Simulation of auditory-neural
transduction: Further studies
Ray Meddis
DepartmentofHurnanSciences,
University
of Technology,
Loughborough
LE I13 TU, England
(Received31July 1986;accepted
for publication3 November1987)
A computationalmodelof mechanicalto neuraltransductionat the hair cell-auditory-nerve
synapseis presented.It producesa streamof events(spikes)that are preciselylocatedin time
in responseto an arbitrary stimulusand is intendedfor useas an input to automaticspeech
recognitionsystemsaswell asa contributionto the theoryof the originof auditory-nervespike
activity.The behaviorof the modelis comparedto datafrom animalstudiesin the following
tests:(a) rate-intensityfunctionsfor adaptedand unadaptedresponding;(b) two-component
short-termadaptation;(c) frequency-limited
phaselockingof events;(d) additivityof
responding
followingstimulus-intensity
increases
anddecreases;
(e) recoveryof spontaneous
activityfollowingstimulusoffset;and (f) recoveryof abilityto respondto a secondstimulus
followingoffsetof a firststimulus.The behaviorof themodelcompares
well with empirical
databut discrepancies
in tests(d) and (f) pointto the needfor furtherdevelopment.
Additionalfunctionsthat havebeensuccessfully
simulatedin previoustestsincluderealistic
interspike-interval
histograms
for silenceandintensesinusoidalstimuli,realisticpoststimulus
periodhistogramsat variousintensitiesand nonmonotonicfunctionsrelatingincrementaland
decrementalresponses
to backgroundstimulusintensity.The modelis computationally
convenientand well suitedto usein automaticrecognitiondevicesthat usemodelsof the
peripheralauditorysystemasinput devices.It is particularlywell suitedto devicesthat require
stimulusphaseinformationto be preservedat low frequencies.
PACS numbers:43.63.Bq,43.63.Pd,43.63.La
INTRODUCTION
decidebetweenequallysuccessful
accounts.
Regrettably,we
have
not
yet
reached
that
stage
because
no
existingmodel
Spikeactivity in auditory-nervefibersis a probabilistic
has
been
shown
to
agree
with
all
of
the
published
results
nonlinearfunction of the instantaneousamplitude of the
already
available.
Moreover,
as
new
results
are
published,
it
acousticstimulus.In recentyears,a numberof increasingly
is
difficult
to
decide
whether
existing
models
can
account
for
sophisticated
computationalmodelsof this processhave
investigation.
beenpresented
thataimto explaintheparticularnonlineari- themexceptafter a full-scalecomputational
ties that occurat the junction betweenthe inner hair cells Armchair evaluationof the issueis normally totally inadeand individualauditory-nervefibers,the point of neurome- quate.The purposeof thisarticleis to reporton a computationalinvestigation
of onemodel(Meddis,1986)in theconchanical transduction (Siebert, 1965; Weiss, 1966; Nilsson,
1975; Schroederand Hall, 1974; Oono and Sujaku, 1975;
text of three recentresearchresults:(a) the effectof stimulus
Eggermont,1973; Geisler et al., 1979; Brachman, 1980;
Ross,1982;Schwidand Geisler, 1982;Smithand Brachman,
1982; Westerman, 1985; Westerman and Smith, 1986;
stants(Westerman and Smith, 1984); (b) the effectof incre-
Cooke, 1986; Meddis, 1986).
Thesemodelsare of interestto hearingresearchersfrom
a numberof pointsof view. They offer a readilytestable
scientific
explanationof theobserved
phenomena
andstimulate the further developmentof theoriesof mechanism.
Modelsthatgeneratesimulatedspiketrainsin response
to an
acousticstimulusare also a necessaryprerequisiteto detailed modeling of physiologicalproee•eu occurring at
"higher"levelsof the system,for example,the cochlearnucleusor psychological
processes
suchas auditoryselective
attention(e.g., Evans,1986;Lyon, 1985). In addition,an
immediatetechnological
applicationof suchspike-generating systems
occursin the designof automaticspeechrecognition devices.In hearinglaboratories,spikegeneratorsare
alreadyin usefor trainingresearchers
and testingapparatus
without the needfor live preparations.
Ideally, a proliferationof modelsshouldstimulatefruitful empirical studiesby suggestingcrucial experimentsto
1056
J. Acoust.Soc.Am.83 (3), March1988
amplitudeon rapid and short-termadaptationtime conmentsand decrementsof stimulusamplitude(Smith et al.,
1985); and (c) the recoveryof rapid and short-term responsecapacity following maskingstimuli (Westerman,
1985).
