Prince SJD, Pointon AD, Cumming BG, Parker AJ.

18 Pages·2001·0.34 MB·English

JNeurophysiol
87:191–208,2002;10.1152/jn.00465.2000.
Quantitative Analysis of the Responses of V1 Neurons to Horizontal
Disparity in Dynamic Random-Dot Stereograms
S.J.D. PRINCE, A. D. POINTON, B. G. CUMMING, AND A. J. PARKER
University Laboratory of Physiology, Oxford OX1 3PT, United Kingdom
Received6June2000;acceptedinfinalform1October2001
Prince,S.J.D.,A.D.Pointon,B.G.Cumming,andA.J.Parker. propertiesofV1neurons,suchasorientationtuningandocular
Quantitative analysis of the responses of V1 neurons to horizontal dominance.
disparity in dynamic random-dot stereograms. J Neurophysiol 87:
Thereareseveralreasonswhyitisimportanttostudythese
191–208, 2002; 10.1152/jn.00465.2000. Horizontal disparity tuning
issues with RDS. First, a change in the disparity of a bar
for dynamic random-dot stereograms was investigated for a large
population of neurons (n (cid:1) 787) in V1 of the awake macaque. stimulus also generates changes in the monocular images,
Disparity sensitivity was quantified using a measure of the discrim- which by themselves may influence neuronal firing. By con-
inabilityofthemaximumandminimumpointsonthedisparitytuning trastthemonocularimagesofrandom-dotstimuliarespatially
curve. This measure and others revealed a continuum of selectivity homogeneous. There is nothing that can be discovered about
ratherthanseparatepopulationsofdisparity-andnondisparity-sensi- the disparity of RDS by inspecting one eye’s image alone.
tive neurons. Although disparity sensitivity was correlated with the Second, random-dot patterns contain a complete spectrum of
degreeofdirectiontuning,itwasnotcorrelatedwithothersignificant
orientations, which permits horizontal disparities to be ex-
neuronal properties, including preferred orientation and ocular dom-
plored regardless of the neuron’s orientation preference. With
inance.InaccordancewiththeGaborenergymodel,tuningcurvesfor
barsorgratings,onlythecomponentofdisparityorthogonalto
horizontal disparity were adequately described by Gabor functions
thestimulusorientationcaninfluencetheneuronregardlessof
whentheneuron’sorientationpreferencewasnearvertical.Forneu-
ronswithorientationpreferencesneartohorizontal,aGaussianfunc- the receptive field properties. Remarkably, there are no pub-
tion was more frequently sufficient. The spatial frequency of the lisheddatathatcompareselectivityforthehorizontaldisparity
Gaborfunctionthatdescribedthedisparitytuningwasweaklycorre- oforientationbroadbandstimuliagainstorientationpreference
lated with measurements of the spatial frequency and orientation in area V1. Third, ever since the initiative of Julesz (1964,
preference of the neuron for drifting sinusoidal gratings. Energy 1971), many psychophysical studies of stereopsis have used
models make several predictions about the relationship between the RDStoisolatebinocularfrommonocularprocesses.Tounder-
response rates to monocular and binocular dot patterns. Few of the
standthephysiologicalsubstrateofsuchbehavior,itisimpor-
predictions were fulfilled exactly, although the observations can be
tant to use equivalent stimuli in a species whose psychophys-
reconciled with the energy model by simple modifications. These
ical performance approaches that of human observers
same modifications also provide an account of the observed contin-
(Harwerth and Boltz 1979; Harwerth et al. 1995; Prince et al.
uum in strength of disparity selectivity. A weak correlation between
the disparity sensitivity of simultaneously recorded single- and mul- 2000; Siderov and Harwerth 1995) and in a brain area where
tiunitdatawererevealedaswellasaweaktendencytoshowsimilar the neuronal performance can potentially account for the pre-
disparitypreferences.Thisiscompatiblewithadegreeoflocalclus- cision of psychophysical performance (Prince et al. 2000).
tering for disparity sensitivity in V1, although this is much weaker We therefore undertook a quantitative survey of the re-
thanthatreportedinareaMT. sponsestoRDSinalargepopulationofV1neurons(n(cid:1)787)
recorded from awake behaving monkeys. This provides a de-
taileddescriptionoftheprevalence,type,andrangeofdispar-
ity tuning in macaque V1 as well as the relationship between
INTRODUCTION
disparity selectivity and other RF properties. We also tested
Selectivityforbinoculardisparitywasinitiallydemonstrated quantitativelymodelsoftheunderlyingmechanismsofdispar-
usingelongatedbarstimuliincatarea17(Barlowetal.1967; ity tuning. Specifically, we sought to determine whether the
Pettigrew et al. 1968) and V1 of the awake monkey (Poggio observed data are compatible with the “energy” model of
andFischer1977).Subsequently,Poggioandcolleagues(Pog- disparity selective neurons (Ohzawa et al. 1990), which was
gio1995;Poggioetal.1985,1988)examinedthesensitivityto developed to describe data from area 17 of the cat. In this
horizontal disparity in random-dot stereograms (RDS) in ma- model, binocular simple cells are modeled as linear filters
caque V1. None of the studies using RDS has attempted to followed by a static output nonlinearity—a half-squaring op-
describe the disparity tuning quantitatively. Consequently, eration(AlbrechtandGeisler1991;Heeger1992;Jagadeeshet
there has been no quantitative analysis of the relationship al.1993;Movshonetal.1978;TolhurstandDean1987,1990).
between disparity selectivity to RDS and other fundamental Disparity selectivity arises because the response of the linear
Addressforreprintrequests:A.J.Parker,UniversityLaboratoryofPhysi- Thecostsofpublicationofthisarticleweredefrayedinpartbythepayment
ology, Parks Road, Oxford OX1 3PT, UK (E-mail: andrew.parker@physiol. ofpagecharges.Thearticlemustthereforebeherebymarked‘‘advertisement’’
ox.ac.uk). inaccordancewith18U.S.C.Section1734solelytoindicatethisfact.
www.jn.org 0022-3077/02$5.00Copyright©2002TheAmericanPhysiologicalSociety 191
192 PRINCE, POINTON, CUMMING, AND PARKER
filterforoneeyeissummedwiththatofasimilarfilterforthe METHODS
othereyebeforethehalf-squaringoperation.Hencetheexpan-
General methods
sive nonlinearity provides an increase infiring rate when both
left-andright-eyefiltersmatchtheimage.Ohzawaetal.(1990) ThemethodsemployedinthisexperimentforrecordingfromV1of
suggest that a complex cell may be constructed by combining theawakebehavingmonkeyhavebeendescribedinfullinCumming
andParker(1999).Alloftheprocedurescarriedoutcompliedwiththe
four simple cells that have different monocular phase profiles
UnitedKingdomHomeOfficeregulationsonanimalexperimentation.
but are all tuned to the same disparity.
