1990 - Greenwood - Cochlear Frequency-position Function for Several Species - 29 Years Later

8/16/2019 1990 - Greenwood - Cochlear Frequency-position Function for Several Species - 29 Years Later

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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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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

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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



6/14

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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



7/14

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

2598



8/14

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).




9/14

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



10/14

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



11/14

•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



12/14

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



13/14

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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1990 - Greenwood - Cochlear Frequency-position Function for Several Species - 29 Years Later