Artifacts in spatiochromatic stimuli due to variations in preretinal absorption and axial chromatic...

1694 J. Opt. Soc. Am. A/Vol. 20, No. 9 /September 2003 Nicolas P. Cottaris

Artifacts in spatiochromatic stimuli due tovariations in preretinal

absorption and axial chromaticaberration: implications for color physiology

Nicolas P. Cottaris

Group in Vision Science, University of California, Berkeley, Berkeley, California 94720

Received October 16, 2002; revised manuscript received May 6, 2003; accepted May 6, 2003

The spatiochromatic receptive-field structure of neurons in the macaque visual system has been studied almostexclusively with stimuli based on the human foveal cone fundamentals of Smith and Pokorny [Vision Res. 15,161 (1975)] and generated on cathode ray tube displays. In the current study the artifacts evoked by cone-isolating, spatially structured stimuli due to variations in the eye’s preretinal absorption characteristics andaxial chromatic aberration are quantified. In addition, the luminance artifacts evoked by nominally isolumi-nant sinusoidal grating stimuli due to the same factors are quantified. The results indicate that the spatio-chromatic stimuli commonly employed to map receptive fields of neurons at eccentricities .10 deg are espe-cially prone to artifacts and that these artifacts are maximal for the high-contrast S-cone-isolating stimuli thatare often used. On the basis of these simulations, a method is introduced that improves spatiochromaticreceptive-field estimates by compensating for response contributions from the incompletely silenced cone mo-saics during cone-isolating stimulation. © 2003 Optical Society of America

OCIS codes: 330.4270, 330.1720.

1. INTRODUCTIONThe L-, M-, and S-cone fundamentals are the spectral re-sponse functions of the long-wavelength (L) -sensitive,middle-wavelength (M) -sensitive, and short-wavelength(S) -sensitive cone receptor mechanisms, measured at thecorneal plane. As such, cone fundamentals differ fromthe cone photopigment absorption spectra because theyinclude the effects of preretinal light absorption by the op-tical media, the crystalline lens, and the macularpigment.1 Cone fundamentals also depend on photopig-ment optical density, which in turn depends on the recep-tor outer segment’s axial length and photopigmentconcentration2 and on photopigment peak absorptionwavelength, which is controlled by the variety of photo-pigment genotypes expressed by the receptor.3

Because of the variation with eccentricity in the opticaldensities of both the macular pigment and the receptors’photopigment, cone fundamentals vary with stimulus ec-centricity. The optical density of the macular pigment ishighest in the foveola, declines steadily throughout thecentral 5 deg, retains a low constant value out to the cen-tral 8–10 deg, and finally completely disappears beyondthe central 17–20 deg.4–6 The optical density of receptorphotopigment also declines with eccentricity because ofthe shortening of the receptor’s outer segment, and thisdecline may be different for different types of receptors.7

Also, because of the lenticular shape of the crystallinelens, the effective light path through the lens is shorter inthe periphery than in central regions. This may result ina reduction of the lens’ optical density in more peripheralregions, but documentation on this is lacking. In addi-tion to changes with eccentricity, there are variations in

1084-7529/2003/091694-20$15.00 ©

the optical density of the preretinal pigments with age,1,8

among species,4,5 and among different individuals.9–11

Finally, the optical density and peak absorption wave-length of the receptors’ photopigment also varies amongindividuals.3,12 It follows that a single set of cone funda-mentals cannot be used to accurately estimate retinalcone contrasts generated by stimuli projected at differentretinal locations or in the eyes of different subjects or spe-cies.

Spatially structured chromatic stimuli, i.e., nonuniformfields, are further subject to a wavelength-specific defocusthat arises from chromatic aberration of the eye. Chro-matic aberration is a result of the inverse relationship be-tween the refractive index of ocular media and lightwavelength.1 Axial chromatic aberration is the mostdominant aberration of the eye and results in a variationof the eye’s focusing plane with wavelength, causingshort-wavelength rays to focus in front of long-wavelength rays, thus differentially affecting the contrastavailable to different types of photoreceptors. Trans-verse, or lateral, chromatic aberration causes rays emit-ted by image locations off the optical axis of the eye tostrike the retinal plane at different locations, dependingon their wavelengths. Transverse chromatic aberrationdepends critically on object location and pupil positionwithin the eye13,14 and is not analyzed here.