Inbrief,extracellularrecordingsweremadefromthestriatecortexof
According to the energy model, the shape of the disparity
two adult monkeys (Macaca mulatta), which had been trained to
tuning curve is determined by the shapes of the monocular
perform attentive fixation while viewing visual stimuli in a Wheat-
receptivefields.Thestrongesttestofthemodelisthereforeto stonestereoscopeforfluidrewards.Single-unitsensitivitytodynamic
perform a quantitative comparison of the shape of the monoc- random-dot stereograms was measured as a function of horizontal
ularsubunitsandthedisparityselectivity.Suchananalysishas disparity. Sinusoidal and bar stimuli were used to characterize a
recently been performed for simple cells in anesthetized cats varietyofotherparameters.
(Anzai et al. 1999b). The analysis has not been performed for
complex cells because the spatial nonlinearity of these cells Apparatus and single-unit recording
makes it difficult to determine the receptive field (RF) profile
Binocular stimuli were presented on two monochrome monitors
ofthesubunits.Inawakeanimals,theanalysisisdifficulteven
(Textronix GMA 201) driven by a split-color signal from a Silicon
for simple cells, owing to the complications created by fixa-
Graphics Indigo computer and viewed using a Wheatstone stereo-
tional eye movements (Livingstone and Tsao 1999). scope.Meanluminancewas188cd.m(cid:2)2,themaximumcontrastwas
Fortunately, there are many other tests of the energy model 99%,andtheframeratewas72Hz.Thescreenswereatadistanceof
that can be applied in the absence of direct measurements of 89cmfromtheeyes,suchthateachpixelsubtended0.98arcmin.For
monocular RF structure. These are based on assuming a spe- asmallnumberofthelaterexperiments,EIZOFlexScanF78monitors
cificfunctionalformfortheunderlyingmonocularsubunits.A wereusedwithameanluminanceof42cd.m(cid:2)2.Thepositionsofboth
suitablefunctionistheGaborfunction,whichhasbeenexten- oftheanimals’eyesweremonitoredusingamagneticscleralsearch
coil system (C-N-C Engineering). To initialize a stimulus presenta-
sively evaluated as a suitable function for describing both
tion,theanimalswererequiredtofixatetowithineither0.4°(monkey
monocular RF profiles of simple cells (Daugman 1985; Jones
Rb)or0.6°(monkeyHg)ofabinocularlypresentedspot.Iftheanimal
and Palmer 1987; Marcelja 1980) in cortical area V1. Under
failedtomaintainfixationwithinthiswindowforthetrialdurationof
the assumption that the Gabor is a correct description for the
2 s, the trial was abandoned and a brief time-out period ensued. For
monocular profiles, the shape of the disparity tuning curve themajorityoftrials,oculomotorcontrolwasmuchtighterthanthese
should be well described by a Gabor function, whose param- limits.
eters should be related to the spatial properties of the neuron Tungsten-in-glass microelectrodes (Merrill and Ainsworth 1972)
(for example, its orientation and spatial frequency tuning). werepassedtransdurallyintotheopercularcortex.Extracellularmea-
SomedeviationsfromtheGabormodelforV1corticalneurons surementsofelectricalactivityincorticalareaV1weremadefromthe
have been previously observed (Hawken and Parker 1987). lefthemisphereformonkeyHg,andbothhemispheresformonkeyRb.
Onisolationofasingleunit,theclassicalminimumresponsefieldwas
Although similar deviations occur within the present data set,
determined, and its orientation preference was measured with a
they are slight.
sweepingbarstimulus(seefollowingtext).Ninety-fivepercentofthe
Theenergymodeliscommonlycombinedwiththeassump-
RFcenterswereateccentricitiesbetween0.99and4.93°.
tionthatitssubunitsaredescribedbyGaborfunctions.Thishas
been used widely (e.g., Fleet et al. 1996a,b; Prince and Eagle
Measurement of disparity tuning functions
2000; Qian 1994; Qian and Zhu 1997) since its inception and
will be referred to here as “Gabor energy model.” All experi- Thedisparitysensitivityofsingleunitswasassessedusingdynamic
mental data used to assess its validity (Anzai et al. 1999a,b, random-dot stereogram patterns. Each stereo-half consisted of equal
1997; DeAngelis et al. 1995; Ohzawa and Freeman 1986a,b; numbers of black and white dots (usually 0.08 (cid:3) 0.08°) presented
Ohzawa et al. 1990, 1996, 1997) have been gathered from V1 againstamidgraybackgroundwithanoveralldensityof25%.Anew
patternofrandomdotswasusedoneachvideoframetoconstructthe
neurons in the anesthetized cat using one-dimensional stimuli
stereograms.Thusina2-spresentation,therewere144frames.Asan
(barsorgratings).Thepresentpapergivesaquantitativesum-
absolute minimum, every measurement of a disparity-tuning curve
mary of the responses of cortical neurons in V1 of the awake
wasbasedonatleasttwotrialsforeachdisparity,whichmeansthat
monkey to RDS patterns and examines how well the energy (cid:1)288differentrandom-dotpatternswerepresentedtoeachneuronfor
model describes these responses. eachdisparitytested.Forpracticalpurposes,thesumofthisstimula-
The accompanying paper (Prince et al. 2002) concentrates on tion is spatially homogeneous in each monocular image. In practice,
those neurons that show strong disparity selectivity. The param- many more trials were acquired for data that were subjected to
etersofthecurvesfittedtothedisparitytuningfunctionsareused detailedquantitativeanalysis(seefollowingtext).
toaddressfourquestionsconcerningthemechanismofdisparity The stereogram stimuli consisted of a central circular region that
selectivity. 1) Is there any evidence for a distinct grouping into variedindisparityandasurroundregionthatwasheldataconstant
disparity.Thecentralregionwassufficientlylargetocoverthemon-
different types of disparity tuning curve? 2) Are interocular dif-
ocularreceptivefieldsofthecellevenatthelargestdisparitytested.
