Characterizing auditory neurons using the Wigner and Rihacek
distributions: A comparison
JosJ. Eggermontand GeoffM. Smith
BehavioralNeuroscience
Research
Group,DepartmentofPsychology,
TheUniversity
of Ca/gary,
2500 University
DriveN. V•.,Calgary,AlbertaT2N1N4, Canada
(Received19June1989;acceptedfor publication29 August 1989)
Becauseof their dynamicproperties,mostsoundscanbestbecharacterizedin the combined
frequency-time(FT) domain.Powerfulfrequency-time
characterizations
are the Wignet
distributionfunction(WDF) andthe Rihacekenergydensityfunction(RDF). In the present
paperseveralnew conceptsare introducedsuchas usingthe WDF to characterizethe tuning
of auditoryneuronsunderwidebandnoisestimulationand a new methodto quantifyphase
lock of auditoryneuronsto a widebandnoise.No appreciable
differences
werefoundbetween
the WDF andRDF in narrow-bandsignalrepresentations.
However,the differences
between
the WDF andRDF increaseasthe bandwidthof the signalincreases.
When signalsare buried
in uncorrelatedbackgroundnoise,the averageFT functionof thesesignalsmay be obtained
throughaveraging
theFT functions
forehchsignalplusnoisesegment.
TheWDF takesat
leasta factor2 morein time to computethan the RDF. The FT functionscanbe usedto
characterize(linear) filtersby averagingFT functionsof input-noisesegments
that precede
thresholdcrossings
of the filter'soutputsignal.Boththe WDF andthe RDF wereusedto
characterizeauditoryneuronsfrom the midbrainin anurans;the WDF alwayshad a smaller
bandwidththan the RDF. By comparingthe spectrumof the reversecorrelationfunctionand
the averagespectrumof the noisesegmentsprecedingthe spikes,a quantificationof the
amountof phaselock of the auditoryneuronto the noiseis obtained.
PACS numbers:43.64.Qh,43.64.Pg,43.64.Wn, 43.64.Bt
INTRODUCTION
signalx(t) an imaginarysignal,i•(t), where.•(t) is defined
A. Frequency-time representations
as
Frequency-time representationsof nonstationary
soundshavebeenintroducedby Gabor (1946) andpopularizedby theinventionof the "soundspectrograph"
(Potteret
al., 1947), whichproduced"visiblespeech."This is a frequency-time
pictureof soundin whichthechangewith time
in the power of relevantfrequencycomponentsis shown.
Typically,the spectrogram
is obtainedby passing
soundsimultaneously
througha bankof filters(eitherwith constant
bandwidthor with proportionalbandwidth)anddisplaying
the intensityof the filter outputasa functionof time. Problemswith the useof physicalfiltersare the arbitrary, often
predetermined,
settingof thebandwidthandthe reciprocal
relationbetweenfrequencyresolutionand temporalresolution.Otherfrequency-time
representations
thatdonotsuffer
from these restrictions and that are all based on a Fourier
transformof a functionalof the signalcan be foundas well.
We will discuss
two of themin thepresentpaper.
It is known that any periodic real signal can be expressedas a Fourier series:
x(t)----ao
+ • (am
cosn2•rftq-b,
sinn2•rft).
(1)
n•l
However,it isalsopossible
to usea complexseriesasa representation:
•(t)= • (am
--ib•)ei•2'•.
n•O
(2)
This complexrepresentation
isformedby addingto the real
246
J. Acoust.Sec.Am.87 (1), January1990
•(t)=_•l•f_'
x(S)
ds,
(3)
whichis to betakenasthe Cauchyprincipal-value
integral,
x(t) and•(t) are saidto bea pair of Hilbert transforms,and
•(t) is calledthe analyticsignalof x (t):
•(t) ----x(t) -F i•(t).
(4)
For example,if x(t) = a cos2•rft, then •(t) = a sin 2•rft
and•(t) = aea•f•.Thespectrum
•(f)
of •(t) equals
zero
for negativefrequencies
and twiceX(f) for positivefrequencies(Papoulis,1977).
We canform thecomplexfrequency-time
function:
R(ft) = • *( f)e - '•'/'•(t),
(5)
where
* stands
forcomplex
conjugate.
Also,R( f t) hasbeen
calledthe "complexenergydensity"(Rihacek, 1968); it is a
complex-valued
functiononthefrequency-time
domain.Direct integration over time or frequency results in the two
marginal densities,the first of which is
I(t)=• fR(ft)df
=•*(t)•(t),
(6)
the temporalintensity,which is alsoequalto the squareof
the envelopeof the signal•(t), and the secondis
J(f)=fR(ft)dt
=•*(f)•(f),
(7)
thespectralintensity.Doubleintegrationoverbothtimeand
0001-4966/90/010246-14500.80
@ 1990Acoustical
Societyof America
246
frequency
results
in theenergyE of thesignalin the (f,t)
B. Frequency-timedistributionsof multifrequency
windowunderstudy:
signals
E=ffR(f,t)dfdt.
Whenmorethanonefrequency
component
issimultaneously
presentin thesignal,crosstermsoccurin boththe
Wignerand Rihacekdistributions.
For instance,whena
tone s(t) and anothersignaln(t) togetherform a signal
(8)
It can be shown that
x(t),
R(f,t)
=f •*(t- r)•(t)e
-'a'•f•
dr
=fRgg
(t,r)e
-,2,•s•
dr.
(12)
(9a) x(t) =s(t) + n(t),
and,when•(t) istheanalyticsignalofx (t), a(t) theanalyt(9b)ic signalofs(t), andv(t) theanalyticsignalof n(t), then
Thus the Rihacekenergydensity,or Rihacekdistribution
Wg(f,t) = W,,(f,t) + Wv(f,t)W•,v(f,t).
function (RDF), aswe will call this from now on, can alsobe
consideredas the Fourier transformof the time-dependent
Thuscrosstermsappearthat havetheform
autocorrelation
function
oftheanalytic
signal,
Rg•(t,r). We
thereforemayalsocall the RDF a time-dependent
spectrum,
whichbecause
of the factthat it is a complexfunctionrepresentsboth intensityand relativephaseinformation.It is an
empiricalfindingthat in additionto the realpart, the imaginary part of the RDF is not very informative.It is noisier
than the real part and can, in fact, be computedfrom it. It
(13)
l0
W•,•(ft)=;[•r(t+-•-r)v*(t---•+ø'*(t-T
(14)
ghost images in the WDF at
canbeshownthatonlytherealpartof theRDF contributes [(f, +f•)/2,(t• + t•)/2]. If thesignal
components
are
to the integralsshownin Eqs. (6) and (7) becausethese not correlatedin the analysiswindow,thenthe crossterms
integralsvanishwhen computedover the imaginarypart
will vanish.Thusthe presence
of crosstermscanbeusedto
(Johannesmaet al., 1981). For thesereasonswe will only
advantagein exploringwhethercomponents
of multifreshowthe real part of the RDF in comparisons
with the
quencysignalsare correlatedin the averageWDFs and
WDF. In caseonewantsto usethe RDF for predictionpurRDFs as,e.g.,usedin auditoryneurophysiology
(Eggerposes,one needsthe phaseinformationcontainedin the
montet al., 1983). It shouldbe appreciated
that, in caseof
imaginarypart (in combination
with the realpart).
signals
withmanycomponents,
as,e.g.,speech,
theWigner
It hasbeencustomaryto takea moresymmetricalform
distribution
canbecomequitenoisyastheresultof themyrof theproductfunction[ in Eq. 9(a) ] astheintegralkernel iad of crossterms (Allard et al., 1988).
and
(Claasenand Mecklenbrauker, 1980 a--c):
result
in
For the Rihacek distribution the cross terms are of the
form
W(f,t)
=f•*(t-•-r)•(t
+-•-r)e-'a'S•
dr.(10)
This resultis knownas the Wignerdistributionfunction
(WDF). TheWDF isrelatedto theRDF bya doubleconvolution (Rihacek, 1968; Cohen, 1987):
W(f,t) = ei4'•']•R(f,t),
Ro•(ft)= [a(t)v*(f)+a*(f)v(t)]e
-'•f'.
