An evaluation of eight computer models of mammalian inner
hair-cell
function
MichaelJ. Hewittand Ray Meddis
Department
ofHumanSciences,
University
of Technology,
Loughborough
LE113TU, UnitedKingdom
(Received10July 1990;revised5 December1990;accepted29 March 1991)
Eightcomputermodelsof auditoryinnerhaircellshavebeenevaluated.
Froman extensive
literatureon mammalianspecies,
a subsetof well-reported
auditory-nerve
properties
in
response
to tone-burststimuliwereselected
andtestedfor in the models.This subsetincluded
testsfor: (a) rate-levelfunctionsfor onsetand steady-stateresponses;(b) two-component
adaptation;(c) recoveryof spontaneous
activity;(d) physiological
forwardmasking;(e)
additivity;and (f) frequency-limited
phaselocking.As modelsof hair-cellfunctioningare
increasingly
usedasthefrontendof speech-recognition
devices,
thecomputational
efficiency
of eachmodelwasalsoconsidered.
The evaluationshowsthat no singlemodelcompletely
replicates
thesubset
of tests.Reasons
aregivenfor ourfavoringtheMeddismodel[R. Meddis,
J. Acoust.Soc.Am. 83, 1056-1063(1988) ] bothin termsof its goodagreementwith
physiological
dataanditscomputational
efficiency.
It is concluded
that thismodelis well
suitedto providethe primaryinputto speechrecognition
devicesandmodelsof central
auditoryprocessing.
PACS numbers:43.64.Ld, 43.64.Pg,43.64.Bt
INTRODUCTION
During recentdecades,the input/output characteristics
of the mammalianhair cell/primary-fibercomplexhave
beenthe focusof extensiveresearch.In response
manycomputationalmodelsof the variouspropertiesof the junction
havebeenproposed.The modelsprovidea convenientenvironmentfor evaluatinganddevelopingtheoriesof spikegeneration.Furthermore,theyareanintegralpartof manyperipheralauditorymodelsandspeechrecognitionsystems.The
problemfacingthedesignerof suchsystems
liesin thechoice
of whichsynapse
modelto incorporateintohis/her composite model. A review of the literature finds many possible
candidates(Siebert, 1965; Weiss, 1966; Eggermont,1973,
1985; Schroederand Hall, 1974; Oono and Sujaku, 1974,
sultsarecomparedto datafromelectrophysiological
studies
for the followingproperties:(a) onsetandsteady-state
rateintensityfunctions;(b) two-componentadaptation;(c) exponentialrecoveryof spontaneous
activity after stimulus
offset;(d) recoveryof functionto respondto a secondstimulus after offsetof a first; (e) additivity;and (f) low-frequency phaselocking. This is the samesubsetof testsusedby
Meddis (1988) to evaluate his model; however, two additional methodsof analysisare introduced.Johnson's(1980)
synchronization
index is introducedto analyzechangesin
maximumsynchronyacrossfrequency.Second,the computational efficiencyof eachmodelis considered.For this purpose,comparativetimesto processa 1-stoneburstare presented.
1975; Geisler et al., 1979; Ross, 1982; Schwid and Geisler,
1982;Smith and Brachman, 1982;Allen, 1983;Cooke, 1986;
Meddis, 1986b, 1988; Westermanand Smith, 1988;Meddis
et al., 1990).
Hair-cell modelscan be thoughtof as variationson a
basicproto-modelconsistingof reservoirsof a transmitter
The modelshaveappearedovera substantial
periodof
time and each lays claim to simulatea differentsubsetof
data. Moreover,many importantempiricalfindingshave
beenpublishedsincetheappearance
of someof the models.
As a result,it is difficultto makemeaningfulcomparisons
betweenthe variousproposals
andit istimelyto reviewtheir
performancein the light of recentdevelopments.
The choiceof anappropriatemodelof synaptic-primary
auditory-nerve(AN) modelsdoesnot only affectour theoreticalperspective.
It couldbe crucialto otherfuturemodeling effortswherethe modeloutputswill serveas inputsto.
simulationsof morecentralprocesses.
Modelingthe central
auditorysystemcanbecomemassively
complex;in suchsystemsinaccuracies
at the input stagewill rapidly propagate
throughoutthe system.
This paperreportsa comparativeinvestigationof eight
many featuresincluded in hair-cell simulations.No model
computationalmodelsof hair-cell function. The model re904
substance that is released across the cell membrane into the
synapticcleft.Figure1showsa generalized
representation
of
containsall of the featuresasshownin the diagram,but the
transmitterflowpathsof almostall the modelscanbe representedby a reducedversionof the proto-model.Early attemptsmodeleda single-reservoir
systemwith lossand replenishmentof transmitter quanta. Later models added
extra reservoirsor complicatedthe principlesof transmitter
flow control.
A reservoirrepresents
a storeof transmittermostprobably locatednear to the baseof the hair cell. This feature is
consistent
with knownphysiological
and anatomicalstructures which are ubiquitousamongreceptor-synapticsystems.Releaseof transmitterin quantaor discretepacketsis
anotherwell-reportedsynapticmechanism.
The introductionof multiple-reservoir
andmultiple-site
modelswas stimulatedby the empiricalfindingsof Furukawaandhiscolleagues(Furukawa etal., 1978;Furukawa
J. Acoust.Soc. Am. 90 (2), Pt. 1, August1991 0001-4966/91/080904-14500.80
@ 1991 AcousticalSociety of America
904
INNER
HAIR
CELL
Schroederand Hall (1974) proposedone of the first
hair-cellmodelsto meetour criteria.A singlereservoirreleasedquantaof transmitterin amountsproportionalto a
IMMEDIATE
s'rORES
SYNAPTIC
stimulus-relatedpermeabilityfunction and the number of
quanta availablefor release.Transmitter releasedfrom the
cell stimulatedthe post-synapticafferentfiber. Once released,transmitterquanta were irretreviablylost from the
system.
A fixed-rate
replenishment
scheme
wasusedto supply the reservoirwith new transmitter.
Oonoand Sujaku's( 1974, 1975) modelshowedcertain
structuralsimilarities
to theSchroeder
andHall proposal.
A
singlereservoirof transmitterwasusedtogetherwith a stimulus-relatedpermeabilityfunction.Unlike the Schroeder
and Hall model,however,only a fractionof the total transmitterwasreleased
pertimeunitwhenthecellwasactivated.
Replenishment
waseffectedaccordingto the formula.
IMEMBRANE
SMI'I-rER
KE
CELL MEMBRANE
FIG. I. Generalizedrepresentation
of hair-cellmodels.Redrawnfrom
Meddis (1986a).
andMatsuura,1978;FurukawaetaL, 1982)workingon the
AN fibersof goldfish.They recordedthestrengths
of successiveexcitatorypost-synaptic
potentials(EPSPs) following
onset of tone bursts imposed on pedestaltones. The
strengths
of the EPSPs(m) were,accordingto Furukawa,
determinedby the productof two independent
parameters
(n and p). After a stimulusincrement,parametern increasedand parameterp remained relatively unchanged.
Furukawa and Matsuura designatedn to be the amount of
transmitteravailablefor releaseandp, the probabilitythat
an availablequantumwouldbe released.
Theseconclusions
havebeenrepresented
in morerecent
modelsby increasing
the numberof reservoirs
andmaking
theiravailabilitydependent
uponstimulusintensity.In such
models, the transmitter lies in reservoirs or sitesclose to the
dq(t)/dt • q(t) [ 1 -- q(t) ]/•,
( 1)
where dq(t)/dt isthe rateof transmitterflowinto the reservoir,q(t) istheamountof transmitterin thereservoirat time
t, and •- is a time constant.
SchwidandGeisler(1982) represented
theconclusions
of Furukawa and Matsuura (1978) with a model that con-
sistedof sixindependentreservoirsof synapticmaterial.The
reservoirswere ordered by increasingthresholdsuch that
transmitterwasreleasedfromonlyonereservoirat low stimulus levelsand from all six at very high stimuluslevels.
