Phenomenological characterization of eardrum transduction
Christopher
A. SheraandGeorgeZweig
Theoretical
Didsion,
LosA/amos
National
Laboratory,
LosA/amos,
NewMexico87545and
Physics
Department,
California
Instituteof Technology,
Pasadena,
California
91125
(Received
3January1989;revised
22January1991;accepted
26February1991)
A phenomenological
description
of thetransduction
effected
by theeardrumispresented.
That
descriptionis providedby a transfermatrix,whoseelementsdefinethosemeasurements
sufficient
to characterize
eardrumtransduction.
Causalityprovides
constraints
on thematrix
elements.
In addition,measurements
of thematrixelements
candetermine
whethertheysatisfy
constraints
imposed
byminimum-phase
behavior
andtheprinciple
of reciprocity.
Those
constraints
maybeusedeitherto reducethenumberof measurements
necessary
to
characterize
theeardrumor to checktheconsistency
of measurements
thatoverdetermine
the
system.
Withinitsregionof validity,thetransfermatrixof theeardrumprovides
a common
groundfor thecomparison
between
theoryandexperiment.
As anexample,
a simplemodelfor
thetransduction
characteristics
of theeardrum,definedcompletely
in termsof measurable
quantities,is presented.
PACS numbers:43.64.Ha, 43.64.Bt
INTRODUCTION
ity. If the eardrumis a completelypassivemechanicaltransducer,
reciprocitymusthold.Thoseconstraints
maybeused
The eardrumbeginsthe process
of auditorytransduceither
to
reduce
the
number
of
measurements
necessary
to
tion,convertingincidentsoundintoa mechanicaloscillation
characterize
the
eardrum
or
to
check
the
consistency
of
meaof the bonesof the middleear. Althoughdetailedmeasuresurementsthat overdeterminethe system.The transfer-mamentsof eardrumsurface-displacement
patterns(Tonndorf
andKhanna,1972;KhannaandTonndorf,1972;yonBally, trix elementsprovidea concisesummaryof the functional
response
of theeardrum--therebyfacilitatingincorporation
1976;L$kbergetal., 1980;Decraemer
etal., 1989)providea
of eardrum transduction characteristics into models of the
qualitativepictureof the complexityof eardrumvibration,
middleear--and providea commongroundfor comparing
the characteristics of that transduction have not been meatheoryand experiment.
sured.Suchexperiments
do not, for example,measurethe
Justas the overallresponse
of the ear to soundcan be
force transmitted to the ossicles and so fail to characterize
factoredintopartsdescribing
theseparate
actionsof theinthoseproperties
of theeardrumdirectlyrelevantto thepro-
cessof hearing.Althougheardrumdynamicshavebeenexploredtheoreticallyat varyinglevelsof complexity(Helmholtz, ! 868;Shaw, 1977;Shawand Stinson,1986;Funnellet
al., 1987;RabbittandHolmes,1986)andqualitativeagreement betweenmeasuredand computedsurface-displacementpatternsobtained,thetransformation
propertiesof the
eardrumhavenotbeenthoroughlyexamined.
Sincemeasurements
are incompleteand eardrummechanicsnot well understood,
the needexistsfor a comprehensive framework that identifies those measurements nee-
ner,middle,
andexternal
ears,sotheresponse
ofthemiddle
ear itself can, under certain conditions(Shera and Zweig,
199!b), befactoredintocomponents
describing
theindividualactionsof itsconstituent
parts.Thosecomponent
factorizationsare convenientlysummarizedby transfermatrices;
the overalltransfermatrixis simplythe productof componentmatrices.This paperrepresents
the firstin a seriesof
papers(SheraandZweig, 1991a-c) devotedto thephenomenologiealdescriptionof eardrumand middle-earmechanics.Subsequent
papersextendthe programbegunherewith
the eardrumto providea phenomenological
descriptionof
the other major componentsof the middleear.
essary and sufficient to characterize the transduction
propertiesof the eardrum.Sucha frameworkwould show
onehow to provewhat is believedknownaboutthe system
I. PHENOMENOLOGICAL
CHARACTERIZATION
OF
(e.g., its analyticityand symmetryproperties),and, once
THE EARDRUM
proven,howto removetheeffectsof thoseconstraints
from
measurements
to exposedirectlythe underlyingdynamics.
The eardrum(illustratedschematicallyin Fig. l ) forms
This paperproposes
sucha frameworkfor the phenomeno- the lateral wall of the tympaniccavity and vibratesin relogicaldescriptionof the transformation
propertiesof the
sponseto incidentsound.Volumedisplacements
of the earcardrum.
Thatdescription
isprovided
bythetransfer
matfix drum createpressurevariationsP,cin the tympaniccavity.
'• (•o) of the eardrum,whichsummarizes
thosedynamical At the sametime, the eardrumexertsa forceF u on the malcharacteristics
of theeardrumimportantfor understanding leus.That forceresultsfrom the mechanicalresponseof the
itsrolein transmittingsoundto themiddleear.The transfer- eardrum to the driving pressuredifference
matrixelementsare necessarily
constrained
by causality;in
P•----P, --P,c
(1)
addition, measurementsof the matrix elements can be used
A
to determinewhetherthey are minimum-phase
functions
andwhethertheeardrumsatisfies
theprincipleof reciproc253
J. Acoust.Soc.Am.90 (1),July1991
betweenthe ear canaland the tympaniccavity.
The measurements of Guinan and Peake (1967) and
0001-4966/91/070253-10500.80
¸ 1991Acoustical
Societyof America
253
Ptc
malleus
Pe-Pt½
Fu
ec
u
ear canal
tympaniccavity
FIG. 1. Schematicdrawingof the eardrumin cross-section
adaptedfrom
Fig. 5-6 of yonB6k6sy(1960).
