1990 - greenwood - cochlear frequency-position function for several species - 29 years later
TRANSCRIPT
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A cochlear frequency-position function for several
species29 years later
Donald D. Greenwood
School fAudiologyndSpeech ciences, niversityfBritishColumbia, ancouver,ritishColumbia
V6T1WS, Canada
( Received 0 May 1989; cceptedor publication 6 January1990)
Accurate ochlearrequency-positionunctions ased n physiologicalatawould acilitate
the nterpretationf physiologicalndpsychoacousticatawithinandacrosspecies.uch
functions ightaid n developingochlearmodels, ndcochlearoordinatesouldprovide
potentially seful pectralransforms f speech ndotheracoustic ignals.n 1961,an almost-
exponentialunction asdevelopedGreenwood,96 b, 1974)by ntegratingnexponential
function itted o a subset f frequencyesolution-integrationstimatescriticalbandwidths).
The resulting requency-positionunctionwas ound o fit cochlearobservationsn human
cadaver arsquitewelland,with changesf constants,hose n elephant, ow,guinea ig,rat,
mouse, nd chicken B•k•sy, 1960), aswell as/n vivo behavioral-anatomical) ata on cats
(Schucknecht,953. Since 961, ewmechanicalndotherphysiologicalatahave ppeared
on the human,cat,guinea ig,chinchilla,monkey, nd gerbil. t is shown ere hat the newer
extended ataon human adaver arsand rom ivinganimalpreparationsrequitewell it by
thesame asicunction. he unction ssentiallyequiresnlyempiricaldjustmentf a single
parameter o setan upper requencyimit, while a "slope"parameter anbe eft constantf
cochlear artitionengths normalizedo 1 or scaledf distances specifiedn physical nits.
Constancyf slope nd orm n deadand ivingearsandacross peciesncreaseshe
probabilityhat he unctionittinghuman adaver atamayapplyaswell o the ivinghuman
ear.Thisprospectncreaseshe unction'salue n plotting uditory ataand n modeling
concernedithspeechndotherbioacousticignals,incet fits heavailable hysiological
datawelland,consequentlyif those ataarecorrect), emainsndependentf, andan
appropriatemeanso examine, sychoacousticataandassumptions.
PACS numbers: 43.64.Kc, 43.64.Bt
INTRODUCTION
Since he late 1960s,more data on the frequency-posi-
tion coordinates of the cochlea have become available for a
numberof species nd supplement arlier data, gatheredby
B•k•sy in the 1940s and Schucknecht n the 1950s. The
newerdata-•on man, cat, chinchilla,guineapig, gerbil,and
monkeymhave ncluded additional species, nd, in some
cases, chieved onsiderableoverage f the cochlear arti-
tion. It may be useful o comparesuchdata again with the
simple requency-positionunctionsdeveloped mpirically
29 years ago from critical bandwidth data in man (Green-
wood, 1961 , 1974). The newerdata ncrease mpiricalsup-
port for a family of suchalmost-exponentialrequency-posi-
tion functions and for a scaling or normalization
relationship, mong hese pecies,hat appearso govern he
slopecoefficient f the function. This means hat these unc-
tions differ essentiallyn only the other main constant.
A review of this development Greenwood, 1974) is
briefly recapitulated. he original requency-positionunc-
tion wasderived rom a critical-band unctionproposedo fit
my critical-band estimates in 1959-1960 (Greenwood,
1961a, b). The developmentof the function assumed hat
critical bandwidths followed an exponential function:
CB = 10 ax b), of distance along hecochlearartition,
and corresponded o a constantdistanceon the basilar mem-
brane. The latter hypothesis ad been advancedand sup-
ported nfluentiallyby Fletcher ( 1940, 1953 and Zwicker et
al. (1957). Correspondencef critical, or other,bandwidths
to equal, althoughunknown,distances n the basilarmem-
branewould imply proportionality o the derivativeof a fre-
quency-position unction of the membrane. The paper of
1961 simply integrated the suggestedcritical-bandwidth
function to obtain a frequency-positionunction, in which
positionon the membranewas expressedn critical-band
units. The length of a critical-bandunit in physicalunits
could henbe determined y dividing he engthof the mem-
brane by the number of critical bands end to end that sub-
tended he audible requency ange.
By coincidence, bout 35 critical bands,according o
this function, subtended bout 35 physicalunits, millime-
ters. The convenient ffectof this correspondenceas hat
the frequency-positionunction hus obtainedcouldbe com-
pareddirectly o B6k6sy's lot of frequency ersus osition
on the membranewithout a changeof slopeconstant o al-
low distancex to be expressedn millimeters. The coinci-
dence hus meant that thesebandwidthswere proportional
to the derivative with a constant of 1 rather than with some
other constant,as would have been mplied by correspon-
dence o a differentdistancen millimeters.The comparison
had two major outcomes: he frequency-positionunction
closelyagreedwith B6k6sy's ochlearcoordinates, nd this
2592 J. Acoust. oc.Am.87 (6), June1990 0001-4966/90/062592-14500.80 @ 1990Acoustical ociety f America 2592
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providedsupport o the two ideas hat the initial critical-
bandwidth estimates and some additional measurements of
frequency eparationsn earlyexperiments n consonantn-
tervals (Mayer, 1894)--later solidlyconfirmedby Plomp
and Levelt (1965) and Plomp and Steeneken 1968)-
might correspondo a constantdistance nd obeyan expo-
nential function. As agreementwith the cochlearmap was
evident, these conclusionswere simply contingent on
whether he physiological ata wereaccurate nd character-
ized equallyboth the quick and the dead.
However,our chief nteresthere s in the frequency-po-
sition unction tself, ndependent f the two hypotheses n-
derlying ts origin,whichmightor mightnot be true or gen-
erallyapplicableo anygivensetof bandwidthestimates.We
focus irst on the most mportant outcome, he closeagree-
ment of the frequency-positionunctionwith B•k•sy's mea-
surements f cochlearcoordinates,he only physicalmea-
surementshen available, nd second n the degree o which
the frequency-positionunction may successfully pply to
physical nd physiological ata from other species.
In 1961, he functionprovideda convenientmathemat-
ical expressionor a cochlear requency-position ap that
was a rational scale or plotting resultsand might assist n
their interpretationwithin and across pecies. he function
fitted not only B•k•sy's human cadaverdata and his similar
results rom six other species ut alsoSchuknecht'sn vivo
cat data, essentially y the changeof only a singleconstant
that determined he upper requencyimit. The other main
constant,which governed lope,couldbe scaledor normal-
ized acrossdead and living preparations,which has tended
to support he map'sapplicability o the living humancoch-
lea. Sincewe necessarilyely still on physiological ata from
living preparations f other species s he mostnearly direct
sourceof inference o man, the relation of the 1961 function,
especiallyts slope onstant, o the augmented ataof recent
and future years will be important, among other reasons,
because f its bearingon the applicabilityof cadaverdata
and the function o living humancochleas.
The frequency-positionunctionobtainedas described
above is
F=A(lOa"--k), (1)
wheresuitableconstants for man) are:A = 165.4 (to yield
frequencyn Hz) and a = 0.06 (if x is expressedn millime-
ters), or 2.1 (if x is expressed s a proportion of basilar
length). The latter constant s an empiricalconstant rising
in the critical-band unction.The integrationconstant was
originally eft at the value 1, but it may sometimes e better
replacedby a number rom about0.8 to 0.9, to set a lower
frequency imit dictatedby conventionor by the best it to
data. Thus the value k = 0.88 would yield the conventional
lower frequency imit of 20 Hz for man, and this value was
almost used n 1961, but we will continue to use 1.0 for man
and otherwise .85 throughout hispaperas wo values hat
seemadequate or the moment or mostof the species on-
sidered ere,althoughLibermanhas ound hat 0.8 bestad-
justs his unction o his ow-frequency atapoints n the cat
and (with appropriateA) sets 0 Hz as ower requencyim-
it. (However, hequestion f the appropriate alue or k may
also be related to the questionof what is the "effective"
lengthof the cochlea, nd what is its effective pical end-
point, so ar as the physics f the cochlea s concerned.
The constant (essentiallyhe slopeof the straightpor-
tion of the frequency-positionunction,when og frequency
is plottedagainst ochlear osition)was oundnot only to be
scalable mong he otherspecies tudiedby Btktsy but in at
leastsomeof those pecies,o agree easonably ith the oga-
rithmic slopeof the volume-compliance radientsmeasured
by Btktsy along their cochlear partitions (Greenwood,
196 b). To say that a is scale elated across wo speciess
simply o say hat, f known or one, t canbe obtained or the
otherby mulitiplicationby a scale actor,determinedn this
case y the ratio of their cochlear artition engths.Or, to say
the same hing, a times basilar ength would be a constant
among cochleas f they were scalerelated n this respect.