Transmitterreleasefrom an individualreservoirwas governedby a permeabilityfunctionrelatedto thestimulusamplitude. Each reservoirreleaseda constantfractionof its
contentsper time unit when activated.Replenishmentof
transmitterwas a sequentialprocessfrom the least-to the
presynaptic
membrane.The sitesare orderedby increasing most-sensitive reservoir. The flow of new transmitter casthreshold.If thestimulusamplitudeissufficientto activatea
cadeddownthe reservoirchain,eachhigherthresholdresergivensite,a fixedproportionpertime unit of that reservoirs voir being filled until the availablereplacementstore was
transmitter
is released.
exhausted.
Meddis (1986b), however,has offereda reinterpreta-
Ross(1982) in an attemptto replicateshort-and longtermadaptationphenomena
describes
a modelconstructed
of physicaland mathematicalelements.Four reservoirsare
senting the amount of transmitter available for release, chainedtogetherin series.MetabolicenergyflowsunidirecMeddisdesignatedp
asthe probabilitythat a givenquantum tionallythroughthe seriesdeterminedby concentrationdifwill successfully
traversethe cleft (and be invariant with
ferencesand fixedpermeabilities.
Energyfrom the finalresstimuluslevel), and n as the numberof quantain the cleft
ervoirwasfed into a Poissongeneratorat a rate governedby
(whichwouldvarywith stimuluslevel). Empiricaldatacol- its concentrationand two permeabilities,one of which is
lectedto date and known conceptsof synapticphysiology fixed,the other relatedto the instantaneous
signalamplicannotresolvethedifferences
betweentheseinterpretations, tude.The Poissongeneratoremittedunitary chargesat an
and the matter remains controversial.
averagerateproportionalto theenergyflowintothe generator.Thesechargeswerefedto a leakyintegratorwhichfired
I. THE MODELS
whenthesummedinputreacheda threshold.After firingthe
Below, we studyeight modelsof hair-cell functioning. integratorwas dischargedand a refractoryperiod introducedwherebyinputsto the unit wereinhibitedfor 0.5 ms.
They werechosenfor studybecausethey were capableof
accepting
an arbitrary• inputsignalanddelivering,
asout- Rossmodeledeightunitseachwith a differentspontaneous
firing rate. For this study, we selectedunit 297-43 (Ross
put, AN eventprobabilities.We restrictedour attentionto
1982,TableI), whichhad a spontaneous
rateof 60.6 spikes
thosemodelsthat involvedintermediateprocessing
which
tionofFurukawa's
data.Insteadofprepresenting
theprobability that a quantumof transmitteris releasedand n repre-
resembledknown physiologicalmechanisms.We also used
only thosemodelsthat had beendevelopedfor useon digital
computersbecausethisisthe mostlikely mediumfor modelers in the immediate future. The main features of each model
are outlined below.
905
d. Acoust.Sec. Am., Vol. 90, No. 2, Pt. 1, August1991
per second.
Brachman (1980) and Smith and Brachman (1982) de-
velopeda synapsemodel specificallyto replicatetwo-componentadaptationand additivity. Half-wave-rectifiedstimulusenvelopeswereusedas inputsto the modelto simulate
M.J. Hewittand R. Meddis:Hair-cellmodelsevaluation
905
high-frequency
tonebursts.Transductionof the input to an
electricalsignalwassimulatedby applyinga modifiedversionof Zwislocki's(1973) generalizedrate-intensityfunction for sensoryreceptors.The final shapeof the function
was similar to the unadaptedrate-intensitycurve determinedin empiricalstudies.The receptorpotentialwasthen
low-passfiltered.This stagewasascribedto the functioning
tic body seenin electronmicrographsof mammalianhair
of the hair-cell membrane on the basis that all membranes
the membrane fluctuated as a function of the instantaneous
haveintrinsicresistance
andcapacitance
that enablethemto
act aslow-passfilters.The filteredreceptorpotentialevoked
amplitudeof theacousticstimulus.A smallfractionof transmitter releasedinto the synapticgapwaslostthroughdiffusion. A further fraction was actively transferredback into
thecellfromthecleft.The remainingtransmitterisleftin the
cleft to stimulatethe post-synaptie
afferentfiber.The transmittertakenbackintothecellisheldin a reprocessing
reservoir for a shorttime beforebeingtransferredback into the
freetransmitterpoolfor later release.
release of transmitter.
In Brachman's model, transmitter available for release
wasstoredin 512 independent
immediatesites.Duringany
one time frame, the numberof sitescontributingto the response(total transmitterreleasedintothesynapticgap) was
determinedby the corresponding
filteredreceptorpotential.
Eachsitereleased
a constantfractionof itscontentspertime
unit whenactivated.Theseimmediatesiteswerereplenished
by a localstoreof transmitter,andthis,in turn, wasreplenishedat a slowerrate by a global reservoirof transmitter.
Flow of' transmitterwas driven by diffusiongradients
between the stores.
cells.
Meddis( 1986b,1988) proposeda three-reservoir
model incorporatinga noveltransmitterreuptakeand resynthe-
sis processloop. The hair cell manufactureda chemical
transmitterthat wasdeliveredto a site adjacentto the cell's
membrane(the freetransmitterpool). The permeabilityof
Allen (1983) combined a variable resistance model of
the generationof the receptorpotentialwith an activelinear
transformto producethe neuralresponse.
The variable-resistancemodeldescribes
the receptorpotentialasa function
of ciliadisplacement.
In Allen'smodelthereceptorpotential
wascalculatedfrom the instantaneous
signalamplitude.A
linear transformwas appliedto the receptorpotentialto
computethe current.The final stagewasto low-passfilter
the current.The low-passfilter wasascribedto the diffusion
The implementationused in this study is Payton's
(1988) modifiedversionof Brachman'soriginalmodel.Essentially,Paytonrenderedthe modelsuitablefor input of
arbitrarystimuli.Thesemodifications
enhanced
theperfor-
of calcium ions from the cell membrane to the site of vesicle
mance of the model, and so it will be referred to as the Brach-
release.
man-Payton model.
Cooke's(1986) peripheralauditory modelincludesa
two-stagehair-cellsimulationthat transformedthe mechan-
ical inputsignalintoAN event-rateoutput.The firststage
simulatedtransductionof soundpressureto the hair-cellreceptorpotentialusingan asymmetricsquare-root
function.
His state-partitionmodel (SPM} simulatedtransmitterreleasefromthehaircellin response
to thereceptorpotential.
Followingthe hypothesized
modelproposedby Furukawa
andMatsuura(1978), Cooke'sconceptual
modelconsists
of
manyreleasesitesspatiallyorderedby increasing
threshold.
Moreover, each site releasesa constant fraction of its avail-
abletransmitterper time unit whenactivated.However,in
orderto achievecomputational
efficiency,
theworkingmodel groupedmanyreleasesitesinto threeseparatecompartments. As a result only two variables(total volume and
numberofsitesactive)wererequiredto describe
eachgroup
of sites.
Sites in the first or "immediate"
state have thresholds
belowthatof theinstantaneous
signalamplitudeandrelease
II. MODEL
EVALUATION
All models were programmed in FORTRAN77 on a
Masscomp5450computer.To ensure,asfar aspossible,
correctimplementation
of the models,we firstreplicatedall of
the figuresin the eightoriginalpublications.
For the purposeof evaluation,simulations
wererun usinga samplestepof 0.05ms (samplerate = 20 kHz). Stimuli were generatedwith a 2.5-ms rise time and were 1-kHz
sine-phase
toneburstsexceptwherestated.All modelswere
programmedasdeterministicsystems,individualspikesusingpseudorandom
numberswerenotgenerated.