-I
Buunenand Viaming (1981) on the cat indicatethat the
eardrumand middleear respondlinearlythroughoutthe
intensityrangeof normalhearingbelowthe activation
threshold
for theacoustic
reflex.Linearityimpliesthatthe
eardrumcanbe completelycharacterizedin termsof its response
to puretones.For simplicity,
all variables
in thispa-
FIG. 2. The eardrumrepresented
asa two-portnetworkthat transformsa
pressure
andvolumevelocityin theearcanalintoa forceandvelocityat the
umbo.The factoredrepresentation
shownin the bottompanelmakesex-
perhavebeenwrittenascomplex
quantities
characterizing plicitthedimensional
transformation
(tra•nsduction)
performed
bytheearof thematrixproduct• A• • in cascade
withan ideal
those
responses.
Forexample,
thevariable
Vu,representing drumandconsists
transformer
of turnsratio.4• whosefunctionis to providea changeof dimension.The idealtransformerisrepresented
by thematrixA•, definedby
the velocityof the umbo,isdefinedin termsof the measured
amplitude
.4(w) andphase•(co) relativeto thepressure
at
the eardrumby
Vu(•o)---.4(co)e
i•(ø').
tive area" of the eardrum (see Sec. I).
(2)
Underconditions
discussed
below,a phenomenological
between
inputandoutputports,isnatural
descrip•on
ofeardrum
mechanics
isprovided
bythetransfer rainsa separation
for
describing
a
cascade
of
systems.
For example,thematrix
matrix
• (co)oftheeardrum.
Illustrated
schematically
in
describing
thecombinedactionof theeardrumandossicular
chainis simplythe productof thematricesrepresenting
the
eardrumand ossiculartransformations
individually(Shera
and Zweig, 1991b).
Equation(3) is the mostgenerallinearequationrelatingthefourvariables
definingtheeardrumtransformation.
The relationis independent
of the natureor positionof any
Fig. 2, T• (co)isdefinedby theequation
A
ee = TU•VuJ
,
(3)
where Ue represents
the volumevelocityof the eardrum.
The inputandoutputvariables
areidentified
by super-and
A
subscriptson the matrix. Stnce
'
T,e is definedin termsof the
pressure
difference
Pc--whichreduces
to Pe whenthemiddle-earcavitiesaresurgically
exposed
andopenedwidelyto
theatmosphere--the
matrixdepends
onlyontheproperties
oftheeardrum;
theeffects
ofthecomplicated
acoustic
geometryofthemiddle-ear
cavities,
discussed
in thenextpaperin
this series(Sheraand Zweig, 199lb), havebeen"factored
OUt."
In the languageof electricalcircuit theory,Eq. (3)
viewsthe eardrumas a two-portnetworkcharacterized
in
termsof transfercoefficients
(e.g., Brillouin, 1946;Friedlandetal., 1961;Lampton,1978).Althoughthephenomenologycouldhavebeenbasedon any of the other standard
characterizations
of two-portnetworks(e.g.,impedance
or
admittance coefficients), the transfer coefficientsof the ear-
drum haveespeciallysimpleinterpretations--as,for example, effectiveareas (seebelow)--and their measurement-basedon manipulationof the ossicularload--is convenient.
In addition,the transfer-matrixrepresentation,
whichmain254
J. Acoust.Soc.Am.,Vol.90, No.1, July1991
so•cesorloads
presented
totheeardrum.
(It isratiossuch
as P•/Ue or F,/V u that dependon the loadingof the eardrum by the middle and inner ears.) Furthermore, the matfix characterizes both "forward"
and "reverse" transduc-
tion; the same elementscharacterize eardrum transduetion
whetherit isdrivenby pressurevariationsin the ear canalor
by the autonomous
motionof the organof Corti. The transfer matrix thusallowsa convenientseparationbetweenthe
boundary,or loading,conditions
asrepresented
by theinput
andoutput
vectors
and•
thedynamics
oftheeardrum
asdescribedby the matrix
The simple2 X 2 transfer-matrix
description
of the eardrum is appropriatewhenthe four variablescomprisingthe
input and output vectorssufficeto determinethe eardrum
transformation.For example,althoughsoundpropagation
in thehumanearcanalisonedimensional
at frequencies
less
than the cutofffrequencyfor higher-ordermodes--which
for a typicalhumanear canalis approximately19 kHz (Ste-
C.A. SheraandG. Zweig:Description
of eardrumtransduction
254
venset al., 1987)--the pressure
maybenonuniformcloseto
the eardrum due to the presenceof evanescentwaves;the
frequency,the velocityof the umboproducedby a known
drivingpressure:
acoustic
forcedriving
theeardrum
mightnotth? beaccurately approximatedby the pressuredifferencePe. The effects of thosenonpropagatingmodesappear to be small,
however,at frequencies
lessthan • 6 kHz (Stinson,1985).
Althoughthemotionof themalleusmaybequitecomplex at high frequencies
(e.g., Decraemeret al., 1989)consisting,
for example,of bothrotationaland translational
components(Donshue, 1989)--the one-dimensional
variablesFu and Vu sufficeto describethe "output" of the eardrum transformationsolongasthe effectivemechanicalinput to the middleear is well approximatedby a singleforce
zc=
Fu = 0 (ossicular load removed)
(7)
Similarly,the elementYecanbefoundfrom the relation
,Y•=•Ue[
v•=0(malleus
held
fixed)
(8)
Were the eardruma rigid plate,fixingthe malleuswould
preventtheumboandtherestof theeardrumfrommoving;
the matrix elementYe would then be zero.
Note that T u cannotbe foundfrom measurements
of
surface-displacement
patterns.
Such
experiments
do
not,
for
andvelocity
coordinate,
2 asit appears
to bea lowfrequenexample,measurethepressure
in thetympaniccavityor the
cies (the preciseregionof validity, however,has yet to be
force
transmitted
to
the
ossicles.