Thus this constantproductcan tselfbe taken as he valueof
a amongsuchspeciesf cochlearpositionor distance s ex-
pressed s a proportionof total partition ength (apex = 0,
stapes= 1 .
From both Btktsy's frequency-position ata and his
compliance ata,especiallyor the humanspecies,he prod-
uct of a timesbasilar engthappeared o be about2.1 in 1961.
As then noted, 10 raised o this power s about 126, or the
factor of about 100 with which Btktsy characterized the
variation in stiffness f the cochlearpartition from end to
end.As described bove, his normalizedslopeconstantwas
quite adequate or functions itting frequency-positionata
available n 1961. As will be seen, he new frequency-posi-
tion data from man, cat, chinchilla,guineapig, monkey,and
perhaps erbil end o confirm his valueof about2.1.
I. MORE RECENT FREQUENCY-POSITION DATA FOR
THE HUMAN SPECIES
We consider irst data from human emporalbones,ob-
tainedwith the Mtssbauer echnique y Skarstein Kringle-
botn et al., 1979). The sevennew data pointswere obtained
from seven resh emporalbones,8 to 24 h after death, at
recordingsitesextending rom about the 2.2- to 6.2-kHz
pointson the cochlear artition.The pointsclosely xtrapo-
lated Btktsy data, which had endedat the 2- kHz point, and
also it the frequency-positionunctionproposed y Green-
wood in 1961. The top panel of Fig. 1 is reprinted rom
Kringlebotnet al. (1979). The samedata, replotted n Bt-
ktsy's 1942 ormat, reappear n the bottompanelof Fig. 1,
with additionalpoints epresentedy x's andcrosses erived
from two plottingsby Btktsy in 1943and 1947of displace-
ment envelopes long the basilar membrane [see also
Fletcher's 1953) and Zwislocki's 1965) plottingsof Bt-
ktsy's hree series f visualobservationsn relation o theo-
reticalcurves.l
Although death could reasonably e expected o have
exerted some nfluence on both setsof data, that effect in the
observations resentedmay havebeen elatively imited. Bt-
ktsy's accounts f proceduresndicate hat, observing giv-
en turn, he not only made observations f amplitudemilli-
meter by millimeter along the partition away from the
positionof maximum amplitude,but that "phasemeasure-
ments provided a sharp definition of position, especially
2593
2593 J. Acoust. Soc. Am., Vol. 87, No. 6, June 1990 Donald D. Greenwood:Cochlear requency-positionunction
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,9, 10
ßvo..rgs•
2O 2O0 2O0O 2OOO0 •Z
100 1000 10000
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mates of cochlear ength [from thoseof Schuknecht nd
Neff ( 1952); Schuknechtand Sutton, (1953) ].
But, in addition, given his revisedestimateof 25 mm
(instead of 22 mm) for the average basilar membrane
length,whichaltered he scale actor,hisdata also ndicated
that the constantA, which determineshe upper requency
limit, wasnot scalable rom the corresponding onstant p-
plicable o humandata.SuchanA, if scaled romman, ed o
a functionparallel to, but displaced rom, the new cat data.
In short, the earlier relation (suggested y Schuknecht's
lengthestimateand parallelbut shifteddata), in which the
cat upper frequency imit indicatedby the displaceddata
equaled he upper requencyimit in man multipliedby the
square f the scale actor,wasshownby the correctedength
andnewdata to havebeenadventitious, s t may well bealso
for the elephant,which, among he eightspecies onsidered
in 1961, was the only other whoseupper frequency imit
suggestedhis relation (Greenwood,1961 ). However, he
parallelcourses f the newcat dataand he displaced urve,
whose value of A was too small, demonstrated that, for an
appropriately elected , generating n empiricallycorrect
upper requencyimit, Function 1 must it the newdataas
well, or better than, it had fit the old.
Hence, as Greenwoodhad done in 1961 for the eight
species onsideredhen, an appropriate alue or the con-
stantA for the catwassimplydeterminedrom thenewdata,
which Liberman found yielded a best-fittingvalue of 456
(for Hz), as compared o the originalvalueof about418 in
1961.Figure 2 presentshe cat data n relation o Function
( 1 , when Liberman best-fittedall constants o the data. In
the final analysis, he 1961 cat functionhas becomea func-
tion well fitting the new data by revisingA upwardby about
9% and by reducingk from 1 to 0.8 to providea better it to
the new ow-frequency ata points,while the expected re-
mainsat 2.1 whendistances to beexpressedsa proportion
0.•
0.6
1.0-
0 --
ß
...,
_
_
.
_
.
i I '1 [ i ii1| ' I I I I Ill I I I I I'l
o.i i.o io 60
CHARACTERISTIC FREQUENCY (kHz)
FIG. 2. Adapted romLiberman 1982). Curve sFunction 1 fitted o the
data,withx expressedsa proportion f totalnormalizedength rom he
apex.Data display elation etween rimary iberCF andcochlearocation
in the cat; eachof the 52 fiber ocationswas normalized o its respective
cochlea.Circle, X, and illed riangle efer o high,medium,and ow rates
of spontaneousischarge. ean engthof cochlea ndbest-fitting onstants
areprovided y Liberman s ollows:Mean engths25 mm;A (for frequen-
cy n Hz) becomes56 (a 9% increaserom he 1961 unctionittingSchu-
knecht's ata); a remains .1 as n man,or becomes .1/25 = 0.084 (rather
than 2.1/22 as in 1961 ifx is to be scaled n mm and referenced o mean
cochlearength;k = 0.8 (rather than 1 .
ofbasilar engthor, if averagemillimetermeasures desired,
while 2.1 is divided by 25 mm (instead of by 22 mm, as in
1961).
B. Other cat data: Spiral ganglion cell CF and
mechanical measurement
Additional requency-positionata had earlierbeenob-
tained in the cat by Kohl16ffel 1974, 1975). The CFs of
single piralganglion ellswere elated o cell ocations,ela-
tive to the baseof the cochlea, y radially projecting rom the
locusof penetrationn the ganglion o thebasilarmembrane.
Thus thesedata alsowere rom living cochleas naffected y
errorsof reconstructionndpresumably lso ongeron aver-
age han the older 22-mm estimateof mean ength.All but
one of 105 cells were 2 to 4 mm from the basal end, and all
but three,when grouped or analysis, ell in 2-dimensional
"bins" (0.5 mm X 5 kHz) whose central CFs were between
20 and 35 kHz. The coordinates f the cells were in quite
goodagreement, hencalculations f position rerelated o
the basal end, with both Liberman's data and Function ( 1
in Fig. 2 and with the original 22-mm function since he
functionsdiffer rather little, relative o the basalend, n plot-
ting high CFs.
Two bodies of mechanical data had been obtained over
the basal 9 mm of the cat cochlea, by Wilson and Evans
(1977) and n a smallbasal egionby Khanna and Leonard
(1982). The data of Wilson and Evans had demonstrated
good agreementn slopewith Schuknecht's1953) basal
data and the 1961 unction.However, he slopeof thesedata,
judgedvisually,may be slightlymoregradual han the 1961
function and Schuknecht's ata), both of which wereplot-
ted on a 22-mm membrane, which would indicate that the
Wilson and Evans slope s in still closeragreementwith
Function ( 1 and Liberman'sdata n Fig. 2, whichpertain o
a 25-mm membrane.This closeness f slopewas showndi-
rectly by Liberman's eplot n his Fig. 8 of the Wilson and
Evans data in comparison o his data in Fig. 2. Although
their slopes parallel o Liberman's atacurve, he mechani-
cal data are displaced oward ower frequencies n the ab-
scissa basally on the ordinate). On the samegraph, the
Khanna and Leonard data agree n displacementwith the
replottedWilson and Evansdata, while Kohl16ffel's eural
data agree,as notedabove,with Liberman'sneuraldata.
In summary, he frequency-positionunction or the cat
appearso bewell andconsistentlyetermined y fivebodies
of data. Liberman'sdata,by covering he wholecochlea nd
establishinghe most reliable estimateof cochlear ength,
well supportshe function's orm while t enhanceshe con-
sistency f the slopeconstant , whoseapparentscalability
across peciess our second ocusof interest.However, he
paralleldisplacementsotedabovebetween omebodiesof
datarequirecomment nd are relevant o the interpretation
of someof the frequency-placeata reported or other spe-
cies.
C. Why might mechanical peak frequencies be
displaced basally from primary-fiber CF data?