(15)
Thusin the RDF the crosstermswill appearat thetime of
s(t) andthefrequency
of n(f), andalsoat thetimeofn(t)
andthefrequency
of s(f). Thisisdistinctlydifferentfrom
thecross-term
distributionin theWDF. Oneparticularcon-
(11)
where
)• stands
fordouble
convolution.
Notethate"ø is
sequence
of suchinteraction
products
canbeseenwhenwe
compute
theWDF andRDF for anasymmetrical
(around
themean)signal
suchasa tone-pip
plusadcshift.Theinteractionproducts
intheRDF showupatfrequency
zeroandat
thefrequency
of thetone-pip,andfollowtheinstantaneous
amplitude
ofthetone-pip:
Thusweobserve
thismodulation
ontopof thedccomponent
andsuperimposed
ontheenvelopeof theRDF aswell[Fig. 1(a) and(b) ]. FortheWDF
the interactionproductoccursat a frequencyhalfway
theanalyticsignalof cos4rrfi.The Wignerdistributionhas
in recentyearsbeenusedextensively(Hermes,1985;Poletti,
1988;Yen, 1987;for a completeliteraturesurveyup to 1985,
seeMecklenbrauker,1987) in (acoustical)signaldescription;in contrast,theRihacekdistributionhasonlybeenused
occasionally(Johannesma
et al., 1981). A reasonfor this
of thetone-pip;it doesnot
preference
maybe that the WDF is a real valuedfunction. betweenthedeandthefrequency
signalproper[Fig. 1(c) ].
The marginaldensitiesare, as for the RDF, equalto the affecttheWDF oftheundistorted
theRDF at frequencyzerois• * (0)•(t), thecomtemporalandspectralintensities
of the signal,andthe dou- Because
willalways
berepresented
atthezerofrequency
ble integrationagainresultsin the signalenergy[replace pletesignal
wheng(0) = 0. Thusartificially
forcingthevalR(f,t) by W(f,t) in Eqs.(6)-(8) ]. Despitethissimilarity, lineexcept
in thespectrum
to zerobeforecalcuthe WDF and the RDF are quite dissimilarin appearance ueofthedccomponent
in the undisespeciallyfor broadbandsignalsas a resultof the difference latingthe analyticsignalresultsin bothcases
torteddistribution
function[Figs.2(a)-(c)]. Smoothing
in the positionof the interactionproducts.The RDF is in
with an appropriate
functioncanalleviatethisproblemas
contrastto the WDF factorizablein separatefrequencyand
time functions.
247
d.Acoust.
Sec.Am.,Vol.87, No.1, January1990
well (Andrieux et al., 1987).
d.J. Eggermont
andG. M. Smith:Characterizing
neuralresponses
247
(b)
aL, 1983). To illustratethe useof the averagefrequencytime distributions,we assumea linear bandpassfilter followedby a threshold-crossing,
spike-generating
mechanism
asan ultimatesimplificationof an auditory-nervefiberunit.
This systemis presented
with Gaussianwidebandnoiseasa
resultof whichspikesaregenerated.We calculatethe R (f,,t)
for eachnoisesegmentof, say,25-msdurationthat precedes
a spike.Averagingthe RDFs for all the noisesegments
that
precedethe spikes,
RDF
=
1
•r
R,(f,t) • •, R. (f,w.
(c)
0
WDF
tirr• (ms)
25J
FIG. i. RihacckandWigherdistribution
functions
ofa 1600-Hztonepip
withadccomponent.
(a) Thesignal;(b) therealpartoftheRihacekdistributionfunction(RDF), and( c) theWigherdistribution
function( WDF ).
Timeis displayed
horizontal
fromleftto right,andfrequency
is running
backwardin the horizontalplane.The interactionproductsbetweenthe
1600Hz andthe dc appearin the RDF on topof thede andthe 1600-Hz
components,
however,
in theWDF midwaybetween
them.
C. Averaging frequency-time distributions
The WDF hasso far only beenappliedto singlesignal
epochs;in contrast,the RDF has beenusedmainly in its
averagedform asan aid in the analysisof nonlinearsystems,
notablythe auditorysystem(for a review,seeEggermontet
la)
1600 Hz
-t),
(]6)
wherethe (w.) are the spikeoccurrence
times,resultsin an
ensemble-averaged
RDF, Re, that characterizesthe frequency-timepropertiesof the linear filter. The useof this
methodhasfirstbeensuggested
by Johannesma
( 1971), introducedinto auditoryneurophysiology
by Hermeset al.
(1981), and subsequently
also usedby Eggermontet aL
(1983) and Eppingand Eggermont(1985). In contrastto
the reversecorrelationmethod(De Boerand Kuyper, 1968;
De Boer and de Jongh,1978), wherealsoGaussianwidebandnoisestimulationisused,but thenoisesignalpreceding
the spikesis averagedto obtainan estimateof the impulse
response
of theauditoryfilter,thefrequency-time
averaging
methoddoesnot requirephaselockof the spikesto the stimulusbecausethe RDF is insensitiveto the absolutephaseof
thesignal.After subtracting
theexpectedvalueof the RDF
(obtainedwith randomtriggers),oneobtainsa frequencytimefunctiondescribing
whatsignalcomponents
theneural
unit extractsfrom the noise.The marginaldensitiesIe (t)
and Je(f) of Re(f,t) representthe averagetemporaland
spectralintensitiesof the noisesignalprecedingthe spikes.
The spectralintensitygivesan indicationaboutthe neuron's
tuning under continuousnoisestimulation.Note that the
averagedRDF is no longerfactorizablein separatetime and
frequencyfunctionsthat have a simplerelationshipto the
signal.On basisof a comparisonof the RDF of the reverse
correlationfunctionand the ensemble-averaged
RDF, one
can,in principle,obtainan estimateof theamountof phase
lock of the neural firings to the noise (Eggermont et
al., 1983). In thepresentpaperwe will explorethisfor simulated conditions and neural data.
D. Relation between the WDF, the RDF, and the
spectrogram
(b)
RDF
A spectrogramof a realsignalx(t) canbeconsideredas
a setof time functions,onefor eachbandpassfilter usedin
theproductionof the spectrogram.
Assumea setof M filters
with impulseresponseh,, (t); then the spectrogramofx(t)
•5
is givenby a setof M temporal
intensity
values,
{Yr,(t)},
m----1 .....M, suchthat
(c)
WOF
{y,•(t)} =
h•(s)x(t--s)ds
.
(17)
Given that a spectrogramis derivedwith a particularfilter
0
time (ms)
set{h,• (t) }, oneisnotabletotransform
sucha spectrogram
256
FIG. 2. Rihacekand Wignet distributionfunctionsofa 1600-Hztonepip.
Samelayoutas Fig. 1.