In addition, whereas surestablished).In the simplestpictureof eardrumdynamics,
thosecoordinatesare naturallyassociated
with the forceapface-displacement
•atterns
depend
ontheloading
oftheearpliedto themanubriumandthevelocityof theumbo.If the drum, the matrix T• doesnot. Althoughsurface-displaceactionof theeardrumismorecomplicated,
otherdefinitions mentpatternsarenotdirectlyrelatedto thefunctionalrole
of theeardrumin theprocess
of hearing,theycanbehelpful
forthosecoordinates
maybemoreusefulfi
in establishing
an understanding
of theinternaldynamics
of
A
the eardrum (Funnell et al., 1987; Rabbitt and Holmes,
A. Measurement and interpretation of the matrix
1986). For example,examinationof displacement
patterns
canaidin determiningwhichof a familyof modelsthat accuDeterminationof the matrix • (w) and its (speciesrately predictthe transfer-matrixelementsprovidesthe
dependent)regionof validity constitutes
a fundamental
mostrealisticdescriptionof the eardrum.
problemin eardrumphenomenology.
The elementsof the
elements
A
transfer matrix
A
B. Constraints
!
A
on the matrix elements
Studyof a complexsystemsuchas the eardrumoften
bestproceeds
by firstidentifyingtheanalyticityandsymme-
Ye
thatcons•ain
thedynamics.
Atevery
angular
can be measuredby manipulatingthe middle-earossicles. tryconditions
frequency
co
the
matrix
T•
has
four
elements,
each
of
which
(The diacriticalhats indicatethat the matrix elementsare
hasbotha realandan imaginarypart.Thoseeightfunctions
measured
withthecavities
opened
widely.
) Forexample,
if
themalleus
isimmobilized
sothatV, = 0, theelement
A• •
is givenby the ratio
of frequency,
however,
arenotall independent.
Twoclasses
of constraints
providerelationsamongthematrixelements.
First are thosethat followfrom generalphysicalprinciples
suchascausality.Secondarethose,suchasreciprocity,
that,
althoughnotbindingin general,mightapplyto theeardrum
gu •u=O (malleus
heldfixed)
of certainqualitativecharacteristics
ofitsdynamics.
Note thatAv (•) hasthedimensions
of an area.Whenthe because
All physicalsystems
areconstrained
bycausality;
a sysmalleus
isimmobilized,
the"effective
areaforfor•e"repretem
cannot
respond
before
it
is
driven.
For
a
linear
system
sentsthat areaby which the pressuredifferenceP• mustbe
characterizedin the frequencydomain,causalityrequires
multiplied
toobtain
the•rce onthemalleus.
The matrix elementAv providesanothereffectivearea, that the real and imaginarypartsof eachtransfer-matrix
•1• Pe
but in this case associat•
(5)
with the transformation of veloc-
element
T•7(co)beHilberttransforms
ofoneanother
(Bode
1945):
ity coordinatesby the eardrum:
Av
=• r•=o
(o•i•
Io•
......
•>
(6)
Whenloading
duetotheossicular
chainhasbeenremoved,
4
the "effectiveareafor velocity"providesthe proportionality
factorbetweenthe velocityof the umboandthe total volume
vel•ity of the eardrum.If the eardrumwerea rigid piston,
botheffective
•eas wouldberealconstants
equa•othearea
efthepiston.
In general,
however,
thetwoareasAy
(o) and
Av(o) aredifferent
complex
functions
off[equency.
Theremaining
elements
of thematrix• •ay bcdeterminedby similarmeasurements.
The elementZ½represents
the no-loadtransferimpedanceof the eardrumfoundby removingthe ossicularload and measuring,as a functionof
255
J. Acoust.Soc.Am.,Vol. 90, No. 1, July1991
-
1f_ø
/-r
©Im[Te(o')]
o'-co do' (9)
and
Im[T/•
(co)
] I
Re
(lO)
Here, the integralsrepresentCauchyprincipal-valueintegrals.Equations(9) and (10) are equivalentto the statementthat considered
asa functionof complexfrequencythe
matrixclem•ntsT,y(to) amanalyti•in th• lowerhalfof th•
frequencyplane. Causality thus providesanalytic constraintson the matrixelements)Thoseconstraints
maybe
usedtocheckmeasurements
oftheTeforinternalconsistenC.A. Sheraand G. Zweig:Description
of eardrumtransduction
255
cy, reducemeasurementuncertainty,and determinefunctional valuesat frequencies
for whichmeasurements
are not
available(Zweig, 1976;Zweig and Konishi, 1987).
In additionto causality,the matrix elementsmaysatisfy
the strongerminimum-phaseconstraint,whichrequiresthat
therealandimaginary
partsof In To(co)beHilberttransformsof oneanother.The responses
describedby eachmatfix elementwouldthenbeproducedby a mechanicalsystem
thatresponds
asrapidlyaspossible
consistent
withitsamplitude characteristicand the constraintof causality(Bode
1945). The matrix elementsof the examplemodel (seeSec.
II) are suchminimum-phase
functions.Althoughcertain
combinationsof matrix elements,suchas the umbo transfer
function,mightbeexpectedto haveevolved"optimal"minimum-phasecharacteristics,
suchteleologicalargumentsdo
not apply to the matrix elementsindividually.
Linearpassivemechanicalsystems
obeythe principleof
reciprocity (Helmholtz, 1860; Maxwell, 1864; Rayleigh,
1896;FoldyandPrimakoff,1945).A two-portsystemissaid
to be reciprocalif it is not possible,giventhe transferfunctionFo•t/ Vi,--where Foutis the generalized
forceproduced
at theoutputportbya generalized
velocityV•, at theinput-to identifywhichof thetwoportsservedastheinputportfor
obtainingthe giventransferfunction.A necessary
andsufficientconditionfor a systemto be reciprocalis that its transfer-matrixelementssatisfythe algebraicrelation
detT=
q- 1.
(11)
The classical
conceptof theeffective
areaconfuses
those
two actionsof the eardrum.For example,the classicaleffective areaActhasbeendefinedto bethe areaof the platethat,
if movedthrougha distanceequalto that traversedby the
umbo, would sweepout the sametotal volumeas the eardrum (Wever and Lawrence, 1954). Act has beenestimated-taking into accountthe conicalgeometryof the membrane-to be roughlytwo-thirdsthe anatomicalarea of the
eardrum (von B6k6sy,1941;Wever and Lawrence,1954).