Although he slopes re the same, he peak requencies
in the mechanical ata are displacedrom Liberman'sdata
2595 J. Acoust. Soc. Am., Vol. 87, No. 6, June 1990 Donald D. Greenwood:Cochlear frequency-positionunction 2595
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relating unit CF to position.Liberman isted wo possible
explanations: ochlearmechanics re disrupted y the pro-
cedures equired o make the mechanicalmeasurements;r,
alternatively, hat the frequency equired o producemaxi-
mum amplitudeat a givenspot s simply ower han the CF
of a primary fiber nnervating he samespot.However, he
displacements about6% of cochlear ength,or 1.5 mm on
the ordinate.This displacement ouldseemmoreeasilyun-
derstoodon the first basis han the second, or the following
reasons.
On the hypothesis f cochleardisruption,a shift of the
maximumamplitudeof a displacement nvelopeoward the
basewith cochlear njury wouldaccount or the requirement
of a lower frequency han normal to place hat maximumat
any givenpoint. Suchshiftsdue o injury, anoxia,and death
have beenseen n severalstudies Kohll/Sffel, 1972b;Rhode,
1973; LePageand Johnstone,1980; LePage 1981; Khanna
and Leonard, 1982; Sellick et al., 1982; Robleset al., 1986),
and a shift of 1.5 mm may not be too large.
As for the secondhypothesis, ote that an increaseof
stimulus frequencysuch that the displacement nvelope
shifted .5mmbasally between and«oct n thecat) before
the stimulus requency eached he CF of the primary neu-
ron innervating hat point wouldbe arge elative o the api-
cal segment f the envelope. ince he apicalsegment f the
envelopen the basalhalf of the cochlea s probablyonly
about0.71 mm in the squirrelmonkey (if cochlear ength s
20 mm, but see ater comment) and about 0.66 mm in the
guineapig ( Greenwood, 974), it isprobably o more han a
scaled distance of about 0.9 mm in the cat. A 1.5-mm shift,
larger han the lengthof the apicalsegment f the envelope,
argues hat the second ypothesis bove, akensingly,would
require hat the envelope ecome maximallyeffective tim-
ulus or a neuron nnervating his point when he envelopes
shiftedso ar basally hat even he apicalfootof the envelope
is about 0.6 mm basal o the point in question.A primary
neuron excited most effectively (at lowest threshold) by a
tone whose argestamplitudeeffectsdo not reach he neur-
on'spoint of innervationwould not easilybe accountedor
with current conceptions.t is conceivable, f course, hat
both alternativehypothesesogethercouldeachaccount or
a part of the displacement f the data curves, ut the second
for only a presumablyimited part.
D. Why might frequency-place correlations of hearing
loss be displaced basally from primary-fiber CF
correlations with place?
Liberman (1982) also summarized data basedon corre-
lationsof the CFs of singleauditoryunitsshowing hreshold
shifts Liberman and Kiang, 1978) with placeof noisedam-
age, n comparisono the data of Schuknecht 1953), which
correlated requencyof hearing osswith cochleardamage.
Both setsof data showa systematic hift toward ower fre-
quencies way rom the curveshown n Fig. 2 based n prim-
nary fiber CF. Liberman notes hat Robertsonet al. (1980)
havesuggestedhat, in cases f chronicsurgicalesionso the
organof Corti, the CFs of fibers rom damaged ites anshift
to ower requencieshatareasmuch s« o «octaway, the
same nterval noted above).
The same basal shift of the displacementmaximum,
owing to cochlear njury, that would account or the dis-
placement f the mechanical ata rom Liberman's rimary-
CF data and the curve n Fig. 2 shouldconstitute robably
the most mportant factor causing a) lower fiber CF with
elevationof fiber hreshold, nd (b) a lowercutoff requency
for hearing oss han the normal CF for the point at which
damagebegins.
Thus, if outer hair-cell (OHC) damageor death inear-
izesand reduces asilarmotion at a givenpoint nnervated
by a primary iberunderstudyand causes basalshiftof the
positionof maximumamplitude hat would normallyoccur
at this point for a tone at the point'snormal CF, then these
changes lso should aise the studied iber's hreshold o a
tone at its normal (and now ex-) CF. Thresholds will be less
affected,however, or tone frequenciesower than the for-
mer CF, frequencies t which he point's esponses normal-
ly linear or more nearly linear. Moreover, tonesof these ow-
er frequencieswill now be required in order to place
maximum displacement mplitudeat this point. Thus, de-
spite higher thresholdsat the lower frequencies,he now
greater elativesensitivity f the basilarpoint and ts inner-
vating iber o tones ower than the former CF will establish
a new and lower CF for both point and fiber. As a result,
also, behavioral threshold to tones at the former CF of units
innervating he damagedpoints shouldbe higher, making
hearing ossbeginat frequenciesower than the normal CF
for the point at which damagebegins.
III. CHINCHILLA: FREQUENCY OF HEARING LOSS
VERSUS POSITION OF DAMAGE
A map of frequencyversusposition or the chinchilla
(Eldredge et al., 1981 is basedon the relation betweenau-
diometric eaturessuchas notchesor abrupt transitions n
sensitivity o correspondingesionsof the organ of Corti.
They discussedhe considerable ariability in data of this
kind in a review of the experimentalmaterials,methods,
problems, nd sources f variability.Although the variabil-
ity in suchdata reducesheir power o indicatedifferencesn
goodness f fit to functionsof different orm, the methodolo-
gy and quantityof data allow confidencehat the coefficient
of the exponentof the simpleexponential unction itted to
the data (i.e., the slopeconstantwhen frequency s logged)
mustclosely pproximate he average lopeof any otherem-
pirical or theoretical urve o be compared ventually o the
data. Six correlationsby Ryan and Dallos (1975) of OHC
losswith the corner requency f hearing oss ranging rom
1-8 kHz) yield locations hat the presentauthor finds fit
within the rangeof variation oundby Eldredgeetal. andare
on averageabout 0.75 mm aboveFunction ( 1 in Fig. 3.
Eldredgeet al. give the average ength of the cochlear
partition n the chinchillaas 18.4 mm and report he chin-
chilla hasnearly he same requency angeasman. The sec-
tion they considered asbasal o a point about5.6 mm from
the apex; he corresponding oint in man is about 10.6 mm
from the apex.This is a restriction n both caseso the fre-
quencyrange aboveabout 500 Hz, the lower limit of the
chinchilladata and the frequency bovewhich B•k•sy's hu-
man data are well known o approximate straight ine on a
2596 J. Acoust. Soc. Am., Vol. 87, No. 6, June 1990 Donald D. Greenwood: Cochlear frequency-position unction 2596
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%
ioo
• 8o
o•60
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I I
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. I .' I , . I,,,,I
Ol
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6
14
12
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6
4
2
I • • 1,•1 I 0
0.2 0.5 1.0 2.0 .5.0 I0 20 k Hz
Frequency in Kilohertz
% 0.1 1 10
X 00...................
80
,,,, •' , , , , ,,,,• , , ...... • ,
O. 1 1 ..10
Fne•uency • n K• I ohentz
FIG. 3. Upper graph:Taken from Eldredgeet al. ( 1981 . Relationof fre-
quencyof hearing oss n chinchilla o placeof cochlear esion,basedon
audiometriceatures uchas notcl•es r abrupt ransitionsn sensitivity.
Straightineexpresseshesimple xponentialelation rovided yEldredge
et al. asasbest-fito theirdata.Dashed ection:egion f no data--apical
30% of cochlea;requency ange epresentedy data iesbetween .5 kHz
to about15kHz. Lowergraph:Straight inerepeatshe same imple xpo-
nential unction boveprovided y Eldredge t al. Dashed ection sabove.
Curved ine represents unction 1 with samea applicableo man or cat
(or 2.1/18.4 fx is to be expressedn mm of an average ochlea, = 163.5
asdetermined y dataof Eldredge t al. (to yield hesameupper requency
limit as heirexponential), ndk is eft at 0.85,whichseems dequateo the
purpose. olidpointprovided y Ruggero ndRobles 1984,personal om-
munication; obles t al., 1985).A parallel hiftof thestraight art of the
curvedownward y about1/3 mm would ncrease pper requencyimit by
about2 kHz. For comparison, similarshiftdownwardby about 1.25mm
would ncreasehe upper requencyimit to about30 kHz.
log-linear lot. The coefiScientf the exponentialhat El-
dredgeet al. fitted to the chinchilladata (basilar distance
expresseds a proportionof total length) was about 5.1
(base or 0.277, f distances expressedn mm). In man,a
simpleexponential itted to the human data above500 Hz
wouldhaveabout he samevalueof 5.1; 5.1 convertso 2.2
(base 10).
Thus an almost-exponentialunction, Function (1),
can alsoeasilybe superimposedn the chinchilladata, with
the slopeconstanta scaled rom man or cat. Its value re-
mains2.1 if distances expressedsa proportionof cochlear
length or is about 0.114 (0.263 base e), if distance s ex-
pressedn mm. The upper requencyimit (20.5 Hz) given
by the exponentialunctionof Eldredge t al. is maintained f
the constant in Eq. ( 1 is set o about163.5,a valuenearly
the sameas he constant or man, whichwouldequallysuf-
fice.