248
J. Acoust.Soc. Am., Vol. 87, No. 1, January1990
into anotherwith, e.g., a better spectralresolution.It is,
however,possible(Johannesma
et al., 1981) to deriveany
type of spectrogramfrom the RDF (or WDF) througha
J.J. Eggermontand G. M. Smith:Characterizingneuralresponses
248
simplemultiplication
in thefrequency
domainandconvolution in the time domain with the RDF (or WDF) of the
impulseresponse
of theselected
filter,H,,(f,s):
I. METHODS
A. Stimulation and recording
Widebandnoisewasgenerated
by transforming
a software-generated
uniformamplitudedistribution
intoa Gaussianoneusingthe "distributionmethod"described
by EckhornandPfpel (1979). In short,a memorysectionof length
Mis filledwith orderedamplitudevaluesdrawnfroma uni-
y(t)=
f fH.(f,s)R(f,t--s)dfds.
(18)
Takingthebandwidthof the ith filteras Aft, thenthe time
resolutionAt• of filter i will be boundedby Gabor's(1953)
inequality:
(19)
Thus,in orderto arriveat a physically
meaningful
interpretationof theRDF or WDF, onehasalwaysto integrateover
anareain thefrequency-time
domainthat isat leastequalto
t•r (e.g.,25Hzby3.2ms).Thismeans
thatsmoothing
ofthe
RDF and WDF is generallyjustifiedsinceincidentalhigh
peaksor dipsmaynotbephysicallymeaningful;
in addition
interferenceproductscanbe removedin thisway. The computationof the WDF overa finitetime windowalreadyresultsin a smoothingin the frequencydomain;so the actual
resultobtainedis a so-calledpseudo-WDF(Claasenand
Mecklenbrauker,
1980a).In theremainderof thispaperwe
will continueto useWDF for thepseudo-WDF.
The frequencyselectivityof the auditorynervoussystem hasmainlybeenexploredusingsimplestimulisuchas
clicks,tone-pips,and continuoustones.Most ethologica!ly
meaningfulsoundsas speechare, however,multifrequency
signals.Tuningof theauditorynervoussystemundermultifrequencystimulationmay bedifferentfrom that undersingletonestimulation;e.g.,the filter maybehavemorelinearly
under stimulationwith continuousnoisethan with pure
tones.Thus predictionsof the responseto complexstimuli
may requirethe characterizationof the auditorysystemusing statisticallystructuredmultifrequencystimuli (Eggermontet al., 1983).The RDF andWDF aretwo potentially
usefulcharacterizations
of thespectrotemporal
propertiesof
the auditoryfilterasexhibitedin the firingpropertiesof the
neuralunits.By virtueof theirdifferentphaserelationships
andtheappearance
of theinteractionproducts,onemightbe
preferableoverthe other (Hermes, 1985).
In the presentpapera comparisonwill be givenfor the
WDF and RDF for elementaryacousticsignalssuchastone
pips,multifrequency
signalssuchas impulseresponses
of
broadbandpass
filters,andfor nonstationary
stochastic
signalssuchasnoisesegments
thatprecedetheoccurrence
of an
actionpotentialin the auditorynervoussystem.The comparisonof theaverageWDF andRDF in applications
such
aslinearsystems
analysisandin auditoryneurophysiology
is
to our knowledgethe firsteverbeenpublished.Specialemphasiswill begivento the interpretationof the two distributionfunctionsandtheirsuitabilityfor characterizing
thefrequency-timepropertiesof the tuningin the auditorysystem.
We alsoelaborateon a measureof phaselock for usewith
noisestimulithat weintroducedrecently( Eggermontet al.,
1983) and will apply this for the first time to neural data. A
form distribution
of random numbers which occur with a
frequencyaccordingto a Gaussianprobabilitydensityfunc-
tion.Subsequently,
theseamplitudevaluesareinterchanged
by samplingaccordingto anotherrandomsequence
to result
in a setof completelyindependent
Gaussiannoisesamples.
The samplingrate was 10 kHz, and the total lengthof the
noisesequence
was3 s. For electrophysiological
recordings
100noisesequences
werepresented,
resultingin a 300-slong
signal. As is well known (Marmarelis and Marmarelis,
1978), estimatingfirst-orderpropertiesof a systemrequires
noisewith a flat spectrumanda bandwidthgreaterthanthat
of the system,andthusa relativelyshortandnonoscillating
impulseresponse.
Whensecond-order
propertiesof the system suchas energydensitiesin frequencytime haveto be
estimated,the noisemustin additionhaveadequatesecondorder propertiesitself.Thus the second-orderautocorrelation functionshouldbe equalto zero everywhere.This requires,amongotherthings,that the amplitudedistribution
of the noiseis symmetric;i.e., the skewness
shouldbe zero.
For the noise used it was assured that the second-order auto-
correlationfunction was essentiallyzero everywhere(Eggermontand Smith, 1988).
The filteractionof anauditoryneuronwassimulatedby
passing
the noisethrougha bandpass
filter (Wavetek753A)
with low- andhigh-frequency
slopesof ! 35 dB/oct. The output of thefilterwaspassedthrougha Schmitt-trigger
setat a
level of two standard deviations (24r) above the mean. The
triggermomentswerestoredin a data file,just as for real
neural data.
Singleunitswererecordedfrom the auditorymidbrain
of the leopardfrog, usingtungstenmicroelectrodes.
Spike
occurrences
in response
to stimulationwith noiseweretimed
with an accuracyof 10/zsandstoredin computermemory.
Detailsare in Eggermont(1989).
B. Computational details
The computation
of theWignerandtheRihacekdistributionsproceedsalongsimilarlines,whetheraveragingis
donefroma spikefile (in thecaseof neuraldata) or a trigger
file (in the case of simulation data). For each event, the
followingstepsareperformed:( 1) Extracta segment
of the
stimulussignal,typicallythe 25 ms precedingthe spikeor
triggerevent;(2) convertthe signalsegmentto its analytic
form [ Eq. (4) ]; ( 3) computetheappropriate
cross-product
matrix,
•( t)• * ( f )
for
the
RDF
or
•( t + r/2 )• * ( t -- r/2 ) fortheWDF, forallvaluesoftandf,
respectively,
r; and (5) repeatthis until all eventsare exhausted while accumulating the results in memory.
When all triggereventsare exhaustedand averagingis
comparison
of thecomputationtimesfor thetwomethodsas
well as the signal-to-noise
ratio to the ensemble-averaged complete,any requiredpost-processing
is performed.To
frequency-timedistributionswill be given.
movefrom the (t,r) domain into a time-frequencyrepresen249
J. ACOuSt.
Sec.Am..VoL87. No.1.January1990
J.J. Eggermont
andG. M. Smith:Characterizing
neuralresponses
249
tation, the Wigner product matrix is Fourier transformed
with respectto •'. The RDF requiresa final demodulation
throughmultiplicationby exp( -- i2rrft).
The calculationof theseaveragedistributionsis very
computationintensive,owingto the largenumberof com-
II. RESULTS
A. Simple signals
A sequence
of two tonepipsdifferingin frequencywill,
aswe haveseen,giveriseto interferenceproductsasa result
plexmultiplications
andadditionsinvolved.Severaloptimi- of the phaserelationshipbetweenthe two stimuli. In this
zationswereimplementedas the softwarewasdeveloped. examplewe useda sequenceof an 800- and a 1600-Hz tone
For example,it wasnotedthat, for theWDF, multiplication pip. The real part of the RDF is shownin Fig. 3(b) and
featuresthe interactionproductsat the time of the 800-Hz
andaveragingneedonlybe donefor positivevaluesof •'. The
productat eachnegativelag was derivedas the complex pip and the frequencyof the 1600-Hz pip [Fig. 3(a)] and
that theinteractionproductsshowa
conjugateof the valueat the corresponding
positivelag, viceversa.Oneobserves
rapid
alternation
between
positiveandnegativevalues.Intesince
grationoveran areafor whichthe productAfAt>lz- will
•(t + •-/2)• *(t - •-/2) = [•(t - •-/2)• *(t + •-/2) ] *.