Definedwith referenceto eardrumvolumedisplacement,
the areaAc•is ofteninappropriately
appliedto thetransformationof pressurebetweenthe eardrumand the oval window (e.g.,Zwislocki,1975). In addition,sinceloadingconditions on the eardrum are not specified,the classical
definition of the effectivearea is ill-defined. Nevertheless,
Ac•provides
a useful
referencearea
with•which
theamplitudesof thetwoeffective
areas`4v (co)and,4• (t0) appearing
in • maybecompared.The nextpaperin thisseries(Shera
A
andZweig•199
lb) disc•usses
therelative
importance
ofthe
two areas,4v (co) and ,4• (co) for middle-earand cochlear
mechanics.
A simpleargumentbasedon reciprocitygivesthe relative sizeof the two effectiveareasat low frequencies.
The
displacement
oftheumbo
•Xuisthenexpected
tobeinphase
withthedrivingpress•ure
Pewhentheossicular
leadisremoved.The elementZe is thusproportionalto l/ice:
Z e -• K•/ico
(K e >0).
(13)
All lineartwo-portsystems
thatcontainonlycapacitors,
rethat at low frequencies
Ye is
sistors,and inductors---or
their mechanicalanalogs--and Analogousargumentssuggest
proportional
to
ice:
no sources
of generalized
forceor velocityare reciprocal
6
(Friedland et al., 1961). For small input signals,the earYe --' ico/Ka
(Ka > 0),
(14)
drum appearsto be a linear,passivemechanicalsystemand
canthusbeexpected
to bereciprocal.
If satisfied,
reciprocity
andthat theeffectiveareas,4v and,4F arebothpositiveand
would allow determination of one matrix element from meareal. Reciprocityrequiresthat
surements of the others:
Av/AF --ZeYe = + 1.
(12)
The eardrum models discussed in Sees. II and III
are all
reciprocal.It is possible,however,that eardrumsin some
animalsmayin theprocess
of transduction
activelyamplify
(or diminish)the energyin incomingsounds.Weretheyto
berealizedin nature,sucheardrums
mightnotbereciprocal.
Discussions of the contribution
,o•o`4v
,o-o1 q-Zeye
-
1
<1.
(15)
lq-K½/Kd
Notethat,4v and,4F havesimilarvaluesat lowfrequencies
if
andonly if Kc/Kd • 1.
II. OSCILLATOR
TRANSDUCTION
MODEL FOR EARDRUM
CHARACTERISTICS
To illustratethe formalism,thissectionpresents
a sim-
C. The effective areas AF and Av
"transformer
lim•r -- lim 1
ple phenomenological
model--the "oscillatormodel" for
of the eardrum to the
the transduction characteristics of the eardrum defined com-
action" of the middle ear often make reference
pletelyin termsof measurablequantities.As such,themodel
to the "effectivearea" of the eardrum(yon B6k6sy,1941; differsfrom other lumped-element
modelsof the eardrum,
Wever et al., 1948; Zwislocki, 1975). As demonstrated whichcontainimpedances
thatwouldbedifficultto measure
above,however,there are two sucheffectiveareas,one asso- on a real eardrum.v In contrastwith other models,no atciatedwith eachof the two principaleffectsof eardrummotempt will be madeto modelthe internaldynamicsor comtion: transmission of a force to the middle-ear ossicles and a
plex oscillationsof the eardrum. Rather, a differentapchangein thevolumeof thetympaniccavity.The areasasso- proach
isadopted
based
onthe•ssumption
thatthreeofthe
ciated with thosetwo actionsof the eardrum (i.e., transforelementsof the transfermatrix T• of the eardrumare "simmationof pressureinto bothforceandvolumevelocity)are
ple" at low frequencies(i.e., are either constant or have
notnecessarily
equal.In.deed,
theconstraint
of reciprocity Laurent expansionsin ice dominated by the first three
(12) implies,
assum•ing
Zc:•0, thatthetwoareascouldbe terms). The remainingmatrix elementis determinedby asequalif andonlyif Yc = 0 (that is,onlyif the eardrumwere
sumingreciprocity.Thoseassumptions,
althoughperhaps
perfectlyrigid).
plausible,haveno firm empiricalbasisandareadoptedsole256
J. Acoust.Sec. Am.,Vol. 90, No. 1, July1991
C.A. Sheraand G. Zweig:Description
of eardrumtransduction
256
ly for purposes
of illustration.
For clarityof exposition
thediscussion
proceeds
byfirst
considering,
and then extending,the transfer-matrixelementsof a simpleplatemodelof theeardrum.The modelis
thusultimatelydefinedby its transfermatrix,withoutreferenceto a particularphysicalmodelof the eardrum.Section
II-A demonstrates,however,that the mechanicaltwo-piston model introducedby Shaw (1977; Shaw and Stinson,
1981) provides,in thelimit impliedby hisparametervalues,
a mechanicalrealizationof the model transfermatrix proposedhere.The matrix elementsfor alternativemechanical
modelsproposedby otherauthorsare discussed
in Sec.III.
A. Matrix
elements
of the oscillator
They found that the eardrum remains in its lowest mode of
oscillationat frequencieslessthan 3 to 4 kHz. At higher
frequencies
the eardrumof the cadaverear breaksup into
complexoscillations.
Althoughtransitionfrequencies
were
higher,KhannaandTonndorf(1972) andDecraemeret al.
(1989) found qualitativelysimilar vibration patternsin
anesthetized
cats.Their findingsareconsistent
with thoseof
othersbasedon human temporalbonepreparations(von
Bally, 1976) and living subjects(L•kberg et al., 1980).
A first approximationto the transfermatrix of the eardrum might thusbe foundby imaginingthe eardrumas a
rigid plate (e.g., Peake and Guinan, 1967; Dallos, 1973;
GeislerandHubbard,1975)withanareaApheldin theear
canalby springsand dampersrepresenting
the annularring.
Sucha modelshouldbea fair approximationat low frequencies where the eardrum vibrates in its lowest mode of oscilla-
tion. The transfermatrix representingthe eardrum would
then have the form
P
.
Teu=0 Ap
=o,
Ye
=•u malleus
held
fixed
reflectingthe fact that the platewasassumed
to be rigid.