The output
of eachmodelisexpressed
in termsof a firingrateandrepresentsthe averagesynapticdrive during a test period.The
synaptic drive or "excitation function" (Gaumond et al.,
1983) wasusedby Westerman(1985) for reportinganimal
data and by Meddis (1988) reportingmodeloutputs.This
practicedisregardedany post-synapticmechanisms(e.g.,
refractoryeffects)includedin certain models.
transmitter. The second or "relax" state consists of sites that
have beendepletedin the recentpast and are beingreplen-
ished.The finalstatelabeled"reserve"containsfullyreplenished sites.Each site can be instantaneously
transferred
betweenstatesindependently
of the numberof sitesin a particular
state. This differs from other n-reservoir
models
where the transfer of sites occurs as a result of concentration
gradientsbetweensites.The replenishment
rate of any one
site was inverselyproportionalto the physicaldistance
betweenthesiteandthereplacement
transmitterpool,modeledasa pointsource.The biologicalcorrelateof thereplacementtransmitterstorewaspostulatedto bethedensesynap906
J. Acoust.Sec. Am., VoL 90, No. 2, Pt. 1, August1991
A. Rate-intensity functions
A steady-staterate-intensity function (RIF)
derived
from typicalAN data is shownin Fig. 2. The spontaneous
rateisa measureof thefiber'sactivityin theabsence
of sound
stimulation.Spontaneous
activityin AN fibershasa trimodal distribution(Liberman, 1978). The majority of fibers
(> 60%) fire with a spontaneousrate of greater than 18
spikesper second(knownashigh spontaneous-rate
fibers).
All of the modelsimplementedin thisstudywereproposed
to replicatethe response
properties
characteristic
of this
groupof fibers.Modelsof low andmediumspontaneous-rate
M.J. Hewittand R. Meddis:Hair-cellmodelsevaluation
906
stimulusintensitycontinueto influence
the onsetresponse.
Asintensityincreases
thesynchrony
of onsetresponse
within thefirstbinalsoincreases.
Smithandcolleagues
conclude
that the underlyingsynapticdrive to the fiberat stimulus
onsetcontinues
to growwithincreases
in stimulus
intensity,
evenwhenthefiringrateislimitedby refractoryeffects.
Methodsto uncovertheprobability
of spikeoccurrence
disregarding
refractoryeffectshavebeendeveloped
(e.g.,
o Onset
800-
4oo
Steadystate
Sponlaneous---•
firing
0
rate
Gray, 1967; Gaumond et al., 1983; Westerman, 1985).
ß
•)
Whenappliedto post-stimulus
time-histogram
responses,
theeffects
of neuralrefractoriness
areshownto beparticularly strongat onsetwherefiringratesare highest;further
evidence
thatthesynaptic
driveat onsetcontinues
to grow
with increasing
intensity,evenwhenthe maximumfiring
2•) 4'0 6'0 8•
Level (dB)
FIG. 2. Onsetandsteady-state
rate-levelfunctions
for an auditory-nerve rate has been reached.
fiberstimulated
by a toneat thefiber'scharacteristic
frequency.
For each model we have determined a reference sound
fibershavebeenproposed(e.g.,Geisler,198I, 1990;Meddis
et al., 1990) but are not evaluated here.
Electrophysiological
studiesdistinguish
a secondrateintensityfunction.The onsetRIF is a rate measuredover a
shortperiodof timefollowingthe onsetof a toneburst.The
functionincreases
monotonicallywith tonelevelandshows
littleor nosignof saturation
at highstimulus
levels(Fig. 2).
The exactshapeof the onset-RIF varieswith the risetimeof thesignalandthedurationof the recordinginterval
(Smith, 1988).In all cases,however,therefractorypropertiesof spikegenerationprovidean upperlimit on the onset
firing rate. The upperlimit is about 1000 spikes/s,which
corresponds
to an absoluterefractoryperiodof 1 ms.
Smith et al. (1983) have shown that even when the on-
set firing rate is limited by refractory effects,increasesin
pressurelevelcorresponding
to the rate-intensitythreshold.
Usingthe methodof Smithand Zwislocki(1975), we define
zerodB asthe levelat whichstraightlinespassthroughthe
steepest
portionsof theonsetandsteady-state
RIFs intersect
a horizontallinedrawnat an ordinatecorresponding
to the
rateof spontaneous
activity(Fig. 2).
Following the methods of Westerman and Smith
(1984), onsetand steady-state
RIFs wereobtainedfor each
modelin responseto 300-mstonebursts.The onsetfunction
represents
the firingrateaveragedoverthe first (or highest,
if not the first) msof the response
andthe steady-state
function represents
the firingrateaveragedoverthe last20 msof
response.
The resultsare presentedin Fig. 3. The empiricaldata
obtainedfrom gerbil (Westerman,1985) are shownin the
top-left panel. This format is commonto all figureswhere
modeledoutputsare comparedto animaldata. All models
Onse/
• 9001
gerbil
data
9001
9001
Allen•,•.
• Steady
state
0/ Meddis
ß13
0/
0
o
O3
•
20
40
900 Cooke
O1
o
2'0 40
0
20
40
0
20
40
9ooJ
Schwid
& 900]
Oono
Sujaku
0/Geisle.••
O/
• &
0
20
40
0
20
40
20
40
Payton
0
20
40
0
20
40
0
Level(dB)
FIG.3.Comparison
ofrate-level
functions
between
themodels
andWesterman's
( 1985
) gerbil
data.Theonset
function
represents
thefiringrateduring
the
first(or highest)
msaftertoneonset.
Thesteady-state
function
represents
theaverage
firingrateoverthelast20msofa 300-ms
toneburst.
907
J. Acoust.SOC.Am.,Vol. 90, No. 2, Pt. 1, August1991
M.J. Hewittand R. Meddis:Hair-cellmodelsevaluation
907
A
0
• O0
200
0
'1O0
200
0
• O0
200
900t
b.Cooke
9001
Schwid
&900]
Oono
&
01•r
•
0
100
01-•
200
0
•
100
01---•
200
0
9001
Hall
•
100
Brachman
-
0
200
L
1 O0
200
0
1 O0
200
0
1 O0
200
Time (ms)
FIG. 4. Post-stimulus
timeexcitation
histograms
forthemodels
compared
toWesterman's
(1985)gerbildataforstimulus
presentation
levels
of l0 and40
dB.
The response
of the Schroeder
andHall modelshowsan
onsetpeak that is far too largeat mediumand high signal
levels.This givesan adaptationresponse
that failsto match
the two-component
functionreportedin the literature.Allen'smodelalsofails to producea two-component
adaptation response.
The remaining modelsshow two-componentadaptaB. Two-component adaptation
tion;however,onlythe Meddismodeldemonstrates
approximately
level-independent
short-term
time
constants
aswell
Followinga periodof silence,a toneburstwill produce
as
level-dependent
rapid
time
constants.
aninitialpeakof activityfollowedbya declinethatisinitialThe modelsproposedby Ross,and Oono and Sujaku
ly rapidthenslower.WestermanandSmith (1984) propose
show
two-component
adaptationonlyat levelsabove10dB.
that the adaptationresponse
consists
of two exponentially
Ross
highlighted
the
insufficient
onsetpeakingat medium
decayingcomponents:
stimuluslevelsasa majorfault of the model.Further analyy(t) = ARe-'/• + Asxe
- t/•,T+ Ass,
(2) sishasshownthat thisdefectwasdue,in part, to the model
neuron that Rossincludedin the original simulation.RewhereAR andASTare the magnitudes
and •'R and rSTthe
decaytimeconstants.
of therapidandshort-termadaptation movalof this componenthasshownthat insufficientonset
components;
Assis a constantrepresenting
the steady-state peakingis only a problemat low (below 10 dB) stimulus
levels.In contrast,Westerman'sgerbil data (Westerman
response.
andSmith, 1984;Westerman,1985) showa prominentonset
The firstcomponent,
knownasrapidadaptation,
hasa
peakat a stimuluslevelof 8 dB.
time constant(rR) of lessthan 10 msand decreases
with
increasing
tonelevelto as low as 1-msfor high-amplitude
producethe predicteddichotomybetweenonsetandsteady
stateRIFs althoughthepreciseformof thefunctionsvaries
betweenmodels.However, it'shouldbe notedthat the onset
functionof the Schroederand Hall modelis veryunrealistic.
The rategrowsfar toorapidly(spikes/sperdB) at medium
and highstimuluslevels.
stimuli (Westerman and Smith, 1984;Yates and Robertson,
1980;Yateset al. 1985).The second
timeconstant(tsT) is
about70-ms(range20-100ms) andis independent
of tone
C. Recovery of spontaneous activity
level (Smith and Zwislocki, 1975; Westermanand Smith,
1984).
beforerecovering
backto spontaneous
rate.The recovery
functionis describedby a singleexponential,with a time
Post-stimulustime histograms(PSTHs) were constructedfrom modelresponses
to 100-mstoneburstsat var-
constantbetween40 and 100 ms (Harris and Dallos, 1979;
Smith, 1977;Westerman, 1985). Yates et al. (1985) noted
iousintensities.