Despite this multitude of factors that affects cone fun-damentals, nearly all studies of color physiology haveused the 2-deg Smith–Pokorny (S–P) cone fundamen-tals15 to construct stimuli that attempt to isolate eitherindividual cone pathways16–20 or individual postrecep-toral mechanisms.21–26 Only a few more-recent studies

2003 Optical Society of America

Nicolas P. Cottaris Vol. 20, No. 9 /September 2003 /J. Opt. Soc. Am. A 1695

have employed stimuli based on fundamentals other thanthe S–P-cone fundamentals.27–29 Very little attention(but see Flitcroft30) has been directed to the types of arti-facts that may be generated by spatiochromatic stimuliunder different conditions of preretinal absorption and asa function of stimulus spatial characteristics. The pur-pose of the present study is to quantify the retinal arti-facts in typical cathode ray tube (CRT)-generated, S–P-based spatiochromatic stimuli that may arise fromvariations in the optical density of the human andmacaque eye inert pigments with eccentricity or age andfrom axial chromatic aberration. We ignore variations inabsorbance spectrum peaks introduced into photorecep-tors that express different genotypes or in hybrid photo-receptors with more than one cone pigment, variations inthe optical density of photopigments, and effects intro-duced by transverse chromatic aberration. The informa-tion obtained is crucial for evaluating the artifacts gener-ated when such spatiochromatic stimuli are viewedthrough nonstandard eyes, for example, in mapping thespatiochromatic receptive-field organization of macaqueneurons at different eccentricities.

2. METHODSA. Generation of Cone-Isolating StimuliThe excitation of a cone mosaic, Econe , by a stimulus withspectral power distribution, SPD(l i), is given by

Econe 5 K(i

SPD~l i!fconeS–P~l i!Dl, (1)

where fconeS–P(l i) is the corresponding S–P cone fundamen-

tal and K is a constant. In the computations of this studywe have Dl 5 5 nm and l i P @390, 395,...780# nm. If wedenote by r, g, and b the red, green, and blue CRT-gunmodulation depths needed to generate this stimulus, wehave

Econe 5 K(i

@rR~l i! 1 gG~l i! 1 bB~l i!#fconeS–P~l i!Dl,

(2)

where R(l i), G(l i) and B(l i) are the SPDs of the CRT’sred, green and blue guns, respectively. We denote

^RL& 5 K(i

R~l i!fLS–P~l i!Dl,

^GL& 5 K(i

G~l i!fLS–P~l i!Dl,

^BL& 5 K(i

B~l i!fLS–P~l i!Dl, (3)

and after rearranging the terms in Eq. (2) we get the fol-lowing equation for this stimulus’ L-, M-, S-cone excita-tion vector, (EL , EM , ES)T:

S EL

EM

ES

D 5 F ^RL& ^GL& ^BL&

^RM& ^GM& ^BM&

^RS& ^GS& ^BS&G 3 S r

gbD , (4)

where ^RM&, ^GM&, ^BM&, ^RS&, ^GS&, ^BS& are de-fined similarly to ^RL&, ^GL&, ^BL& [Eq. (3)]. Therefore,to generate a stimulus with an arbitrary cone-excitationvector, one needs to modulate the red, green, and blueCRT guns at the following modulation depths:

S rgbD 5 F ^RL& ^GL& ^BL&

^RM& ^GM& ^BM&

^RS& ^GS& ^BS&G21

3 S EL

EM

ES

D . (5)

Note that such a stimulus will be realizable on a particu-lar CRT if and only if the resulting gun modulationdepths are between 0.0 and 1.0.