ferences in RF position or phase used to generate selectivity for
The surround region was present to mask the monocular shifts in
nonzerodisparities?3)WhatrangeofdisparitiesissignaledbyV1
stimuluspositionthataccompanytheintroductionofdisparity,andto
neurons?4)Isthedisparityencodinglimitedbytheperiodicityof
provide a reference for simultaneous psychophysical judgments (see
thetuningcurve(size-disparitycorrelation)?Together,thesetwo Princeetal.2000fordetails).Thedisparityofthesurroundhasbeen
papersprovideacomprehensive,quantitativeaccountoftheprop- demonstratednottoinfluencethemeanfiringrateofunitsinV1(see
ertiesofdisparity-selectiveneuronsinprimateV1. CummingandParker1999).Aninitialtestofdisparityselectivitywas
JNeurophysiol•VOL87•JANUARY2002•www.jn.org
RESPONSES OF V1 NEURONS TO HORIZONTAL DISPARITY 193
carried out using five stimuli with disparities varying from (cid:2)0.4 to outtotestwhethertheGaborfunctionexplainedsignificantlymoreof
0.4°.Ifthemeanfiringratewas(cid:4)10spikes/sforalldisparities,then thevariationinthecurvesthantheGaussianfunction.Becausethese
thedatawerediscarded.Ifthecelldidnotmodulateitsfiringratewith regressions are nonlinear in their parameters, a separate numerical
disparity, then measurements of spatial properties were made and simulationwascarriedouttoconfirmthattheFtesthadanappropri-
another unit was sought. For a subset of a cells, a wider range of aterejectionrate.
disparitieswassampledattheoutsettoensurethatcellstunedonlyto Many statistical procedures, including regression analysis, rely on
largedisparitieswerenotmissed.Inallcaseswheredisparitytuning anassumptionofhomogeneityofvariance.Thisposesaproblemfor
wasfound,therewassomemodulationintherange(cid:5)0.4°,evenifthe neuronal firing data, in which the variance is known to be approxi-
maximumresponsewasforagreaterdisparity. matelyproportionaltomeanfiringrate(e.g.,Dean1981;Tolhurstet
Ifthecellmodulateditsfiringratewithdisparity,furthermeasure- al. 1981). Taking the square root of measured firing provides a
mentsofdisparitysensitivityweremade,andthestimulusdisparities variance stabilizing transformation. This should remove the mean:
wereadjustedtocovertherangeoverwhichmodulationoccurred.In variance dependency and de-skew the data (see Armitage and Berry
general, the disparity tuning curves analyzed here were used for a 1994; Snedecor and Cochran 1989), and APPENDIX A shows that the
variety of other studies, which influenced the choice of sampling. transform achieves this for our dataset. All statistical analysis and
Thusthequantityandrangeofdatagatheredvariedwidelyfromcell curvefittinginthispaperwashenceperformedon(cid:12)firingrate.When
tocell.Thenumberofdisparitylevelssampledvariedfrom5to34, analyticfunctionssuchasGaborcurvesarefit,thesquarerootofthe
andthetotalnumberoftrialsvariedfrom10to958.Forsomecells, function is fitted to the square root of the data. The result is that of
responses to binocularly uncorrelated random-dot stereograms were fittingaGaborfunctioninwhichdataathighfiringratesareweighted
alsomeasured.Disparitytuningfunctionswererecordedfromatotal lessbecausethosefiringratesareknowntobemorevariable.
of787V1corticalcells,ofwhich489werefrommonkeyRband298
werefrommonkeyHg.
Measurement of spatial properties
Sensitivity to disparity was first characterized with a binocular
interaction index or BII (Ohzawa and Freeman 1986b; Smith et al. Orientationpreferencewasassessedusingbinocularsweepingbar
1997b),whichmeasuresthedegreetowhichthefiringmodulateswith stimuli.Meanfiringratewasmeasuredatanumberoforientationsand
disparity. A BII of near 1 indicates that disparity variations can a Gaussian curve was fitted to the resulting tuning profile. The
modulate the firing rate from zero to the maximum rate on the position of the peak of this curve was taken to be the preferred
disparitytuningcurve.ABIIofnear0indicatesthatdisparityhardly orientation. For many units, orientation preference was also tested
changesthefiringrateatall.Thebinocularinteractionindexisdefined withbinocular,driftingsinusoidalgratingsoftheoptimalspatial,and
as temporalfrequency.AGaussiancurvewasalsofittedtothisorienta-
tionresponsecurve,andthepeakpositionwastakenasameasureof
BII(cid:2)Rmax(cid:3)Rmin (1) the preferred orientation. Where both bar and grating stimuli were
Rmax(cid:4)Rmin usedtomeasurethecell’sorientationpreference,resultsweregener-
ally in close agreement. In these cases, the sinusoidal grating data
whereR isthefiringrateatthepreferreddisparity(i.e.,thegreatest
max wereused.Forbothstimuli,thehalf-widthathalfheightofthefitted
response on the tuning curve) and R is the firing rate at the least
min Gaussiancurvewastakenasameasureoftheorientationbandwidth.
preferreddisparity(theminimumresponseonthetuningcurve).
Preferred spatial frequency was assessed by presenting a drifting
TheGaborenergymodelpredictsthatdisparitytuningcurveswill
gratingpatchatthepreferredorientation.Theneuronalresponsewas
have the form of either a one dimensional Gabor function or a
measuredatanumberofspatialfrequencies(usually5),andaGauss-
Gaussiancurve(seeAPPENDIXB).Welltunedcellswerefitwithboth
ian in log frequency was fit to the resulting data. The peak of this
ofthesemodelsusinganonlinearleastsquaresalgorithm(Numerical
Gaussianwastakenasameasureofthespatialfrequencypreference
AlgorithmsGroup,Oxford).Aone-dimensionalGaborfunctionmay
ofthecell.WhenthepeakpositionofthefittedGaussianwasabove
bedescribedbytheequation
or below the data range, the data were deemed not to permit the
G(cid:6)d(cid:7)(cid:2)Pos(cid:8)R (cid:4)Aexp(cid:9)(cid:2)(cid:6)d(cid:3)d(cid:7)2/2(cid:5)2(cid:10)(cid:1)cos(cid:6)2(cid:6)f(cid:6)d(cid:3)d(cid:7)(cid:4)(cid:7)(cid:7)(cid:11) (2) designation of a preferred spatial frequency. It should be noted that
mean 0 0
spatialfrequencysamplingwasusuallysparse(typicallyat1,2,4,8,
where R is the mean height of the curve (binocular baseline and 16 cpd), and hence our estimates of the preferred spatial fre-
mean
firing),Aistheamplitude,disthestimulusdisparity,d isthemean quencyforluminancegratingsarelessprecisethanestimatesofother
0
positionofthecurveindisparity(disparityoffset),and(cid:5)isthewidth parameters. Examples of disparity, orientation and spatial frequency
of the function. The frequency and phase of the Gabor function are tuning curves are shown in Fig. 1. The spatial frequency preference
controlledbytheparametersfand(cid:7),respectively.Notethatphaseis was usually measured after characterizing disparity selectivity. In
defined relative to the disparity offset. Hence the phase parameter manycases,theunitisolationwaslostbeforethisstagewasreached
describes the symmetry of the tuning profile relative to the mean so there are many units for which no spatial frequency tuning is
position of the Gaussian envelope. The operation Pos denotes half- available.