The article will also presentan unintended,emergent
propertyof themodelwhichisthesimulation
of Roseetal.'s
(1967) observationof phase-locked
respondingand its restrictionto low-frequencyacousticstimuli. The high-frequencylimit on phaselockingis normallyascribedto lowpassfiltering characteristicsof the hair cell membraneas
manifest in the decline of the ac/dc ratio of inner hair cell
potentialsas stimulusfrequencyrises(Sellick and Russell,
1980;Palmer and Russell, 1986). In the model, it arisesas a
consequence
of delaysin removingtransmitterfrom the hair
cell-nerve fiberjunction.
I. THE MODEL
The modelhasbeenfully describedelsewhere(Meddis,
1986;modelB) butissummarized
in Fig. 1. It canbefully
0001-4966/88/031056-08500.80
@ 1988 Acoustical
Societyof America
1056
m
Free Transmitter
Pool
I
Reuptake
Reprocessing
Store
dq= y(1-q(t ))+ xw(t)-k(t )q(t )
The model is summarizedby three differentialequationsthat aregivenin Fig. 1. For the purposes
of computation, dt is normallysetto 0.00005s exceptwhenexplicitly
stated.The threeequationsare evaluated;therefore,20 000
timesper secondandthe quantitiesq(t), c(t), and w(t) are
changedafter eachiteration.The modelhassevenparameters,y, x, l, r, g, A, and B, that can be set by the modeler
(Table I).
dt
dc
•- = k(t ) q(t)-)c( t)- re(t)
II. METHODS
OF EVALUATION
dw
•- = rc(t)-xw(t)
FIG. 1. Flowdiagramfor transmittersubstance
anddifferentialequations
definingthe model.Takenfrom Meddis(1986), modelB, Fig.10.
understood
in termsof theproduction,movement,anddissi-
pationoftransmitter
substance
in theregionof thehaircellauditory-nerve
fibersynapse.
An amountq(t) of transmitter
In a previousarticle(Meddis, 1986), it wasshownthat
the model could realisticallysimulatemammalianadapted
spike-rate/intensity
functionin the auditorynerve,appropriateintervalandperiodhistograms
in response
to sinusoidal stimulation,andsuitableintensityrelatedratechanges
in
response
to increases
and decreases
in stimulusintensity.
Subsequent
research,
to bedescribed
below,showed
thatthe
modelcouldreproduce
Smith's(1977)observation
thatthe
adaptationresponse
followingsuddenstimulusincrementis
characterizedby the sum of two exponentialdecayfunc-
existsinside the cell wall near the junction. A fraction
k (t)q (t) dt ofthistransmitterisreleased,
between
timet, and
timet Jr dt acrossthemembraneintothecleft.A permeability factork(t) is a nonlinearfunctionof the instantaneous tions. It alsoindicatedthat the period histogramsdemonphase-locking
response
that
amplitudeof the signalafter mechanicaleffectshavebeen strateda frequency-dependent
was
analogous
to
Rose
et
al.'s
(1967)
observations.
takenintoaccount(althoughmechanicaleffectsareignored
Theseobservations
werefollowedby a periodof paramin this article),
etermanipulationthat aimedto fit themodel'sresponding
to
k(t) =g[S(t) + A]/[S(t) + A +B],
detailedpublishednumericalaccountsof auditoryfiberacfor [s(t) +•t] >0,
tivity (WestermanandSmith,1984).Whena usefulconfiguration
of parameters
hadbeenestablished,
the modelwas
k(t)=O, for[S(t)+AI<0,
(1)
furthertestedagainsttwo recentlypublished
resultsinvolvwhereA andB are parameters
of the modeland$(t) is the
ing the "additivityprinciple"and the effectsof masking
instantaneous
amplitudeof the signal.
stimulion the subsequent
recoveryof response
capability.
A fraction lc(t)dt of the amount c(t) of transmitter in
The followingaccountisnot a historicalrecordof thesedethe cleft is subjectto chemicaldestructionor lossthrough velopments
but a demonstration
of thestrengths
andweakdiffusion.Another fractionrc(t)dt is takenbackup into the
nesses
of the modelusingthe final configurationof paramcell.The restremainsin the cleftto stimulatethe postsynapo eters.Thesevaluesaregivenin TableI alongside
the values
tic membrane.It is assumed,
for the sakeof simplicity,that
usedin Meddis(1986). Optimizingtheconfiguration
of paspikeoccurrence
in theauditorynerveislinearly,probabilis- rameters
isa problematic
affairwithnoguarantee
of finding
tically relatedto the residueof transmittersubstance
in the
an idealsetof values.An accountof the process
of developcleft.Accordingly,the quantityc(t) is to beidentifiedwith
ingthemwill bepostponed
untilaftertheexposition
of the
the "excitation function" of Gaumond et al. (1983), Gau-
mondet al. (1982), or Gray's (1967) "recoveredprobability," i.e., the probabilityof spikeemissiondisregardingrefractoryeffects.Resultsbeloware expressed
in termsof the
excitationfunctionbecauseWesterman(1985) haspresented the results of his observations in these terms and this
performance
of thecurrentmodel.Unlessotherwise
stated,
the stimuli used to evaluate the model are 1-kHz sinusoidal
TABLE I. Valuesofparameters
usedin thisanda previous
evaluation
ofthe
model (Meddis, 1986).