Given a stimulus with cone-excitation vector(EL , EM , ES)T (background), one can modulate a pedes-tal stimulus around this background in such a way as tochange the excitation in only one of the three cone typeswhile leaving the excitations of the remaining two conetypes unaffected. Such stimuli are called cone isolating,and this method of isolating a differential response from atargeted cone pathway is named silent substitution.31 Acone-isolating stimulus is fully specified by the cone exci-tations of the background stimulus and its contrast value.The L-, M-, S-cone-excitation vector of an L-cone-isolatingstimulus with contrast CL is given by

S EL

EM

ES

D 5 F 1 1 CL 0 0

0 1 0

0 0 1G 3 S EL

EM

ES

D . (6)

In the present study we analyze cone-isolating stimulithat are modulated against a background with an LMS-cone-excitation vector of (46.26, 24.18, 0.62)T. Thisbackground has CIE’31 chromaticity coordinates of x315 0.310, y31 5 0.316, and a CIE’31 luminance of Y315 70.0 cd/m2 (Judd–Vos luminance 70.44 cd/m2). Theanalyzed L-, M-, and S-cone-isolating stimuli have the fol-lowing cone-contrast vectors: (0.20, 0.00, 0.00)T, (0.00,0.24, 0.00)T and (0.00, 0.00, 0.84)T, respectively. Thesecontrast values are close to the maximum for cone-isolating stimuli modulated around chromatically neutral(grayish) backgrounds and generated on typical CRT dis-plays.

B. Generation of Isoluminant StimuliIsoluminant stimuli are designed to have the same effecton the luminous-efficiency function, Vl , as the back-ground stimulus against which they are modulated. Thisrequires that the direction of cone-excitation change (frombackground to stimulus) of isoluminant stimuli and thedirection defined by the observer’s Vl be orthogonal.Similar to cone fundamentals, the shape of Vl depends onthe preretinal absorption conditions and eccentricity, butunlike cone fundamentals, it also depends on the chro-matic adaptation point,32 spatial and temporal factors1

and the relative numbers of L and M cones.33 Because ofthis specificity and the large variation among individualobservers, Vl is measured separately for each individualand for each particular experimental paradigm. Inphysiological experiments with macaque monkeys, how-ever, the experimenter usually cannot measure the ani-mal’s luminous-efficiency function, and it is as-


sumed that the macaque Vl has the same shape as thestandard human Vl , as supported by an early study.34

Because the focus of the present study is on physiologi-cal studies, the isoluminant stimuli examined here areconstructed to be nominally isoluminant, i.e., orthogonalto the standard human luminosity function,35,36

VlJudd–Vos(l). Since

VlJudd–Vos~l! 5 fL

S–P~l! 1 fMS–P~l!, (7)

the excitation of the VlJudd–Vos luminance mechanism will

be E lum 5 EL 1 EM . Therefore any stimulus whose L-,M-, S-cone-excitation vector is

~ELstim , EM

stim , ESstim!T

5 ~EL 1 DELM , EM 2 DELM , ES 1 DES!T, (8)

and that is modulated against a background with(EL, EM, ES)T cone-excitation vector, will yield a differ-ential luminance response of DE lum 5 (EL

stim 1 EMstim)

2 (EL 1 EM) 5 0; i.e., it will be nominally isoluminant.The two orthogonal axes, DELM , DES , which are also or-thogonal to the luminance differential excitation axis(DE lum) define the isoluminant plane. On this plane allstimuli have the following L-, M-, S-cone-contrast vectors:(DELM /EL, 2DELM /EM, DES /ES)T. This is the basisunderlying the generation of the isoluminant color spacesproposed by MacLeod and Boynton37 and by Derringtonet al.21

For the present choice of background (x31 5 0.310,y31 5 0.316, Judd–Vos ’78 luminance570.44 cd/m2) andCRT display (NEC 5FGe), the maximum attainable conecontrasts in the isoluminant plane are 0.0796 (L),20.1523 (M), and 0.8620 (S).

C. Human and Macaque Preretinal Absorption SpectraThe human lens pigment density spectrum used in thepresent simulations corresponds to a small pupil (3-mmdiameter) and is the one employed by Stockman et al.38

This spectrum includes absorption by the other opticalmedia as well. The human macular density spectrum isalso the one employed by Stockman et al.38 and is basedon measurements made by Bone et al.39 (Both the lensand the macular pigment density spectra were obtainedfrom http://cvrl.ioo.ucl.ac.uk/.)