waverectification.Fortheenergymodel,thebinocularbaselinefiring Ocular dominance was determined by presenting monocular drift-
rate about which modulations occur (R ) is the response of the inggratingsofthepreferredspatialfrequency,orientation,anddirec-
mean
neurontodotsthatarebinocularlyuncorrelated. tiontoeacheyealone.Forsomecells,monocularrandom-dotstimuli
Three constraints were placed on the Gabor fitting. First, the am- were interleaved in the main disparity tuning measurement, and a
plitudeparameterAwasrestrictedtobelessthantheobservedrange further estimate of ocular dominance was produced from these. For
of firing rates. This ensured that the fitted curve was limited to a both stimuli, the eye that was not being tested viewed a blank, dark
plausible range of firing rates. Note that for the Gabor function, the screen.Theoculardominanceindex(ODI)wasdefinedbyLeVayand
largestpossiblerangeoffiringratesisR (cid:5)A,sotherestrictionon Voigt (1988) as the response of the ipsilateral eye to a monocular
mean
Aallowsforthefittedmodulationtobeuptotwicetheexperimentally stimulus divided by the sum of the ipsi- and contralateral responses
observed range. Second, an upper limit on the frequency of the (seeEq.3).Hence,cellsthathaveanoculardominanceindexnear1
sinusoidal component (f) was set so that it did not exceed the limit havealargeipsilateralresponse,cellswithanoculardominanceindex
constrainedbythedatasampling.Third,thedisparityoffset(d )was near0havealargecontra-lateralresponse,andcellswithanODIof
0
constrainedtobewithintherangeofthedatasamples.Foreachcurve, near0.5arewellbalanced.Thiscanbere-expressedasamonocularity
a Gaussian function was also fit. This is defined identically to the indexMI,where0istotallybinocularand1istotallymonocular(see
Gabor with the cosine term omitted. A sequential F test was carried Eq. 4). It should be noted that, unlike LeVay and Voigt (1988),
JNeurophysiol•VOL87•JANUARY2002•www.jn.org
194 PRINCE, POINTON, CUMMING, AND PARKER
butstimuluscyclesduringwhichasaccadewasmadewerediscarded.
Cells in which the F1:F0 harmonic ratio was greater than one were
classifiedassimple,followingSkottunetal.(1991).
RESULTS
Prevalence of disparity tuning
In this section we address two questions. First, we consider
whetherthedegreeoftuningforhorizontaldisparityisdistrib-
uted continuously or whether there is evidence for a distinct
population of disparity selective cells. Second, we examine
whether disparity selectivity is correlated with other neuronal
properties. To answer these questions, disparity selectivity
must be adequately characterized.
The most common way to assess disparity selectivity has
been to employ a “relative modulation” index. For example,
Ohzawa and Freeman (1986a) and Smith et al. (1997b) mea-
sureddisparitytuningusingdriftingsinusoidalgratingstimuli.
They fitted sinusoidal functions to the tuning curves and de-
finedtheBIItobetheratiooftheamplitudetothemeanfiring
level. In this paper, we use a related measure, also referred to
astheBII,whichissuitableforusewithdatafromrandom-dot
stereograms (see Eq. 1 in METHODS).
There are several potential difficulties with the BII because
it takes no account of the variability in firing, nor the depen-
denceofthisvariabilityonthefiringrate.Forthesereasonswe
developed a different, statistical index that estimates the dis-
criminability of the maximum and minimum points on the
disparity tuning profile. We call this the disparity discrimina-
tion index or DDI
(cid:6)R (cid:3)R (cid:7)
DDI(cid:2) max min (6)
(cid:6)R (cid:3)R (cid:7)(cid:4)2.RMS
max min error
where,asbefore,R isthegreatestresponseonthemeasured
FIG.1. Sampleexperimentaldataforoneneuron(Rb526).Theerrorbars max
represent the standard error of the firing rate for the given stimulus. A: the tuningcurve,andR isthesmallestresponse.RMS isthe
min error
firing rate as a function of the disparity of random-dot stereograms. B: the square root of the residual variance around the means across
firing rate of the same neuron to drifting sinusoidal luminance patterns as a the whole tuning curve. All of these calculations were per-
functionofspatialfrequencyatthecell’spreferredorientation.C:thefiring formed on (cid:12)firing rate.
rate for the same neuron as a function of the orientation of a sweeping bar
stimulus. TheDDIessentiallycomparesthedifferenceinfiringtothe
preferred and least-preferred disparities to the within-stimulus
spontaneous rates were not measured explicitly in the present work
variationinneuronalfiring.Ifthedisparitytuningcurvemod-
and have not been subtracted from these measures. However, exam-
ulatesagreatdealandtheresponsetoeachparticulardisparity
inationofthemeanfiringrateintheprestimulusperiodsuggeststhat
onthecurveisstatisticallyreliable,thenthisindexwillbenear
spontaneous rates were almost always small compared with the re-
sponsestorandom-dotpatterns one. If the firing rate is not modulated by disparity, then the
fluctuations in the disparity tuning curve will be due to noise
ODI(cid:2) I (3) and this index will be small. Better estimates of the term
I(cid:4)C RMS are of course achieved by increasing the number of
error
MI(cid:2)2(cid:1)ODI(cid:3)0.5(cid:1) (4) stimulus presentations. However, increasing the duration over
which rates are measured systematically reduces RMS ,
error
Directionpreferencewastestedbypresentingamonoculardrifting leadingtolargevaluesoftheDDI.Itisthereforeimportantthat
grating of the preferred temporal frequency, spatial frequency, and
comparisonsofthistypeofmeasurearemadebetweendatasets
orientationtothedominanteyeandcomparingtheresponseinthetwo
with the same the interval of time over which firing rates are
possibledriftdirections.ThedirectiontuningindexorDTIquantifies
measured,sincechangesinRMS alterthevalueoftheDDI,
thedifferenceinfiringtothepreferredandnulldirections.Itisdefined error
evenifR andR donotchange.TheDDIindexisrelated
as the difference between the responses in the preferred and null max min
directionsdividedbytheirsum to the ability of an ideal observer to perform a disparity
discriminationtask,givenonlythemeasuredfiringratesofthe
DTI(cid:2)Rpref(cid:3)Rnull (5) neuron at the preferred and least-preferred disparities. See
Rpref(cid:4)Rnull Prince et al. (2000) for a further discussion of neuronal dis-
Cells were classified as simple or complex using the method of crimination of horizontal disparities.