methodavoidsthe needto overlaythe model'sperformance
with additional,possiblycontroversial,
assumptions
concerningtherecoveryfunctionof auditory-nerve
activity.
Transmitter taken back into the cleft is not immediately
Meddis, 1986
A
8
5
B
320
300
g
I 660
availablefor releaseagainbut is delayedin a reprocessing y
store.A fraction xw(t)dt of the amount of transmitter w(t)
in thisstoreis continuously
transferredto the freetransmitter pool.The transmitteroriginatesin a manufacturing
base
or "factory"that replenishes
the free transmitterpool at a
ratey [ tn -- q(t) ], wheretn is the (approximate)maximum
l
r
x
dt
16.67
500
12 500
I 000
0.00005
10013
11.11
I 250
16 667
250
0.00005
Time constants (ms)
T•
60
T,
2
quantified
versionof the model,rn is setto unityand all
T,
0.08
transmitter amounts are construed as fractions of the total
Tx
1
amount of transmitter to be found in the pool. In the un-
New values
198
0.40
0.152
15.08
possibleamount.
1057
J. Acoust. Sec. Am., Vol. 83, No. 3, March 1988
Ray Meddis: Model of auditory-neural transduction
1057
stimuli with either an instantaneous rise time or a rise time of
2.5 ms. The outputof the model,its excitationfunction,is
based on the cleft contents c(t). The cleft contents are al-
waysaveragedoveronewholecycleof a 1-kHz signalthen
multipliedby a factor 69 080 in order to estimatethe approximatefiring rate in events(spikes) per secondfor that
cycle.This valuewasbasedon fittingthe functionsby eyein
Fig. 2 in orderto arrive at a compromise,goodfit between
modelresultsand empiricaldata.
A. Rate intensity
Usingthe parametersin Table I, column2, a new setof
rate-intensity
curveswasproducedandisgivenin Fig. 2. The
resultsare comparedwith Westermanand Smith's (1984)
resultsfor a singlefiber (E8F2) with a centerfrequencyof
1170Hz. Figure2 distinguishes
two rate-intensity
functions.
The steady-statefunction representsthe firing rate after
adaptationto the stimulustoneandis sampled300 msafter
the toneonset.The onsetfunctionrepresents
the firingrate
in the 1-msperiodwiththehighestrateof firingimmediately
followingtoneonset.
WestermanandSmithgive0 dB as"AV threshold."In
the absenceof a more precisedefinition,we have defined0
dB for the modelasthe pointat whichtheonsetandsteadystatefunctionsdiverge.
B. Rapid and short term adaptation
Westermanand Smith (1984) alsostudiedthe adaptation functionsof fibersto brief tonebursts.Figure 3 showsa
recoveredPST histogramfor the samefiber (E8F2) in response
to toneburstsat 63 dB abovethreshold.Resultsof the
modelin response
to the samestimuliare superimposed
on
their data.
Westermanand Smith characterizedthe adaptation
functionasthe sumof two exponentialdecayfunctionsplus
a constant.The morerapiddeclinehad a time constantless
than 10 ms and the slower (short-term) declinehad a very
muchslowertime constantin the regionof 70 ms. Figure4
shows their estimate of the two time constants for the same
fiber as a functionof the intensityof the tone pulse.The
short-term time constant remains steady acrossa 40-dB
range. The rapid time constant,however,showsa steady
declinefrom approximately8 to 1.5 ms.
The resultsof the model (dottedline) aresuperimposed
on the empiricalvaluesin Fig. 4. The methodof fittingthe
exponentials
to themodelresultsis givenin the Appendix.
Becausethe model results take the form of smooth curves,
the process
of curvedfittingis relativelystraightforward.
Time constants were the same for both instantaneous and
2.5-ms rise times.