The macaque lens pigment density spectrum was esti-mated by scaling its human counterpart to obtain a den-sity of 1.22 at 400 nm.40 The macaque macular pigmentdensity spectrum was estimated by scaling a normalizedmacaque pigment density estimate5 to obtain a value of0.29 at 460 nm.40

D. Axial Chromatic Aberration ModelThe spatio-spectro optical transfer function (OTF) of themacaque eye is modeled as done by Marimont andWandell.41 In this model, the spatio-spectro opticaltransfer function, OTF( f, l), incorporates axial chro-matic aberration, H( f, l), combined with wavelength-independent aberrations, K( f ), and is given by the fol-lowing term:

OTF~ f, l! 5 H~ f, l!K~ f !. (9)

H( f, l) describes axial chromatic aberration for adiffraction-limited optical system with a circular apertureand is computed as follows42,43:

H~ f, l! 54

paE

0

A1 2 ~s/2!2

sinFaSA1 2 y2 2usu

2 D Gdy ,

(10)

where

s 5 clf

Dop, a 5

4p

lu f u

DoD~l!

Do 1 D~l!,

D~l! 5 q1 2q2

l/1000 2 q3. (11)

In Eqs. (9)–(11), l is expressed in nanometers, f is ex-pressed in cycles per degree, c is the inverse magnificationfactor of the eye in degrees per meter, Do is the dioptricpower of the unaccommodated eye in diopters, p is the ra-dius of the entrance pupil in meters, and D(l) is the rela-tive defocus, in diopters, as a function of wavelength.D(l) is zero for the in-focus wavelength, which is deter-mined by the values of the q1 , q2 , and q3 parameters.The term K( f ) is a scale factor that captures the effects ofwavelength-independent aberrations as a function of spa-tial frequency. In the human eye, this factor is describedby the term44:

K~ f ! 5 0.3481 1 0.6519 exp~20.1212f !. (12)

We adapted this spatio-spectro OTF model to themacaque eye by setting the entrance-pupil radius of theeye to 1.5 mm, the eye’s inverse magnification factor to4063.53 deg/m, and the eye’s unaccommodated power to70.92 diopters.45 We assumed that the eye is focused inthe yellow region of the visible spectrum (580 nm), whichis close to the dominant wavelength of the retinoscopelight beams that are commonly used for refraction. TheOTF was focused at 580 nm by setting q1 5 1.7312,q2 5 0.63346, and q3 5 0.21410. The wavelength-independent aberrations were assumed to be similar tothose of the human eye.

3. RESULTSThe results of the present study pertain to stimuli gener-ated on a NEC 5FGe CRT display. However, since thephosphors used by most CRT manufacturers are verysimilar, our results should apply to a wide gamut of CRTdisplay systems.

The spectral power distributions of the NEC 5FGe red,green, and blue phosphors are displayed in Fig. 1(a). TheSPDs of the 20% L-, the 24% M-, and the 84% S-cone-isolating stimuli are depicted by dark-gray curves in Figs.1(b)–1(d) together with the SPD of the background, de-picted by the light-gray curves.

A. Cone FundamentalsThe S–P 2-deg human cone fundamentals are based onlens and macular pigment optical density spectra for astandard observer. Here we examine the effects of varia-tions in the optical density of these inert pigments onstimulus retinal spatiochromatic composition. Toward


this end, we derive new sets of cone fundamentals thatare based on various assumptions about the lens andmacular pigments, and we use these fundamentals to es-timate retinal cone contrasts in S–P-based cone-isolatingstimuli and retinal luminance artifacts in Judd–VosVl-based nominally isoluminant stimuli.

The transmission spectrum of a pigmented medium,T(l), i.e., the fraction of light transmitted through themedium as a function of wavelength of the incident light,is related to the optical density spectrum of the medium,D(l), as follows1:

T~l! 5 102D~l!. (13)

Figure 2 shows the standard transmission spectra of thecrystalline lens and of the macular pigment for humaneyes (solid curves) and the corresponding spectra formacaque eyes (dashed curves), as derived in Section 2.C.The S–P 2-deg human cone fundamentals are depicted assolid curves in the top right panel of Fig. 3 (this set of fun-damentals is labeled b).