Figure4 showstheresultfor eachmodelfor
that the amountof depression
of firingat toneoffset,and the
exactnatureof recovery,depended
onthestimuluslevel,and
not on the absolutevalueof spontaneous
rate. They reporta
stimulus levels of 10 and 40 dB above threshold. Time con-
stantsof adaptationderivedfrom the responses
andtypical
AN data (Westerman,1985) are shownin Fig. 5. The method usedfor fittingtheexponentials
to themodelresponses
is
givenin Meddis (1988).
908
J. Acoust.Sec.Am.,Vol.90, No.2, Pt.1, August1991
At the offsetof a tone burst, AN firing briefly ceases
"typical"recoverytimeconstantof 20 ms,accompanied
by
a small amountof slowerrecoveryfollowingthe offsetof
more intense stimuli.
M.J. HewittandR. Morris:Hair-cellmodelsevaluation
908
TABLE I. Timeconstants
ofrecovery
tospontaneous
rateaftertheoffsetof
a 40-dB tone burst.
Physiological
forwardmaskingparadigmshavefurther
helpedour understanding
of AN recoveryprocesses.
The
work of Smith (1977) andHarris andDallos (1979) investigatedthe roeeveryfrom adaptationof singleAN fibersin
gerbilandchinchilla,respectively.
In brief,it wasestablished
that AN recoveryfrom maskingstimuli followeda single
exponentialcurve.Both studiesnoted,however,that the responseat stimulusonsetrecoveredat a fasterrate than did
the total response.
Thisinitial,fastercomponent
of recoverywasstudiedin
detail by Westerman(1985) usingthe paradigmshownin
Fig. 6. The response
to the probewasmeasuredasa decrement whencomparedto the response
in the absenceof the
preceding
maskingtone.Usinganalysiswindowsof 1and20
Time constant
Model
( ms)
Schroeder and Hall
9.2
Oonoand Sujaku
7.9
Allen
2.3
Schwid and Geisler
9.4
Ross
57.6
Brachman-Payton
4.0
Cooke
8. I
Meddis
30.4
Chinchilla ( + 20 dB)
Harris and Dallo• ( 1979}
Gerbil ( -I- 40 dB)
Westerman (1985)
37.0
47.0
Guineapig
D. Recovery of adaptation functions
ms (onset and short-term measures}, Westerman fitted two
20.0
recoverytime constantsto thedata;the fastcomponentwith
Yates et al. (1985)
a time constant of between 20 and 50 ms and the second,
slowercomponentwith a time constantof greaterthan 150
ms.
An examination of Table I suggeststhat all but two
models (Ross, and Meddis) would fail to mimic the data of
acThe time constantsof recoveryto spontaneous-firing Westerman.In mostcases,the recoveryof spontaneous
rateswerederivedfrom themodelPSTHs (Fig. 4) andare
tivity followingtoneoffsetis far too rapid.Figure7 confirms
the poor modelingof recoveryprocesses.
The modelsproshownin TableI alongwith data from threespecies.
The
modelsof MeddisandRossshowthemostrealisticrecovery posedby Rossand Meddisquantitativelypredictthe slower
short-termrecoveryrate but alsoincorrectlypredictan onof spontaneous
activity.'However,
thecomplete
cessation
of
firing at tone offsetis not shownby any model.Moreover, setrecoveryrateof equalslope.It is worth noting,however,
the remainingmodelsshowveryunrealistic
recoveryprop- that the data pointsbetween0 and 20 ms of Westerman's
erties.Quantitatively,the process
occursin the modelsat a
resultsdoseemto departfroma singleexponentialimprovement, although, he chose to draw a single best-fit line
muchfasterratethanthat derivedempirically.
gerbildata
10
c
o
o
M
30
50
001
10
Allen
30
50
001
-
Cooke
10
30
001
Schwid&• -- :
Geisler
50
10
30
& Hall
10
50
30
•001
Brach•
Ross*
10
-
[ Sujaku
*
50
1001
-Schroeder
10ono•t
Pg•yton
3O
5O
10
-
3O
5O
Level (dB)
FIG. 5. Time constantsof adaptationfitted to model responses
comparedto Westerman's(1985) data fitted to derivedexcitationfunctions.The twocomponent
adaptationresponse
equation(2) couldnotbefittedto the responses
fromtheSchroeder
andHall model,or to Allen'smodel.Asterisksindicate
single-component
adaptationat 10 dB.
000
J. Acouat. Sea. Am., VoL 00, No. 2, Pt. 1. August 1991
M.J. Hewitt and R. Meddis: Hair-cell models evaluation
909
are required to quantify the exact nature of the recovery
functions.
Level
(dB)
E. Additivity
Smith and Zwislocki (1975) have establishedthat the
0
processof short-termadaptationis additivein nature.They
showedthat the increasein firing rate elicitedby a stimulus
amplitudeincrementis independentof the stateof adaptation of the fiber.Similaradditivitywasfoundfor responses
to decrementsin intensity applied after adapting tones
(Smith, 1977;Abbas, 1979).
Later,Smithetal. (1985) studiedadditivitywith analysiswindowsdesignedto emphasizethe propertiesof rapid
adaptation.The stimulusparadigmsusedby Smithand his
colleagues
are shownin Fig. 8. Large-windowanalysis(20ms) confirmedthe earlier findingsthat adaptationwasaddi-
500
tive for stimulus increments and decrements. The effect was
alsoshownto bevalidfor small-window(1-ms) responses
to
Time (ms)
stimulusincrements.In contrast, the small-window deereFIG. 6. (a) Forwardmaskingparadigmusedby Westerman(1985) to
study recoveryof responseafter adaptation.Masker duration:300 ms;
probeduration:30ms;maskerandprobeat43 dB,At variedbetween
0 and
200 ms. (b) Schematicof corresponding
AN fiberresponses.
mentalresponse
decreased
withincreasing
timedelay,andin
proportionto the decrease
in firing rate producedby the
pedestal.Normalizedmodel responses
to the addifivity
paradigms
areshownin Fig. 9, togetherwith the empirical
data of Smith et al. (1985).
throughthe points.The samedeviationcan be seenin the
onsetresponses
of the Meddismodel.The issueis not clearcut, however, as a more detailed examination of Wester-
man'sdata showlargevariationsin the recoverytimesmeasuredfromhisfiberpopulation.It seemsthat additionaldata
103
•l)
t=
•
103
I ] term
0
gerbil
data 1
100 200
0
ß•
O "-"
u}
IVieddis1 •
100 200
103
(n '•.
•- o)
All of the computedresponsesshowed spike-rate
changesthat werequantitativelyverydifferentfrom the empiricaldataof Smithetai.; however,it wasevidentthatsome
of themodelsshowedthecorrecttrends.The multiple-reservoir modelsproposedby Brachman-Payton,and Schwid
and Geislerqualitativelyreproducethe dataof Smithet al.
(1985). The small- and large-windowincrementresponse
Cooke
11 7',,_
0
100
$chwid&
Geisler
1'
200
0
100
200
rr 103
,
•03
t
Schroeder
o
] 00
•00
o
• oo
Allen.
0
200
103
t• Oono
&
103
t• Brachman
Sujaku
0
100
0
100
1I
2oo
100
200
Payto?_
200
Time after masker (ms)
FIG. 7. Recoveryof response
components
after adaptation.Comparisonof modelsand Westcrman's(1985) data (includinghis "best-fit"lines). The
decrement
isthedifference
betweentheresponse
of an unadapted
fiberto a 43-dBtoneandtheresponse
of a fiberto theprobeaftertheoffsetof a 300-ms,43dB maskingtoneof the samefrequency.The onsetdecrements
arebasedon the maximall-ms firingrateafter probeonset.The short-termdecrements
are
basedon the ratebetween10and 30 msafter the probeonset.