Also shown n Fig. 3 (solid circle) is the determination
by Robles t al. (1985) ofa CF of 8.35kHz for a pointabout
3.5 mm from the baseof the cochlea.This point deviates
apically rom Function (1) by only about0.13 mm. How-
ever, the chinchilla data, like the cat data of Schuknecht
(1953) and Liberman and Kiang (1978), are basedon a
correlationof hearing oss utoff requencies ndpositions f
cochlear esions. f the chinchilla curve correlatingcutoff
frequencyof hearing oss o position urned out to be dis-
placedupward on the ordinate (from one relatingunit CF,
or cochlear-pointCF, to position)by an averagedistance f
aboutone-thirdof a millimeter, an equalcompensatory hift
of Function (1) downward n order to representprimary
fiber-CF data would increase he upper frequency imit by
only about 2 kHz. The shifted curve would also still pass
close o the mechanical bservation, eviatingby only about
0.19 mm, this time passing elow t. In short, he chinchilla
data, even if systematicallydisplaced somewhat from a
curve that might be basedon primary-CF data or from a
curve hat describedhe true positions f displacement nve-
lope peaks,not only provide a well-definedslopebut prob-
ably are quiteclose o a curve hat might be basedon further
direct determinations ike those of Robles et al. (1985).
IV. SUMMARY OF GUINEA PIG FREQUENCY-POSITION
COORDINATES
A. Frequency-position data
The two functionspublished n relation to the early
guineapig data (Greenwood, 196 b) were both of the form
of Function (1), although B6k6sy'sdata curve, showing
some eversedcurvaturenear the apex, was not. One func-
tion took into accountonly B6k6sy'sdata from dead coch-
leas and extrapolated he straight upper end of the curve.
The other attempted o fit B6k6sy's ata as well as possible,
while itting also he approximately 0-kHz upper requency
limit reported rom living cochleas y Pestalozza nd Davis
(1956). The slopeconstants f the two functionsbracketed
the value of 0.1135 that would be obtainedby scaling rom
the value applicable o man.
Later, Kohl16ffel 1971 provideda precise etermina-
tion of the point of maximum CM responseo tones rom
13.5 to 14.5 kHz and republished orrelationsof cochlear
damagewith exposure one frequencyby Smith and Wever
(1949) and by Neubert and Wiistenfeld (1955), all of which
were n betteragreementwith the functionyielding he high-
er frequency imit. Kohl16ffel's 1972a,b) laserstudyof the
basilarmembranen deadguineapigpreparations nd wo n
vivopreparations Kohl16ffel,1972c), alsosupported oth
the approximate orrectness f a slopescaled rom the value
for man and the higherupper requency imit. All these e-
sultsoverall are in still closeragreementwith the function
discussedelowand appearwith it in Fig. 4.
Wilson and Johnstone ( 1972, 1975) summarized their
own data from the basal 4 mm of the cochlea and those of
others, including Kohl16ffel's 1972c) in vivo determina-
2597 J. Acoust.$oc. Am., Vol. 87, No. 6, June 1990 DonaldD. Greenwood:Cochlear requency-positionunction 2597
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0.1
X -
ElO-
0 -
-
_
$ -
(J 5_
0
o
o. 1
FIG. 4. Frequency ersus ochlearposition n the guineapig. nset graph:
Data as plottedby Kohll/Sffel 1971 . Solid square:Kohll/Sffel's oint of
maximum responseo 13.5- to 14.5-kHz tones,basedon CM responsese-
cordedby a 12-electrode rray ( 150-/zm nterelectrode pacing).Solidcir-
cle representsnferred ocationof 15 kHz maximumbasedon a 0.22-mm
basalshift of CM minimum. Short vertical ine represents amage o the
cochleawith exposure o a 10-kHz tone (Smith and Wever, 1949). Long
vertical ine ndicatesegionof swollen air cellsafter ongexposureo a 15-
kHz tone NeubertandWiistenfeld,1955 . Opensquare: ohll/Sffel's oint
of maximum mechanical esponse in vivo) to 28 kHz, repeated n main
graph.Main graph:Solid points epresentmechanical nd inner hair-cell
data. Open points are based on cochlearmicrophonicmeasurements.
Crosses( ) representhe CFs of primaryneurons. olidsquare:Kohl16f-
fel ( 1972c)--28-kHz peak requency f mechanicalesponset point 1.5 o
1.7 mm from base n two in vivopreparations. olid riangles:Wilson and
Johnstone 1975)m13 cutoff requencies f mechanical esponse urves;
five symbolson or touching ine, with four below and four above.Solid
circles: ussell ndSellick 1978, 988 able fdatapointsmpersonalom-
munication)mCFsof 14 ndividual nner hair cells;six of thesepointsare
virtually on the line and, in the cluster, at about 15 mm, there are seven
points, iveof whichare among he sixon the ine.Open nverted riangels:
Schmiedtand Zwislocki ( 1978)--peak frequencies f responseunctions
basedon cochlearmicrophonic ata. Open circles:Dallos' cochlearmicro-
phonicdata as plottedby Wilson and Johnstone 1975). Crosses: obert-
son and Manley (1974)m13 CFs of spiral ganglion ellsversusposition,
points hat are nearlyparalleland about0.5 mm above he ine;onepoint s
undera circle.Wilson'sand Johnstone's eak requencieslso ollow the
slopewell, displaced n average omewhat o lower frequenciesWilson
(1972) ]. B6k6sy's uinea igcurve snearlyparallel o the ine rom about5
to 12 mm from apexand displaced bout 1.9 mm basally upward) on the
ordinate. The curve is Function (1), where ,4 =0.35, a= 2.1/18.5,
k =0.85.
tionsof the 28-kHz point and estimatesheybasedon Dal-
los' CM measurements t a numberof loci. They fitted these
dataon a log-linear lot with a straight inewhose lopewas
0.1204commonog unitsper mm (2.5 mm/oct) and whose
basal nterceptwas45 kHz (Wilson and Johnstone, 975).
This slope onstant onvertso 2.22 (base10), or 5.13 (base
e) (if distancesexpressedsa proportion fbasilarength),
close o the values or man and chinchilla f simpleexponen-
tials are used.Their own peak frequencies lso ollow the
sameslopewell, somewhat picallydisplaced n average o
lower frequency oints Wilson and Johnstone, 972).
If, instead, n equationof the form of Eq. (1) is used,
thesedata canbe quitewell fitted with the scaled lopecon-
stant of 2.1/18.5 = 0.1135 (2.1 for proportionaldistance)
and the value of about 0.35 for the constant A. This function
yields an upper frequency imit of about 43.8 kHz, and
agrees qually loselywith mostof the data.
The determinationsof inner hair-cell (IHC) CFs by
Russelland Sellick ( 1978, 1988, personalcommunication),
at positionsromabout1 o 4.5 mm from hebase, realso n
goodagreementwith the Wilson and Johnstone ata and
Function ( 1 . Immediatelyon the low-frequency ideof the
main concentration f IHC points,of which six are on the
line, Kohl16ffel's measurementsof the 14- and 15-kHz
points romhisCM recordingslso all on he unction. wo
points representpeak frequencies f responseunctions
basedon cochlearmicrophonic ata of Schmiedt nd Zwis-
locki (1977). An additionalgroup of pointswaspublished
by Robertson nd Manley ( 1974); the CF of a cellrecorded
in the spiralganglionwaspairedwith the cochlear oint at
the endof a line radially projected bout700p to the coch-
lear partition.Thesepointsare nearlyparalleland slightly
basal o the function. Robertsonet al. (1980) later plotted
the CFs versus basilar location of 102 neurons between
about 1.5 and 5 mm from the base.They agreequite well
with Wilson and Johnstone's 1975) simple exponential;
about 62% of the pointsare apical to the function.The
points gree bout quallywellwith theFunction 1 in Fig.
4; about58% of the pointsare basal o the function.Quite
consistent ith functionslope s the reportof Johnstone nd
Taylor (1970) of a shift n peak frequency rom 19 to 16.5
kHz, with a 0.5-mm shift of the Mfssbauer source rom 1.5
to 2 mm from the base.These ocationswould be displaced
basally rom the calculated urveby about1.67mm, but the
calculated eparation f the 19- and 16.5-kHzpoints s 0.53
mm, in goodagreementwith the shift of their source.
