursts were followed by a negative polarity tone burst which had triple their magnitude. All

four responses were summed together.

In Figure �5-15 the resulting TEOAE spectral response to tone bursts (at frequencies: 1KHz,

2KHz, 2.5KHz, 3KHz, 3.5KHz, 4KHz, 4.5KHz, 5KHz and 6KHz) is demonstrated.

(a)

(b)

Figure �5-15 – Linear vs. Nonlinear tone burst response spectrum. (a) The OAE spectrum of a cochlea with

normal OHCs, i.e. E(�)=0.5. (b) The OAE spectrum from a cochlea with dysfunctional OHCs ,i.e.

E(�)=0.2. The tone burst frequencies are: 1 KHz, 2 KHz, 2.5 KHz, 3 KHz, 3.5 KHz, 4 KHz, 4.5 KHz, 5

KHz and 6 KHz.

Figure �5-15a demonstrates a cochlea with fully active OHCs (E(�)=0.5) while Figure �5-15b is

taken from simulations of a cochlea with partially activated OHCs (E(�)=0.2). In Figure �5-15a

we can see the stimulus spectrum mirrored in the normal ear response. This is in conjunction

with research stating that TEOAEs detected in normal ears mirror the spectral properties of

the stimulus (Glattke & Robinette, 2002 - Figure �5-16b). Our simulations show that impaired

ears do not mirror the stimulus spectrum in their TEOAEs and that their TEOAEs are filled

only by noise.

By comparing our simulated TEOAE spectrum to “nonlinear” TEOAE responses taken from

the literature (Figure �5-16a) we can see several similarities. The picture is taken from the ILO

software and is a TEOAE tested in a healthy normal ear. The bumps in the spectrum correlate

with the spectrum of TEOAEs from differential tone bursts. The low frequency noise (seen in

red) is in correlation with the 1-2 KHz “noise” in our simulations.

61

(a)

(b)

Figure �5-16 – “Nonlinear” TEOAEs from the literature. (a) The ILO spectral screen for a normal hearing

ear. (b) The stimulus spectral properties are demonstrated mirrored in the TEOAE.

Localized � "roughness"

We have already described the simulated click responses concerning a cochlea with “smooth”

�(x) (OHC gain). We have also gone over the influence of � magnitude on the response when

�(x) has a random distribution. Here localized “roughness” is injected into an otherwise

smooth �(x). This alters the models behavior and the results are quite different, instead of a

short multi-frequency response we see a narrow band long-term effect. The stimulus used was

a click, the E(�) was set at 0.5 and the localized “roughness” used was 210� �� .

By localizing the “roughness” the OAEs generated have a dominant semi-sine wave

frequency. The location of the injected “roughness” controlled the frequency emitted by the

cochlea.

The correlation between the OAE frequency emitted and the section into which the

“roughness” was injected is demonstrated in Figure �5-17. Four different “roughness”

locations are depicted generating four different OAE frequencies. Each figure shows the OAE

generated (top left), the OAE spectrum (top right) and the time-place representation from

within the cochlea (bottom plot). In each of the simulations in Figure �5-17 the OAE emission

is clearly seen resonating as a “standing-wave” inside the cochlea. It seems that the localized

“roughness” transforms the cochlea into a tuned resonant cavity. The interaction between the

reflected energy (by the “roughness” impedance mismatch) and the OHC gain generates a

“long lasting” OAE frequency.

62

Figure �5-17a has roughness along sections 150-200 and generated OAEs with a mean

frequency of 2250 Hz, Figure �5-17b has roughness at sections 200-250 and a frequency of

1250Hz, Figure �5-17c has roughness at sections 250-300 and a frequency of 950Hz and

Figure �5-17d has roughness at sections 300-350 and a low frequency that resides under the

noise floor of the simulation environment and can only be seen in the time-place

representation.

(a)

(b)

(c)

(d)

Figure �5-17 – Localized � “roughness”. The figures demonstrate the simulation output when localized

“roughness” was added along different sections of the BM partition. The top left plot in each figure shows

the OAE created while the top right plot shows the OAE spectrum. The bottom time-place representation

clearly depicts the resonating energy within the cochlear.

Although it would seem that the "long lasting" resonance should change the simulated

audiogram created by their unique �(x) distribution, this is not so. Because audiograms are

measured with sine waves over a long period of time, the resonating wave dies down and does

not influence the simulated audiogram by more that 3 [dB] (results not shown).

63

� mean

In order to study the influence of E(�) on the simulated TEOAE spectral properties we ran

multiple model simulations with different mean values. The results are shown in Figure �5-18.

In Figure �5-18a we see the simulated spectral properties versus E(�) and in Figure �5-18b the

corresponding audiograms are plotted.

(a) (b)

Figure �5-18 – Nonlinear spectral response vs. E(�) and their corresponding audiograms. (a) The figure

demonstrates the spectral click response vs. the decrease of E(�). (b) The audiograms generated by the

same E(�) show no significant deviation from constant �(x) audiograms.

Each spectral line was obtained from 20 different runs of the model, while �(x) was randomly

selected having a constant E(�) and 610� �� . The average spectrum is plotted for each

constant E(�) revealing a frequency “migration” with the decrease of E(�). The OAE

magnitude is lowered by more than 100 dB while the amount of different frequencies in each

OAE is reduced drastically. The maximal frequency component changes from 8 KHz in

E(�)=0.5 to around 1.5K Hz at E(�)=0. Thus, it is clear that low values of E(�) yield OAEs

without high frequency components. This estimate is in conjunction with the traveling wave

peak moving toward the stapes with the decrease of E(�).

In clinical tests OAEs are separated into bands of frequencies and each band is checked for

the presence of the OAE energy. Our results suggest that with the onset of hearing loss

(degradation of the OHC gain factor) the high frequency bands are the first OAE components

that are lost.