However, measurementsin the cat made with the ossicles
immobilized
by gl•ng themto thecavitywalls(Lynch,
1981) indicatethat Yeisnonzeroandprobablycompliantat
lowfrequencies,fi
assuggested
byEq. (14). Moregenerally,
we assumethat at higherfrequencies
this elementhasthe
form of a simpleresonator:
?e= Yd=--{iCOMd
+ Rd+ Kd/iCo)
-'
(16)
constant:
•V
• • •Se
ossicular
load
removed
m real positiveconstant •A•.
The elementZ• is taken,againasin the platemodel,to
Z• •Z• •i•M•
+ R• + K•/i•.
I• theplateapproximation
thetwoeffective
areas
Ar
= 1/( 1 + Z• Y• ),
(22)
Note that the parametersof the modelprovidephenomenologicalcharacterizations
of thedynamics;theydo not corresponddirectlyto particularanatomicalstructures,
but summadze their collective behavior.
With thoseassumptions,
the effectiveareafor forceAv,
whichrelatesthepressure
acrossthemembrane
to theforce
on the malleusat the umbo, may be detemined from the
other elementsby assumingreciprocity:
A
det• = + 1 •
(18)
held
fixed
= (I + Z•Y•)/Av.
(17)
andAv are identicalandequalto the areaof the plate.Since
reciprocityimpliesthat
(21)
have the form of a harmonic oscillator:
malleus
containsparametersrepresenting
the effectivemass,damping, and compliance(per unit area) of the plate and the
structuresholdingit in place.
•v/•v
(20)
1/Y• by the first three termsin a Laurent expansionin i•.
With the ossicularloadremoved,the eardrummightbe
expectedto remain in its lowestmode of oscillationat frequencieshigher than those observedby Tonndoff and
Khanna. In the lowestmodethe volumevelocityof the eardram and the velocity of the umbo shouldbe roughly in
phase and propo•ional. The effectivearea for velocity is
thusapproximated,as in the plate model,by a real positive
The impedance
Zp=iO)J•[p
+Rp + Kp/i•
(19)
WritingYe(•) in thatformis equivalent
to representing
model
Measurementsindicatethat at low frequenciesmostof
the eardrumvibratesin phaseas a unit, muchlike a simple
piston.For example,Tonndorf and Khanna (1970; 1972)
havemeasuredthe surfacedisplacement
of eardrumsin human cadaversin responseto tonesof severalfrequencies.
^
^ ue
(23)
Thus,although
Av is constant
•n[he model,AF canvary
substantially
in regionswhereZ• Y• is not constant.The
model transfer matfix then has the fore
• •
YaA•
•
1
•
0 • '
(24)
where the factofization, which separatesthe dimensional
and dynamical transformationseffectedby the eardrum,
to therepresentation
of Fig. 2 (b). The transfer
theplateapproximatio•remai•
approximately
valid•in corresponds
matrix
can
be
factored
further
to
allow the topologyof the
thesense
thattheareashlandAv havesimilar
values•at
equivalentcircuit to be obtainedby inspecfrequencies
for which IZ½Y½I • 1. If that inequalityis satis- corresponding
fied, eardrum transductioncan be consideredeffectively tion (see,for example,Table I of Lampton1978):
"pistonlike"even in the presenceof complexsurface-displacement patterns.
The plateapproximationmaybeextendedby examining
the transfer-matrixelementsindividually. Note that in the
plate model the matrix element
257
d. Acoust.Soc. Am.,Vol.90, No. 1, July1991
The modelof Eq. (24) canbe thusrepresented(seeFig. 3)
by a series
impedance
Z• • • andan idealtransformer
(of
C.A. Shoraand G. Zwoig:Description
of eardrumtransduction
257
ee
I:A,
,••l•
•
Ptc
mal
U½
•
•
Pc- P•½
FIG. 3. Networkrepresentation
of theoscillatormodeldefinedby Eq. (24).
turnsratioAt ) separatedby a shuntadmittanceAeYd.Fac-
torization
in thatformfollo•ws
immediately
fromtheassumedconstancyof the areaAv: were the first two matrices
FIG. 4. Schematic
illustrationof the compound-eardrum
modelin which
theeardrumisrepresented
bytwocoupledplatesoneofwhichisattachedto
in the productto beinterchanged,
the areaAF, andnotAv,
the malleus.FollowingShaw( 1977,1981), the impedances
Z• andZo are
wouldhavetheconstantvalueAe.
shown
as
harmonic
oscillators
and
the
coupling
impedance
Z,
asa springSimple,qualitativeargumentsgiverelationsamongsevdampercombination.
eral of the parameters.First note that the area of the eardrum that movesis presumablygreaterwhenthe ossicular
loadingis removedthan when the malleusis fixed:
callyin Fig. 4, represents
the eardrumby two coupledplates
held in the ear canalwith springsand dampers.The malleus
(Ao= area in motion)Io•icutar
load
...... d
> (A© = area in motion)lmaueu
• he•d
•xea'
is rigi^dly
attached
to thecenterplate.Thepressure
differ-
(26)
encePcbetweenthe earcanalandtympaniccavitydrivesthe
The subscripts
"0" and "o•" denotetheno-loadandinfinitemotionof the platesand producesa forceF, on the malleus,
load conditions,respectively.To the extent, however,that
whichis displaceda distanceX•. The motionof the plates
thetotalmassin motionisproportionalto thecorresponding
alsochangesthe volumeof and hencethe pressurein the
area, one expects
(Me - mass0/Ao) = (Ma -- masso•/Ao•).
tympanic
cavity.If theequatio•ns
of motion
arewrittenin
(27)
termsof thepressure
difference
P•, thosetwo effectsof plate
motionare uncoupled.
The transfermatrixT• (to) for the compoundeardrum
followsimmediatelyfrom the equationsof motion(the matrix is writtenmostsuccinctlyasa productof factors):
Consistent
withthemodel's
originasan•extensi•n
of the
plate model,one requiresthat the areasAF and Av be approximatelyequalat low frequencies.