Thus he various odies f dataplotted n Fig. 4, despite
any residualuncertainties, eem o be reasonablyit by the
functionsuperimposedn them, which possesseshe same
normalized lopeconstant sed or the precedinghreespe-
cies. t isalsoveryclose o the average lope f thebasal wo-
thirdsof B6k6sy's ata,whichare shiftedbasally, way rom
the in vivo function, in the way describedby Kohllfffel
(1972b).2
B. Shape of the functionsdegree of curvature
It may well be, of course, hat the degreeof curvature
near heapexmaydiffer romonespecieso another epend-
ingon heirevolutionary pecializations,ot o mention hat
the form of the required unctionmay differ somewhat s
well. Since he function itted here to the data is empirical,
there s nothing o argue hat the samevalueof k is suitable
for everyspecies, venamong hose or which he general
form may providean adequate it to the data.
However,as to the suggestionr possibilityhat a sim-
ple exponential,hat is, a straight ine on a log-frequency
versuspositionplot, may be adequate or a givenspecies
everywheren the cochlea,a second onsiderationmust be
raised. If evidence exists that very-low-frequency ones
causemostof the cochleaof the givenspecies,ncluding he
apical egion, o vibratealmost n phase nd if it is known
(viz., Andersonet al., 1971 or theoreticallypredicted hat
phase ravel time to a point is an exponentialunctionof
2598
J. Acoust. Soc. Am., Vol. 87, No. 6, June 1990
Donald D. Greenwood' Cochlear frequency-positionunction
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distanceraveled o that point, hen requency annotalsobe
laid out exponentially n the apical region (Greenwood,
1977). To make this more intuitively clear, consider hat
phase t a point ssimply heratioof phaseravel ime o that
point over he tone'speriod. f we know how two of these
variables hange,we know how the third changes. latten-
ing of the spatialphase urveoccurs e.g., n man, cat, and
squirrelmonkey)becausehaseravel imeat theposition f
maximum amplitude, for example, obeysan exponential
functionof distanceover he apical three-fourths f the par-
tition, whereas he frequency-positionunctiondoesnot de-
creaseexponentiallyat progressivelymore apical points--
where octavesbecome crowded. Hence, the period (fre-
quency'seciprocal)of the tone eachingmaximumat those
progressively oreapicalpoints ncreases ore hanexpon-
entially, in the denominatorof our ratio. In effect, a low-
frequencyonedoes ot travel o points ar enough own he
cochlea or its period (ratio's denominator) to constitutea
smallor constantproportionof the tone'sphase ravel times
to apicalpoints (ratio's numerator). Rather, the reverse s
true, leading o smaller valuesof accumulatedphase (i.e.,
flattenedspatialcurves) or low-frequencyones.
Thus, n the guineapig, if spatialphasecurvesbecome
flatter n the apical egion han elsewhere, hile he relation
of phase ravel time to positions f maximumamplitude n
that region s knownor believed o be exponentiallyelated
to cochlearposition, hen frequencyn the guineapig will
not be laid out logarithmically ver the whole cochleabut
will resemblenstead he patternseen n man, elephant,and
cat. The same easoningmay be applied o the chinchilla, n
the regionbelow500 Hz whereFig. 3 lacksdata,and o the
gerbil,where data are scant.
monkey might be about 2.1/23 = 0.091 for a and 0.370 for
A, yieldingan upper requency imit of 46 kHz.
Estimates f frequency f hearing ossversus he posi-
tion of IHC loss or four M. Nernestrinamonkeys avebeen
reportedby Stebbins nd Moody (1979). These our mon-
keyshad an averageengthof cochleaof 25.6 mm (Stebbins
and Moody, 1988, respectivepersonalcommunications).
The cutoff requencies f hearing osswere correlatedwith
the position of 50% IHC loss to the nearest0.5 mm and
supplied o me with the individualcochlear engths.These
four positions re replottedn Fig. 5, after irstnormalizing
them with respect o individualcochlear engthsand then
converting hem to positions elative to the mean cochlear
length among these our animals (0.63 mm less han the
meanof 52 monkeysof the samespecies).
Function (1), using the constants a = 0.082 and
A = 0.36 (to yieldkHz), calculates curve yingon average
1.25 mm apical (lower on the ordinate) to the four data
pointsplottedasopensquares.When the four pointsarealso
plottedassolidcircles1.25mm moreapically, hey demon-
stratea goodagreementn slopeand alsosimply llustrate
the suggestion ere hat the frequency-positionunction or
thesemonkeysprobably ies apical to the positions f the
squares,o proceed o the 45-kHz upper requencyimit in-
dicatedby Stebbins' ther data cited above.The displace-
ment of the curve 1.25 mm from the locations of 50% IHC
loss s comparable o the displacement f Liberman'spri-
mary-fiberCF data (in Fig. 2) from the hearing ossversus
placecorrelations ited by Liberman (1982). As in the case
of the cat hearing-lossata, t is reasonableo expect hat, as
a tone's requency s raisedand progressivelyhifts he dis-
placementmaximum oward he regionof missing uterand
v. MONKEY: FREQUENCY OF HEARING LOSS VERSUS
POSITION OF 50% IHC LOSS
Data have become available on a number of monkey
species.tebbins1970) andStebbinstal. ( 1973 reportan
upper requencyimit of about 5 kHz amonghemacaques.
Stebbins nd Moody (respective ersonal ommunications,
1986, 1988) reportmeancochlearengths or seven pecies
ofmacaquesanging rom23.05-26.26mm. f theslope on-
stanta scalesrom man o these pecies,his requencyimit
and thesebasilar engthswould suggestheseconstants:
A = 0.36 for all and a = 0.09 to 0.08, respectively.
Beecher (1974a,b) has reported he upper frequency
limit of the squirreland owl monkeyso be about46 kHz.
For the squirrelmonkey, garashiet al. (1968) report a
cochlear artition engthof 20 mm, but also eporta 22-mm
length or thecat, andan 8-mm ength or therat. However,
these estimatesuse the method of reconstructionused by
Guild ( 1921 and Schuknecht 1953), which underestimate
membraneength. As noted earlier, Liberman ound the
length or the cat o be25 mm when hissource f errorwas
avoided, nd B6k6sy eported length or the rat of 9.7 mm.
Ifcochlearpartition ength or the squirrelmonkey s under-
estimated y a percentagerrorcomparableo the error for
the cat, henbasilarmembraneength n thismonkeymay be
almost23 mm, rather than 20, mm. If it were, he applicable
constantsor the requency-positionunctionor thesquirrel
• 0.1 1 10
, ,,l•l I I I il i i i i [ I I i , i i i i I I , i
•25•
c
•20 o o
_
0
L15
E
•10
o 5
._
•0,
0.1 1 10
Frectuency in K• I ohertz
FIG. 5. Relation of cutoff frequencyof hearing oss n M. Nemestrina o
placeof 50% lossof innerhair cells Stebbins nd Moody, 1979, 1986and
1988, personal ommunications). pen squares: ochlear ocationsex-
pressedsproportions f cochlearength n eachmonkey, efore xpression
with respecto the meancochlearength 25.6 mm) in these our monkeys.
Opencircles: ame oints hifted ownon heordinate y 1.25mm. Curve s
Function ( 1 , whereA = 0.36, a = 2.1/25.6, k = 0.85. ConstantA is set o
yieldupper requencyimit of 45 kHz (Stebbins, 970). Shifted ircles re
to illustratea fairly closeagreement f data slope o the function.Original
squares howa displacementf correlations f frequencyossversus lace
of 50% IHC loss rom a function hat yields he upper requencyimit indi-
catedby behavioraldata and that might represent ositionof maximum
cochlear-displacementmplitudeversus requency see ext).
2599 J. Acoust. Soc. Am., Vol. 87, No. 6, June 1990 Donald D. Greenwood:Cochlear frequency-positionunction 2599
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inner hair cells, he frequency t which ossbeginswill be
lower than the normal CF for the pointsat which damage
begins.
Vl. FREQUENCY-POSITION DATA FROM OTHER
SPECIES
For otherspeciesor whichwe possessrequency-posi-
tion data, he data (elephant, ow, at, andmouse)areolder
(B6k6sy,1960) and were considered arlier (Greenwood,
1961 ) or arenewbut sparse--Mongolian erbil Sokolich
et al., 1976). Perhaps therdataexist hat shouldbe treated
here, but this survey s not intended o be exhaustive, nd
only thesespecies ill be considered.
To review heolddata,elephant ndcowwerequitewell
fit in 1961by functions mploying caled lopeconstants,
equal o 0.035 and0.055, respectively,r 2.1 whenbasilar
distances normalizedas a proportion.B6k6sy'smouse nd
rat data were hensatisfactorilyit by functionswith exactly
scaled lopes. owever, he atterdataallowedof consider-
able latitude in both main constants,chiefly constant A,
which or a given lope onstant asempirically etermined
by approximateonformancef thecurve o B6k6sy'sata.