As already seen, in the beginning of this chapter, changes in E(�) cause a considerable change

to the simulated audiogram (Figure �5-18b). By lowering E(�) the threshold level decreases

64

substantially reaching a minimum at frequencies around 5 KHz. A healthy ear (E(�)=0.5) sets

the baseline for all simulated audiograms, generating a straight line across the plot. When

dysfunctional OHC ears (E(�)=0) are estimated the damage amounts to around 60dB in

frequencies associated with human speech (4 - 6 KHz).

Nonuniformity and Energy Explosion

In Figure �5-19 the total energy inside the cochlea is plotted versus different distributions of

�(x). When � is increased above a threshold an energy explosion occurs inside the cochlea.

Figure �5-19 – Total cochlear energy vs. different � s.

The energy explosion is demonstrated as seen inside the cochlea in Figure �5-20. Figure �5-20

demonstrates a simulation run in the linear time domain model where E(�)=0.5 and 410� �� .

Figure �5-20 – Energy explosion that occurs when a large � is used.

65

The energy explosion represents the speed of the BM partition increasing indefinitely.

Without anything to restrain the energy (in the linear model) the BM velocity increases to

infinity. This phenomenon is not realistic because in human ears a one time high intensity

stimulus does not trigger indefinite ringing.

The energy explosion can be explained if backward-traveling energy, reaching the base of the

cochlea, is partially reflected back into the cochlea by the middle ear boundary condition. The

reflected energy becomes a forward-traveling wave and generates new backward-traveling

waves that are also reflected back into the cochlea by the middle ear. At frequencies for which

the total phase change due to round-trip wave travel is an integral number of cycles, standing

waves can build up within the cochlea, which is then acting, in effect, as a tuned resonant

cavity.

Up to this point the model was treated as a Linear Time-Invariant (LTI) system. In such a

system for an input that includes sinusoids the output can be described as a sum of the same

sinusoids with phase and amplitude changes. Up to the inclusion of nonuniformities in the

cochlear equations the system acted as a LTI system, all equations were linear and the system

was time-invariant. After we include a large enough nonuniformity into the model, the model

stops to act as a LTI system. It is not time-invariant anymore. This fact is obvious when we

pay attention to the system stimulus as simulation time progresses. At the beginning of the

simulation only our stimulus excites the cochlear model. As time progresses energy returning

off impedance mismatches reaches the cochlear base and is reflected by the middle ear

boundary condition back into the cochlea. This energy is added to the cochlear stimulus,

which means that the stimulus to the cochlear model is the outside stimulus contribution plus

the contribution of the twice reflected energy from within the cochlea itself. This fact breaks

the time-invariant property and prevents us from treating the complete system as an LTI

system.

Introducing Nonlinearity

Because no active damping is present in the linear model, the energy buildup leads to an

energy explosion inside the cochlea. This energy buildup starts locally and draws all the BM

sections into the process. In order to constrict the energy explosion we added nonlinearity into

the model.

66

There are many known nonlinear cochlear phenomena. For example combination tones and

emitted distortion products (ADP). It is clear that every realistic cochlear model must include

nonlinear terms. In the study by Elbaum and Furst (2005) different nonlinear functions were

introduced into the BM-OHC cochlear model. The nonlinearities were inserted in three

different locations: the BM damping, the BM stiffness and the OHC electromotility

components. The research purpose was to analyze different functions integrated in a variety of

model components and to check the generation of Combination Tones (CTs) and their

characteristics. Their results show the insensitiveness of the model to the location of the

nonlinearity. They concluded that it is reasonable to assume that there are a number of sources

for nonlinearity inside the cochlea, one nonlinear source causing linear CT amplitude growth

and an additional nonlinear source causing saturation effects.

Thus we included the nonlinearity in the BM damping factor by assuming:

20 1

( , )( , ) ( )*(1 *[ ] )bm x t

r x t r xdt

��

�� � (5.3)

Eq. 5.3 is substituted in Eq. 4.10.

By implementing nonlinearity (in the damping factor) we constrict the energy buildup which

still occurs, but now reaches an upper limit (Figure �5-21). With the nonlinearity in place the

OAE resembles stabilized amplitude standing waves. From this point on the model described

includes nonlinearity in the BM damping factor plus the nonuniformity in the OHC gain �(x)

parameter.

Figure �5-21 - Large � simulation combined with constricting nonlinear damping. The nonlinear damping

in the BM partition limits the energy generated inside the cochlea.

67

Once the standing wave is initiated inside the cochlea it never decreases. In simulations

conducted the process continued without losing amplitude for over 1000ms. The process

reaches a steady state and becomes the models “noise floor”. It seems that the nonuniformity

creates noise inside the cochlea that might cause a significant hearing loss.

By applying different magnitudes of � to the nonlinear model we can generate a � versus total

cochlear energy graph (Figure �5-22). The model was run with multiple �'s (having E(�)=0.5)

and different damping factors (�1 in equation 5.3). In Figure �5-22 we can see that with the

nonlinear equation in place the energy buildup reaches a saturation level. Our conclusion is

that above a threshold of 610� �� the energy explosion is initiated and the total cochlear

energy stabilizes, due to the nonlinearity in the damping factor. In Figure �5-22 two different

damping factors (�1 in Eq. 5.3) generating two different energy stabilization levels are

demonstrated. We can clearly see that the damping factor (�1) from equation 5.3 controls the

energy stabilization magnitude (the “noise floor” amplitude).

(a) (b)

Figure �5-22 - � vs. Total cochlear energy. The � magnitude was varied while E(�)=0.5. The total energy

that developed inside the cochlea was calculated in response to a click stimulus. (a) damping factor �1=10

(b) damping factor �1=1e-5.

Stimulus magnitude influence on nonlinear model responses

When working with a nonlinear cochlear model (for example where E(�)=0.5 and 610� �� )

we can divide the stimulus magnitude into three distinct levels. The magnitudes are separated

according to the kind of cochlear response they generate.

68

For small stimulation magnitudes the models' response is similar to a linear models response.

The OAE generated are well defined and the reflected energy is easily separated from the

background noise in the time-place representation (Figure �5-23a).