We comparedthe computationtimes for RDFs and
WDFs that werecomparablewith respectto resolutionand
rangein the time and frequencydomain.We tried to avoid
buildingany hardware-specific
biasesinto our benchmarks.
For example,it would be quite straightforwardto codethe
RDF calculationto involve linear, sequentialaccessto
successive
elementsof boththe inputsignalandthe output
matrix,allowingthecompilerto generateefficient,registerbased,processor
instructions.
The Wignerproductcalculation tends to involve very scatteredmemory references,
which can becometroublesomeon a paged-memorymachinesuchasour Micro VAX II. We had the luxury of allocaring3-5 Mbytesof RAM to avoidpagingduringtesting,
allowingthe datafor eachcalculationto beresidentin memory at all times. However, we observedexecutiontimes for
the WDF
to increase more than a hundredfold
when mem-
ory waslimited,whilethe RDF sufferedlessfrom a shortage
of memory.Computationtimemaybea significant
consider-
ationwhenonehasto relyonsharedcomputingresources
or
on small-memorysystems.
alwaysresult in a zero value; thus the interactionterms do
not representreal power.For the WDF [Fig. 3(c) ] the interactionproductappearshalfwaybetweenthe two signals,
bothin thetimeandthefrequencydomains.Again,integration (smoothing)overthe appropriatearea in the interferenceregionwill resultin zeropower.
B. Average frequency-time representations of a tonepip in background noise
For a situationwherea sequence
of identicaltone-pips
is
presentin uncorrelated
background
noisewith peakamplitudeequalto that of the tone-pip,averagingthe frequencytimerepresentation
over30 tonepipsresultsin analmostfull
recoveryof thefrequency-time
functionof the tonepip without noise.In Fig. 4(a) we presentonesegmentof the toneplus-noise
signal.In Fig. 4(b) we showthe real part of the
RDF. Oneobserves
that therealpart looksaboutthesameas
that for a tonepip in the absence
of noise[ e.g.,Fig. 2 (b) ];
the small difference is the result of a modulation
of the am-
(a)
The net result of our benchmark test is that execution
RDF and33 s for theWDF. The additionaltimepertrigger
was 1.88 s for the RDF and 3.39 s for the WDF. Thus, for a
Hz
z
times are quite differentfor RDF and WDF: A 256X 128
RDF takes7.13 s;a WDF takesabout36 sexecutiontime. In
orderto separatethe computationaloverheadfrom the time
per trigger,we comparedalsocomputingtimesfor 30 and
752 triggers.We calculatedoverheadtimesof 5.8 s for the
{b)
RDF
(c)
WDF
typical500-neural-spikes
datafile, the computations
of the
averageRDF and the WDF take about 15 and 29 min, re-
spectively,CPU time on a Micro VAX II with 5 Mbytesof
RAM.
For very large numbersof triggerscomparedto the
number of noisesamplesin the pseudorandom
noisesequenceit may be beneficialto constructa periodhistogram
asa firststepandthento computean RDF or WDF for those
time binswhichcontainspikes,andto multiplythemby the
numberof spikesin the time bin. In our casewith a 30 000samplenoisesegmentthat wasrepeated100times,this was
not a desiredapproach.One hasto keepin mind that reducing the sequence
lengthcannotbe donewithout interfering
with thesecond-order
autocorrelation
propertiesof thenoise
and thus is not alwaysadvisable(see Eggermontet al.,
anda 1600-Hztonepip.The timescaleistwicethatin previous
figures.The
interactionproductsin theRDF areclearlyvisibleandin a differentplace
1983).
than for the WDF.
250
d. Acoust.Sec. Am.,Vol.87, No. 1, January1990
0
time {ms)
51.2
FIG. 3. Rihacekand Wignerdistributionfunctionsof a sequence
of an 800-
J.d. Eggermontand G. M. Smith:Characterizing
neuralresponses
250
(a)
When the signal-to-noise
ratio is decreased
by a factor5.5,
onecanstill observethe presence
of thetone-pipin theaveragedfrequency-time
representations
after30averages
[Fig.
5(a)-(c)
].
C. Use of frequency-time representations in the white-
noise analysisof linear systems
(b)
ROF
To explorethe useof frequency-timerepresentations
in
the characterizationof linear systems,we presenteda linear
filter (slopes135dB/octave) with a sequence
of 100 identical noisesamples,each3 s longand sampledat 10 kHz. The
N =30
outputof the filter wasmixedwith noisefrom an independent source (Wavetek 132) in order to introduce some ran-
(c)
0
WDF
timetms)
N=30
25.6
FIG. 4. AverageRihacekand Wignerdistributionfunctionsfor a 1600-Hz
tonepipin noisewiththesamepeakamplitude.(a) Onetone-pipin noise.
After 30averages,
whichtheoretically
improves
thesignal-to-noise
ratioby
abouta factor5.5,theRDF andWDF emergeclearlyfromthenoise.
domnessin the threshold-crossing
patternobtainedby passing the noisethrougha Schmitttriggerand alsoto improve
the averagingprocedure.Figure 6(a) and (b) showsdot
displaysof the Schmitt-triggeroutputfor the filterednoise.
In (a) we observethe resultfor a filter with both the high
andlow cutofffrequencyat 800 Hz; the timebaseof the dot
displayis 3 s (equalto thelengthof thenoisesequence),
and
eachline represents
the triggersfor a sampleof the noise.
One observes
the consistent,althoughsomewhatstochastic,
sequence
of triggerevents.Figure6(b) showsthe resultfor
the filter with the cutofffrequencies
setat 600 and 1200Hz.
Two typesof analysisare commonin the white-noise
plitudewith a frequencyequalto that of the tonepip. As we
have seen, this can be the result of an interaction with a
(a)
800 Hz P-P filter
correlatedlow-frequency
componentin the noiseor a small
dc componentsuperimposed
on the noise.The WDF [Fig.
4(c) ] doesshowthisinteractionproductat a lowfrequency.
(a)
(b)
600-1200
(b)
i' "."
i,
(c)
0
Hz BP filter
RDF N :30
'r
".•i'
:.:
.
•
'.' : :"','
.!"
. ß ."
.. '
,ß
. ßi. , '
ß!
: !
, • . •.• .' , ,.;.,
..,.,-
,....,
.,,.,;
: :
ß
. ß
WDF N=30
time (ms)
25.6
....
..
:, '. {,,:.! ,..,. ,' ='..;, .:
time Is)
FIG. 5. Average Rihacek and Wigner distribution functions of a 1600-Hz
FIG. 6. Dot displays for level crossingsof noise filtered by (a) a 800-Hz
tonepip in noisewith about5.5 timeslargerpeakamplitude.(a) One tone
pip in noise.Theoretically,30 averages
shouldnow resultin a signal-to-
bandpass
filterand (b) a 600- to 1200-Hzbandpass
filter.The horizontal
timebaseis 3 s,whichis identicalto thelengthof thenoisesequence
used.
Vertically,theresponses
forthe100noisepresentations
areshown;eachdot
represents
onelevelcrossing.
noise ratio of about 1, and one can observe that the RDF and WDF in this
casebarelyemergefrom the noise.
251
J. Acoust.
Soc.Am.,Vol.87, No.1, January1990
J.J. Eggermont
andG. M. Smith:Characterizing
neuralresponses
251
(a)
(a)
800Hz
BPfilter
/1J••,n/•'A • •
(b)
•
RDF N=752
25
(c)
25.6
800HzBP
filter
+jitter
:?',.•/•.• .....