C. Phase locking
From an early stageit was clear that the ability of the
model's excitation function to reflect the fine structure of the
stimuluswas limited by the rate at which the transmitter
could be cleared from the cleft. In the model this is affected
1200
by two routes:(a) dissipationand chemicaldestructionin
the cleftand (b) reuptakeinto thecell.Whentheseare slow
relativeto the stimulusfrequency,phaselockingwill beless
evident.The process
will alsobeaffectedby theabilityof the
haircellpermeability
functionto respondquicklyenoughto
Instantaneous
rise-time
lOOO
800
lO00
2.5 ms
600
rise-time
400
200
Steady state
0
10
20
30
40
500
Relative Intensity
(dB)
Time (ms)
FIG. 2. Comparisonof rate/intensityfunctionsbetweenmodel behavior
(dotted) andWesterman's
( 1985,p. 74) gerbildata (solidline). In thecase
of the model,I ms (onsetfunction) refersto the excitationfunctionduring
the first (or highest)millisecondaftertoneonset.The steady-state
function
is based on the excitation
1058
function
300 ms after tone onset.
J. Acoust.Sec. Am., Vol. 83, No. 3, March 1988
FIG. 3. Poststimulustime excitation function for the model (dotted line)
comparedto Westerman's( 1985,p. 72) derivedexcitationfunction(solid
line) for thesamefiberusedin Fig. 2. The stimulusfor themodelwasa 43dB, 300-ms,1-kHz tone againsta backgroundof silence.
Ray Moddis:Modelof auditory-neuraltransduction
1058
•0o
kHz wereusedasstimulianddt reducedto 0.01ms.Figure5
givesthesynchronization
coefficient
for thecomputersimulationasa functionof frequency.
Johnson
(1980) alsoshowedthat,for a givenfrequency/,
synchronization
increasedwith stimulusamplitudeover a
limitedrange.This rangewasnot the sameasthe dynamic
rangeof the adaptedfiringrate of the fiberbut commenced
itsupswingwellbeforethe firingraterisesabovethe spontaneouslevel.The modelsuccessfully
mimicsthis effect.Figure 6 showsbothsynchronization
andratemeasuresderived
from the model'sperformance
asa functionof stimulusamplitude.
Short term
lO
D. Additivity test
The increasein firingrate followinga stimulusamplitude incrementhasbeenshownto be independent
of the
stateof adaptationof the fiber (Smith and Zwislocki, 1975;
Rapid
0
10
20
Relative Intensity
30
Smith, 1977; Smith etal., 1985). This effectis true for onset
( 1-ms window) and short-term ( 10-ms window) measures
40
of rateincrease.
The effectis alsovalid,followingstimulus
(dB)
FIG. 4. Time constants
for two additiveexponential
components
fittedto
the excitationfunctionfollowingbrief tonebursts.The toneburstsfor the
model(dottedline) werel-kHz sinusolds
(2.5-msrisetime). Theintensity
ofthetoneburstsisrelativeto threshold.
Theempiricalvalues(solidline)
aretakenfromWesterman
andSmith( 1984,Fig. 8).
amplitudedecrement,for short-termmeasuresof rate decreasebut notfor onsetratemeasures.
Followingthemethod
of Smithet al. (1985), a 1-kHz pedestaltone, 13 dB above
thresholdwas presentedto the model followedby a 6-dB
increment or decrementat 0, 10, 20, and 30 ms after the
onsetof thepedestal.The increment/decrement
in rateisthe
differencebetweenthe response
to the pedestalplusincre-
theinstantaneous
amplitude
ofthesignal.
Thislatterprocess
isnotsimulated
heresothattheformerprocess
canbestud-
ment and the pedestalalone.
ied in isolation.
Figure5 showsRoseet al.'s ( 1967) synchronization
coefficient
expressed
asa functionof stimulusfrequency.
This
coefficient
isbasedonperiodhistograms
andrepresents
the
"mostpopulous"
halfof thehistogram
asa percentage
of its
total area.A valueof 50% indicatesno phaselocking.A
well-replicated
findingis that synchronization
measures
declinein strengthbetween1 and 5 kHz.
90
Synchronizalion
8O
To testthe model, sinusoidalstimuli of 1, 2, 3, 4, and 5
lOO
Synchronization
Coefficient
%
6O
70
60
5O
5O
-70
I
2
3
Frequency
4
(kHz)
FIG. 5. Synchronization
coefficientas a functionof stimulusfrequency.
Modelbehavior(dottedline) wascomputedusingdt = 0.01ms.Empirical
data (solid line) are taken from Rose et al. (1967).