For larger stimulus energies, where the model is not linear anymore, the stimulation

magnitudes divide into two phenomena. Medium stimulation magnitudes generate an

emission pressure that is under the cochlear “noise floor”. The emission pressure is dominated

by the standing wave energy being emitted by the cochlea and the TEOAE we are interested

in is not seen. Inside the cochlea a large energy mass is being generated blocking the view on

the TEOAE of interest (Figure �5-23b).

For high stimulus magnitudes the emission signal we are interested in is well above the “noise

floor”. The simulated ear canal pressure signal is dominated by the TEOAE high amplitude

response and we can easily see the BM partition responding to the strong energy stimulus.

The high energy input activates the BM partition so intensely that the velocity created by the

stimulus is well above the movement created by the standing waves inside the cochlea (Figure

�5-23c).

(a)

(b)

(c)

Figure �5-23 – Different stimulus magnitudes in the nonlinear model. (a) A low level sine wave stimulus

response creating a “linear” response. (b) A medium energy sine wave response, under the “noise floor”.

(c) A high energy sine wave response, dominating the movement of the BM partition.

In the medium stimulus magnitude range the cochlear response can be partially retrieved by

averaging a large amount of cochlear responses. The “noise” created by the standing wave

cancels itself out and we are left with an almost “normal” cochlear response (Figure �5-24).

This response is similar to the low energy linear response.

69

(a) (b)

Figure �5-24 – Signal in noise. (a) A plot of 100 simulations with different random selections of �(x) (all

with the same �). (b) The average of the same 100 OAEs demonstrates the cancellation of the random

noise and reveals the “low-level-linear-response” we would get from a cochlea with a “normal” �.

Figure �5-24 demonstrates the averaging of 100 cochlear responses to a medium sized input.

As we can see in Figure �5-24b, the averaged signal of 100 responses resembles the “linear”

low level response created by the model. More than 100 responses are needed in order to

achieve the fine details of the low level response and all its properties.

70

6. Discussion

The human ear is partially still a mystery, and there are several cochlear phenomena that are

not yet fully understood. This research is the first step towards the implementation of a model

capable of simulating different emissions generated by the human ear.

The model presented in this thesis is an extension of the one-dimensional model created by

Cohen and Furst (2004). Their model was enhanced in order to create OtoAcoustic Emissions.

A simple middle ear model and an outer ear canal were integrated into the cochlear model to

form an entire human ear system. The main assumption of the model is that the middle ear is

equivalent to a simple mechanical piston translating the tympanic membrane motion to oval

window displacement. The model was tested and its ability to generate TEOAEs was verified.

In this enhanced model the stimulus is injected through changes in ear canal pressure and the

output is recorded as OAE pressure (also in the ear canal).

The model algorithm solution was implemented in the time-domain. A frequency-domain

solution was also developed for steady-state stimuli and both solutions yielded exactly the

same results. However, when nonuniformity was introduced the time-domain solution

“exploded” while the energy in the frequency-domain solution was unchanged. The

nonuniformity reveals the fact that the model is not time-invariant. Thus, a frequency-domain

solution is not justified.

The time-place representation which is obtained by the time-domain solution reveals temporal

phenomena in the cochlear responses. The time-place cochlear representation changes with

each sample of the input signal; hence, it enables discrimination between short time events.

Thus, the time-domain model enables response prediction for both click and tone burst signals

(transient signals).

The model of Cohen and Furst stands out from the rest of the cochlear models because of it’s

incorporation of an OHC model into the BM model. This special feature allows us to

distinguish between normal functioning cochleae and dysfunctional cochleae. This is done by

changing the OHC gain factor and influencing the OHC functionality along the cochlea. The

influence of OHC functionality on TEOAE generation is clearly seen in the difference

between normal and impaired ears time-place representations.

71

Other models have shown that some kind of “roughness” must exist in the model in order to

generate TEOAEs (Talmadge et al. 1998, Zweig and Shera 1995, Shera and Guinan 2003).

These models did not incorporate an OHC model and so had to insert the roughness in an

artificial way into other model parameters, i.e. Talmadge et al. (1998) incorporated the

“roughness” into their place-frequency map.

The first research goal was to verify whether the cochlear model by Cohen & Furst could be

enhanced into a model capable of emulating TEOAEs. We succeeded in generating a model

capable of simulating TEOAEs. The TEOAEs were generated after nonuniformity

(“roughness”) was introduced into the OHC gain parameter. Without “roughness” the

emission pressure waves generated resemble stimulus artifacts, with no TEOAE

characteristics. Only the addition of minute impedance mismatches caused the generation of

TEOAEs with realistic properties. It seems that a totally “smooth” human ear is not capable of

creating TEOAEs, and some kind of “roughness” must exist somewhere inside the inner ear.

We have shown that the addition of “roughness” in several of the model parameters creates

the TEOAE phenomena. There is no conclusive evidence as to where the “roughness” is

located in the real human ear. The main conclusion of this research is that a “too smooth”

cochlea is unnatural and the cochlea must have some kind of nonuniformity in order for the

cochlea to be able to emit any kind of sound.

In the present model the OHC mode of activity is characterized by one parameter �(x), which

can vary along the cochlear partition and is regarded as the OHC functionality. This research

focuses on how TEOAE responses vary due to variations in �(x). We refer to a cochlea with

E(�)=0 as a dysfunctional cochlea (passive/dead cochlea - without active OHCs). A cochlea

with E(�)=0.5 is regarded as a normal cochlea (healthy with functioning OHCs). The

dysfunctional cochlea reproduces the behavior of the basilar membrane typical of postmortem

measurements and the healthy cochlea reproduces the high tuning basilar membrane motion

of an active cochlea.

When the OHC gain factor is normal, i.e. E(�)=0.5, cochlear displacement and velocity are

enhanced in the vicinity of the characteristic frequencies. The enhancement is mostly

significant for frequencies above 1 KHz (the cutoff frequency of the OHC membrane).