The datadid not permitverysecure xtrapolationo an up-
per requencyimit.Pairsof functionseemingo delimit he
permissiblerequencyimitsand o brackethescale-related
slope alue llustrated hesepoints.
Now, more ecentestimates f the upper requencyim-
its in mouseof 120 kHz (Ehret, 1975) and in rat of 80 kHz
(Kelly and Masterson, 977) yieldA constants f about
0.960and0.640, espectively,iven caled lope onstantsf
0.3 for the mouseand 0.216 for the rat. The resulting unc-
tions are more securelydeterminedand remain in good
agreement ith B6k6sy's atacurves s regards lope.B6-
k6sy'smouse ata curve s nearlyparallelbut displaced a-
salward,consistentwith Kohl16ffel's 1972a,b) findings,by
about1.4mm. B6k6sy'sat datacloselyit thenew unction,
coinciding t 200 and5000Hz, andarenevermore han0.5
mm from it.
A tentative requency-positionunction or the Mongo-
lian gerbilcanbecompared ith only hreepairsof empiri-
cal frequency-positionoordinatesSokolich t al., 1976).
In thesedata, the maximumCM is plottedversus lectrode
position. wo of the threeCM peaks, t 0.5 and2 kHz, are
rather clearly ndicatedand would seem o warrant greatest
weight n fitting,but the mostapicalpointof these wo is
based nonlya single erbil.The averageength f thegerbil
cochleas reportedby Sokolich t al. as 12.1 mm, which
would indicate a scaled constant a of about 0.174. This con-
stantandanA constant f 0.4 yieldan upper requencyimit
of about50 khz, which s n fairly reasonablegreementwith
their plot of their data.To bring his unction nto a better
agreement ith themostapicalpointa subtractiveonstant
of about 0.35 rather than 0.85 is needed. However, the data
of Ryan and Bone (1978) suggesthat 500 Hz may be asso-
ciatedwith a point2 mm (or slightlymore) from the apex,
which is more consistent with k = 0.85 or 1.
The accuracyof the slightly variant gerbil functions
shown n Fig. 6, if more frequency-positionata were ob-
tained, s, of course,uncertain, and they seemnot to agreeas
oo looo loooo
::• 12 .... I ........ I ........ i , , ,
,_
x ß
O 6 _
O 2
._
r-I 0
:1O0 1000 0000
Frequency in Hertz
FIG. 6. Data points:requency ersus ositionn theMongolian erbil So-
kolichet al., 1976), based n frequency f maximumCM versus lectrode
position. ean ength f gerbil ochleasreported s12.1mm.Solid urves:
,4---0.400 to yield a 50-kHz upper frequency imit; a= 2.1 (or 2.1/
12.1 = 0.174 to scale in mm: k = 0.85 or 0.35 to bring unction nto better
agreement ith the mostapicalpoint (but seeRyan andBone,1978).
closelywith the existing ataas n the othercases onsid-
ered,althoughheshort engthof membraneepresentedy
the ordinatemakesa millimeterdeviation of a point from
the ine) in the gerbilappear bout wiceas argeas n the
cat.However, he pointof thiscomparisons not that three
datapoints ictate curveof the ormof Function 1 , but
that hepoints rereasonablyonsistent ith t. The figure
also s intended o illustrate hat, for an increasinglympor-
tant laboratory pecies,more frequency-positionata are
needed.
VII. RECENT DEVELOPMENT: A PSYCHOACOUSTIC
FREQUENCY-POSITION FUNCTION IN EQUIVALENT
RECTANGULAR BANDWIDTH (ERB) UNITS FOR
HUMANS
A recent requency-positionunction or man hasbeen
developedby Moore and Glasberg (1983) and Moore
(1986), alsoby integrating critical-band ERB) function,
the samemeansemployedby Greenwood n 1961,but based
on more recentbodiesof data obtainedby Houtgast (1977),
Patterson (1976), Weber (1977), Fidell et al. (1983), Pat-
terson et al. (1982), and Shailer and Moore (1983). The
ERB valuesbasedon thosedata are similar n size (slightly
smaller) and slope, over the fitted part of the frequency
range, o critical bandwidthsas expressed y Greenwood's
( 1961b, 1974) critical-band function.
Moore and Glasberg (1983) assumed, n effect, that
their ERBs followeda second-order olynomial unctionof
frequency. hey fitted the function o data over he frequen-
cy range rom 125 Hz-6.5 kHz, integrated,and obtaineda
frequency-positionERB-rate) function, when ERB units
were assumedo correspondo a constantdistance n mm,
specifically bout0.9 mm. Greenwood's1961 requency-po-
sition function had indicated to them that, in the above fre-
2600 J. Acoust.Soc. Am., Vol. 87, No. 6, June 1990 Donald D. Greenwood:Cochlear requency-positionunction 2600
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quency egion, the estimatedERB valuescorresponded
closely o 0.9 mm (Moore, personal ommunciation, 983).
The ERB-rate functionobtained y integratingwasquanti-
tativelysimiliar n a part of the fitted frequency egion o
Greenwood's1961 frequency-positionunction n Fig. 1.
Given that Function (1) reasonably pproximateshe B•-
k•sy-Skarstein ochlearmap, it may be useful o consider
briefly herelationof thebandwidth stimatesbove o equal
distances ccording o Function (1) and to compare he
ERB-rate unction o the cochlearmapspresentedn earlier
figures.
The bandwidth estimates f Houtgast and Patterson
hadcloselyollowed heslope f the 1961critical-band urve
and the two-toneconsonance ata originallycompared o
the atter.Hence, heyalsocorrespondloselyo equalbasi-
lar distancesalculated y Function 1), as seen n Fig. 7,
where he data fall close o superimposedurves epresent-
ing requencyntervals orrespondingo particular onstant
distances.n addition, he da(aof Shailerand Moore (1983)
andFidellet al. (1983), citedabove, gree athercloselyo a
constant istance f 0.9 mm, in the main graphof Fig. 8. It
wasunclearhow Moore and Glasberg 1983, their Fig. 1
had plottedWeber'sdata,whichwereobtained t fivespec-
trum levels rom 10 to 50 dB SPL, and variednoticeably t
the two highestevels.Therefore, ll Weber'sdata are plot-
ted n the nsetgraphof Fig. 8, and,at the three owest evels,
the requencyntervals orresponduitecloselyo a distance
of 0.53 mm.
lOO lOOO lOOOO
i J i i i i , I i • , , , iii i i i i i
ß 1oo 1ooo 10000
N •ooo
I 1000
._
ß lOO
o
lO
, i , , [ i , , , , , i i [ ,
1O0 i 0[00 10000
FrecLuency in Hz
FIG. 8. Main graph:The circlesare the data of Shailerand Moore (1983).
the squares re the data ofFidell et al. (1983). Both setsof bandwidth esti-
mates reobtainedrom temporalgapdetection ata.The curve epresents
frequencyntervals orrespondingo 0.9 mm. Inset graph:data of Weber
(1977), analogouso Patterson'sn Fig. 7. Obtained t fivespectrumevels
from 10 o 50 dB SPL.The curve epresentsrequencyntervals orrespond-
ing to a distance f 0.53 mm. The uppermost ircles epresentesults t the
50-dBspectrumevel,and he 0.9-mmcurve not shown)passesust above
those ircles t 1000and4000Hz, by nearly he same mount. he squares
indicate ataobtained t the 40-dBspectrumevel.The remaining ataat
the 30-, 20-, and 10-dB evels unchup, with the brokencrossingines ndi-
cating ittle or no systematic ffectof level. As in Fig. 7, the curve s not
fitted to the data, but rather is proportional o the derivativeof Function
( 1 , as in Greenwood (1974).
N
I 1000
._
ß lOO
o
lo
loo lOOO 10000
, i , t i I iJ i , , , 'Nil] I i i t till
• .... ,,o,o.... .Lo,oo... .o.0,oo
,
i I , , , 1,1 i ' ] ' ' ''1
1000 10000
Frequency in Hz
FIG. 7. Estimatesof auditory-filterbandwidths.Main graph: Houtgast
( 1977); upper set of estimatesare of Gaussianbandwidthsderived from
ripple-resolutionata obtainedn simultaneous asking xperiments. he
uppercalculated urve epresentshe frequencyntervalcorrespondingo
1.1 mm, according o Function ( 1 . Lower curve and points:estimates f
filterbandwidths erived rom ripple-resolutionataobtainedn pulsation
thresholdmeasurements.he calculated urve epresentshe requencyn-
terval correspondingo 0.65 mm as above. nset graph:Patterson 1976);
uppersetof pointsare estimates f Gaussian ilter bandwidthsderived rom
notched-noiseimultaneous asking xperiments. he Patterson ointsat
500 and 2000 Hz nearly coincidewith Houtgast's pper set of pointsat
those requencies.he uppercalculated urve epresentshe frequencyn-
tervalcorrespondingo 1.16mm, as above.Lower curveand points:Esti-
matesof equivalent ectangular ilter bandwidths erived rom samedata.