Equation (15) then
impliesthat
K• <K•.
(28)
Measurement
of the matrix elements would allow these rela-
^e
T u --
tionsto be testedexperimentally.
Zo + (1
ZoZc
X(zo+z
1 (2+ •c)Z•
+•c-•(Zo
+Z•))
B. Comparison with simple mechanical models
x(,0
Further insightinto the interpretationof the oscillator
modelcanbe gainedby comparisonwith the transfermatricesof simplemechanicalmodelsof the eardrum. Recall that
Here, Z m representsthe impedanceof the plate attachedto
theoscillator
modelreduces
to therig•-plate(single-pis- themalleusandZ o the impedance
of the otherplate:
ton) model when the matrix elementY• is zero. A more
complicated"two-piston"model has been introducedby
Shaw( 1977;Shawand Stinson,1981) in an attemptto accountfor measuredmembranesurface-displacement
patterns.Two limitingformsof the two-pistonmodel,distinguishedprimarilyby the strengthof the couplingbetween
the pistons,are discussedbelow. When the couplingis
strong,the transfercharacteristics
of the two-pistonmodel
are shownto reduceto an equationof the form (24). The
systemthen providesa mechanicalrealizationof the model
transfer matrix given above. When the couplingis weak,
however,the two-pistonmodelhastransfercharacteristics
similarto othersimpleeardrummodelsproposed
in theliterature (e.g., Matthews, 1980; Neely, 1981).
The compound-eardrummodel, illustrated schemati258
d. Acoust.Soc. Am., Vol. 90, No. 1, July 1991
P•
(30)
Zrn•A3Vu
Zc=0;Fu=O
and
Zo=A-•o
z•=o
Z• represents
the couplingimpedancebetweenthe plates
and •cthe ratio of the plate area•:
•'=Ao/A •.
(31)
In Shaw'smodel Z•, and Zo have the form of harmonic
oscillatorsand the interplatecouplingimpedanceZ• con-
sists
ofa spring-damper
combination.
9 A straightforward
calculationgivesdet • = + 1; as expected,the systemis
reciprocal.
The transfer-matrixelementsof the compound-eardrum model are, in general,complicatedcombinationsof
the impedances
Zm, Zo, andZ½that definethe model.In the
C.A. Shera and G. Zweig:Descriptionof eardrumtransduction
258
III. PREDICTIONS
limit impliedby Shawand Stinson's( 1981) parametervalues,however,the matrix elementsare, as shownbelow, con-
OF THE OSCILLATOR
MODEL
To illustrate how the transfermatrix providesa com-
siderablysimplified.Althoughthe matrix elementsthemselvescan be readilycomparedwith experiment,because
definitions(30) require that determinationof the impedancesZ mandZ o bemadein the limit Z c-• 0, thosequanti-
monground
forcomparing
theory
andexperime•nt,
thiss•ection discusses
predictionsconcerningthe areasAv andAv,
first for the oscillator model and then for several other mod-
elsthat makedefinite,testablepredictionsaboutthe transductioneffectedby the eardrum.
ties would be difficult to measure on a real eardrum.
To find a mechanical realization for the oscillator mod-
el, it is instructiveto examine the structure of the matrix
elements(29) in the limit in which the area of the plate
A. The ratio of the areas
The deviationof the ratioAF/Av from unityprovidesa
coupleddirectlyto themalleusisonlya smallfractionof the
measure of the extent to which eardrum transduction cannot
total area of the eardrum (i.e., •c• 1). Note that if the cou-
becharacterized
asresultingfrom theactionof a singlepis-
plingbetweenthe umboand the adjacentmembrane(i.e.,
betweenthe innerandouterplates)is strongrelativeto the
stiffness
of themajorityof themembrane,
thenIZol< IZcl-
ton. For the oscillator model
= 1
Assume then that
1
'
(35)
Izm
IlZcl
<"lzcl}
IZol
1).
<32>where,forexample,
(o•= K•x/• e and6e--co•R•/K•.
At
thisratioapproaches
In essence,
theseassumptions
imply that the two resonant low frequencies
normalmodesof theunloadedcompoundeardrumarewide•o =lim(AF/Av ) = 1/( 1 + Ke/K • ),
ly separated
in frequencyso that at low and intermediate
frequencies
thehighermodemaybe neglected.
In addition,
since•c• 1, the motionof the plate attachedto the malleus
makeslittle contributionto the total volumevelocityof the
1.25
eardrum.Thelimitingmodeloutlinedhereisessentially
that
impliedby theparameters
valuesusedby ShawandStinson
(1981).lø
•)
_1
(36)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
62.5
I
125
I
250
I
500
I
1000
I
2000
I
4000
i it
I
I_.l
1.00
A
Underassumptions
(32), thetransfer
matrix• forthe
compound
eardrumreduces
to anespecially
simpleform:
'--
0.75
E
I II
0.50
limT.•0
Zd-l
strong
Ao'
(33)
coupling
0.25
asgivenby theoscillatormodel[Eq. (25 ) ]. The strongme-
0
chanical
coupling
between
theplates
ensures
thatthey•move •-•
togetherwhenthe ossicularloadis removed.Hence,Av is
approximatelyconstant.The coupling--whichis not so
strong
thatit prevents
bothplates
frommoving
when
motion
o•
-30
of themalleusisblocked--allowsthemovingplateto applya
force
tothemall•s,thereby
giving
risetoafrequency
variation in the areaAF.