1059
J. Acoust.Sec. Am., Vol. 83, No. 3, March 1988
0
60
5
Amplitude d I• re threshold
FIG. 6. Synchronizationcoefficientasa functionof stimulusamplitudefor
a l-kHz tone (solid line). Steady-statefiring rate of the modelasa function
of amplitudefor the samestimulus(dotted line).
Ray Meddis:Model of auditory-neuraltransduction
1059
250
200
(a)+6clB
e
150
(c)-6clB
ß•,,
FIG. 7. Effectsof prior adaptationon response
to 6-dBincrements
anddecrements
of a 1-kHz pedestaltonepresented
at 13dB
abovethreshold.Here, (a) and (c) are rap-
100
(b)+6rib
m
--...'-.ß.
Onset
(1ms)
id effects based on the first millisecond after
stimuluschange,
and(b) and(d) areshortterm effects derived from the first 10 ms
5O
afterstimuluschange.Empiricalstudiesobtain horizontal lines for (a), (b), and (d).
Shortterm,(10 ms)
Delay
(ms)
The resultsdo not agree with their results (Fig. 7).
Their studyshowedhorizontalfunctionsfor 6-dBincrement
CF = 842 Hz). Clearly, the ability to producea brief response
recoversmorequicklythanthe abilityto sustainthat
(I- and 10-ms window) and for 6-dB decrement (10-ms
window). The short-term decrement function (1-ms win-
response.
dow) wasshownto reducewith increasingdelay.The model
doesshowthe requiredresponses
followingstimulusdecrementsbut isclearlydiscrepantduringthefirst 10msof delay
for stimulus increments.
E. Recovery of function
Followingan intensemaskingtone,spontaneous
firing
of the fiberis brieflysuppressed
beforeslowlyrecovering
to
normalspontaneous
levels.Westerman(1985) givesmean
recoverytime constantsof 40 ms (standarddeviationof 25
ms) basedon 12 fibers.The model,usingthe specifiedfinal
parameters,hasa recoverytime constantof 46 ms.
In the first millisecondfollowingstimulusoffset (50
dB), theexcitationfunctionof thecomputermodelfallsto a
valueequivalent
toa rateof9 spikes/s.
Thiscontrasts
witha
periodof totalsuppression
thatiscommonly
observed.
The
modelis,therefore,unableto explaina totalsuppression
of
spikeactivity.Alternatively,
wemaybeseeking
anexplanationof thedeadperiodin thewrongplace.The totalpoststimulus
suppression
mayreflectpostsynaptic
fatiguethatis
not represented
in the modelat all.
The dashedlinesin Fig. 8 showthe resultsof applying
this experimentalparadigmto the model.There are some
cleardiscrepancies
betweenthe model'sbehaviorand the
empiricaldata.However,thedashed
linesareapproximately
straightfor muchof theirlengthand,therefore,indicatean
exponentialimprovementin the capacityto respond.Between0 and 20 ms,thereis an upturnin the modelresults
tooo
lOO
lO
Westerman (1985) also measuredrecoveryof the ca-
pacityto respondto a secondstimulusby presenting
30-ms
testtonesat varyingintervalsafterthe cessation
of a 300-ms
durationmaskingtone.Bothtoneswere43 dB abovethreshold. The responsewas measuredas a decrementwhen com-
paredto theresponse
in theabsence
of a preceding
masking
tone. The onset rate is based on the first millisecond after the
o
1 DO
200
Time after masker
testtoneonset.The short-termresponse
isbasedon the period 20-30 ms after the onset of the test tone. Westerman's
resultsaregivenassolidlinesin Fig. 8. The useof a logarithmic scalefor theresponse
decrementmeasuremeansthat the
two straight line functionsobtainedrepresentexponential
recoveryin both cases.Westermanfound two recoverytime
constants,49 ms for the onsetresponseand 68 ms for the
short-term responsefor this particular fiber (E27F13,
1060
J. Acoust. Sec. Am., Vol. 83, No. 3, March 1988
FIG. 8. Recoveryof responsefollowingadaptation.The decrementis the
differencebetweenthe responseof an unadaptedfiber to a 43-dB (re:
threshold) 1-kHz tone and the responseof a fiber soonafter the offsetof a
300-ms,43-dB maskingtoneof the samefrequency.The onsetdecrements
are basedon the maximal 1-msfiring rate after test tone onset.The shortterm decrements are based on the rate between 10 and 30 ms after test tone
onset.Solidlinesaretakenfrom Westerman( 1985,p. 52, Fig. 21). Dotted
linesrepresentthe response
of the model.