•---•-•,•
(b)
RDF N=752
:
S
WDF N=752
time (ms)
l
(c)
25.6
0
WDFN=752
time (ms}
0
FIG. 7. Impulseresponse,
RDF, andWDF for the800-Hzbandpass
filter.
Timeisrunningfromrightto leftin thiscasebecause
thefunctions
shown
relateto thenoisepreceding
thetriggersthatwereshownin Fig.6(a). The
numberof averages
is 752 in all threecases.
Despitethe largenumberof
averages,
oneobserves
a strongperiodicstructurein the RDF andsomewhatlessin theWDF surrounding
theregionof interest.
FIG. 8. Impulseresponse,
RDF, andWDF for the 800-Hzbandpass
filter
with a uniform 4- 2-msjitter appliedto eachtrigger.The impulseresponse,
whosecomputation
reliesheavilyonphaselock,isabolished.
However,the
RDF and WDF are immuneto the absence
of phaselockandshowa clear
response
at 800Hz. Notethattheperiodicstructurein theRDF andWDF
hasnowdisappeared.
approachto linearor nonlinearsystems
analysis(Marmare-
time is to be consideredastime beforethe trigger.
For the 1-oct-widefilter (600-1200 Hz), the impulse
response
and the RDF and WDF are shownin Fig. 10(a)-
lis and Marmarelis, 1978): the (first-order) cross-correlation method and the Wiener method (which estimates as
(c). Especiallyfromthehalf-amplitude
contourplots[ Fig.
11(a) ] the wideningof the filter becomesobvious(cf. Fig.
correlation: a so-called reverse correlation function (De
9). Again, the resemblance
of both representations
is clear,
Boerand Kuyper, 1968) or time-reversed
impulseresponse althoughthe WDF is narrowerin the frequencydomain
thantheRDF. The signal-to-noise
ratioissomewhat
higher
for the 800-Hz bandpass
filter. The longlastingimpulseresponsereflectsthe steepfilter slopesas well as the narrow- for the RDF than for the WDF. For the 2-oct-wide filter
nessof the filter. Figure 7(b) and (c) showsthe RDF and
the WDF for a time windowof 25.6 ms beforethe triggers.
1250
Oneobserves
botha peakat the resonance
frequencyof the
1500 Hz BP filter
filter and anotheronewith somewhatlongerlatency(time
beforea trigger)at abouthalf thisfrequency.
Modulationof
112.5
theamplitudecanbeseenbothin theRDF andWDF. Serial
correlations
in thenoisesegments
beforethe triggersto the
iooo
filterednoisearemostlikelythe causeof thismodulation.In
orderto testthis assumption,
the spikesin the triggerfile
frequency(Hz)
werejittered uniformlyover -I- 2 ms.This completelyabol875
ishedthe first-ordercrosscorrelation[Fig. 8(a) ], however,
preservedthe RDF and WDF and actuallyeliminatedmost
of the modulation[Fig. 8(b) and (c)]. The peak values
750
decreased
by about 10% in eachcase,and the WDF appears
manycrosscorrelationsof higherorderasneededor possible). In Fig. 7(a) we showthe resultof thefirst-ordercross
smootherthan the RDF. The low-frequencycomponentin
both the RDF and WDF is mostlikely the resultof trigger-
ing only of the positivelevel crossings
of the filterednoise
(Eggermontet al., 1983). Figure 9 comparescontoursfor
theRDF andWDF at half thepeakvalue;thereisa tendency
for the WDF (dark shading) to be somewhatnarrower in
thefrequencydomainthan the RDF at beginningandendof
the impulseresponse.Note the time reversalin the graphs;
252
J. Acoust.Sec. Am., Vol. 87, No. 1, January1990
12.8
time (ms)
,
'
0
625
FIG. 9. Comparison
of theRDF (light shading)andWDF (dark shading)
in a contourplotsat thehalf-power
levelforthefunctions
shownin Fig. 8.
The time windownowonly covers12.8msbeforethe triggers,andthe fre-
quencywindowdisplayed
isfrom625-1250Hz. In general,theresultsare
quitecomparable;
thereisa tendency
fortheRDF tobebroaderat theonset
and offsetof the response.
J.J. Eggermontand G. M. Smith:Characterizingneuralresponses
252
(a)
(400-1600 Hz), the frequency-time
representations
further
broadenin the frequencydomainand shortenin the time
domain. The differences between the RDF and WDF be-
comemoreobvious[Fig. 12(a) ], the latter is morepeaked
and againnarrowerin the frequencydomain.
D. Measurement and quantification of phase lock for
(b)
noise stimulation
RDF N=1227
Justas one can calculatefrequency-time
functionsfor
tonepips,onecancalculatethemfor the first-orderreverse
correlationfunction(or impulseresponse
in caseof a linear
deterministicsystem).Figure 10(a) showsthe reversecorrelationfunctionfor the 1-octbandpass
filter. Figure 11(b)
showsa comparisonat the 50% contourline betweenthe
WDF
(
128
tin• (ms}
0
FIG. 10.Impulseresponse,
RDF, andWDF for the600-to 1200-Hzband-
passfilter.Thetimebasecovers12.8msbeforethetriggers
shownin Fig.
6(b}.
2.5
and RDF
of the reverse correlation function. Com-
parisonwith theaverageWDF andRDF [Fig. 11(a) ] suggeststhat thereare hardlyany differences
betweenthe FT
functionof the impulseresponse
and the averageFT function. A completelydifferentsituationis foundfor the2-octwidefilter:Comparethe 50%-contourplotsfor theaverage
FT functions
in Fig. 12(a) andthosefor theFT functionsof
the impulseresponse
in Fig. 12(b). First of all, it is noted
600-1200 Hz BPfilter
&O0--1600 Hz BPfilter
2.5
(a)
(a)
2O
1.5
1.5
1.0
1.o
0.5
O5
(b)
(b)
2.0
1.5
1.5
1,0
0.5
O5
5.•,
time (ms)
time (ms)
FIG. 11. Comparisonof the half-amplitudecontoursfor the averageRDF
(light shading)andWDF (dark shading)of (a) the 600-to 1200-Hzbandpassfilterand (b) for theRDF, respectively,
WDF of the impulseresponse
asshownin Fig. 10.The time basecovers6.4 msbeforethe triggers;frequency rangesfrom 0-2.5 kHz. One observesthat the WDFs are somewhatnarrowerin the frequencydomainand somewhatbroaderin the time domain.
In addition,the averageRDF andWDF resemble
quiteclosely,respective-
ly, theRDF andWDF of the impulseresponse.
253
J. Acoust.
Soc.Am.,Vol.87, No.1, January1990
0
FIG. 12.Comparisonof thehalf-amplitudecontoursfor the averageRDF
(light shading)and WDF (dark shading)of (a) a 400- to 1600-Hz bandpassfilter and (b) for the RDF, respectively,
W DF of its impulseresponse.
The time basecovers6.4 msbeforethe triggers;frequencyrangesfrom 0-2.5
kHz. Again, the averageWDF is somewhatnarrowerin the frequencydo-
mainthantheaverageRDF, whichseemsto be splitup in varioussubregions.The WDF of thefilter'simpulseresponse
hasa distinctboomerang
shapeanddiffersconsiderably
fromtheRDF of thesameimpulseresponse.
J.J. Eggermont
andG. M. Smith:Characterizing
neuralresponses
253
mum are probablynot valid and are thereforenot shown.