Frequencies between 3 and 6 KHz receive the most amplification by the OHC models

contribution to the traveling wave. Changes to the OHC gain factor cause the decrease of the

72

basilar membrane motion peek. The most significant energy component is reflected within the

region of maximum displacement (the traveling wave peek), thus the OHC gain factor has a

large influence on the generation of TEOAEs. A low OHC gain does not enable the

generation of OAEs with enough energy to reach the ear canal.

In contrast to incorporating an OHC model in the cochlear model, other researchers have to

“hand manufacture” the gain profile in their models. Nobili et al. (2003) had to manually

create a cochlear amplifier (CA) gain profile for their model (Figure �6-1) by comparing

psychoacoustic data from subjects with normal hearing to data from patients with acquired

hearing loss of cochlear origin. In our model the gain profile is inherent in the design. The

OHC model is the one responsible for the heightened area in the 1-5 KHz frequencies.

Figure �6-1 - The profile of the Cochlear Amplifier by Nobili et al.

The fact that the cochlear model has OHC functionality incorporated into it makes the

insertion of roughness/inhomogeneity much more intuitive. Because TEOAEs are thought to

originate as a side-effect of the cochlear amplifier (Kemp 2002) which has become associated

with the OHC motility (Liberman et al. 2002, Liberman et al. 2004) we decided to locate the

“roughness” in our model in the OHC gain factor. Although there is no evidence in the

auditory physics literature that there is OHC gain nonuniformity in the human ear, it is only

natural that minute differences are present between neighboring OHCs. It is not conceivable

that the biological tissue is totally uniform throughout the cochlea and small perturbations will

be enough to cause energy reflections. Even a small bending of the OHCs in relation to one-

another, or inhomogeneities in the OHC forces due to random, cell-to-cell, variations in the

number of somatic motor proteins will suffice to create the inhomogeneities incorporated into

73

our model (changes of less than 0.0001% were incorporated into the OHC gain factor). These

small changes between adjacent sections of the BM partition are sufficient to create reflection

source OAEs.

The assumption of random distributed OHC gain along the cochlea seems realistic since in

humans the loss of sensitivity in 4 KHz was found independently of the type of noise they

were exposed to (Saunders et al 1985; Moore 1998). Cohen and Furst (2004) showed that a

random �(x) along the cochlea will generate simulated audiograms with a maximum threshold

at 4 KHz.

Hearing-impaired people who suffer from OHC loss exhibit a significant degradation in the

performance of all known nonlinear phenomena such as two-tone suppression, combination

tones, and cochlear otoacoustic emissions (Moore 1998). This research provides model

predictions of the influence of nonuniformity on the TEOAE generation in the human ear. We

have shown that decreased OHC functionality leads to degradation in cochlear reflection

source otoacoustic emissions.

Simulated audiograms were generated, on the basis of the basilar membrane velocity, for

‘normal’ ears and ears with OHC loss. Our results show that the mean OHC gain magnitude

has a substantial influence on the simulated audiogram. Even a relatively small decrease in

E(�) will create a hearing threshold change in the modeled ear. On the other hand, the

modeled "roughness" has very little influence on the audiograms obtained (in the range

tested). This enhances the notion that the goal of the human ear is to transform sound waves

to neural excitation patterns and not the creation of TEOAEs. TEOAEs are only a side-effect

(an artifact) of the normal workings of the human ear.

In contrast to audiograms, TEOAEs are very small signals that are influenced considerably by

both the “roughness” magnitude and the mean OHC gain magnitude. The mean OHC gain

controls the formation of traveling wave peeks and the “roughness” magnitude controls the

proportion of energy reflected along the cochlea. Without both, TEOAEs will not be created.

By comparing the generated TEOAE from our model with recorded TEOAEs from the

literature we see a resemblance in the time domain characteristics but the clinical recorded

74

TEOAE spectrum is different from our simulated results. The literature details that the

recorded TEOAE spectrum has frequency components between 0 and 7 KHz, with a

maximum at about 1.7 KHz. In comparison, our generated TEOAEs lack low frequency

components. We were not able to generate frequencies below 1.5 KHz in our simulated

OAEs.

Our results clearly show that the OAE frequency is dependent on the “roughness” place. In

the some time-place representations we can clearly see a particular BM section resonating and

“creating” the OAEs. This happens when we inject localized “roughness” into the BM

parameters.

The coherent reflection model (Zweig and Shera, 1995) predicts that the TEOAE evoked by a

click comprises a sum of waves scattered by perturbations located throughout the peak of the

traveling wave. In our model we can see that the TEOAE arises from a distributed region,

roughly equal in extent to the width of the traveling wave envelope.

The two techniques used in the clinic to improve the OAE recorded SNR are Averaging and

the Derived Nonlinear Technique (DNT). Both methods were simulated and the results

resembled clinical responses. When averaging was applied, the tail of the acoustical stimulus

waveform (up to 6ms after stimulus onset) was seen interfering with the early parts of the

response. When we implemented the DNT method, the stimulus linear artifact was completely

canceled out (in the linear model). When run on a model with nonlinearity, the low

frequencies (mainly stimulus artifact) in the calculated DNT were significantly smaller and

we were able to get better results for the high frequency components. In real practice the

stimulus artifact is not completely canceled, but is considerably attenuated (about 40 dB)

disclosing the early part of the OAE response which otherwise would be mixed with parts of

the stimulus artifact.

When the nonuniformity magnitude reached a threshold, the system behavior became

unstable. Large “roughness” in conjunction with medium sized stimuli created very large

reflected waves. The reflected energy created an increasing standing wave inside the cochlea

that led to an energy explosion. In order to prevent the energy explosion from taking place,

75

nonlinearity was added to the damping factor of the BM partition. The nonlinearity

constricted the amplitude of the standing wave and thus stopped the energy “explosion”.

Other researchers had to incorporate nonlinearities into their models in order to solve the

energy explosion problem as well. Talmadge et al. (1998) incorporated “stabilizing

nonlinearity” into their model in order to overcome this nonrealistic phenomenon.