The calculated urve representshe frequency nterval correspondingo
0.89 mm, as above.
Note that the calculatedcurves n Figs. 7 and 8 are used
as "measuringsticks" to assesshe bandwidths'confor-
manceor nonconformanceo constantdistances sgivenby
Function ( 1 and the cochlearmap n Fig. 1. The curves re
neither fitted to the bandwidth data nor presented o draw
support rom them, since hey have an independentstatus
providedby the cochleardata. However,beyond ndicating
the bandwidths' relation to cochlear distance, the calculated
curvesserveas more than adequatedescriptive unctions or
these data.
As for the derivedERB-rate (frequency-position)unc-
tion, Moore (1986, his Fig. 5) has compared t directly to
someof B6k6sy's nd to Skarstein's ata (Kringlebotnet al.,
1979). Moore states hat thebestcorrespondences obtained
if eachERB bandwidthcorrespondso about 0.89 mm and
readjustshis constantsaccordingly rom those originally
published.However, his Fig. 5 restricts tself to the range
from 400-6500 Hz by omitting our of B6k6sy's ight points
from the Kringlebotn igure,all of thosebelow400 Hz, two
of which are at and above 100 Hz in the same ange of fre-
quencies verwhich he ERB curvewas itted o the empiri-
cal bandwidth estimates.
However, the comparisonof the ERB-rate function to
basilar coordinates s as close as his figure indicatesonly
above 400 Hz, in the selected egion and in the largely
straight sectionof the function, ust prior to its increasing
convexity. n Fig. 9 of this paper,Moore's function s com-
paredover he wholehuman requency ange o the omitted
B•k6sypointsand to Function ( 1 of Fig. 1. The ERB-rate
function exhibitsa lesseragreementwith B6k6sy'sdata in
2601 J. Acoust.Soc. Am., Vol. 87, No. 6, June 1990 DonaldD. Greenwood:Cochlear requency-positionunction 2601
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•35
• -
_
c30 -
.-- _
--
-
x25 -
--
20-
E 2
o
L15-
_
• -
o10
C -
O -
ßm-, 5-
Itl -
• o
loo lOOO
ß
/,.?•
ß
' ' ' ' ' ''1 ' ' ' ' ' ' ''1 '
lOO lOOO
Freckoency
10000
_
-
_
_
.
_
T , ' ' ' ''
lOOOO
in Hz
FIG. 9. The solidcurve s Function ( 1 from Fig. 1. The dashed urve s the
ERB-rate functionof Moore and Glasberg 1983) and Moore (1986), as
modifiedby Moore for direct comparisonwith B6k6sy's 1960) and Skar-
stein's Kringlebotnet al., 1979) data, which relatepositionof maximum
amplitude o frequency.
the lower frequencies.A lesseragreementwould also be
shown by Zwicker's critical-band-rate function (Zwicker
and Terhardt, 1980) if it werecompared o thesephysiologi-
cal data. The comparisons lso show hat the curvaturebe-
yond the high-frequency nd (6.5 kHz) of the rangeof the
originalpsychoacousticata eads,not surprisingly,o high-
er than usualestimates f the upper requencyimit of hear-
ing and to an unrealistic orm, if we can generalize rom
other species.
In several pecies,he evidence ince he 1940s ndicates
that, over most of the basilar membrane, and most of the
frequency ange, log frequencyversusbasilar position s
nearly a straight ine. Thus the almost-exponentialorm of
the 1961 function, and the constancyof normalized slope
among severalspecies,ndicates hat it more plausiblyap-
proximates he form of the function hat shouldbe theoreti-
cally derived. Zwislocki (1965) has long sincederived an
almost-exponentialunction; omeothermodels ndempiri-
cal curveshave used simple exponentials. or the species
considered ere, he 1961almost-exponentialrequency-po-
sition function seems t this moment as satisfactory n form
as a description f physiological ata as t did 29 yearsago.
SUMMARY AND DISCUSSION
A. Present status of the cochlear frequency-position
function
Sincepossible ochlear requency-positionunctions re
chieflydependent ltimatelyon the accuracy f the available
physiologicalrequency-positionata, it has beenunfortu-
nate that those data were initially, with the exceptionof
those from the cat, only from dead specimens.Partly
counter-weightinghat fact were the observationshat the
slopeconstantcould be scaledor normalizedacross pecies,
importantly ncluding he live cat, and that, in man, the up-
per frequency imit was consistentwith behavioral esti-
mates.
In all of the comparisons f more recentdata, t hasbeen
possibleo scale he slopeconstant exactly rom the origi-
nal slopeconstantof 0.06 found to be suitable or man, by
multiplying t by the ratio of basilar engths man's divided
by that of the other species). hat is, the productof a times
basilar ength is approximatelyconstantamong hesespe-
cies,specifically .1; thus, f distance long he membrane s
expressedsproportional ength, rom 0 to 1, a canequal2.1
in all thesecases if physicaldistance s required,divide2.1
by basilar ength). Obviously, t isnot shownnor certain hat
the slopeparameter s completelyconstant.However, t is
clearenough hat it is very similar among hesespecies nd s
not likely to vary much among hem if more definitivedata
are obtained.Apparently hesecochleas re sufficiently en-
eralized or this relation o hold, although here seems o be
little reason o expectsucha convenient elation and func-
tion to apply unmodified o more specialized ochleas, uch
as thoseof certain bats with an enlargedpatch of cochlear
partitiondevoted o a particuIar requency r thoseof a bur-
rowing rodent like the "mountain beaver" (Aplodontia
Rufa) that is reported o be specializedor low frequencies
(Merzenich et al., 1973 .
Hence, he functionshown n 1961 o fit quite accurate-
ly the data then available rom eight species asbeenshown
here o fit closelyconsiderable dditionaldata from someof
the same pecies--human, at, and guineapig--as well from
the chinchillaand a species f macaque at least n respect o
slopeand upper frequency imit), on which data had not
previously eenavailable.Most of theseadditionaldata are
from ivingspecimens.he basic unction s certainlya plau-
siblecandidate o fit both gerbil data and data from other
macaque ndsquirrelmonkeys,f moredata rom these pe-
cies become available.
B. Desirability of further physiological support for
function (1) for homo sapiens in particular
Since or humanbeingsour cochleardata mustremain
indirect,onereason or surveying he interspeciesompari-
sonsn thispaperhasbeen or theirbearing n thedegree f
confidencehat may, or may not, be ustified n the possibil-
ity that the frequency-versus-positionata from humanca-
davers pplyalso o livingears.Since he samesimple orm
of functionhasbeenshown o fit reasonablywell the data of
up to tenspecies, tilizinga single onstant for normalized
slope, he useof this requency-positionunction or the iv-
inghumanearseemso bereinforced. he case f thecat, or
which data are mostcomplete, eems ihgly o be the most
supportive t this time (Liberman, 1982), with the guinea
pig a close econd incedata of basically our kindssupport
the sameslope.Whateveruncertainitiesemain,at least he
comparisons avebeenbroughtup to date.
But it would be valuable, or the most nearly direct rel-
evance o homo sapiens,o obtaindetailed requency-place
correlationsrom primates.The squirrelmonkeyandM. Ne-
mestrina are reasonable candidates. In the first instance, we
already have the mechanicaldata of Rhode (1971, 1978).
Study of the squirrel monkey could permit any necessary
revisionof cochlear ength,as well as of estimatesmadeof
the cochlearoci of Rhode's ecording ites nd the longitu-
2602 J. Acoust.Soc. Am., Vol. 87, No. 6, June 1990 DonaldD. Greenwood:Cochlear requency-positionunction 2602
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dinal extentof the apicalsegment nd peak regionof the
displacement nvelope Greenwood, 1974). In the second
instance, ehave hedataof $tebbins ndMoody (1979) in
Fig. 5, which would afford he opportunity o clarify the
relationof basalheating ossdata to, still nonexistent, ata
from the monkey hat would relate primary fiber CF to
cochlearocationusing he horseradisheroxidaseHRP)
techniquesmployed y Liberman 1982).