(n -60
Anotherclassof eardrummodels--those
withconstant n
A•--arises when the mechanicalcouplingbetweenthe
platesis weakandtheplatesmoveindependently.•l
The
-5o
transfer matrix then reduces to
Frequency(Hz)
^
lim •
wca•
Zd-
•
=
0
Am
. (34)
coupling
Note that relative to the oscillatormodel, and its realization
asa pairof stronglycoupledplates,thepositions
of thematricescorresponding
to seriesandshuntimpedances
arein-
FIG. 5.Amplitudeandphase
oftheratio•(w) • Av/Av forthreemodels
of
theeardrum:(--), theoscillatormodelofSec.II usingtheparametervalues
wJ2•r = I kHz, wa/2rr = 5 kHz, 6• = 1.5,6a = 0.5,and•o =0.95; (---),
themodelof Kringlebotn(1988) for thehumaneardrum;(' ß ß), themodel of Matthews(1980) for thecateardrumwith parametervaluesfromNeely ( 1981). Theparametervaluesusedfor theoscillatormodelaretypicalof
thosefoundby fittingmeasurements
of the inputimpedance
in cadavers
(Merchant et al., 1988;Rosowskiet al., 1990) to a modelof the human
terchanged.Sinceonly a singleplate contributesto the force
middle ear (gitera and •Zwei•5
• 100I e). The modelof l•[atthevogIMeelydoeg
onthemalleus,
compound-eardr•um
models
withweakcou-
takentobe•ro•ortional
tothe•
pi•ston
area.Notethatsince
t•hemodels
are
plingbetweenthe plateshaveA• approximatelyconstant
(e.g., Matthews,1980;Neely, 1981).
259
J. Acoust.Soc.Am.,Vol.90, No.1, July1991
notexplicitlyspecifythemassof thepistonattachedto themalleus;it was
reciprocal,
AK/A
v = 1/(1+ Z•Y•). InthefirsttwomodelsAy
isconstant;
in the third, A•.
C.A. SheraandG. Zweig:Description
of eardrumtransduction
259
IV. SUMMARY
as givenby Eq. (15). Note that if K e < K d as arguedabove,
then 1/2 < So< 1, in agreementwith Eq. (15). Combining
Eqs. (27) and (28) yieldsthe further inequality
coe<COd.
(37)
AlthoughAv is constant,
a localmaximumin IAF(CO)loc-
cursat roughly
COp•ak
--X/•-•
whenI1-- •ol'• 1 and
CO½/COd
• 1.
Figure5 showstheratioof the effectiveareas•(CO)asa
functionof frequencyfor the oscillatormodel.Shownfor
comparison
are the predictionsof otherlumped-parameter
modelsof the eardrum for which •(CO) is well defined
(Kringlebotn, 1988;Matthews, 1980;Neely, 1981). Eachof
themodelspredicts
that ]•(CO)[ isessentially
constant
andof
O( 1) at lowfrequencies
but risesto a maximumat frequenciesabove 1 kHz. (With the given parametervalues,the
oscillator
modelpredicts
a [•(CO)[ closest
to unity;thatmodel isthusthemost"platelike"at lowfrequencies.
) The position, magnitude,and sharpness
of the peak,however,vary
The paper presentsa framework,basedon the matrix
T•, for the phenomenological
descriptionof eardrumtransduction.Measurements
determiningthematrixelementsare
defined.The measurements
areinvasiveandinvolvemanipulation of the middle-earossiclesto vary the load on the
eardrum.(Measurements
of eardrumsurface-displacement
patterns do not determinethe transfer-matrixelements.)
Within its regionof validity, the transfermatrix of the eardrumconstitutes
a commongroundwheretheoryandexperiment may be systematicallycompared.In addition: (1)
Analyticityand symmetryconditionsthat may placeconstraints on the measurements of the matrix elements are
identified.Those constraintsmay be usedeither to reduce
the numberof measurements
necessary
to characterizethe
eardrumor to checkthe consistency
of measurements
that
overdetermine
the system.Causalityconstrains
the analytic
structureof the matrix elementsindividually.Reciprocity
and minimum-phase
behaviorare constraints
whoseapplicabilityto the eardrumdependson the natureof its dynamics.The principleof reciprocity,if foundto be applicable,
dramatically
between
themodels.
I,,.n
themodel
ofKringle- provides
algebraic
const?nts
amon3•
theelements.
(2) Two
bothAv is constant;
the peakin [AF(CO)]resultsprimarily
from a sharpresonance
in the impedance
representing
the
structures,principallythe annularring, that suspendthe
complexeffective
areas,A• (CO)andAv (CO),
comprise
twoof
thematrixelements.
Thosetwoareasareassociated,
respectively, with the transmissionof forceto the m•ddle-earossi-
eardrum
in theearca•al.Themodelof Ma•hews-Neely clesand with the changein volumeof the tympaniccavity
predicts
thattheareaAF isconstant
butthat [Av(CO)]hasa
broad, shallow minimum--producinga maximum in
If(CO)[--at frequencies
wherethetwo (mechanically
uncoupled) platesrepresenting
the eardrummovewith opposite
resultingfrom motionof the eardrum.Reciprocityimplies
that thosetwo areascannotbe equal.(3) For purposes
of
illustration an oscillator model of the transfer characteristics
of the eardrumis presented.The oscillatormodelusesreci-
phase.
In manyeardrummodels(e.g., Onchi, 1961;Zwislocki,
1962;Shaw and Stinson,1981) the matrix elementsare not
procity,andtheassum•ed
constancy
of theareaAv(CO),
to
determinethe elementA F (CO)in termsof otherswhoseform
can be approximatedby a Laurentexpansionin ico.The
modelmatrix elementsare minimum-phase
functions.
determined
because
someeardrumimpedances
areeitherincompletely
specified
or lumpedtogetherwithparameters
describingthe middle-earossicles.
Althoughno quantitative
comparison
ispossible,
theratio ]•(CO)
I predicted
by Shaw
and Stinson's(1981) two-pistonmodelcanbe expectedto
ACKNOWLEDGMENTS
The authors thank William Peake and John Rosowski
resemble that found for the oscillator model.
for theircomments
onanearlierversionof themanuscript.
This work wassupportedby theTheoreticalDivisionof Los
Although
other
authors
donotgi•eexplicit
expressions
for theelements
of thetransfermatrixT• of theeardrum,the
AlamosNational Laboratoryanda NationalScienceFounimplicit assumptions
underlyingtheir work can sometimes dationGraduateFellowship
to C. A. S.
beinferredfrom the topologiesof their middle-earnetworks.
Most authors(e.g., Zwislocki, 1962;Lutman and Martin,
Equations
similar
toEqs.