Cummingetal.(1999):responsestodriftinggratingswereanalyzed Figure 2 compares these two measures of disparity selectivity
JNeurophysiol•VOL87•JANUARY2002•www.jn.org
RESPONSES OF V1 NEURONS TO HORIZONTAL DISPARITY 195
ANOVAisused,431/787(55%)ofourV1neuronsweresignif-
icantatthe5%level.Figure2showsthattheDDIisquiteclosely
relatedtothestatisticalclassification:valuesofDDIsmallerthan
0.4areveryrarelytheresultofsignificantmodulation,andvalues
ofDDI(cid:13)0.6arealmostinvariablytheresultofsignificantmod-
ulation.Evenso,somecaseswheretheDDIis(cid:13)0.6andstatisti-
callysignificantactuallyrepresentveryweaktuning(seeFig.3B).
WithamorestringentcriterionthatacceptsvaluesofDDI(cid:13)0.8,
most tuning curves were strongly modulated by disparity and
couldbereliablyquantifiedatlaterstages.
Comparing selectivity for disparity with other neuronal
properties
The preceding section establishes the DDI as a valid mea-
sure of the strength of disparity tuning. Figure 4 shows the
relationship between the DDI and other neuronal properties.
FIG.2. Comparisonof2measuresofthestrengthofdisparityselectivity,
binocular interaction index (BII, top) and the disparity discrimination index
(DDI, bottom). Both measures show a continuum in the extent of disparity
selectivity.A:afrequencydistribution(n(cid:1)787)oftheBII((cid:1) ,neuronsthat
havestatisticallysignificantdisparitytuningona1-wayANOVA,P(cid:4)0.05).
B:theBIIisnegativelycorrelatedwiththemeanfiringrate,unliketheDDI.
Forthisreason,somelargevaluesoftheBII,whichoccurwithlowfiringrates,are
notstatisticallysignificant(1).C:thefrequencydistributionoftheBII,detailsas
forA.TheDDIismuchmorecloselyrelatedtowhetherornotdisparitytuningis
significant.D:theDDIisnotcorrelatedwithmeanfiringrate.
inawaythatrevealsthreeadvantagesoftheDDI.First,theBIIis
negativelycorrelatedwiththemeanfiringrate,whichtheDDIis
not.Thereasonisthatneuronswithlowfiringratescanspuriously
acquireahighvalueofBIIsimplyduetorandomfluctuationsof
anotherwiseweakresponse.Second,andconveniently,theDDI
is more or less normally distributed close to a Gaussian in its
frequencydistribution.Third,andmostimportantly,theDDIisa
betterindicatorofwhetherdisparity-inducedmodulationisstatis-
ticallyreliable.Forallthesereasons,itismoreappropriatetouse
the DDI when examining the correlation of disparity selectivity
with other neuronal properties. Figure 3 presents examples of
disparity tuning curves with low (A), medium (B), and high (C)
disparity tuning indices. Note that error bars on these plots rep-
resenttheSDsofthefiringrate.Acrosstheentirepopulation,the
orderofDDIvaluesaccordedwellwithjudgmentsbyeyeofthe
strengthofdisparitytuning.
Figure2showsthedistributionsofbothmeasuresofdispar-
ity selectivity (BII and DDI) for the whole population (787
neurons). There is no evidence of two distinct populations
despite the large sample. Rather there is a continuum in the
strengthofdisparitytuning.Asimilarresultusinggratingshas
been reported both in the cat (Ohzawa and Freeman 1986a,b)
and the monkey (Smith et al. 1997b). We also quantified
disparity tuning in a number of other ways, including the
maximumrateofchangeoffiringwithdisparityandtheFratio
from a one-way ANOVA. None of these showed a separation
FIG.3. ThreeexamplesofdisparitytuningcurvesillustratingtheDDI.Error
into two populations. barsontheseplotsrepresenttheSDsoffiringrate.A:acellwithalowDDI(0.16).
Becausethedistributionofdisparityselectivityisunimodal, Thesmallchangesinmeanfiringratecanbeattributedtorandomfluctuations.
the proportion of neurons that are deemed to be disparity B:acellwithaDDIof0.60,somewhatlargerthanthepopulationmean(0.54).
Althoughthiscellmodulatesitsresponsesignificantlywithdisparity,itcouldnot
selective will depend on the criterion used. For example, 378/
bedescribedasstronglytuned.C:acellthatishighlytunedforbinoculardisparity
787 (48%) neurons showed significant modulation at the 5%
andhasahighvalueofDDI(0.88).Thiscellmodulatesitsresponseswithdisparity
levelonaKruskal-Wallistest.Ontheotherhand,ifaone-way agreatdealrelativetothevarianceoffiring.
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196 PRINCE, POINTON, CUMMING, AND PARKER
FIG.4. TherelationshipbetweentheDDIandotherneuronal
properties.Ineachcase,anonparametrictestforcorrelationwas
applied(Spearman’srank).A:DDIiscorrelatedwithdirection
tuningindex(n(cid:1)185,r (cid:1)0.23,P(cid:8)0.002).B:DDIisnot
s
correlatedwiththeneuron’soculardominance(n(cid:1)185,r (cid:1)
s
0.03,NS,usingthemonocularityindex).C:DDIisnotsignif-
icantly correlated with orientation bandwidth (n (cid:1) 391, r (cid:1)
s
(cid:2)0.09,P(cid:8)0.07).D:DDIisnotcorrelatedwiththepreferred
stimulusorientation(n(cid:1)391,r (cid:1)0.057,NS).E:DDIisnot
s
significantlycorrelatedwiththepreferredspatialfrequency(n(cid:1)
268,r (cid:1)0.067,NS).F:theDDIisnotcorrelatedwitheccen-
s
tricity(n(cid:1)787,r (cid:1)0.03,NS).Thenumberofpointsoneach
s
graph varies because not all parameters could be successfully
measuredforeachcell.(cid:1) ,datafromHg;(cid:1),datafromRb.
Figure4Ashowsaweak,butsignificant,positivecorrelation is not observed in our dataset. The correlation between orien-
(r (cid:1) 0.23, P (cid:8) 0.002) between the DDI and direction selec- tation bandwidth and DDI in Fig. 4C is weak compared with
s
tivity. Inspection of the plot indicates that it is uncommon for that found by Smith et al. (1997b) and not statistically signif-
neurons to show a combination of strong direction selectivity icant (r(cid:1) (cid:2)0.09, P (cid:4) 0.07). This difference may reflect the
s
and weak disparity sensitivity. Figure 4B plots the disparity fact that Smith et al. (1997b) used the BII, a measure that
discrimination index as a function of the degree of monocu- depends on mean firing rate. Indeed in our data, a stronger,
larity.Thereisnotendencyforneuronsthatrespondequallyto statistically significant correlation was found between the BII
monocular stimulation in each eye to exhibit a greater sensi- andorientationbandwidth(r (cid:1)(cid:2)0.13,P(cid:4)0.012).However,
s
tivitytodisparity,againinagreement with earlierquantitative we also found that orientation bandwidth was negatively cor-
datafromanesthetizedanimals.AlthoughSmithetal.(1997b) related with the mean firing rate.
found that simple cells with ocular imbalance tended to have Figure 4E indicates that there is no tendency for disparity
lower disparity sensitivity than those that were balanced, they selectivitytovarywiththepreferredstimulusspatialfrequency
found no relationship for complex cells. In area 17 of the cat, (r (cid:1)0.07,n.s.).Figure4Fshowsthatthereisnotendencyfor
s
Ohzawa and Freeman (1986b) reported that the degree of the strength of disparity selectivity to change as a function of
binocular interaction in simple cells did not depend on their eccentricity (r (cid:1) 0.03, n.s.). Indeed, the general lack of
s
ocular dominance. structure in these data sets is a little surprising at first glance.