Figure 13(c) and (d) showsthe same, but basedon the
WDFs. Onenoticesthat thereareonlystatisticaldifferences
that the "centerof gravity" for the FT functionof the impulseresponseis at a lower frequencythan for the correspondingaverageFT function.Second,the WDF for the
reversecorrelationhasa boomerangshape,while the averageWDF is morerestrictedin the time domain.
The averagefrequency-timefunction doesnot depend
on phaselock;however,the averagesignalbeforea spikeas
determinedwith reversecorrelationonly differsfrom noise
whenthereis phaselock. The differencebetweenthe averagedFT functionand the FT functionof the averagesignal
will thereforebea measurefor theamountof suchphaselock
(Eggermontetal., 1983). Sincephaselockin auditoryneuronsispredominantlydeterminedby thefrequency,it maybe
characterizedby the ratio of the spectralintensitiesof the
betweenthe spectraand the indexof phaselockc(f) for
both representations
with the exceptionat frequencies
around500 Hz, wherebothJ, (f) andJ(f) are relatively
smallandof the samesize,resultingin a c(f) of 1. Onecan
seethat c(f) decreases
monotonically
to disappear
around
1.5kHz. Sincethespikes
weresubjected
toa uniformjitter of
0.25 msin additionto thestochastics
resultingfromtheadded uncorrelatednoiseto the outputof the filter, this lossof
phaselock is understandable.
For the 800-Hz bandpass
filter,thec(f) wasonlymeaningful
at onefrequency
bin;the
resultsfor the2-octbandpass
filterwerecomparable
to that
two FT functions:
for the l-oct filter.
c(f)
= J• (f)/J(f),
(20)
where ,I• (f) is the spectralintensityof the averagesignal
beforethe spike (the impulseresponse),and J(f) is the
spectralintensity of the averageWDF, respectively,the
averageRDF. For noise-freeestimatesof the spectralintensities,c(f) will benon-negativeandsmallerthanor equalto
1, and a functionof both spectralsensitivityand phaselock.
We will comparethe estimatesof c(f) derivedfrom the
WDF and RDF for the narrow-band, and the 1- and 2-octwide filters.
In orderto reducethe fluctuationsin the averagedspectrum, we haveuseda three-pointsmoothing.Figure 13(a)
showsthe magnitudeof the averagespectrum(light shading) and the spectrumof the impulseresponse(dark shad-
ing) basedontheRDFs for the 1-octfilter;Fig. 13(b) shows
c(f). It shouldbepointedoutthat thec(f) valuesresulting
from divisionof spectralvaluelessthan 10% of the maxi-
E. Frequency-time characterization of auditory units in
the midbrain of the leopard frog
We investigatedauditory midbrain neuronsthat respondedin a sustainedway to the repeatedpresentationof
the 3-s noisesequenceat differentstimuluslevels.The resultsare comparedfor similarity, frequency-timearea (at
half amplitude), signal-to-noise
ratio, and the amount of
phaselock.
An exampleis shownin Fig. 14(a)-(c) for 70-dB-SPL
noise stimulation. The RDF and WDF
locationin the frequency-timeplanewith a latencyof about
25 ms;however,the WDF is narrowerin the frequencydo-
main as shownin the half-amplitudecontourplot [Fig.
14(a) ]. The SNRs (definedat the ratio of peak value to
standarddeviation(s.d.) of the background)are, respec-
(a)
600-1200
1875
CFZ,91
Hz BP filler
.(a) ' iiiiii!l'
have about the same
1632
i•OF
•1390
•11/,7
906
664
31.6
time (ms}
12.8
5
z}
0
1
2
3
z,
5
WDF N=1991
frequency(kHz}
FIG. 13. Calculationof the amountof phaselock c(f) by dividingthe
spectrumof the impulseresponse
by theaveragespectrumof the noisesegmentspreceding
thetriggers[ Eq. (20) ]. (a) The averagespectrumandthe
spectrumof the impulseresponsebasedon the RDF, and (c) the same
basedon the WDF, The maximum in each graph is that of the averagespectrum. This explainswhy the spectraof the impulseresponseare not com-
51.2
time (ms)
0
pletelyidentical;that in (c) containsa componentaround500 Hz, whichis
setto zeroin (a). (b) and (c) The c(f) valuesthat areaboutthesame,with
FIG. 14.Comparisonof the averageRDF and WDF ( 1991averages)for a
neuronin the auditorymidbrainof theleopardfrog. (b) and (c) The RDF
theexception
forthecomponent
around500Hz, whichisartificiallyhighas
andWDF, respectively,
and(a) the50% contourplots.Also,for thencur-
a resultof the divisionof two relativelysmallnumbers.
onal data, one observesthat the WDF is narrower than the RDF.
254
J. Acoust.Soc. Am., Vol. 87, No. 1, January 1990
J.d. Eggermontand G. M. Smith:Characterizingneural responses
254
tively,7.7 and7.0, in favorof the RDF. At lowerintensity
levelsthefindingsweresimilar,andthebestfrequencyesti-
(a•
mated from both the WDF and RDF is 1300 Hz. For tonal
stimulithe unit appeared
to bedoubletunedwith bestfrequencies
of, respectively,
644and1400Hz.
For one neuron we obtained the WDF and RDF for 15
different
noisepresentations,
covering
eightintensity
levels,
sevenofwhichwerepresented
twice.Thestandarddeviation
(s.d.) of the fluctuations
in the WDF wassignificantly
larg-
er (t]4 = 7.45,p = 0.0001) thanin theRDF; however,
there
was a near-perfect
correlationbetweenthe s.d.'sfor the
WDF (.V)andRDF (x):y = 1.29x-- 10.22;
r• = 0.996.The
SNR for the WDF appearedto be significantlysmaller
1
2
3
•*
5
(t]4 ----2.38,p -- 0.016)thanfortheRDF, andtheSNRsfor
frequency(kHz}
WDF (y) and RDF(x) are not so perfectlycorrelated:
y----0.57x+ 1.99;r2 = 0.431.For the RDF the response FIG. 16.(a) Comparison
oftheaverage
spectrum
ofthenoise
preceding
the
(lightshading)
andthespectrum
ofthereverse
correlation
function
areaappearedto be clearlydetectable
whenthe SNR was spikes
(darkshading)
and(b) theamount
ofphase
lockshown
asi. At theCF
aboveabout5; for the W-DF, there was not as clear a critee(f) ----0.35.thevalueat 190Hz isartificially
highbydivision
oftworelarion in this respect.
tivelysmallnumbers
andistruncated
to 1.
Figure 15 shows(a) the reversecorrelationfunctionfor
a low CF neuronwith phaselock and (b) RDF, and (c)
WDF. Boththe FT functionsareextremelynarrowandcenteredaround115Hz. The spectrafor the averageFT (light
resulted
in a valueof 1.Forcomparison,
however,
wehaveto
shading)and the FT of the average(dark shading) are takeintoaccountthatthesynchronization
factorresultsin
shownin Fig. 16(a) togetherwith the c(f) in Fig. 16(b).
an averageoverall frequencies.
Oneobserves
therelativelysmallvalueofc(f) = 0.35at the
BF(115 Hz) and value of about I for the 190-Hz compo- III. DISCUSSION
nent.Calculatingthe amountof stimuluslock throughthe
Wehavepresented
a heuristic
introduction
totheuseof
shiftedautocoincidence
function(Eggermont,1989) results theaverage
RDF andWDF, theirinterpretation,
andusein
for a 8-msbin width (about one period at the BF) in 269
analyzing
responses
ofauditory
midbrain
neurons
tocontinsynchronized
spikesout of 1094resultingin a synchroniza- uousnoise.For simplesignalsthe WDF and RDF are not
tion factor of about 0.25. Perfect stimulus lock would have
CF688U•
strikingly
different;
however,
forsignals
containing
relatively widelyseparated
frequencycomponents,
the different
formandlocationof theinteraction
productsin theFT plane
makethe WDF moreuseful.Broadband
signalsalsohave
differentRDFs andWDFs; in general,the WDF appearsto
be narrower in its FT distribution than the RDF.