910
J. Acoust.Sec. Am.,VoL90, No. 2, Pt. 1, August1991
M.J. Hewittand R. Moddis:Hair-cellmodelsevaluation
910
1. Changes with frequency
Level
(dB)
4O
2O
I Control
20
ms
I 0ms 30
ms
I 10
ms 40
ms
0 / ..... • m
.•
m
/
o
60 Time(ms) o
60
(b)
Control
20 ms
o ms
30 ms
10 ms
40 ms
Periodhistograms(PHs) showthatAN-fiber responses
phase-lock
to thepositivehalf-cycleof low-frequency
tones.
At high signalfrequencies,
however,the periodhistogram
showsno relationshipto the signal'sphasecharacteristics.
Roseetal. (1967) quantifiedthelossof synchrony
with the
synchronization
coefficient(the densityof the mostpopuloushalf of the periodhistogramdividedby total density).
Johnson (1980) studied first-order synchronyeffects
over a rangeof frequenciesusingthe synchronizationindex
(SI). Using the Fourier transformof the periodhistogram,
Johnsoncalculatedthe SI by takingthe responsemagnitude
at the stimulusfrequencydividedby the magnitudeat dc.
Below 1 kHz, the SI is independent
of stimulusfrequency.
Above 1-kHz, the data are modeledby SI = I -f/6, where
f= frequencyin kHz.
Johnson'smethodassumes
that the periodhistogramis
sinusoidal,an assumptionthat is not alwaysjustifiedespeciallyat low frequencies
andhighsignallevels.Accordingly,
it is helpfulto useboth measures.
To measuremodel synchronizationcoefficientsand indices, stimuli of 1, 2, 3, 4, and 5 kHz were used and the
Level
samplerate increasedto 100 kHz. The resultsare presented
in Figs. 10 and 11. Only three modelsshowfrequency-dependentsynchronization
measures.
The modelsproposedby
Meddis and Allen simulate the falloff of synchronization
with increasingfrequencyaccurately.One small deviation
from the empiricaldata occursin the 5-kHz casewherealmost completelossof synchronization
shouldbe evident.
The measuresof synchronizationfor the Brachman-Payton
(dB)
o
so Time(ms)o
6o
model are similar, but the rate of falloffat 3 kHz and aboveis
too shallow.
FIG. 8. Schematicrepresentation
of the paradigmsfor studyingadditivity.
(a) Decrements
of 6 dB appliedat varioustimedelaysaftertheonsetera
pedestal.(b) Incrementsof 6 dB appliedto a pedestal.In both (a) and (b),
thepedestal
wasof 60-msdurationand 13dB abovethreshold.
The control
stimuluswasthe pedestalalone.
2. Changes with level
Johnson(1980) alsousedthe SI to monitor synchroni-
zationchangeswith level.Doubleordinateplotsof SI and
firingratechanges
overa rangeof stimulusintensities
show
themodulationof spontaneous
activityby aninputstimulus
occurs at intensities below rate threshold. From such data,
the modulationthresholdfirst notedby Roseet al. ( 1971)
functionsare horizontal(i.e., additive),asis the large-windowdecrementresponse
function;finally,thesmall-window
decrementfunctiondecreases
with increasingdelay.
canbederived.Figure12showsthatall modelssuccessfully
replicatethiseffect.
Cooke's model is additive for stimulus increments but
fails to producea large-windowtime-independent
decrementalresponse.
The modelsproposed
by OonoandSujaku,
Allen, and Meddissuccessfully
replicatethreefunctionsbut
the small window incrementalresponseis nonadditiveand
discrepantwith the empiricaldata at very shorttime delays
( < 10 ms).
The modelsproposedby Schroederand Hall, and Ross
predicttime-dependent
incrementalresponses
for bothlarge
and small windowsand are clearly nonadditivein nature.
G. Computational efficiency
As stated in the Introduction, hair-cell models are in-
corporatedinto modelsof speechrecognitionand central
auditoryprocessing.:
Modelersofsuchsystems
haveto consider the trade-offbetweenspeedand accuracyof the haircell modeltheyselect.To completethisstudy,we havemeasuredthe relativecomputationalefficiencyof eachmodel.
Comparativetimestakenfor eachmodelto processa 1 s
toneburst (includingprecedingandfollowingperiodsof silence) presentedat 40 dB above thresholdare shown in Ta-
F. Low-frequency phase locking
This section deals with the fine-time
structure
of AN
responsesto single-frequencytonesat different levels and
frequencies.
911
J. Acoust.Sec. Am., Vol. 90, No. 2, Pt. 1, August1991
ble II. No attemptwasmadeto optimizethe computercode
of any model.An examinationof the resultsshownegligible
differences
betweenmostmodels.Exceptionsto thisare the
multiple-reservoirmodelsproposedby Schwidand Geisler
M.J. Howittand R. Moddis:Hair-coilmodelsevaluation
911
•'
•
o
ß
LW
0
40
Meddis
o
0
Allen
0
40
0
4O
SchWid
&
0
40
0
Oono &
Geisler 0
4O
Sujaku
0
4O
Hall
FIG. 9. Modelresponses
andtheemBrachman
0
40
0
0
40
-
. . P.a?on. .
0
40
pirical data of Smith eta/. (1985) to
theadditivitystimuli.(a) Responses
to decrements in level as a function
oftimedelay.(b) Responses
tostimulus increments as a function of de-
Timedelay (ms)
lay. The small window (SW) is
basedontheresponse
duringthefirst
msafterthestimulus
change,
andthe
large-window (LW) measure is
based on the first 20 ms after the
stimulus
change.
0
• '--
0
•O
_.c(•
•
0 0
gerbil
data
40
0
Meddis
40
0 0
Cooke
40
Geisler
40 0 0
Schwid
&
0 0
,•m1.0,• Schroeder
& 1.0
•
0
-'
0
....
Sujaku
40
1.0
Brachman -
Ross
0
40
Allen
40
0
Payton
0
40
o
40
Timedelay (ms)
1
3
5
50],Meddis
1
3
5
lOO]•• :
3
5
1001,
Oono
&
50]Geisler
1
3
FIG. 10. Modelsynchronization
coefficients
as a functionof frequency
Schwid
&
Cooke
1
compared to squirrel monkey data
50JSujaku
5
1
3
from Rose et al. (1967). No syn-
5
chronization data for Cooke's enve-
lope model.
1001
100].
Schroeder&
501Hall
3
5
501Ross3
100
•......•....•
1 Brachman-
5
50]Payton
1
3
5
Frequency (kHz)
912
J. Acoust.Soc. Am.,Vol. 90, No, 2, Pt. 1, August1991
M.J. Howitt and R. Mealdis: Hair-cell models evaluation
912
x
1.0
cat
dat•
&
1.0
i • IOono
ISchroeder
& 1.01 1.0
Brac•._..•..._
'
.1
1
3
5
1
3
3
5
Schwid &
Cooke
.lJSujaku
.1 Geisler
1
3
5
3
5
1
.1 Ross
.1 Hall
1
3
5
1
3
5
.1]Payton
1
3
5
Frequency (kHz)
FIG. 11.Modelsynchronization
indices
asa functionof frequency
compared
toJohnson's
(1980)catdata.
(1982) and Brachman-Payton(Payton, 1988), which are
relativelyslow.
Although one particular implementationof a model
may beoptimizedto run fasterthan another,it is clearthat
tracking the contents of discrete immediate sites (e.g.,
Brachman-Payton,Schwidand Geisler) reducescomputational efficiency.The return for the increasedcomputation
time requiredto run multiple-sitemodelsis the time-independentresponseto incrementalstimuli.
III. DISCUSSION
Thispaperhasreporteda detailedcomputational
investigationof eight hair-cell models.The propertiesof each
model are summarized
in Table III.
A. Adaptation
AN-fiber responses
adaptin response
to a constant-level
stimulus.This propertyis not exclusiveto the hair-cell/AN
1'øt
cat
data
_•'
x
0•.
•
-40
0
40
-40
0
40
-40
0
40
Schwid
&j•'
Cooke
Oono&
_,,"
i1.00
Geisle•_•
ß
-40
0
40
-40
0
40
0
Payton
.
-40
0
40
03
-40 0
40
Brachman
-40
0 u_
0
40
Level (dB)
FIG. 12.Synchronization
indexasa functionof stimulus
levelfor modelresponses
andempiricaldatafromcat (Johnson,
1980),together
withnormalized
steady-staterate-levelfunctions.