C. Relation of the cochlear frequency-position function
to psychoacoustic data
Although the frequency-positionunctionconsidered
herewasobtained riginallyby ntegration f an exponential
function fitted to some of the critical-band estimates avail-
able n 1961, t was hencomparedo cochlear ata for the
kindof confirmationhat such function equireso serve s
a cochlear requency-positionunction, rather than as a
purelypsychoacousticalonstruct, oweveruseful he atter
might be. Psychoacousticata cannotbe known n advance
actually o reflect,unconfoundedy any other factor, only
the separationf cochlear isplacement axima,or a "spa-
tial" factor,howevernamed.That hypothesisn respect o
any given body of data requiresassessmentnd may not
apply, without any fault in thosedata. For example,more
than oneoperative actormay co-varywith change n a sin-
gle independent ariable,producingpotentiallydifferential
effectsn differentperformanceasks.Although he validity
of any test of correspondencef frequency-resolutionsti-
mates o constant istances n an independent hysiological
scalewill depend n the accuracy f the physiological ata, f
the atter are soundand f somepsychoacoustic easures o
correspondo equaldistances nd othersdo not, the latter
measuresmay ipsofacto e especiallynteresting nd useful
to study o determinewhat other actorsactuallyoperate n
the various erformanceasks n question. his point,made
in 1961,still seems oninvidious nd unexceptionable.
Thus, if the future yields accurate cochlear data that
continue o reinforce he frequency-positionunction, the
supportprovidedby the conformance f a givensetof psy-
choacoustic bandwidths to a constant cochlear distance will
turn out to be for the equaldistance ypothesis, sapplied o
thosedata,rather than for the frequency-positionunction.
At this ime, the supportmay be regarded s o someextent
for both only if, and to the extent hat (a) the physiological
data may be regardedas doubtfulon their own, and/or (b)
the equal distancehypothesismay be independently up-
portable,deductively r evidentially.Moreover,othersetsof
psychophysicalstimates r measures f frequency esolu-
tion may not conform o equaldistances, ithout any neces-
saryadverse earingon the accuracyof the frequency-posi-
tion functionor on their own repeatability nd validity as
data. Given he performanceask n question nd the nature
of the system,he equaldistance ypothesis ay simplynot
be correct n a givencase.
D. Uses and advantages of physiologically accurate
basilar-frequency ransformationsof data and complex
spectra
Further n oioo rimatedatamightpermitus o inferany
advisablemodificationsof Function ( 1 constants or homo
sapiens,and/or to confirm its form. However, hesitation
about plotting data on physiologically upportedcochlear
frequency cales hilewaiting or furthersupport eems n-
necessarilyautious,or man and a numberof otherspecies
considered. he easeof plotting computer-stored ata on
any ransformed cale swell ason he original ndependent-
variablescale rgues or the useofcochlearcoordinate cales
whenevert may prove nteresting.Moreover,where here s
willingnesso plot on purely psychoacousticallyenerated
scales,hereshould e ittle reluctanceo usephysiologically
supportedscales.
Given the increasingnterest n speech ecognition l-
gorithmsand hypothetical euralnetworks o process udi-
tory-nerve nput, t may be especiallyo the point to plot the
spectral nalyses f speech ounds n a physiologicallyup-
ported requency-positioncale ather than on scales ased
only on psychoacousticata, since he latter will havebeen
influencedy whateveractors ther hanspatialmayhave
contributed o the originalmeasurements.
To usescales nown not to conformaccurately o exist-
ing physiological ata s, on the onehand, eitheran implicit
argument hat the physiological ata are defective or more
likely to be defective han the psychoacousticata and as-
sumptions sed o arrive at the scale)or, on the other hand,
an argument hat the psychoacousticcale eflects omees-
eontlal "norportrayal"r intogrativenqvt-hnat-nllqtit-quiv-
alence by presumablyumping the effectsof all operative
factors) that the user explicitly wants to incorporateas a
desiredpreprocessingxpected o further his psychophys-
ical ends.The first argument hat physiological ata are the
weakest ink is becomingncreasinglyessplausible s ech-
niques mprove and as similaritiesamongsomespecies e-
come clearer. Is it likely that a psychoacousticallyased
cochlear cale or the cat wouldnowbeproposedf the requi-
site data for the cat existed but led to visible conflict with
Liberman'sHRP data and the description y Function ( 1
of those data?
An argumentof the second ype for usinga frequency-
positionscale hat is known not to agreewith existingphy-
siologicalspecifications f cochlear requencycoordinates
shouldbe madeexplicit.However,whatever he scale's p-
propriatenessor its own ends, n modelingauditory pro-
cesses,t would not seema genuine enefit hat lumped ac-
tors influencing eal post-cochlear uditory processing ill
be already embedded n the assumedspectral ransforma-
tionsbasedon psychoacousticcales iffering rom physio-
logicaldata. It seems rguablymore conservative nd flexi-
ble o usea cochlear requency cale hat is,so ar aspossible,
physiologicallyoundedand to incorporateexplicitly nto
later analysis f the spatially ransformed pectrum ny sub-
sequent eal or hypothetical hysiological r other process-
ing desired--based n inferencesrom either physiology r
psychoacoustics.oreover, differingcochlearscales, ased
on differingsetsof psychoacousticiscriminative nd-prod-
uct data, cannot all reflect actual cochlear coordinates with
equal ndependencerom other actorsnor the optimalspec-
tral transform.
In somecontrast, he physiological ata available or
homo sapiens, lthough n vivodata are lacking and unob-
2603 J. Acoust.Soc.Am.,Vol.87, No.6, June1990 DonaldD. Greenwood: ochlear requency-positionunction 2603
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tainable,havebeenextended nd remain n closeagreement
with Function ( 1 , which appearso havebeenstrengthened
in respecto its slope onstant y the atter'sapplicability,n
scaledor normalized orm, to severalother species rom
which in vivo data have been obtained. The increased inde-
pendence ffordedFunction (1) by this physiologicalup-
port suggestshat a potentiallymportantuse ies n describ-
ing systematically he relations of various sets of
psychoacousticallyignificant andwidthso cochlearoca-
tion and distance,as done, in part, in Figs. 7 and 8. Con-
straints n the applicability f the equaldistance ypothesis,
longspared testagainst he available ochlear ata,maybe
expectedn someperformanceasks.Caseby case nalysis,
given entative r "working"acceptancef thecochlearmap
in Fig. 1, might clarify responsibleactorsand other func-
tional relations of interest.
ACKNOWLEDGMENTS
I would like to thank Dr. J. E. Hind and the faculty of
the Department of Neurophysiology t the University of
Wisconsin,where this paper was begun n the summer of
1985, and where its ancestor was written in 1960, for their
provision f a congenial laceof work and heir hospitality.
would like to thank those who have commented on the
manuscript nd especially o thank John Nicol for essential
computerassistance.Work supported y NSERC, Canada.
•Althoughheagreementf theB6k6sy ndSkarsteinatapointss close,
satisfactionmight seem o be temperedby recalling hat in Bredberg's
(1968) studyof a largecorpus f human emporal ones,he engthof the
organof Corti exhibited total variationof about28% of the mean stan-
darddeviation otreported).This igurewasquitesimilar o Hardy's 33%
(1938), who also eporteda standarddeviationof 6.8%, and very similar
to the total variationof lengthwithin a numberof the macaque pecies,
where he total rangeand standarddeviationwerealmost29% and 6% to
7% of the mean, espectivelyfiguresobtained rom data personally om-
municatedby Stebbins,1986). In chinchilla, Bohne and Carr (1979) re-
port the rangeand standarddeviationof cochlear engths s 26% and al-
most 5% of the mean, respectively.However, they also report that the
dimensions f the chinchillacochlea,suchas ts width, are very similar at
correspondingormalized oints,whichwould tend o keepconstantn-
traspecies arameters uchas the normalizedslopea in Function ( 1 . In
cat, Liberman (1982) finds hat normalizingbasilar ength educes cross-
cat variabilityofunit-CF versus lace, .e., ndicates lsoa common aram-
eter A and upper requency imit.
2However,here ppearsobea complicationn theWilson ndJohnstone
data, in that the frequencieshey measured ver the basal4 mm of the
partitionare cutoff requenciesas defined n their paper) and not peak
frequencies.hey report he peakvalueswouldbe about10% loweror up
to 20% lower, depending pon whether or not an additionalcorrection
wasalsonecessaryor drainageof the scala ympani.However, t may also
be notedhere hat a countervailing orrection n the other directionmay
benecessaryfa progressiveasalpeakshiftwith timeanddeterioration f
the preparationoccurred,of the type reportedby other experimenters
(Kohll/Sffel, 1972b;Rhode, 1973; Khanna and Leonard, 1982; Sellick et
al., 1982;Robleset al., 1986). Hence, f, owing o experimentalrauma,
the measured utoff requenciesre oo ow andshould e raised--before
then makinga downwardcorrection o peakvaluesand for the effects f
drainagesthen he opposed orrections ould o someunknownextent
counteract achother. Although estimation f the net resultmay well be
too uncertain o be useful, n the event tself we observe hat the upper
frequency imit of about 43 to 45 kHz that is consistentwith their cutoff
frequenciess alsoconsistentwith the other n vivodata.
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