We conclude that adding nonuniformity to the model (by itself) is not enough and a

constricting nonlinearity must be added as well. Without the nonlinearity, loud signals would

bring the system to an unreal energy explosion. We must remember that biological systems

are nonlinear in nature and our model was built on linear equations for simplicity. It is most

likely that several nonlinearities contribute to the real human ears’ energy constriction. In our

tests to constrict the energy explosion we tried just one type of nonlinearity and we placed it

in just one place inside our simplistic human ear model.

With the nonuniformity and nonlinearity in place, the energy that escapes basally and reaches

the middle ear boundary is partially reflected back into the cochlea. This energy forms a new

traveling wave that re-stimulates the OHC gain mechanism. Under conditions of high

amplification and endless recirculation of the traveling wave sustained oscillations inside the

cochlea are sustained. These oscillations create “spontaneous” OAEs in the ear canal. Unlike

clinical recorded SOAEs that have one or more pure tones, our simulated spontaneous OAEs

have a broad spectrum of frequencies and follow the traveling waves peek envelope

(controlled by the OHC contribution to the BM motion). These standing waves resemble the

noise floor of the system, once they are triggered (by a loud noise or by large “roughness”

along the BM partition) they never die down.

For these spontaneous emissions to occur, strong OHC amplification must coexist with at

least one distributed irregularity (“roughness”) in one of the BM partition parameters. The

energy reflected by the middle ear must also be sufficient to sustain a continuous oscillation

along a section of the BM, after re-amplification and re-emission.

It is not only spatial imperfections that can generate OAEs. If the forces exerted by OHCs on

the BM do not exactly follow the stimulus waveform (i.e. if the OHC electromotility is

76

“nonlinear”), they will add distortion signals to the forward traveling wave, which are one

cause of aural combination tones.

Research from the recent years has shown that there are several sources to DPOAEs and

TEOAEs, each contributing part of the energy (Shera 2004). The sources for OAE production

can be separated into two main categories: reflection and nonlinear. Reflection sources are

distributed inhomogeneities along the cochlea while the nonlinear OAE sources can be

distributed in several different places in the cochlea and the organ of Corti. In our study we

implemented nonlinearity in the BM partition damping factor. We have seen that our

implemented nonlinearity does not generate DPOAEs in the ear canal.

The work by Elbaum and Furst (2005) tested several types of nonlinearities and their impact

on the cochlear response. Their work concentrated on the nonlinearities responsible for the

creation of DPOAEs inside the cochlea. The nonlinearity implemented in our work did not

succeed in generating DPOAEs in the ear canal, although DPs were created locally along the

BM partition.

Impact on the field of OAE model research

1) Our study has shown that Nonuniformity alone can not explain the different phenomena

encountered in the cochlea. The minimum need is for Nonuniformity with constraining

Nonlinearity in order to achieve a more realistic outcome. Without the incorporation of

restricting nonlinearity into the model we can show that the model is too unstable and tends to

“explode” with unconstrained energy release.

2) During the course of this study several different cochlear models were investigated. By

comparing our own model to them we learned many new things about the data collected.

Several mistakes were found in the different models. Their conclusions were “corrected” into

a better understanding of the data they generate. By studying thoroughly the effects of

choosing ‘wrong parameters’ for the model, we have now a robust and easy to use model

which can form the base for future research.

3) Considering that our model is a simplified one-dimensional model of the human ear, the

created TEOAEs have a good resemblance to clinical recorded TEOAEs. By varying the

OHC gain factor we were able to predict several TEOAE phenomena of healthy and impaired

77

ears. The models’ simplicity and speed enables hypothesis testing in a relatively short time

opposed to the more elaborate 3D models (Bohnke and Arnold, 1999) which take a long time

to set up and run. In addition, this model serves as the start of development of a unified, more

realistic human ear model capable of simulating all OAE types.

Future research possibilities

This unique research has been focused on generating TEOAEs by adding a middle ear model

to the already existing cochlear model. In order to harvest better results the existing cochlear

model was rewritten from scratch. The model now is more robust and more user friendly. It

incorporates a fourth order Runge-Kutta technique and a robust variable step size algorithm.

Many different stimuli with many cochlear configurations were tried out during the course of

the study. During simulated data analysis we have observed very interesting results that could

benefit from further study.

These are several future research directions that should be considered:

1) Noam Elenbaum has been studying the effects of different nonlinearities on DPOAE

generation. His study and results could be combined into this model and its’ existing

nonuniformities and nonlinearities in order to produce ONE COMPLETE model with the

capability of generating TEOAEs, SOAEs and DPOAEs.

2) Our simulated TEOAEs do not exactly match all the properties of clinically measured

OAEs. Further study is needed in order to try and find the source of the missing low

frequencies in our simulated TEOAE spectrum. Parameter changes conducted during this

study have shown that low frequencies can not be produced by the model as is, and an

explanation is needed. Changes to the different model parameters could result in more

realistic OAEs. Changes to the middle ear parameters (or a change of the middle ear model

completely) and the M, R, S parameters should continue to be investigated.

78

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Appendix A – Auditory research and model history

Auditory research history

In the late 19th Century, von Helmholtz (1862-1885) conceived a sophisticated theory of

hearing that invoked the presence of highly selective resonators in the cochlea. Approximately

50 years later, von Bekesy (1928) discovered the cochlear traveling wave phenomenon and

fashioned a theory that disputed von Helmholtz' “resonance” theory. Although the traveling

wave phenomenon discovered by von Bekesy is believed to be the key element in the

cochlea's analysis of sound, the initial observations of von Bekesy revealed responses that

were neither sharply tuned nor of sufficient sensitivity to respond to threshold levels of

stimulation. The difficulty extrapolating from von Bekesy's observations to real-life listening

situations can be appreciated when one considers that his observations were confined to either

cadaver ears or to mechanical models constructed to replicate conditions encountered in

cadaver ears.