Ray Meddis: Model of auditory-neural transduction
1060
suggesting
a departurefroma simpleexponential
improvement.This is not necessarily
inconsistent
with Westerman's
data pointseventhoughhe choseto fit a singlestraightline
timeconstants;
y, thereplenishment
factor,affectsspontaneousandadaptedfiringratesaswellastheshort-termadapta-
throughout.I haveredrawnhis "best-fitlines" to illustrate
the cleft into the cell, affectsphaselockingand the shortterm adaptationtime constant;x, the rate of transmitterreprocessing,
affectsonly the adaptationtime constants;
l, the
rate of lossof transmitterfrom the cleft--and, hence,from
the wholesystem--influences
all firingratesandthe shortterm adaptationtime constant.
this possibility.Perhapsa more detailedanalysisof additional data will resolve this issue.
Thereare two importantdiscrepancies
that deserveattention. First, Westermanfound differenttime constantsfor
the recoveryof onsetandshort-termresponding.
For seven
tion time constant;r, the rate of return of transmitter from
fibers studied in detail, all had faster time constants for the
recovery of the onset response.The model, by contrast,
IV. DISCUSSION
showsequivalenttimeconstants
forbothrecoveryprocesses.
For Fig. 8, Westermangivestime constantsof 49 and 68 ms
Two possibleusesof hair cell modelswere identifiedin
the
Introduction.
First, theyarea readilytestablestructural
for onset and short-term functions. The time constant for
account
of
what
is
actuallyhappening
at thehaircell-audibothfunctions
usingthecomputermodelwasapproximately
tory-nerve
synapse.
Second,
they
can
be
usedasa generator
thesameashisshort-termrecoveryfunction.For a sampleof
of trainsof spikesto act as input to other modelsof, for
sevenfibers,Westermangivesmeanrecoveryfunctionsof 48
example,
cochlear
nucleus
functioning,
binauralhearing,sems (s.d. = 25 ms) for onsetand 169 ms (s.d. = 79 ms) for
lective
attention,
speech
recognition,
etc.
Differentcriteria
short term.
of usefulness
applyin thesevariouscases.
Certainly,weaker
criteriamustapplyto the modelasan inputdeviceto other
III. OPTIMIZING PARAMETERS
modelsbecausethe development
of theoriesconcerning
The preceding
exposition
isbasedonsimulations
using "higher" processes
cannotwait until all the problemsof
anunchanging
parameter
set.Similarly,therescaling
ofcleft characterizingthe "lower" processes
have been solved.
contentsto indicatepotentialfiringrateusedthesamescale Compromiseis unavoidable.
factorthroughout.While it is encouraging
that the model
The currentmodelis clearlyverysuitablefor useas a
wasableto fit the empiricaldata aswell asit did, we haveno
spikegenerator
because
the cleftcontents,
whenmultiplied
guaranteethat its performancecould not have been imby a suitableconstant,can be viewed as a statementof the
provedwith a bettersetof parameters.
The methodof paprobabilitythat a spikewill occurat that time. A random
rameter optimization used here was the laborious "hillnumbergeneratorcan,therefore,beusedto decidewhethera
climbing"approachof changingoneparameterat a timeand
spikedoesoccurat thattime.The modelwill acceptanarbinotingthe effect.If the effectwasbeneficial,
thissetof patrary stimulussampledat any rate that wouldnormallybe
rameterswasusedasa new startingpoint;otherwise,it was
acceptablein acousticanalysis.In response,it producesa
necessaryto revert to the previousset and make a different
streamof spikespreciselylocatedin time. Rate measures
can
change.
bederivedfromthisoutput,asrequired.Moreimportantly,
The benefitof eachnewparameterchangewasassessed knowingthe precisetimingof eacheventis a specialvirtue
in termsin a numberof dependent
measures
thatcompared
modelperformance
withtargetvaluesderivedfromempiricaldata.The measures
andthetargetsgivenin bracketsare
asfollows:( 1) ratioofspontaneous
to 100-dBadaptedfiring
rate (0.2); (2) dynamicrange(30 dB); (3) rapid adaptationtimeconstant
nearthreshold( 8 ms); (4) rapidadaptationtimeconstants
at 40 dB (2 ms); (5) short-termadaptation time constantnear threshold(75 ms); (6) short-term
adaptationtimeconstantat 100dB( 75 ms); ( 7) phaselocking (valuesgivenby Roseet al., 1967).
It wasnotpossible
to matchall of thetargetsexactlyand
theparameter
setusedrepresents
ajudicious
compromise.
It
is quitelikelythatthis'setcouldbeimprovedupon.