1. Deriving Macaque Cone FundamentalsIt is clear from Fig. 2 that the preretinal absorption char-acteristics of macaque eyes are not identical to those ofhumans, in either the shape or the magnitude of theirtransmission spectra. Therefore the S–P cone funda-mentals do not accurately estimate the cone contrasts of achromatic stimulus imaged on a macaque retina. Tocompute a set of cone fundamentals that are more appro-priate for the macaque visual system, we multiplied theS–P cone fundamentals, wavelength by wavelength, by

Fig. 1. Spectral power distributions (SPDs) of phosphors in anNEC 5FGe CRT display and of Smith–Pokorny15-based cone-isolating stimuli generated on this CRT display. (a) SPDs of red,green, and blue phosphors are plotted as light-gray, dark-gray,and black curves, respectively. SPDs of the (b) L-cone isolating,(c) M-cone isolating, and (d) S-cone isolating stimuli are plottedas dark-gray curves. In (b)–(d), light-gray curves depict theSPD of the background against which these cone-isolatingstimuli are modulated.

the ratio of the macaque transmission spectrum to the hu-man transmission spectrum:

fconemacaque~l! 5 fcone

S–P~l! 3T lens

macaque~l!Tmac. pigm.macaque ~l!

T lenshuman~l!Tmac. pigm.

human ~l!. (14)

In this way, the effects of human preretinal absorption,which are included in the estimation of the S–P funda-mentals, are canceled, and the effects of macaque prereti-nal absorption are introduced. The so derived macaquecone fundamentals are depicted as dashed curves in thetop right panel of Fig. 3 (this set of fundamentals is la-beled b8). As expected, almost all of the differences oc-cur at wavelengths below 550 nm, where the transmis-sion spectra of the two species are most discrepant.

2. Deriving Cone Fundamentals for Extreme Cases ofMacular Pigment DensitiesIt is known that large variations in macular pigment den-sity exist both among different individuals and within thesame individual with increasing distance from thefovea.1,4,5,6,9,11 In a recent study on variations in humanmacular pigmentation,11 it was found that the standarddeviation in peak macular pigment density among 32 sub-jects was 0.45 of the mean value, with a range from 0.30up to 2.39 times the mean value. Therefore, to obtain arealistic range of the effects that may result from differ-ences in macular pigment density, we derived two newsets of human and macaque cone fundamentals, assum-ing a macular pigment having either double or half thenormal optical density. This was accomplished by multi-plying, wavelength by wavelength, the ‘‘standard’’ human(S–P) or macaque cone fundamentals by the ratio of thetransmission spectra of the extreme-density macular pig-ment to that of the standard-density macular pigment:

fconeextreme~l! 5 fcone

standard~l! 3Tmac. pigm.

extreme ~l!

Tmac. pigm.standard ~l!

. (15)

The so computed cone fundamentals are labeled a, a8 andc, c8 (double and half macular pigment density, respec-tively) in Fig. 3.

At eccentricities >10 deg the macular pigment is com-pletely absent. The photopigment optical density also

Fig. 2. Transmission spectra of crystalline lens (left) and ofmacular pigment (right). Human data are plotted as solidcurves, macaque data as dashed curves.


Fig. 3. L- (light-gray), M- (black), and S- (dark-gray) cone fun-damentals with which retinal cone contrasts are evaluated.Solid and dashed curves depict human and macaque versions, re-spectively. For (g, h) and (g* , h* ) fundamentals, solid curves de-pict the Stockman et al.49 fundamentals, and dotted curves de-pict the Stockman and Sharpe50 fundamentals. Letter codingof each cone fundamental set is shown inside the panels (seeTable 1).

declines with eccentricity12,38 because of the reduction inthe axial length of the photoreceptor outer segment.2,46

Although this decline in photopigment optical density re-duces the sensitivity of photoreceptors, thus affectingmeasurements that depend on cone excitation, it does notaffect cone-contrast measurements that are normalizedfor the mean excitation (cone excitation to the backgroundstimulus). To estimate cone contrasts in the periphery(eccentricity >10 deg) we derived a new set of human andmacaque cone fundamentals by assuming complete ab-sence of the macular pigment. This was accomplished bymultiplying, wavelength by wavelength, the standard hu-man (S–P) or macaque cone fundamentals by the inverseof the transmission spectra of the corresponding standardmacular pigment:

fconeperiphery~l! 5 fcone

standard~l! 31

Tmac. pigm.standard ~l!