In discovering the traveling wave, von Bekesy observed that the cadaver cochlea has a very

poor ‘imaging’ quality. He found the traveling wave peak in response to a pure tone stimulus

to extend over a third or more of the entire cochlear length. On the contrary, in the healthy

living cochlea, the TW peak for low-level pure tone stimulation is much sharper. The TW

peak covers less than 1 mm and a shift in frequency of just one-third of an octave moves the

TW peak to stimulate an entirely different set of sensory cells.

The discrepancies between the contemporary traveling wave demonstration and the tuning

and sensitivity of the ear, were clearly described by Thomas Gold some 20 years after von

Bekesy's discovery. Gold suggested that parts of von Helmholtz' resonance theory were in

keeping with the results of listening studies conducted with persons who had normal hearing

(Gold, 1948). The idea of an amplifier to overcome physical limitations was first proposed by

Gold. At first his ideas were not well accepted, but after the discovery of hair cell motility by

Brownell they became a credible possibility.

By the early 1970s, sophisticated investigations of cochlear mechanics in living ears at near-

threshold stimulus intensities revealed that healthy systems were capable of better frequency

85

resolution and threshold sensitivity than were predicted on the basis of von Bekesy's

observations (Rhode, 1971).

Approximately 50 years after von Bekesy's discovery, David Kemp reported that sound

energy produced by the ear could be detected in the ear canal (Kemp, 1978). Called

otoacoustic emissions (OAEs), these sounds offer evidence that the ear contains a source of

energy and that the energy may fuel the sharp tuning and exquisite threshold sensitivity von

Bekesy was unable to see in his experiments. Soon after Kemp's discovery there appeared

reports that cochlear outer hair cells (OHC) were capable of movement in response to

electrical stimuli provided in vitro (Brownell, 1983). As the 1980s dawned, Hallowell Davis

(1983) coined the phrase ‘cochlear amplifier’ to describe the phenomenon in which the inner

ear responds to and resolves near-threshold stimuli.

Ren showed that the ear emits sound through the cochlear fluids as compression waves. He

used a scanning-laser interferometer and found forward-traveling waves but no backward-

traveling waves. He also noted that the stapes vibrates earlier than the basilar membrane.

These results show that the ear emits sound through the cochlear fluid as compression waves

rather than along the basilar membrane (as backward-traveling waves) (Ren, 2004).

OHC research progress

The organ of Corti houses two types of hair cells, the inner (IHC) and outer (OHC) hair cell.

Both cell types transduce mechanical stimuli into electrical signals by modulating a standing

cationic current in response to stereocilia displacement (forward transduction). This current

induces a receptor potential across the basolateral membrane of the cell, the depolarizing

phase of which may promote the release of neurotransmitterers (Santos-Sacchi, 2003).

Only after Brownell’s observation of the twitching of the OHC in response to electrical

stimulation did a potential mechanism for the highly selective and sensitive responses of the

mammalian auditory system to high frequency acoustic stimulation appear. Following his

discovery of reverse transduction, a re-evaluation of the classical concepts of mammalian

hearing has been underway. Current theories that are concerned with the basis of the cochlear

amplifier envision an acoustically evoked cycle by cycle feedback process between the OHCs

and the basilar membrane. The acoustically evoked electrical responses of the OHCs are

86

assumed to affect rapid length changes by these cells, which boost the mechanical input to the

IHC. The IHCs are the receptor cells that receive up to 95% of the afferent innervations.

In 2000 Dallos and colleagues identified the OHC lateral membrane motor, a protein of 744

amino acid residues that they named ‘prestin’. This gene is specifically expressed in outer hair

cells. The mechanical response of outer hair cells to voltage change is accompanied by a

'gating current', which is manifested as nonlinear capacitance. In their study they also

demonstrate this nonlinear capacitance in transfected kidney cells. They concluded that

prestin is the motor protein of the cochlear outer hair cell (Zheng et al., 2000).

Further studies showed that prestin expressing cells were electromotile with motility

magnitudes approaching 0.2 ,m. Actual force measurements that were carried out with an

atomic force microscope showed that prestin generates significant mechanical force and that

this force is independent of frequency up to at least 20 kHz (Dallos and Fakler, 2002).

A study conducted by Liberman et al. showed that targeted deletion of prestin in mice results

in loss of outer hair cell electromotility in vitro and a 40–60 dB loss of cochlear sensitivity in

vivo. These results suggest that prestin is indeed the motor protein, that there is a simple and

direct coupling between electromotility and cochlear amplification, and that there is no need

to invoke additional active processes to explain cochlear sensitivity in the mammalian ear

(Liberman et al. 2002, Cheatham et al. 2004).

Furthermore by working with prestin knockout mice Liberman et al. gathered evidence

suggesting that OHC stereocilia transduction is normal in prestin null ears. The round window

CM data shows that, in the absence of prestin, nonlinearities in OHC stereocilia transduction

are still producing a distortion component at 2f1-f2 in the OHC receptor currents. This fact

has no explanation according to the conventional theory (i.e., that nonlinearities in forward

transduction in OHC stereocilia produce a distortion-frequency component in receptor current

which is then reverse-transduced and amplified via OHC somatic motility into distortion-

frequency vibrations of the organ of Corti).

Liberman argues that because nonlinearities in prestin-based motility have been eliminated by

the targeted deletion the only remaining known nonlinearities are in OHC stereocilia. Thus, a

simple view of the persistent DPOAEs in the prestin-null mouse is that distortions in the

organ of Corti motion arise from the direct coupling of the mechanical nonlinearities of OHC

87

stereocilia bundles. This means that the mammalian stereocilia must be sufficiently well

coupled to the motion of the cochlear partition that they can drive the middle ear to produce

DPOAEs in the ear canal. In his studies DPOAE amplitudes fell within a few minutes after

death, clearly demonstrating a biological origin for the phenomena, i.e. an “active” process

depending on endocochlear potential.

Middle ear research

In the last century there have been many attempts to characterize the human middle ear

whether by measurements or by proposed mathematical models (surveyed in Puria 2003).