It is not a simplematter to identifyindividualparametersof the modelwith the dependentmeasuresused.If it
were,thenthediscovery
of anoptimumparametersetwould
havebeenverymucheasier.Changinganyoneparameterin
isolationtypicallyaffectsall measures.
However,individual
for thoseanalysis
systems
thatdependuponthetimeintervalsbetweenspikes(e.g.,Moore,1982)or, moregenerally,
which involveany kind of phasesensitivity(Patterson,
1987).The simplicityof themodelallowsfor rapidnumericalevaluation.A recentimplementation
onan 8-bit6502 ( 1MHz) processor
runsat 10timesrealtimewhenusinga 20kHz samplingrate and we expectto producea real-time
implementation
usinga morepowerfulprocessor
in thenear
future.Moreover,themodelappearsto mimicall of themajor propertiesof auditory-nerve
response.
Suchdefectsas
havebeenrevealedso far are unlikelyto affectadversely
research
progress
for systems
usingthismodelasan input
device.
The failingsof the model,however,are muchmorecritical whenevaluatingits potentialas a structuralaccountof
eventstaking placeat the point wherethe auditorynerve
meetsthe hair cell. It is not possibleto makea directcomparisonwith other publishedaccountsbecausethis is the
parameterstypicallyaffectsomemeasuresmorethan others: firsttime that thisparticularsetof experimentalparadigms
.4, whichoccursin the permeabilityequation,affectsthe hasbeensimulated
asa complete
set.However,therangeof
spontaneous
firing rate and the responsethreshold;B affects phenomenasuccessfully
simulatedsuggests
that the model
everythingexcept the adaptationtime constants;g indefectsare relativelyminor. Theseinvolvetwo anomalies.
fluences the rate of outflow of transmitter from the cell into
First, the onsetresponse
showsan unrealisticsensitivityto
thecleftandthusaffects
all firingratesandrapidadaptation levelof adaptationduringthe first 10 ms of the adaptation
1061
J. Acoust.Soc.Am.,Vol.83, No.3, March1988
RayMeddis:Modelof auditory-neural
transduction
1061
process.
Second,recoveryof theabilityto respondto a new
stimulusfollowingan intensemaskingstimulusshowsimportantdiscrepancies
with empiricalresults.
The failure to replicatethe additivity effectfor onset
responses
is an importantproblem.A numberof existing
models (Sehwid and Geisler, 1982; Smith and Brachman,
1982;Cooke,1986;WestermanandSmith, 1986) haveexpli-
citly directedtheirmodelingeffortstowardsexplainingadditivityby suggesting
that stimulusintensitymodulatesthe
amountof transmittereligiblefor release,i.e., the stimulus
controls the volume of the free-transmitter
reservoir. Wes-
terman and Smith's (1986) most recentaccountproposes
that both the volumeand the membranepermeabilityare
stimulusdependent.
If thesemodelsin anexplicitsimulation
canbeshownto reproducethe additivityeffectsin the paradigmillustratedabove,a casecouldbemadefor introducing
multiplereleasesitesor variablevolumereservoirsinto the
modelcurrentlyunderdiscussion.
The attemptto simulateWesterman's(1985) function
for therecoveryof the abilityto respondto a stimulusafter
previousintensestimulationalsorevealeddiscrepancies.
In
addition,Westerman(1985) findstwo differentexponential
recoveryfunctions,one for effectsmeasuredimmediately
after the onset of the test stimulus and one for the increase
measuredduringa period10-30 msafterwards.For thebulk
of the recoveryperiodthe modelgenerates
only onerate of
recoverythat is the samefor both measures.Westerman and
Smith(1986) haveshownthat thiseffectcanbemodeledby
introducingan additionaltransmitterreservoirbetweenthe
globalstore fraetory) and the free-transmitter
pool. Ross
(1982) usedtwo additional reservoirsin cascade.Some such
amendmentto the modelmay be requiredif it is not found
possibleto solvethe problemby parametermanipulation.
One problemwith the currentpositionis that Westerman givesmeantime constantsof 48 and 169 ms for rapid
and short-termeffects,respectively.However,the standard
deviationsfor thesetime constantsare veryhigh indeed(25
and 79 ms). Somerapid recoverytime constantsfor certain
animals must be slower than some short-term time constants
for other animals. Moreover, some short-term time ari-
srants,in excessof 350 ms, appearto have beenestimated
overonly200-mstimeperiods.Insofarastheremayberoom
for reevaluatingWesterman'spioneeringfindings,it may be
wiseto delayradicalrevisionof the computermodel.