AlthoughtheRDF for a singlesignalcanbewrittenas
the demodulated
productof the (analytic)signaland its
spectrum
andthuscanbecalledseparable
in themathematical sense(whichthe WDF is not), thelocationof the inter-
actionproducts
makestheWDF separable
in thesense
that
these
interaction
products
donotinterfere
withtheWDFsof
theindividualsignalcomponents
(whichdoesnotapplyto
the RDF).
It wasfoundthattheaverage
RDF andWDF of a signal
buriedin uncorrelated
noi• isindistinguishable
fromthatof
thepuresignalwhenthepeaknoiselevels
donotexceed
that
of thesignal.Thiscanbeusedtorecover
hiddensignals
from
noiseandisappliedto theestimation
of theFT functionof
theauditoryfilter.The estimation
of theaverage
FT functiondoesnotrequirephaselockof theneuralfiringsto the
noise.By calculating
theratioof thespectral
density
of the
reverse
correlation
functionandthatof theaverageFT func76.8
ti•e (ms)
256
FIG. 15. (a) Reversecorrelationfunctionand (b) its RDFand (c) WDF
(c) ofa low-frequency
neuron(CF = 115H z) fromtheauditorymidbrain
of the leopardfrog.The timebaserunsfrom 76.8to 25.6 msbeforethe
spikes;
thefrequency
scalein (b) and(c) coverstherangefrom0-600 Hz.
255
d. Acoust_Sec. Am., Vol. 87, No. 1, January 1990
tion. a metric is obtainedto quantify the amountof phase
lockof spikesto continuousnoise.
The resultsobtainedin this studycanbe summarizedas
follows.
(1) There are no appreciabledifferences
in narrow-
d.J. Egoermontand G. M. Smith:Characterizingneural responses
255
band (e.g., tone-pip-likesignals) signal representations the neurons. For a second-orderanalysissuch as used to
between the RDF and the WDF.
derivethe RDF or WDF, the samecautionapplies,but now
(2) The interactionproductsemergingin the FT func- alsorequiresthat the second-order
autocorrelation
of the
stimulus is zero. All these conditions were fulfilled in our
tionsfor multifrequency
signalssuchastwo-tonesignalsand
signalplusdc shiftare not aslikely to affectthe representa- case;yet the finitedurationof the pseudorandom
noisesegtion of the individual signal componentsin the WDF as
ment, 30 000 samples,may causethe backgroundnoiseoutmuch as those in the RDF.
side the FT window for the RDF
(3) The WDF and RDF becomemore different as the
bandwidthof the signalincreases.
(4) When signalsare buried in uncorrelatedback-
groundnoise,theFT functionof thesignalmaybeobtained
throughaveragingthe FT functionsfor each signalplus
noisesegment.
(5) For an equalnumberof triggers,the RDF is signifi-
or WDF
to be not com-
pletelyindependentof the FT function,especiallyin cases
wherethe spikesare stronglyphaselockedto the stimulus.
The standarddeviationof thebackgroundnoisein both
theWDF andRDF decreased
linearlywiththesquareroot
of the numberof averages,indicatingthat the background
noiseis uncorrelatedwith the spikesproducedby the neural
or simulatedunit. However, the SNR definedas the ratio of
the peakvalueof the FT functionto the standarddeviation
of thebackground
appeared
to belargelyindependent
of the
Thisindicates
thatthepeakvalueof the
gers,theWDF takesabouta factor2 moretimeto compute numberof averages.
than the RDF.
FT distributionsdoesnot increaselinearly with the number
The FT functions can be used to characterize linear
of triggersasexpectedfrom a modelwherethe signalconfiltersby averagingFT functionsof input-noisesegments sistsof FT functionplusuncorrelated
noise.It may well be
that precedethresholdcrossings
of the filter'soutputsignal. that the numberof spikesgeneratedby the auditorymidIn contrast to the use of the cross-correlationmethod, this
brain neurons,and therefore the number of independent
doesnot requirephaselock. The methodcanbe usedin the
averages
thatcouldbecarriedout,wasnotsufficiently
large.
auditorynervoussystem,and thereis no preferencefor using The WDF appearsto havea significantly
smallerSNR than
the WDF over the RDF in the casespresented.By comparthe RDF; this couldbe causedby the fact that the WDF is
ing the spectraldensitiesestimatedfrom the FT functionsof
obtainedby a Fouriertransformof an autocorrelation
functhe reversecorrelationfunction and the averageFT function and the RDF is obtainedby averagingtime-dependent
tion, a quantificationof the amount of phaselock of the
spectra.It is known(Oppenheimand Schafer,1975) that
auditoryneuronto the noiseis obtained.
the varianceof the powerspectrumestimateobtainedby
Fourier transformation of an autocorrelation function (a
In this discussionwe will emphasizethe statisticalrequirements
forthenoiseusedin theanalysis,
theeffectof the periodogram)
hasa variance
proportional
to thesquareof
interactionproductson the outcomeof the WDF andRDF
theactualpowerspectrum
of thesignal.Fouriertransformanalysisand its implicationsfor applicationto the auditory ing firstand thenaveragingcouldremediatethis to some
system,and the estimateof the amountof neuralphaselock
extent,especially
whenthereare largenumbersof spikes,
for noise stimuli.
however,at the expenseof sharplyincreased
computation
cantly lessnoisythan the WDF.
(6) For data fileswith more than a few hundredtrig-
times.
A. Signal-to-noise ratio, effect stimulus sequence
statistics
The techniqueof reversecorrelationusesas a starting
pointthetriggeror neuraleventandlooksbackto thepartof
the stimulusthat causedit. Therefore,this techniqueis, in
principle,applicableto all typesof stimuli.However,when
the stimulushasa stronginternalcorrelationstructure,part
of this structurewill showup in the averagedresult. For
example,stimulatingwith a species-specific
vocalization
with a strongperiodiccomponentwill producea strongperiodicaveraged
result,notnecessarily
identicalto thestimulus
or to the impulseresponse
of the system.In principle,a deconvolution
of the result with
the autocorrelation
of the
stimulus can remediate this problem (Aertsen and Johannesma,1981). Only in the casethat the stimulushas a
delta-function-shaped
autocorrelationfunction,e.g., white
noise,is a correctionnot necessary.The examplecreatedto
illustrate the useof the RDF and WDF in linear filter analysisanddiscussed
in Fig. 7 relatesto thisproblemby showing
a periodicstructurein the FT functionsthat doesnot relate
to propertiesof the filter.In auditoryneuronsfrom the midbrainof the grassfrog,
however,thisdoesnot seemto occur,
probablyasa resultof the considerable
jitter in thefiringof
256
J. Acoust.Soc. Am., Vol. 87, No. 1, January 1990
B. Interaction products and consequences for the
analysis of broadband signals
We haveseenthat the positionof the interactionproductsis quitedifferentfor the WDF and the RDF, we have
alsoobservedthat the two FT functionsshowan increasing
differencewhenthebandwidthof the signalislargerasin the
caseof theimpulseresponse
of the2-octfilter.Undercertain
conditions
the placement
of theinteractionproductsin the
WDF seemsto enhancethe narrownessof the representa-
tion; however,the RDF broadensthat representation.