913
J. Acoust.Soc.Am.,Vol. 90, No. 2, Pt. 1, August1991
M.J. Hewittand R. Meddis:Hair-cellmodelsevaluation
913
TABLE II. Timetakenby eachmodelto process
a I-s toneburstpresented
at 40 dB.
Model
Time (s)
Schroeder and Hall
2.8
Oono and Sujaku
2.0
Allen
Schwid and Geisler
3.2
42.0
Ross
2.0
Brachman-Payton
320.0
Cooke
Meddis
1.2
4.0
modelas only a fractionof the availabletransmitterwas
released
pertimeunitwhenthecellwasactivated.
Theprincipleof fractionalreleaseis inferredfromthe workof FurukawaandMatsuura(1978), andwasadoptedinto thelater
multiple-reservoir
models(SchwidandGeisler,1982;Smith
and Brachman,1982;Cooke, 1986). Although both shorttermandrapidtimeconstants
canbefittedto thepost-stimulustimehistograms
of thesemodels,a morecarefulconsiderationof two-component
adaptationwasshownby Meddis
(1988), Ross(1982), and Smith and Brachman(1982).
The Meddismodelquantitatively
simulates
Westerman
andSmith'srapidadaptation
databyfastreuptake
of transmitterfromthecleftbythepresynaptic
membrane.
Thislimits the amount of free transmitter available for release.
complex,but is inherentin all sensory-receptor
systems.
Availableexperimentalevidencesuggests
that spikeactivity
in AN fibersis initiated by the releaseof a transmitter substancefrom the hair cellinto the cleftthat synapses
with the
fiber.The hair-cell receptorpotentialthat is likely to cause
the releaseof transmittershowsno adaptation(Russelland
Sellick, 1978; Goodman et al., 1982). It is, therefore, believedthat adaptationis due tO a progressive
rundownof
Transmittertakenupfromthecleftissubjectto a reprocessingdelaybeforebeingavailable
forrelease
again.Thisdelay
ta. As the probabilityof releasewasdirectlyrelatedto stimulus intensity,the rate of adaptationvariedwith stimulusintensityin a way not consistentwith neuraldata (e.g., Kiang
While muchtime hasbeendevotedto modelingadaptation, the simulationof recoveryprocesses
at the offsetof a
tonebursthaslargelybeenneglected.The presenceof two
differentexponential
recoveryfunctionsisnotfoundin any
modeledoutputs.However,WestermanandSmith's(1988)
analyticalmodelprovides
someinteresting
insightsto recovery processes.
The modelis basedon the configuration
of
reservoirsproposedby Brachman,but differsin that the
process,
aswellasglobalreplenishment
ofthefreetransmitter store (largely), determines
the short-termadaptation
time constant.There is no direct evidenceto supportthis
detailof themodel;it wassimplyincorporated
asa possible
explanation.
However,no othermodelhasaddressed
the
issueof whathappens
to thetransmitterin thecleft.
transmitter substance within the cell. Other available eviThe modelsproposedby Brachman-Payton
and Ross
dence(e.g., Furukawaand Matsuura, 1978) substantiates simulaterapidadaptation
by relyingon smallvolumereserthis viewpoint.
voirslyingcloseto thepresynaptic
membrane.
DuringstimIn general,the modelsevaluatedhereshowan adapta- ulation, thesereservoirs(Brachman'simmediatesitesand
tion of responsethat can be explainedby the dynamicsof
reservoirv4 of Ross)becomedepletedof their transmitter
depletionand replenishment
of transmittersubstance.
The
with a time constantmatchedto that of rapid adaptation.
mechanismsby which the specificadaptationcomponents These immediatestoresare replenishedby larger local
are modeled, however, deservefurther comment.
stores.Short-termadaptationissimulatedby theslowerdeAdaptationin the single-reservoir
modelproposedby
pletionof the localstores.
Schroederand Hall (1974) is producedby the depletionof
transmitterquantafromthereservoir;
theprocess
continues
B. Recovery
until thelossof quantaisbalancedby the replacement
quan-
et al., 1965).
The proposalof Oono and Sujaku (1974) assumeda
single-reservoirmodel with transmitter-releaseproperties
similar to those of the Schroeder and Hall model. However,
more realisticadaptationpropertiesare producedby the
TABLE IlL Summaryof theeightmodelsresponse
properties.
Number
Number
Accept
of
of
arbitrary
reservoirs parametersstimulus
Model
Two-
Additive
Physiological
Low-
component(to stimulusforwardmasking frequency
adaptation increments)response
phase
locking
Computationally
efficient
Schroeder and
Hall
1
3
yes
no
no
no
no
yes
OonoandSujaku
Allen
1
I
3
10
yes
yes
yes
no
no
no
no
yes
no
no
yes
yes
6
4
15
9
yes
yes
yes
yes
yes
no
no
no
no
no
no
yes
19
6
8
yes
no
yes
yes
yes
yes
yes
yes
no
no
no
no
yes
no
yes
no
yes
yes
Schwid and
Geisler
Ross
Brachman-
Payton
Cooke
Meddis
914
1024
3
3
J. Acoust.Sec.Am.,Vol.90, No.2, Pt.1, August1991
M.J. HewittandR. Meddis:Hair-cellmodelsevaluation
914
model parameterswere determineddirectly from experimental data (see Westerman and Smith, 1988, for details).
Their analysissuggests
that two-component
recoverycanbe
simulatedby the Brachmanconfiguration
if the ratesof re-
plenishment
to thelocalandimmediate
storesareapproximatelyequal.Thisimpliesthatreplenishment
of transmitter
is not preferentialto onestore.
C. Additivity
An importantconstraint
onmostmodelsisthefailureto
replicateadditivityphenomena.
The mostcrucialassumption madeby modelswith additivepropertiesis the existence
of manyreleasesitesspatiallyorderedby increasing
threshold (cf. Furukawa and Matsuura, 1978). All other model
configurations
evaluatedin thisstudycouldnot mimicthe
main additivity data.
response
becamesmallin comparisonwith the dc response.
The attenuationof the ac responsewas ascribedto the capacitanceof the cell membrane.Later, Palmerand Russell
(1986) establishedfrom direct measurementsof the membranetime constantand the ac/dc ratiosthat phaselocking
did indeeddeclinedirectlyin proportionto thedeclinein the
innerhair cells'ac/dc ratio,suggesting
that the limitingfactor for phaselockingis thehair-celltimeconstant.
E. Analytical models of hair cell-functioning
Our studyhasconfineditselfto modelsthat wereimplemented by their authors on digital computers. However,
someanalyticmodelsdeservesomecomment.For example,
Eggermont's(1985) model, which features processes
knownto operateat theneuromuscular
junction,showsrealisticadaptationand recoveryproperties.
Transmitteris releasedfrom an hypothesized
hair cellat a rateproportional
to the amount of transmitter available for release and a stim-
D. Synchronization
Anotherfailurecommonto manymodelsis the absence
of reduced responsesynchronizationto high-frequency
stimuli. Only the modelsproposedby Meddis, Allen, and
Brachman-Paytonshowthe effect.
In the Meddismodel,phaselockingis affectedby the
rate at which transmitter can be clearedfrom the synaptic
cleft. The ratesof transmitterlossand re-uptakefrom the
cleftare independentof stimulusfrequency.When the stimulusfrequencyis highrelativeto theseratesthe ability of the
systemto reflect the waveform'sfine structure is thus reduced.
Allen used a low-pass filter in his model to reduce
synchronyat high frequencies.The sourceof the filter was
ascribed to the diffusion of calcium ions from the cell wall to
thesynapticregionof thecellwheretransmittervesicles
reside.
The Brachmanmodeluseda low-passfilter to simulate
the intrinsicresistance
and capacitance
of cell membranes.
Payton (1988) examinedthe synchronyresponsesof the
Brachmanmodelbeforeand after adaptationcomparingthe
resultsto Johnson'ssynchronizationdata. The low-passfilter implemented produced results that showed excellent
agreementwith the empiricallyderivedSI = 1 -f/6. However, the adaptationresponseimposedchangesthat were
characteristicof high-passfiltering,which resultedin the
lossof well-matchedindices.To regainappropriatesynchronization data, Payton introduceda second-orderlow-pass
filter after the adaptationstageof the model. Excellentre-
ulus (envelope)amplituderelatedpermeabilityfunction.