(3),withaspecific
form
for•, were
obtained
1979;
Matthews,
1980;Lynch,1981) implicitly
assume
that
by Esser(1947) byconsidering
a simplemodelof theeardrum.However,
Esserseemsto haverealizedneitherthat equations
of that form follow
immediatelyfrom the linearityof the mechanics,
nor that the coefficients
in hismodelwererelatedby theprincipleof reciprocity.
theeffective
areafor forceAF(CO)isa realconstant
independentof frequency.Onchi ( 1961) and Kringlebotn( 1988),
bycontrast,
maketheimplicitassumption
thattheareafor
Although
atsome
frequencies
themalleus
mayvibrate
inallthreespatial
velocityAv (CO)is a real constant.For the reasonsoutlined
above,that approximationwasexplicitlyadoptedin the os-
dlmension$--$u•½atin• that a •impl• •,r•n•f•r-matrin d•a•ripfion ia not
cillatormodeloutlined
in Sec.II. In •histwo-pis•nmodel,
Shaw(1977) effectively
assumes
thatAv (CO)andAF (CO)are
complexfunctionswith a frequencyvariationdependenton
thedetailedinteractionof thepistons.As shownabove,how-
possible--itmaybethattheeffectiveinputto themiddleearmaystillbe
onedimensional.
Forexample,
vibrationin theplaneof thedisplacement
of thestapes
maybemostsignificant
for coupling
energyintotheinner
ear.
Althoughpresumably
concentrated
at the umbo•the malleusis most
that Av (CO)is approximatelyconstant.The transfer-matrix
elementsfor more detailedtheoreticaldescriptions
of the
eardrum (e.g., Funnell et al., 1987; Rabbitt and Holmes,
firmly attachedto the eardrumat the umbo(Graham et aL, 1978;Shuknechtand Gulya, 1986)--the total forceF• on the manubriumresults
fromforcesdistributed
all alongthetympanicmembrane-malleus
attachmentand can thusbe expressed
as an integraloverthe areaof contact
betweentheeardrumandthemalleus(cf. TonndorfandKhanna,1970;
KhannaandTonndorf,1972). If mallearmotionis predominantly
rota-
1986) are not available.
tionaltheforceF. canbeconsidered
aneffective
force,actingat thereal-
ever,•Shaw
andStinson's
(1981) parameter
valuesimply
260
J.Acoust.
Soc.Am.,Vol.90,No.1,July1991
C.A. SheraandG.Zweig:
Description
ofeardrum
transduction
260
learmoment
arm1manddefined
interms
ofthetotaltorque
bytheequa-
Bode,H. (1945). NetworkAnalysis
andFeedback
AmplifierDesign(Van
tion
Nostrand-Reinhold,
Princeton,N J).
Brillouin,L. (1946). WaoePropagation
in Periodic
Structures
(McGrawHill, New York ).
If, inaddition,
themalleus
doesnotrotateasa rigidbody,but,forexam-
Buunen,
T. J. F.,andViaming,
M. S.M. G. (1981)."Laser-Doppler
veloc-
ple,flexesnearthetip of themanubrium,it maybemoreusefulto define
ity meterappliedto tympanicmembranevibrationsin cat," 1. Acoust.
Soc. Am. 69, 744-750.
¾utobethe"effective
velocity"
oftheumbo,defined
intermsoftheangu-
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&hofthehead
ofthemalleus
(measured
attheaxisofrota-
Dallos,P. (1975). TheAuditory
Peripher9•.
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Decraemer, W. F., Khanna, S. M., and Funnell, W. R. ]. (1989). "Interfer-
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Donahue, K. M. (19891. "Human middle-ear mallear motion: Models and
4NotethattheforceFuismeasured
relative
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byforces
fromtheligaments
ofthemid-
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M. S. thesis,MIT, Cambridge,MA.
die-earossicles)
present
whenthemiddleearisquiescent
andwhichhelps
Esser, M. H. (1947).
to maintainthe configuration
of theeardrum.Because
staticforcesfrom
theligaments
helpmaintaintheequilibriumpositionaboutwhichtheear-
Foldy,L. L., andPrimakoff,H. (1945). "A generaltheoryofpassive
linear
electroacoustic
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andtheelectroacoustic
reciprocity
theorem.
drum vibrates, one cannot establish a no-load condition on the eardrum
bysimplyremoving
themalleusaltogether.
Onecan,however,establish
a
no-loadcondition
byemploying
a feedback
loopandnull-detector:
In additiontodrivingtheeardrumacoustically,
oneapplies
anadditional
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directlyto themalleus
sothatthetotalmeasured
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I," J. Aeoust. Soc. Am. 17, 109-120.
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•Thedispersion
relations
(9) and(10) arevalidwhenthematrixelement
T,•(co)•0 as•o• •o. Whenthatisnotthecase,modified,
or subtracted,
dispersion
relationsexist(Bode,1945).In addition,thesubtracted
form
ofthedispersion
relations
maybecomputationally
moreconvenient
if, for
222.
the high-frequency
behaviorpoorlydetermined(Zweig and Konishi,
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membrane-malleus
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6McMillan( 1946) demonstrates
violation
ofreciprocity
ina passive
linear
431.
electromechanical
system
containing
somewhat
moreexoticcomponents.
7Forexample,
Kringlebotn
(1988)provides
noprescription
formeasuring
(Shawand Stixson,1986) accuratelydescribeeardrumtransduetion.
SLynch(1981)measured
theinputadmittance
Y,•= U½/P•at theeardrum with the malleus"blocked"(so that F'• = 0) and found
dampedfrequencyresponseof a finite-elementmodelof the eat eardrum," J. Acoust. Soc. Am. 81, 1851-1859.
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oftheelement
Tij(•) isknown
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(andnoequations
defining)theimpedance
associated
with thestructures
suspending
theeardrumin theearcanal.In addition,sincetheequations
definingtheplateandcouplingimpedances
havenoobvious
analogs
for
anactualeardrum,it wouldbedifficult--without
comparing
matrixelements•to determinethe extentto whichtwo- or three-piston
models
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ShawandStinson's
( 1981) parameter
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