In
addition,the WDF alwaysseemto havea smootherappearanee than the RDF_ In order to illugtrate thi•, we have con-
structed a quasi-FM signal that producesa boomerang-
shapedWDF whichmimicsthatfor theimpulseresponse
of
the 2-oct-widefilter. The signalconsistsof the sumof seven
tonepipswith a gamma-function
envelope,anequalnumber
of periods(and thereforelastinglongerfor lowerfrequencies), and with frequenciesthat are equallyspacedin the
logarithmicsense
overthefrequencyrangeof 400-1600H z.
The signal,itsRDF, andtheWDF areshownin Fig. 17(c)
ascontourplots.Oneobserves
thattheWDF for theseven-
pipsignalis muchnarrowerthantheRDF. Thegenesis
of
J.J. Eggermontand G. M. Smith:Characterizingneural responses
256
(a)
2pF,s
(c)
2.5
7pips
2.5
RDF
2.0
2.0
1.5
1.5
1.0
1.o
0.5
o5
[
WDF
• 2.5
WDF
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0
I
0
time {ms)
25B
time (ms)
(b)
2.5
3•p•
','•
2.0
FIG. 17.In (a), (b), and (c), theRDF andWDF for a summationof two,
three,andseventonepipsareshown.Thetonepipsall startat timezeroand
1.5
havea frequency-dependent
duration,
whichis25msfor400Hz, 15msfor
800 Hz, 10msfor 1600Hz, and proportional
valuesfor the intermediate
frequencies,
givingthemaboutan equalnumberof periods.The signal
waveformisshownontop;then,therealpartof theRDF isshownfollowed
by WDF. Thecontourlinesareat 0.25,0.5,and0.75between
zeroandthe
maximumof the frequency-time
function.For the two-pipcombination
[400 and 1600Hz, (a) ], oneobserves
that theinteractionproductsappear
ontopoftherepresentation
ofthetwotonepipfrequencies;
however,
forthe
WDF theinteraction
productarehalfwaybetween
thetwofrequency
components.
For thethree-pip
combination
[400,800,and1600Hz, (b) ], one
startstoobserve
theliningupoftheinteraction
products
intheWDF, while
the RDF appearsto be fragmented.For the seven-pip
combination,
the
WDF appears
considerably
narrowerthantheRDF, probably
asa resultof
the constructive
lineupof the interactionproducts.
1.0
O5
I
I
WDF
2.0
1.5
1.0
0.5
0
257
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I
time {m•
I
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J. Acoust.Soc. Am., Vol. 87, No. 1, January 1990
d. J. Eggermontand G. M. Smith:Characterizingneural responses
257
theseFT functionscanbe seenwhenwe analyzethe sumof a
400- and a 1600-Hz tone pip, and then that of the sum of
400-, 800-, and 1600-Hz tone pips [Fig. 17(a) and (b)].
One observesthe constructivealignmentof the crossterms
for the WDF and the destructivealignmentthereoffor the
RDF. Consequently,the WDF seemsbetter to conveythe
generalimpressionthat one obtainsfrom the signalwaveform: a sweepfrom high to low frequency,resultingfrom the
factthat thecenterof gravityfor the tone-pipsshiftsto longer latenciesfor lower frequencies.One couldarguethat the
WDF representsthe energydensitysurroundingthe group
delay of the individual signalcomponents:the seventone
pips.
Sincethe RDF and the WDF are relatedby a double
convolution[Eq. ( 11) ] with the analyticsignalof cos4•rœt,
a linearFM signal,it istheoreticallypossible
to calculateone
distributionfrom the other. This can easilybe verifiedby
inspectingFig. 18 wherewe show(a) the real partsof the
RDF, (b) the linear FM signalacting as the convolution
kernel, and (c) the calculated WDF on basis of a double
convolutionof the RDF for a sequenceof a 400-, 800-, and
1200-Hz tone-pipwith the linear FM signal.Note that the
interactionproductsin the WDF appearalongthe main diagonal;in fact, the interactionproductof the 1200-and400Hz pipsresultsontop of the 800-Hz pip.At low frequencies,
therealandtheimaginarypartof theanalyticsignalchange
onlyveryslowlyovertime;thustheWDF at lowfrequencies
will appearasa smoothedversionof the RDF (cf. Fig. 8).
C. Estimation of amount of phase lock for broadband
stimulation
I
0
I
Estimationof the amount of phaselock for harmonic
signalscanbe doneon basisof the vectorstrengthcomputed
asthe ratio of the magnitudeof the fundamentalcomponent
and the dc componentin the spectrumof the periodhistogram.An alternativeisto computethespectrumof theinterspike-intervalhistogram;the ratio of the magnitudesof the
fundamentalto the dc componentis the squareof the vector
strength (Javel, 1988). The first method is obviouslynot
applicableto noisesignals,because
no periodhistogramcan
be made.The secondmethodbasedon the interspike-interval histogramcouldin principlebe applicable;however,it
turnsout that generallya nonperiodicintervalhistogramis
obtained.This holdsaswell for the filter data (cf. Fig. 6) as
for the real neuraldata reportedon in thispaper.The only
alternativemethodto the estimationof the c(f) as introducedin this paperis that basedon the shiftedautocoincidencefunctionof the spikesfor identicalstimuluspresentations (Eggermont, 1989). This procedurecomparesthe
peaknumberof coincidences
in the cross-coincidence
histogrambetweenthe response
to a stimulusandthe response
to
a secondpresentationof that stimulusto the number of
spikes.When there is perfectstimuluslock, the numberof
coincidencesin the central bin is equal to the number of
I
time (ms]
25.6
time (ms}
12.8
(b) 0.5
-0•_12.8
(c) 1.6
•
spikesin therecordandtheratioisequalto one;in casethere
is no connection with the stimulus, the number of coinci1.2
dencesin the centralbin will be smallandconsequently
also
the ratio with thenumberof spikes.A drawbackof themethod is that when appliedto multifrequencystimuli an average
valueover all frequencycomponentsis obtained.
One canintuitivelyagreethat thec(f) measuredefined
asin Eq. (20) is comparableto the vectorstrength;for perfectphaselockof spikesto a puretone,themagnitudeof the
0.8
in the WDF; one setof interactionproductsappearson top of the middle
averagespectrumwill beequalto thatof thespectrum
of the
average.Whenthereis somejitter, the magnitude
of the
spectralcomponentin the spectrumof the averagedecreases,while that in the averagedspectrumremainsthe
same;as a consequence,
c(f) decreases.For a sufficient
numberof completelyrandomspikes,the averagesignalwill
approachzeroandthusthe magnitudeof the spectralcomponentaswell;hence,c(f) = O.The resultsobtainedwith
tone-pip representation.
this methodfor simulatedand real neuraldata in the present
0
i
0
I
I
time (ms)
I
i
I
{
256
FIG. 18. Calculationof (c) a WDF from (a) an RDF througha double
convolutionwith exp(t2•rft). (b) The convolutionkernelis only shownfor
half the timerangeandhalf thefrequency
range,12.8msaroundzerolag
time,andfor positiveand negativefrequencies
up to 800Hz. Oneobserves
that thesixinteractionproductsin theRDF areall linedup on thediagonal
258
J. Acoust.Soc. Am.,Vol. 87, No. 1, January1990
J.J. Eggermontand G. M. Smith:Characterizing
neuralresponses
258
paper are encouragingand extendthe quantificationof
synchronybetweenspikesand stimulusphaseinto broadband and noiselike stimuli.
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Epping,W. J. M., andEggermont,J. J. (1985). "Single-unitcharacteristics
in the auditorymidbrainof the immobilizedgrassfrog,"Hear. Res.18,
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