Havingdweltat lengthon the difficultieswith the model, it isusefulto rehearsethemanyphenomena
that themodel has successfully
simulated.This summaryis alsodrawn
from a previousreport (Meddis, 1986) that usedthe same
decrements
in stimulation
intensityasa functionof adaptation level for both onset and short-term measures and for
short-termmeasuresafter stimulusincrement;and (8) real-
istic rate of recoveryof spontaneous
firing ratesfollowing
intense stimulation.
An interestingfeatureof the modelis its relianceon
transmittermovementdelaysto generatethe familiarpropertywherebyphaselockingislimitedbystimulus
frequency.
Little attentionhasbeengivento thispossibility
whichexists
whetheror not we acceptthe ideaof transmitterreuptake
intothehaircell.Evenona puredissipation
anddestruction
principle,theremustbe somedelayin clearingtransmitter
from the cleft. Recent work on the close association between
the receptorpotentialsof innerhair cellsandphase-locking
indicesis troubledby two difficulties:
First, directextrapolation from receptorpotentialseemsto underestimate
phaselockingability,and,second,that the greatvariationamong
species
in phase-locking
abilitymayimposetoogreata strain
on the theory (Palmer and Russell,1986). While the model's successful use of transmitter movement as a basis for
generatingthe phenomenon
doesnot grantit the statusof a
true explanation,it doesrequirethat thispossibilitybetaken
into account in future discussion of the matter.
The reuptakeprinciplewasoriginallyadoptedfor reasonsof computationalexpediency.However,a recentstudy
(Siegeland Brownell, 1986) has shownthat this process
may indeed be at work. They offer evidenceof membrane
recyclingat the inner hair cell synapse.Someof the membranerecoveredfrom the cleft (presumablyby a processof
invagination)appearsto be usedin the formationof new
synapticvesicles.This providescircumstantialevidence,at
least,that a fastrouteexistsfor the reuptakeof largemoleculesfrom the cleftinto the presynapticregion.
While the modelhasmanyinterestingand satisfactory
features,it isacceptedthat it maybenefitfrommodifications
basedon other publishedmodels.Unfortunately,no direct
comparisonhasyet beenmadethat would allow a summary
of therespective
strengths
of the differentmodels.It is proposedthat thesetof testsdescribed
in thisarticlecouldprovidea minimumsubsetfor comparativetestingof all current
modelsand a project to do this is currently planned.For
sucha comparisonto be fully effective,someobjectiveand
preferablyautomaticmethodfor optimizingthe parameters
in eachmodelisclearlycalledfor--especiallyfor thosemodelsthat do not allow for analyticsolutions.This problemis
underactivediscussion
in many areasof scientificendeavor
(Kirkpatrick et al., 1983) and someof the proposedsolutionswill be activelyexploredin this context.
modelwithonlyparameterchanges.
Phenomena
successfully simulatedare as follows:(1) steady-state
rate/intensity APPENDIX
functions;(2) poststimulusperiod histogramsat various
The adaptationcurve of the excitationfunction followlevelsof stimulusintensity;(3) interspike-intervalhistoinga stimulusincrementwasdescribed
in termsof theequagramsfor silenceand 70-dB, 1-kHz sinusoid;(4) nonmono- tion
tonic functionsrelating incrementaland deerementalreY, = a + be- ,/r, + ce- ,/r.,
(A1)
sponsesto stimulation amplitude changesas a function of
backgroundstimulationintensity;(5) adaptationfunctions
that can be describedas the sumof two exponentialdecay
functions;(6) realisticsynchronization
coefficients
decayingasa functionof frequency;(7) realisticeffectscausedby
1062
J. Acoust.Soc. Am., Vol. 83, No. 3, March 1988
where¾,istheexcitationfunctionat timet, e istheexponential constant,and a, b, c, T•, and T2 are valuesto be discovered by the method. To begin, we need to find a minimum
valueof Y ( Ymi,), SOthat Ycan be rescaledthus,
Ray Meddis:Modelof auditory-neuraltransduction
1062
Y = Y-- Y•,i,-
(A2)
Sinceweknowthatadaptation,
for ourpurposes,
isvirtually
completeaftera quarterof a second,we canusea valuenot
much smaller than the excitation function after 300 ms of
adaptationhaveelapsedfor Y.,i,. Here, Y•,,, is alsotakenas
our estimateof the parametera.
We assumethat the firsttimeconstant(T•) is unlikely
to be greaterthan 10ms.As a result,we do not expectthis
firstprocessto make muchcontributionto the functionafter
40 ms. We, therefore,computeT2 on the basisof values
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of therapidadaptationfunction
usingthetwo datapointsat 1 and2 ms:
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data, then the time constantsT• and T2 mustbe estimated
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Probabili-
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