The four middle ear measurements carried out are: ear canal impedance, stapes displacement

to ear canal pressure ratio, vestibule pressure to ear canal pressure ratio (middle ear pressure

gain) and reverse middle ear pressure gain (from the cochlea, through the middle ear, and into

the ear canal).

The most extensive measurement of middle ear characteristics was recently done by Puria

(2003). The goal of this work was to allow a full characterization of the human middle ear and

to provide an empirical basis for understanding how the middle ear modifies OAEs generated

by the cochlea and measured in the ear canal. The forward and reverse middle ear pressure

gain measurements are used to quantify the effect the middle ear has on ear canal

measurements of otoacoustic emissions. In addition, the cochlear input impedance and the

reverse impedance are used to quantify the stapes reflection coefficient for OAEs.

Middle and inner ears from seven human cadaver temporal bones were stimulated in the

forward direction by an ear canal sound source, and in the reverse direction by an inner ear

sound source. The forward middle ear pressure gain, the cochlear input impedance, the

reverse middle ear pressure gain, and the reverse middle ear impedance were calculated from

measurements obtained for the first time from the same preparation. These measurements

were used to fully characterize the middle ear as a two-port system.

Presently, the effect of the middle ear on otoacoustic emissions (OAEs) is quantified by

calculating the product of forward and reverse middle ear pressure gain.

88

Figure 7-1 – The effect of the middle ear on OAEs, taken from Puria 2003

In the 2–6.8 kHz region, the roundtrip middle ear pressure gain decreases with a slope of -22

dB/oct, while OAEs (both click evoked and distortion products) tend to be independent of

frequency. This suggests a steep slope in vestibule pressure from 2 kHz to at least 4 kHz for

click evoked OAEs and to at least 6.8 kHz for distortion product OAEs. Contrary to common

assumptions, the measurements indicate that the emission generator mechanism is frequency

dependent.

Voss and Shera used DPOAEs in order to measure the middle ear forward and reverse middle

ear transmission in cat. They used DPOAEs to drive the middle ear “in reverse” without

opening the inner ear of the cats used (Voss and Shera, 2004). The technique allows

measurement of DPOAEs, middle ear input impedance, and forward and reverse middle ear

transfer functions in the same animal. Their results generally agree with the middle ear model

by Puria and Allen (1998). The reverse transfer function is shown to depend on the acoustic

load in the ear canal, and the measurements are used to compute the round-trip middle-ear

gain and delay.

Dynamic analysis of the ossicles shows that the isolated ossicles act as a rigid body in the

audible frequency range. For every measured ossicle the first natural frequency is far away

89

from the audible frequency range (above 30 KHz), so that the ossicles should be considered as

rigid bodies without any energy absorption due to structure bending within the audible

frequencies (Ferrazzini et al. 2002).

Modeling history of the Cochlea

The first recognized model of the cochlea was published by Helmholtz in 1862 in an appendix

of “On Sensation of Tone”. Helmholtz linked the cochlea to a bank of highly tuned

resonators, which were selective for different frequencies, much like a piano or a harp, with

each resonator representing a different place on the basilar membrane. The model he proposed

was not very satisfying since many important features were left out. The most important of

which includes the cochlear fluid which couples the mechanical resonators together. But,

given the publication date, it is an impressive contribution by this early great master of

physics and psychophysics.

The next major contribution was made by Wegel and Lane, and stands in a class of its own

even today. The paper was the first to quantitatively describe the details of the upward spread

of masking, and proposes a “modern” model of the cochlea. If Wegel and Lane had been able

to solve their model's equations, they would have predicted cochlear traveling waves.

It was the experimental observations of the Hungarian researcher G. Von Bekesy, starting in

1928 on human cadavers cochleae, which unveiled the physical nature of the basilar

membrane traveling wave. Von Bekesy, found that the cochlea is analogous to a “dispersive”

transmission line where different frequency components, which make up the input signal,

travel at different speeds along the basilar membrane, thereby isolating those various

frequency components at different places along the basilar membrane. He properly named this

dispersive wave a “traveling wave”. He observed the traveling wave using stroboscopic light

in dead human cochlea at sound levels well above the pain threshold (above 140 dB SPL).

These high sound pressure levels were required to obtain displacement levels that were

observable under his microscope. Von Bekesy's pioneering experiments were considered so

important that in 1961 he received the Nobel Prize.

Over the intervening years these experiments have been greatly improved, but Von Bekesy's

fundamental observations of the traveling wave still stand. Today, we find that the traveling

wave has a more sharply defined location on the basilar membrane for pure tone input than

90

observed by Von Bekesy. In fact, according to measurements made over the last 20 years, the

response of the basilar membrane to a pure tone can change in amplitude by more than five

orders of magnitude per millimeter of distance along the basilar membrane.

Zwislocki was the first to quantitatively analyze Wegel and Lane's cochlear model, explaining

Von Bekesy's traveling wave. Wegel and Lane's cochlear model is constructed from cascade

sections of inductors, capacitors, and resistors; which represent the mass of the fluids of the

cochlea and the basilar membrane mass, partition, resistance and stiffness, respectively. The

aspects of the vertical and width dimensions of each section were suppressed, which means

that each variable was taken as a constant inside the section.

In 1976, Zweig and colleagues noted that an approximate, but accurate, solution for the one

dimensional model could be obtained using a well known method in physics called the

Liouville Green or “WKB” approximation. The results of Zweig et al. were similar to Rhode's

contemporary neural tuning curve responses.

The most common model today is the transmission line model, also called the one

dimensional model. The one dimensional model is built from cascade sections of inductors,

capacitors and resistors, which represent the mass of the fluids of the cochlea, partition

resistance and stiffness, respectively.

1D vs. 2D, 3D

In the 70's, several two-dimensional model solutions became available. Rank was the first to

formulate and consider a two-dimensional model. The 2D model argues that the long wave

approximation is not fulfilled in the region of maximum response of the membrane.