Transmitterquantain the cleftbind with freereceptorsites
on thepost-synaptic
afferent-nerve
fiber.After activationof
sitesby quanta,thereceptorsitesareconvertedto an inactive
or occupiedstate.Inactivation occursat a rate that depends
on thenumberofreceptorsthat canbeoccupiedaswellason
the numberthat are occupied.Enzymaticactionon the receptor-transmitter
complexfreesreceptorsites.The receptor siteis only freeto combinewith transmitterquantaafter
a periodof recovery.The numberof receptor-transmitter
complexes
represents
thepost-synaptic
excitatorypotential.
Eggermont's
modeldiffersfrom otherproposals
in that
it considerstwo adaptationmechanisms,
the first locatedin
the hair-cell/synapse
and the secondin the auditorynerve.
In brief, it has the following properties:(i) additive responses
to stimulusincrements;
(ii) short-termadaptation
time constants
independent
of stimuluslevel;and (iii) forwardmaskingresponses
similarto thosefoundin theempirical studiesof Harris and Dallos (1979) and Smith (1977).
A differentapproachto modelingAN datawastakenby
Westermanand Smith (1988). Usingexperimentaldata on
gerbilAN responses,
theywereableto derivemodelparameterssuitedto a three-stagesynapticmodelsuchas the one
proposedby Brachman. To accurately model two-stage
adaptationin accordance
with empiricaldata,theypropose
that both the volumeand membranepermeabilityof the reservoirsshould be stimuluslevel dependent.This differs
fromall previoussynapticmodelswhereit isconsidered
that
only onefunctionis requiredto be leveldependent.
sults were obtained with the time constant of the second-
order filter setto 0.05 ms. Payton removedthe secondfilter
from her final implementationdue to lack of physiological
evidence for such a filter. It does, however, raise further
questions
aboutthe natureof the filteringrequiredto simulate synchronydata.
A numberof factorshavebeenproposedto accountfor
thelossof synchronization
in AN discharges
with increasing
frequency.
Anderson( 1973) presented
a quantitative
model
basedonthe premiseof temporaljitter withinthe periphery.
Russelland Sellick (1978) noted from their inner hair-cell
recordings
that phaselockingdisappeared
whenthe cell'sac
915
J. Acoust. Soc. Am., Vol. 90, No. 2, Pt. 1, August 1991
F. Biological correlates
Many of the modelsevaluatedin this studycould be
describedas phenomenological.
The modelshavebeendesignedto replicatecertainresponse
propertiesof AN fibers
and then the processing
stepsare correlatedwith known
physiological
processes.
Thisapproachhasresultedin models that are computationally
convenientand alreadywell
suitedto applicationssuchasspeechrecognitiondevicesthat
usemodelsof the peripheralauditorysystemas input devices.However,the modelsdo not representthe biological
processes
accuratelyenoughto suggest
crucialexperimental
M.d. Hewitt and R. Meddis: Hair-cell models evaluation
915
designsto evaluatetheoriesof AN spikegeneration.Future
hair-cell/AN modelswill needto consider,for example,the
role of intracellularcalcium;potassium,and calcium-activated potassiumchannels;transmitterre-uptakeand postsynapticmechanisms.Even though suchmodelswould be
complexand computationallyexpensive,they will providea
meansof assessing
the relativemeritsof any theoryof AN
spikegeneration.
Eventhough
it maynothavebeentheintention
oftheauthors
whoproposedthe models,most,if not all, of the simulationsevaluatedherehave
beenusedto providethe primaryinput to largersystems(e.g., Payton,
1988).
Abbas,P. J. (1979). "Effectsof stimulusfrequencyon adaptationin auditory-nervefibers,"J. Acoust. Soe.Am. 65, 162-165.
Allen, J. (1983). "A hair cell modelof neuralresponse,"in Proceedings
of
theIUTAM/ICA Symposium,
editedby E. deBoerandM. A. Viergever
(Martinus,The Hague,and Delft U. P., Delft, The Netherlands),pp.
193-202.
G. Nonlinearity
It is well known that the release of transmitter
at the
Anderson,D. J. (1973). "Quantitativemodel for the effectsof stimulus
frequencyuponsynchronization
of auditorydischarges,"
J. Acoust.Soc.
Am. 54, 361-364.
hair-cell/synapse
is precededby a numberof nonlinearities. Brachman,M. L. (1980). "Dynamicresponsecharacteristics
of singleauFor example,input-outputfunctionsfor basilarmembrane
ditory-nervefibers," SpecialRep. ISR-S-19, Institute for SensoryResearch,SyracuseUniversity,Syracuse,NY 13210.
vibrationshowsaturationof response
at the bestfrequency
Cooke,M.P. (1986). "A computermodelof peripheralauditoryprocessfor levelsas low as 30 dB SPL (Sellick et al., 1982;Yates et
ing," SpeechCommun.$, 261-281.
el., 1990). The nonlinearpreprocessing
would havehad an
Eggermont,J. J. (1973). "Analoguemodellingof cochlearadaptation,"
important role in shapingthe neural responses
reportedin
Kybernetic14, 117-126.
the literature, and, will have to be accountedfor in future
hair-cellmodelingefforts.All existingmodelsappearto assumethat the basilarmembraneresponse
is linear.Nonlinear permeabilityfunctions,however,may concealeffects
thatmightbebetterattributedto mechanicalprocesses
prior
to the hair-cell transduction.
IV. CONCLUSIONS
Eggermont,J. J. (1985). "Peripheralauditoryadaptationand fatigue:A
model orientated review," Hear. Res. 18, 57-71.
Furukawa,T., Hayashida,U., and Matsuura,S.S. (1978). "Quantalanalysisof the sizeof post-synapticpotentialsat synapses
betweenhair cells
and afferentnervefibersin goldfish,"J. Physiol.(Lond.) 276, 211-226.
Furukawa,T., and Matsuura,S.S. (1978). "Adaptiverundownof excitatory post-synaptic
potentialsat synapses
betweenhair cellsand eighth
nervein the goldfish,"J. Physiol. (Lond.) 276, 193-209.
Furukawa, T., Mikuki, K., and Matsuura, S. (1982). "Quantal analysisof a
decrementalresponse
at hair cell-afferentfiber synapsein the goldfish
sacculus,"J. Physiol.(Lond.)322, 181-195.
We haveexaminedthe relativemeritsof eightcomputa- Gaumond,R. P., Kim, D. O., and Molnar, C. E. (1983). "ResponSe
of
tionalhair-cellmodelsby comparingtheoutputswith mamcochlearnervefibersto briefacousticstimuli:Role of discharge-history
effects,"J. Acoust. Soe. Am. 74, 1392-1398.
malian auditory-nervedata. The evaluationshowsthat the
Geisler,
C. D. (1981). "A modelfor discharge
patternsof auditory-nerve
Meddismodelshowsonly minordiscrepancies
with the emfibers,"Brain Res.212, !98-201.
piricaldata.The Meddismodelis alsooneof the mostcomGeisler,C. D. (1990). "Evidencefor expansive
powerfunctions
in thegenputationallyefficientand is well suitedto providethe prierationof the discharges
of 'low- and medium-spontaneous'
auditorymary input to larger systemssuch as speech-recognition nerve fiber," Hear. Res. 44, 1-12.
Geisler, C. D., Le, S., and Schwid, H. (1979). "Further studies on the
devicesand modelsof central-auditoryprocessing.
Schroeder-Hall hair-cell model," J. Acoust. Soc. Am. 65, 985-990.
One important constrainton the Meddis model is the
Goodman, D. A., Smith, R. L., and Chamberlain, S.C. (1982). "Intracellufailure to replicatethe additivity effectfor onsetresponses.
lar andextracellularresponses
in the organof Corti of the gerbil,"Hear.
Res. 7, 161-179.
This problemcouldbe overcomeby introducingmultiplerelease sites in accordance with Furukawa
and Matsuura's
conceptualmodel.However,this would havea detrimental
effecton the model'srelativelyfast run time.