Figure4Dshowsnocorrelationbetweendisparityselectivity Asaprecaution,were-examinedalloftheserelationshipsafter
and preferred orientation—cells that are highly sensitive to settingatightercriterionontheaveragefiringrateachievedat
disparity are found at all orientations. In practice, only a the most preferred disparity on the tuning curve. Previously,
modestrelationshipbetweenorientationpreferenceanddispar- this had been 10 impulses/s (see METHODS). Raising this value
ity sensitivity in random-dot stereograms is predicted by the to 40 impulses/s did not substantially alter the conclusions.
binocularenergymodel(seeAPPENDIXAandFig.12).Eventhis We also examined the relationship between disparity sensi-
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RESPONSES OF V1 NEURONS TO HORIZONTAL DISPARITY 197
tivity and classification as simple or complex. Cells were namic random-dot stereograms and vary their response as a
classified by the method of Cumming et al. (1999) (see METH- function of the disparity of such patterns.
ODS). Of 226 neurons to which this analysis could be applied, For many cells, disparity tuning was also measured for
57 were classified as simple. It has previously been claimed drifting sinusoidal gratings as a function of the interocular
(e.g., Poggio et al. 1985) that simple cells do not respond to phase difference. The spatial frequency, temporal frequency,
random-dot stereograms and also rarely show disparity selec- orientation and drift direction of these gratings were matched
tothepreferredvaluesforeachunit.Thedisparitydiscrimina-
tivity to RDS. It is important to consider these two issues
tion index for sinusoidal gratings was significantly correlated
separately. There is a significant negative correlation between
meanfiringratetoRDSpatternsandF1:F0ratio(r (cid:1)(cid:2)0.377, with the disparity discrimination index for random-dot stereo-
P(cid:4)(cid:4)0.0001),confirmingthatsimplecellstendtoshavelower grams(rs(cid:1)0.28;P(cid:8)0.00015,n(cid:1)176).Onereasonwhythis
correlation might be less than perfect is that if the spatial
firingratesonaveragethancomplexcellsinresponsetoRDS.
propertiesofthegratingortheRDSpatternarenotoptimalfor
This is unsurprising because simple cells can only respond to
the neuron, this may limit the DDI. A second reason is that
thosedotpatternsthathappentomatchtheirmonocularphase
mis-samplingineithertuningcurveinevitablyaffectsthevalue
preferences. Complex cells may be stimulated by all dot pat-
of DDI that is measured. Nonetheless, neurons that exhibit
terns.AsaconsequenceoftherelationshipbetweentheF1:F0
disparitytuningtodynamicrandom-dotpatternsalsogenerally
ratio and the response rates to RDS, it is important to use a
exhibit tuning to sinusoidal grating stimuli.
measureofdisparityselectivitythatisnotinfluencedbymean
firingratewhencomparingsimpleandcomplexcells.TheDDI
Description of disparity tuning curves
has this property, and we found no relationship between the
F1:F0 ratio and DDI (r (cid:1) 0.026, NS). Fig. 5C shows an Tosummarizethepopulationofdisparitytuningcurves,itis
s
example tuning curve from one simple cell that is strongly useful to fit analytic functions to the disparity tuning data and
selective for disparity in these random-dot stereograms. We examine the resulting parameters. The energy model predicts
conclude that both simple and complex cells respond to dy- that the disparity tuning profile for dynamic random-dot ste-
FIG.5. ExamplesofGaborfitstodisparitytuningcurves.Two
cellsinmonkeyHg(AandB)and2inRb(CandD)showhow
welltheGaborfunctionswereabletodescribethedatainmost
cases.EandF:evenwherethefitisrelativelypoor,theimportant
featuresofthedisparitytuningarewellcapturedbyGaborfit:68%
ofallfitsaccountedforalargerfractionofthevariancethanthefit
inF.EandFillustratetheonlyrepeatabledeviationweobserved
betweenthedataandthefit.Insuchcases,thewidthsoftheside
flanksarebroaderthanthewidthsofthepeaks.
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198 PRINCE, POINTON, CUMMING, AND PARKER
reogramswilltaketheformofthehorizontalcross-correlation accountedforbythefitinFig.5F.Nonethelessthebest-fitting
between the left- and right-eye receptive field shapes (see Gabor functions in Fig. 5, E and F, still capture the essential
APPENDIX B). We did not measure directly the shape of the features of these tuning curves. In particular, the fitted phase
monocularreceptivefields.Previousmeasuresofthisproperty provides an accurate reflection of the degree of symmetry in
haveconcludedthatareasonabledescriptionofthemonocular the tuning curve, and the fitted position of the Gaussian enve-
receptive fields can be delivered by Gabor functions, which lopedescribeswellthecenterofthedisparityrangeoverwhich
modulation occurs.
consistofasinusoidmultipliedbyaGaussianenvelope(Daug-
One parameter that needs to be interpreted with care is the
man 1985; Marcelja 1980; but see also Hawken and Parker
fittedfrequency.Figure6showsanexamplewithtwodifferent
1987). Under these circumstances, the energy model predicts
fits to the same disparity tuning data. The sinusoidal and
thatthedisparitytuningfunctionsshouldbewelldescribedby
GaussiancomponentsofthefittedGaborsareshownseparately
a Gabor function, which is the function we therefore fit to the
in the bottom two panels. Although the fitted frequencies are
data (as described in METHODS).
verydifferent,thesecombinewithequallydifferentGaussians
The requirements for this stage of the analysis were that
toproduceverysimilarlookingGaborfunctions.Toprovidea
neurons should be strongly modulated by disparity and their
fitted measure that fairly reflected the spatial scale of the
modulation should have been reliably characterized. The DDI
modulation in disparity, we adopted a procedure in which the
provides a good estimate of the extent to which neuronal
frequency of the Gabor function was derived directly from
activityismodulatedbydisparity,butitprovidesnostatistical
the data. We examined the Fourier spectra of the disparity
assurance that there was reliable stimulus-related variance
tuningcurvesaftertheDCcomponenthadbeenremoved(Fig.