The 2D model is considered to be theoretically more natural than the long wave theories. In

spite of the better results gained with the 2D model the long wave models has gained more

appreciation because they are easy to understand and simple to solve numerically.

In 3D models, the pressure and fluid flows can vary across the width of the cochlear partition.

This pressure variation is consistent with the notion that the arcuate and pectinate regions of

the basilar membrane have different properties.

91

Givelberg, Rajan and Bunn have constructed a comprehensive 3D computational model of the

cochlea using the immersed boundary method. Their pure tone experiments capture the most

important properties of the cochlear macro-mechanics. Even after extensive optimization and

parallelization a typical experiment with 2 milliseconds of simulated time takes approximately

18 hours on an HP Superdome computer (Givelberg et al., 2001).

Bohnke and Arnold developed a 3D finite element mechanical model of the cochlea including

the fluid structure couplings. The model allows the evaluation of the passive mechanical

behavior of the human cochlea with arbitrary input pressure at the stapes footplate including

all kinds of slow and fast waves in the lymph and the cochlear partition. The models linear

solutions fit early experiments which studied the wave propagation in the cochlea of human

cadavers (Bohnke and Arnold, 1999).

As we can see three dimensional models are very strenuous on computer power and do not

include nonlinear or nonuniformities of any sort (not yet).

Both, 2D and 3D models are more complex and involve complicated mathematics, thus harder

to solve numerically. This is reasonable since the 1D formulations have fewer components to

deal with. The 1D model simulations have gained more appreciation because they require less

memory and fewer computations than the 2D and 3D models, and yet are successful in

predicting a large number of phenomena. Moreover, in a 1D model, parameters can be easily

chosen using methods that make sense anatomically, physiologically, and mechanically.

Enhanced one dimensional cochlear models

With the discoveries of the nonlinear compressive basilar membrane, the inner hair cell

responses, the otoacoustic emission, and the outer hair cell motility the models mimicking the

cochlea became much more complex. The 2D and 3D models became too heavy to compute

and a paradigm shift was seen toward the 1D extended model. Using simplifying assumptions

we can collapse the 3D model formulations to a 1D model formulation. Thus a new branch of

1D extended models evolved which could better represent the complex systems under

observation, i.e. the scala media and the organ of Corti.

The large number of nonlinear phenomena discoveries, which began in the 70's, revealed the

necessity to incorporate nonlinear elements into the cochlear models.

92

Hubbard and Hall both used nonlinear damping that increased with the increase of cochlear

partition velocity. Furst and Goldstein tested a nonlinear damping model versus a nonlinear

damping and stiffness model.

A model that includes a representation of the electrical characteristics of the scala media and

the outer hair cells (OHCs) was represented by Hubbard et al. Their model used a standard 1D

model combined with a model of hydromechanical changes in the organ of Corti. A nonlinear

conductance that varies as a function of the basilar membrane displacement was also added.

Nobili et al suggested a model made of an array of nonlinear oscillators, each of which is

coupled instantly to all the others through hydrodynamic forces transmitted by the fluid in the

cochlea. Nonlinearity in this model is expressed by a sigmoid function operating on the

basilar membrane velocity, and feed back pressure difference on the basilar membrane. In

another work by Nobili, Mammano and Ashmore, a shearing viscosity term was added. This

term represents the viscous forces acting on one oscillator section, caused by possible

different velocities of adjacent oscillators.

The models by Talmadge, Long, and Tubis incorporate nonlinearity in terms of a Van Der Pol

oscillator added to the damping factor. They incorporate the concept of delayed operation, as

suggested by Zweig (both fast and slow feedback). Their model produces SOAEs although no

stimulus exists.

A model of outer hair cell motility that cooperates with a cochlear model was suggested by

Geisler. He used the basilar membrane and reticular lamina as two ‘free bodies’ and the outer

hair cell force frequency transform was described as an all pass filter with a constant delay.

Kolston et al. suggested that the sharpening effect of cochlear amplifiers may be due to

variable impedance. Their impedance was connected in parallel to the partition mass,

stiffness, and damping.

The work of Talmadge et al. showed that a class of nonlinear active cochlear models can

successfully describe a broad body of data on the quasi-periodic variation with frequency of

otoacoustic emission fine structure and the microstructure of the hearing threshold (Talmadge

et al., 1998).

93

Their model is based on a 1D macromechanical model using time delayed stiffness and

simplified models of the middle and outer ears. The cochlear model incorporates a frequency

map of the form suggested by Greenwood and the cochlear nonlinearity is modeled as a

quadratic (“Van der Pol” type) nonlinear damping function. Slow and fast feedback time

delayed stiffness was used in the model. Roughness (distributed randomness) was applied to

the place-frequency map. Talmadge et al.’s conclusion was that random spatial variations of

almost any of the cochlear parameter will give rise to the effects seen in their model.

The advantage of this model is that it allows one to make specific predictions regarding

various modeling assumptions (the effect of nonlinearity or distributed roughness), as well as

to directly test in a time domain cochlear model the predictions of the theoretical framework

laid out.

The key elements of the models are tall and broad cochlear traveling wave activity patterns

and cochlear wave reflections at the base of the cochlea and around the tonotopic place of the

traveling wave, with the latter being due to distributed cochlear inhomogeneities in

conjunction with the tall and broad activity pattern and distributed nonlinear cochlear

response.

Spontaneous emissions may arise and are associated with instability modes of the linear active

component of the cochlear mechanics. The cochlear nonlinearity provides the requisite

stabilization for converting the instabilities into limit cycle oscillations corresponding to

actual SOAEs.

The fine structures for SEOAEs, TEOAEs, threshold microstructure, DPOAEs and the

frequency spacing of neighboring spontaneous emissions are mainly determined by the

parameters for basal reflectance, apical reflectance around the tonotopic place, and the ratio of

the left and right basis functions and their spatial derivatives. The model predicts that

psychoacoustic and SOAE/SEOAE/TEOAE fine structure spacing should be similar, but that

the DPOAE spacing should be wider. The models also account for the band-pass character of

DPOAEs.

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