In thispaperwe havenot considered
the response
characteristics
of low spontaneous-rate
fibers,or, themodelsthat
have been proposedto replicate their properties. Both
Geisler (1981, 1990) and Meddiset el. (1990) have published,preliminaryfindingson the responsepropertiesof
their low spontaneous-rate
models.Work is currently in
progressin our laboratoryto makea moredetailedstudyof
the response
propertiesof the modelsandwill bethe subject
of a future paper.
ACKNOWLEDGMENTS
We are indebted to Jont Allen, Martin Cooke, Dan
Geisler,and Karen Payton for generouslysupplyingprogramlistingsand/or modelparameters.We are gratefulto
Bob Smith for helpfuldiscussions.
We thank Trevor Shackleton, StrangeRoss, and an anonymousreviewerfor valuablecommentson an earlierversionof this paper.
•Oneexception
to thisisCooke's
modelwhichrequires
theinputof halfwave-rectified
stimulusenvelopes.
916
J. Acoust.Soc. Am., Vol. 90, No. 2, Pt. 1, August1991
Gray, P. R. (1967). "Conditionalprobabilityanalysesof the spikeactivity
of singleneurons,"Biophys.J. 7, 759-777.
Harris,D. M., andDallos,P. (1979)."Forward
masking
ofauditory
nerve
fiberresponses,"
J. Neurophysiol.42, 1083-1107.
Johnson,D. H. (1980). "The relationship
betweenspikerate andsynchronyin responses
ofauditory-nerve
fibersto singletones,"J.Acoust.Soe.
Am. 68, 1115-1122.
Kiang, N. Y.-S., Watanabe,T., Thomas, E. C., and Clark, L. F. (1965).
"Dischargepatternsof singlefibersin the cat's auditory nerve," Res.
Monogr.No. 35, MITP, Cambridge,MA.
Liberman,M. C. (1978). "Auditory-nerveresponse
from catsraisedin a
low-noisechamber," J. Acoust. Soc. Am. 63, 442-455.
Meddis,R. (1986a). "Exploitingphysiologicalcodesin ASR," Proc. Inst.
Acoust. 8, 581-588.
Meddis, R. (1986b). "Simulation of mechanical to neural transduction in
the auditoryreceptor,"J. Acoust.Soc.Am. 79, 702-711.
Mealdis,R. (1988). "Simulation of auditory-neuraltransduction:Further
studies,"J. Acoust. Soc. Am. 83, 1056-1063.
Meddis,R., Hewitt,M. J.,andShackleton,
T. M. (1990). "Implementation
detailsof a computationalmodelof the inner hair-cell/auditory-nerve
synapse,"J. Acoust.Soc.Am. 87, 1813-1816.
Oono,Y., andSujaku,Y. (1974). "A probabilistic
modelfor discharge
patternsof auditorynervefibers,"Trans.Inst. Elect.Comm.Eng. (Japan)
57, 35-36.
Oono, Y., and Sujaku,Y. (1975). "A modelfor automaticgain control
observedin the firingof primaryauditoryneurons,"Trans. Inst. Elect.
Comm. Eng. 58, 61-62.
Palmer,A. R., and Russell,I. J. (1986). "Phase-locking
in the cochlearnerveof the guineapig and its relationto the receptorpotentialof inner
M.J. Hewitt and R. Meddis:Hair-cell models evaluation
916
hair cells," Hear. Res. 24, 1-15.
Payton,
K. L. (1988)."Vowelprocessing
byamodel
oftheauditory
periphery: A comparison
to eight-nerveresponses,"
J. Acoust.Soc.Am. 83,
145-162.
Rose,J. E., Brugge,J. F., Anderson,D. J., andHind,J. E. (1967). "Phaselockingresponse
tolow-frequency
tonesin singleauditorynervefibersof
thesquirrelmonkey,"J. Neurophysiol.
30, 769-793.
Rose,J. E., Hind, J. E., Anderson,D. J., andBrugge,J. F. (1971). "Some
effectsof stimulusintensityon response
of auditorynervefibersin the
squirrelmonkey,"J. Neurophysiol.34, 685-699.
Ross,S. (1982). "A modelof the hair cell-primaryfiber complex,"J.
Acoust. Soc. Am. 71,926-941.
Russell,I. J., and Sellick, P.M. (1978). "Intracellular studiesof hair cellsin
the mammaliancochlea,"J. Physiol.(Lond.) 264, 261-290.
Schroeder,M. R., and Hall, J. L. (1974). "Model for mechanicalto neural
transduction
in the auditoryreceptor,"J. Acoust.Soc.Am. 55, 10551060.
Smith,R. L., Brachman,
M. L., andGoodman,D. A. (1983). "Intensity
functionsand dynamicresponses
from the cochleato the cochlearnu-
cleus,"in Hearing--Physiological
Bases
andPsychophysics,
editedby R.
KlinkeandR. Hartmann(Springer-Verlag,
NewYork), pp. 112-118.
Smith,R. L., Brachman,
M. L., andFrisina,R. D. (1985)."Sensitivity
of
auditorynervefiberstochanges
in intensity:
A dichotomy
between
decrementsand increments,"J. Acoust. Soc.Am. 78, 1310-1316.
Smith,R. L., andZwislocki,
J.J. (1975)."Short-term
adaptation
andincrementalresponses
of singleauditory-nerve
fibers,"Biol.Cybernet.17,
169-182.
Weiss,T. F. (1966). "A modelof theperipheral
auditorysystem,"
Kybernetic 3, 153-157.
Westerman,
L. A. (1985). "Adaptationandrecoveryof auditorynerveresponses,"
SpecialRep. ISR-S-24.Institutefor SensoryResearch,
SyracuseUniversity,Syracuse,NY.
Westerman,
L. A., andSmith,R. L. (1984.)."Rapidandshort-term
adaptation in auditorynerveresponses,"
Hear. Res.15, 249-260.
Schwid,H. A., and Geisler,C. D. (1982). "Multiple reservoirmodelof
neurotransmitterreleaseby cochlearinner hair cell," J. Acoust.Soc.Am.
72, 1435-1440.
Westerman, L. A., and Smith, R. L. (1988). "A diffusionmodel of the tran-
sientresponse
of the cochlearinnerhair cell synapse,"
J. Acoust.Soc.
Am. 83, 2266-2276.
Sellick, P.M., Patuzzi, R., and Johnstone,B. M. (1982). "Measurement of
Yates,G. K., andRobertson,
D. (1980). "Very rapidadaptationin audi-
basilarmembranemotion in the guineapig usingthe M•3ssbauer
technique,"J. Acoust.Soc.Am. 72, 131-141.
Sicbert,W. M. (1965). "Someimplicationsof the stochasticbehaviorof
primary auditoryneurons,"Kybernetic2, 206-215.
tory ganglioncell," in Psychophysical,
PhysiologicalandBehavioralStudiesin Hearing,editedby G. vanden Brink and F. A. Bilsen(Delft U. P.,
Delft, The Netherlands).
Smith,R. L. (1977). "Short-termadaptationin singleauditory-nerve
fibers:somepost-stimulatory
effects,"J. Neurophysiol.
40, 1098-1112.
Smith,R. L. (1988). "Encodingofsoundintensitybyauditoryneurons,"
in
Neurobiological
Basisof Hearing,editedby G. M. Edelman,W. E. Gall,
and W. M. Cowan(Wiley, New York), pp. 243-274.
Smith,R. L., andBrachman,M. L. (1982). "Adaptationin auditorynerve
fibers:A revisedmodel,"Biol. Cybern.44, 107-129.
917
J. Acoust.
•oc. Am.,Vol.gO,No.2, Pt.1, August1991
Yates,G. K., Robertson,
D., andJohnstone,
B. M. (1985). "Very rapid
adaptationin theguineapigauditorynerve,"Hear. Res.17, 1-12.
Yates,G. K., Winter, I. M., and Robertson,D. (1990). "Basilarmembrane
nonlinearitydeterminesauditory nerve rate-intensityfunctionsand
cochleardynamicrange,"Hear. Res.45, 203-222.
Zwislocki,J. J. (1973). "On intensitycharacteristics
of sensory
receptors:
A generalized
function,"Kybernetic12, 169-183.
M.J. HewittandR. Meddis:
Hair-•ellmodelsevaluation
917