withinthetuningprofileasawhole.Weusedanothermetricto
6). The frequency of the sinusoidal component of the Gabor
select neurons for further analysis, the F , given by
index
wassettobetheequaltoFouriercomponentwiththegreatest
MS energy,whichwetermthe“disparityfrequency”ofthetuning
Findex(cid:2)MS (cid:4)treMatmSent (7) curve (following Ohzawa et al. 1997). The remaining five
error treatment
parameters of the Gabor curve were then refit. In all cases the
where MS and MS are the familiar “mean-square” shape of the new fit was extremely similar to the previous fit.
error treatment
terms from a one-way ANOVA (performed on (cid:12)firingrate). However,the“disparityfrequency”correspondedmuchbetter
MS isthemean“withindisparityvariance,”andMS with the scale of the tuning curves as assessed “by eye.”
error treatment
is the mean “between disparity variance.” The F is maxi- We also investigated whether a more economic model than
index
mized when the mean firing rate exhibits consistent stimulus- a Gabor function would suffice. Many curves could be well
related changes at several different disparities. Hence, this describedbyafour-parameterGaussianmodel.Wefoundthat
criterionrejectssomecellsthathadahighDDIbutinadequate for 142/253 cells, the addition of the frequency (f) and phase
samplingoftheirtuningcurves.AlthoughtheDDIprovidesthe ((cid:7))componentsfortheGabormodeldidnotprovideasignif-
best estimate of how strongly disparity modulates a neuron’s icantimprovementinthefit(sequentialFtest,P(cid:1)0.05—see
activity, the Findex identifies more accurately neurons that had METHODSfordetails).Fig.7,AandB,demonstratestwocurves
bothstrongmodulationandareliablecharacterizationoftheir that are well described by Gaussian functions. There is no
tuningfunction:strongmodulationbydisparityisessentiallya suggestionofasinusoidalcomponentinthesedisparitytuning
property of the neuron itself, whereas the reliable character- curves.Figure7Cpresentsanexampleofaborderlinecase,in
izationofthetuningfunctionismoreanindicatorofthequality which the Gabor model is significantly better than a Gaussian
oftheexperimentaldatagatheredonaparticularneuron.Using modelbutonlyatthe5%level.Thedisparitytuningcurvesin
the F captures both these criteria, so neurons that had an Figs.1Aand5AprovideexampleswhereGaussianmodelsare
index
F of(cid:13)0.8wereadmittedtofurtheranalysis(338/787cells, insufficient.Notethatafailuretodemonstratestatisticallythat
index
44%). Tuning curves that had been sampled at fewer than aGaborfitisnecessarydoesnotguaranteethattheunderlying
sevendisparitieswereinanycaserejectedfromfurthermodel- tuning is truly Gaussian, only that we cannot reject that pos-
based analysis. This yielded 253 cells (136 from monkey Rb sibility. If the underlying Gabor shape had relatively shallow
and 117 from monkey Hg). side lobes (i.e., the frequency is low relative to the SD), then
Six disparity tuning profiles and their associated Gabor fits our sampling may not have been fine enough to detect these
are illustrated in Fig. 5. The Gabor function described the reliably.
disparity tuning data extremely well: for 163/253 neurons the For neurons that are adequately described by Gaussian tun-
fitaccountedfor(cid:1)90%ofthevarianceattributabletodisparity ingprofiles,thefrequencyterminEq.4ispoorlyconstrained:
and for 233/253 neurons the fit accounted for (cid:1)75% of the provided the period of the cosine term is large relative to the
variance. Only a few disparity tuning functions were not well SDoftheGaussian,ithaslittleinfluence.Thisdoesnotmean
fit by Gabors. For most of these cases, the disparity tuning that the measure of “disparity frequency” is uninterpretable.
curves appeared to lack any coherent form. The only system- Somewhatparadoxically,Gaussiantuningcurvesgiverisetoa
atic deviation from the Gabor model that was noted is illus- peak“disparityfrequency”inthecontinuousFouriertransform
tratedinFig.5,EandF.Forthesecurves,thesidelobesofthe (see Fig. 6D) of the disparity tuning data, because the DC
disparity tuning function are noticeably wider than the central component is removed prior to the transform. A similar issue
peak.ThisisthesameformofdeviationfromtheGabormodel arises with neurons whose disparity tuning is odd symmetric
notedbyHawkenandParker(1987).However,itseffectinthis and broadly tuned for disparity, for which a low-frequency,
datasetofdisparitytuningcurvesisslight.Withinthisdataset, odd-symmetricGaboristhemostappropriatefunctionalform.
theexamplesinFig.5,EandF,arerelativelypoorfits:68%of The disparity frequency gives a consistent measure of the
allfitsaccountedforalargerfractionofthevariancethanthat spatialscaleofthedisparitymodulationthatcanbeappliedto
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RESPONSES OF V1 NEURONS TO HORIZONTAL DISPARITY 199
FIG.6. TheapproachtofittingGaborfunctions.Thesolidand
dashedfittedcurvesinAappearalmostidentical,buttheparam-
etersoftheunderlyingGaborfunctionsarequitedifferent.Cand
D:thecorrespondingenvelopeandsinewavecomponents.The
amplitude,SD,andfrequencyparametersoftheGaborfunctions
aredifferent,buttheresultingcurveisverysimilar.Tinychanges
in the dataset can potentially produce radically different values
for these 3 parameters. These problems can be addressed by
directestimationoffrequency/scaleparameter.TheDCcompo-
nentofthetuningcurveisremovedbysubtractingthemeanfiring
ratefromthefiringrateateachdisparity.TheFouriertransform
ofthecurveiscalculateddirectlybymultiplyingthedatapoints
with pairs of orthogonal sinusoidal functions. The peak of the
Fourier amplitude spectrum (B) is used to fix the frequency
parameter for fitted curves, which is termed the “disparity fre-
quency.”ThedashedcurveinAshowstheGaborthathasbeen
refitafterthefrequencyparameterhasbeenconstrainedtoequal
this estimate. The underlying parameters are illustrated in D.
The peak of the Fourier transform provides a good estimate of
thescaleofthemodulationofthefiringratewithdisparity.The
tuning curve in E shows an different example which is almost
Gaussianinshape.ThereisstillapeakintheFouriertransform
F because the overall mean firing rate was subtracted prior to
calculating the Fourier transform. Estimation of the disparity
frequencybythismeansfixedthespatialscaleofthetuningcurve
forbothGaussianandGaborshapedcurves.
both Gabor-shaped and Gaussian tuning curves. For this rea- model.Itwouldbedifficulttoreconciletheenergymodelwith
son, all subsequent analysis is performed on Gabor fits in disparity tuning curves that looked very different from Gabor
whichthefrequencytermwasnotafreeparameter,butwasset functions. Of course, this does not demonstrate that the shape
to the disparity frequency. ofeachdisparitytuningfunctionisexplainedbythemonocular
For those cells where the Gaussian model is sufficient, the RFstructureofthatparticularcell.Becausewedidnotmeasure
Gabor still yields an equally good description even though monocularline-weightingfunctions,werelyonothermeasures
some of the parameters of the Gabor are poorly constrained. totestthelinkbetweenthespatialpropertiesoftheRFandthe
Two important parameters are still well constrained, even in disparity tuning function.
thesecases.Thefirstisthehorizontalpositionterm.Thecurve Beforeattemptingsuchananalysis,itisimportanttorestrict
in Fig. 7A is both Gaussian in shape and requires a nonzero thesampletoneuronsthatarewelltunedandwelldescribedby
positionterm.Thesecondisthephaseterm,whichisinevitably the Gabor fits. As described in the preceding text, we initially
near zero or (cid:6), because the Gaussian shape is symmetrical. fit Gabor functions only to the 253 neurons with F (cid:13) 0.8
index
When applying population analyses to the shape of tuning that had been sampled at seven or more disparities. Next, we
curves, the shape of the fitted Gabor is used even for cells rejected 13 cells for which the Gabor fit accounted for (cid:4)75%
where a Gaussian would have been adequate. ofthebetween-disparitiesvariance.Also,weexcluded48cells
for which our measurements of disparity tuning covered (cid:4)2
SDsofthefittedGabor(whichistosaythesampleddisparities
Relating disparity tuning to spatial properties
didnotadequatelyconstrainthefit).Allofthesecasesalsohad
The preceding section demonstrates that Gabor functions a minimum of three samples per period of the disparity fre-
provide highly accurate descriptions of the shape of disparity quency(i.e.,thesamplingwasalwaysabovetheNyquistlimit).
tuning functions in primate V1. Given that many underlying Finally we removed a further 12 cells by hand for which the
monocular tuning curves are often well described by Gabor Gabordidnotprovidearealisticdescriptionofthevariationin
functions,thisrepresentsasuccessfulpredictionoftheenergy thecurve.Aftertheserefinements,180tuningcurvesremained,
JNeurophysiol•VOL87•JANUARY2002•www.jn.org
200 PRINCE, POINTON, CUMMING, AND PARKER
Gaussian.Figure8showstheratioplottedasafunctionofthe
preferred orientation. There is a marked absence of points in
the upper left quadrant, indicating that cells with orientations
near horizontal tend to have more Gaussian tuning curves as
predictedbytheenergymodel.Therelationshipisstatistically
significant:forcellswithorientationswithin45°ofhorizontal,
the spread of values for this parameter (cid:5)/(cid:9)was significantly
smaller than for those with orientations within 45° of vertical
(F test, P (cid:4) 0.005). Nonetheless, some cells with vertically
oriented receptive fields also exhibit disparity tuning curves
that are well described by a Gaussian. This is what might be
expectedifthespatialfrequencytuningofthesecellswasvery
broad.
In fact, it is notable that the mean number of cycles per SD
of the Gabor envelope is only 0.25 for the whole population.
Hence, the Fourier amplitude spectra of the majority of the
tuning curves are effectively low-pass. This is surprising be-
cause the Gabor energy model predicts that the bandwidth of
the tuning curves should strictly be narrower than that of the
underlying RFs measured monocularly with sinusoidal grat-
ings (APPENDIX B). For sinusoidal luminance gratings, primate
V1 neurons recorded under anesthesia typically have spatial
frequency bandwidths (full width at half height) of (cid:14)1.5 oc-
taves (DeValois et al. 1982). This corresponds to 0.38 cycles/
SD,whichissubstantiallylargerthanthemajorityofvaluesin
Fig.8,evenforverticallyorientedcells.Thesedataarehighly
suggestive of a discrepancy between the bandwidth of the
disparity frequency and the tuning for sinusoidal luminance
gratings.However,directlycomparabledataofsufficientqual-
ity were not available on sufficient neurons to evaluate this
hypothesis fully.
The second prediction of the model is that the frequency at
which the disparity tuning function modulates (disparity fre-
quency) should equal the spatial frequency of a horizontal
section through the RF (or RF subunits, for complex cells).
FIG.7. Is the Gabor model necessary? A and B: 2 curves for which a This is closely related to the spatial frequency of a horizontal
Gaussiancurveprovidesanadequatedescriptionofthedata.C:anexample cross section through the preferred grating stimulus (the “hor-
thatisslightlybetterdescribedbyaGaborthanaGaussian(0.01(cid:4)P(cid:4)0.05).
TheraisedflanksofthiscurverequiresthesinusoidalcomponentoftheGabor
function.
for which the fit of the Gabor function was both extremely
goodandadequatelyconstrained.Notethatthisgroupincludes
tuning curves that could be described by Gaussians, as the
Gabor function still provides a good fit to the data with a
Gaussian form.
TheGaborenergymodelmakesclearpredictionsabouthow
the form of the disparity tuning curve should depend on pre-
ferred orientation and spatial frequency (see APPENDIX B for a
detailed discussion). First, the sinusoidal component of the
Gabor curve should increase in scale as the preferred orienta- FIG.8. The number of cycles of the Gabor function per SD (cid:5)of the
tion of the neuron moves from vertical to horizontal (when envelopeisplottedasafunctionofthepreferredorientationoftheunit.Asthe
preferred spatial frequency is held constant). When the orien- orientationbecomesclosertohorizontal,themeannumberofcyclesdecreases.
tation approaches horizontal, the period of the sinusoid be- Solidsquares((cid:1) ),cellsforwhichaGabormodeldidnotprovideasignificant
improvement over the Gaussian model at a 5% level. The energy model
comes very broad, and the tuning profile either becomes
predictsthatthenumberofcyclesofthesinusoidwithintheGaborenvelope
Gaussian or flat, depending on the original symmetry of the willdecreasewithsin((cid:10))where(cid:10)istheanglebetweenthepreferredorienta-
curve (see APPENDIX B, Fig. 12). tionandhorizontal.Forhorizontalreceptivefields(RFs),thedisparitytuning
This prediction was tested by taking the ratio of the wave- curve should be Gaussian, but even in this case, our curve fitting assigns a
length of the sinusoid, (cid:9)(cid:1) 1/f, to the SD parameter, (cid:5). This frequencyterm,basedontheFouriertransform.Thisproduces0.1–0.2cycles
in 1 SD (depending on how the Gaussian is sampled). Neurons with near
provides an estimate of the number of cycles in the tuning
horizontalorientationalsohavevaluesinthisrange,indicatingthattheirtuning
curve. As this ratio decreases, the curve becomes more nearly curvesareapproximatelyGaussian.
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