. (16)

The so computed cone fundamentals are labeled as d, d8in Fig. 3.

3. Deriving Cone Fundamentals for Extreme Cases ofLens Pigment DensitiesIndividual differences in the density of the lens pigmentcan also be large,10,38 with a standard deviation (for ob-servers of similar age) of ;25% of the mean lens density.10

Moreover, lens density increases with age of theobserver47,48 and may decrease at more-peripheral re-gions. To obtain a realistic range of the effects that mayresult from variations in lens pigment density due tointer-individual variations and due to age/eccentricityvariations, we derived two new sets of human andmacaque cone fundamentals, assuming a lens pigmentwith either double or half the normal optical density.This was accomplished by multiplying, wavelength bywavelength, the standard human (S–P) or macaque (de-rived in the previous subsection) cone fundamentals bythe ratio of the transmission spectra of the extreme-density lens pigment to that of the standard-densitymacular pigment:

fconeextreme~l! 5 fcone

standard~l! 3T lens

extreme~l!

T lensstandard~l!

. (17)

The so computed cone fundamentals are labeled e, e8 andf, f 8 (double and half lens pigment density, respectively)in Fig. 3.

4. Other Cone FundamentalsTo make our study more complete, we also computed reti-nal cone contrasts with two sets of cone fundamentalsthat have been proposed recently by Stockman et al.49

and by Stockman and Sharpe.50 These fundamentalshave been used to design stimuli used in some of the mostrecent physiological studies.27–29 The 2-deg and 10-degversions of these fundamentals are labeled g, h andg* , h* , respectively, in Fig. 3. Note that since these fun-damentals are normalized to unity, their relative mag-


nitudes do not encode the relative numerosities of the un-derlying cone mosaics.

B. Retinal Artifacts in Cone-Isolating Stimuli

1. Spatially Uniform StimuliSince chromatic aberration does not alter the spectralcomposition of zero-spatial-frequency image components,the retinal images of spatially uniform chromatic stimuliare fully determined by the effects of preretinal absorp-tion alone. The retinal cone excitations generated by achromatic stimulus for each of the 16 analyzed cone fun-damentals, f i, i 5 a, a8,... g* , h* , are computed as inEq. (1), and the corresponding cone contrasts are given by

Cconef i

5Econe

stim,fi2 Econe

f i

Econef i

. (18)

The retinal L-, M-, and S-cone-contrast values for spa-tially uniform L-, M-, and S-cone-isolating stimuli are dis-played graphically in Fig. 4. In this figure, retinal con-trasts visible to the targeted mosaics are shown in theleft-column panels and the residual contrasts visible tothe nontargeted mosaics are shown in the middle- andright-column panels. Within each panel, each of the 16bars represents the retinal contrast assuming a differentpreretinal absorption condition and is computed by using

the corresponding set of cone fundamentals (refer to Table1 for identification of cone fundamentals).

To quantify the degree of contamination in cone-isolating stimuli, we computed a relative-contaminationmetric for each of the two nontargeted mosaics as follows:

econef i

5Ccone

f i

uCtarg.conef i

u3 100%. (19)

A 0% contamination value indicates no contrast visible toa nontargeted cone mosaic (i.e., a completely silent sub-stitution for that cone mosaic), whereas a 100% contami-nation value indicates that the retinal stimulus is equallyvisible to both the targeted and the nontargeted cone mo-saics. The results of this analysis are tabulated in Table1.

A number of observations are notable. Since the ex-amined cone-isolating stimuli are based on the S–P conefundamentals, the retinal cone-contrast estimates com-puted by using the S–P set of fundamentals (b) have val-ues that are identical to the specified ones, with 0% con-tamination. The retinal-contrast estimates based on thederived standard macaque fundamentals (b8) also haveretinal contrasts that are relatively close to the specifiedones, with the M-cone-isolating stimulus yielding thegreatest contamination, a total of 9.7% (which arises

Fig. 4. Retinal cone-contrast estimates for spatially uniform (a) 20% L-, (b) 24% M-, and (c) 84% S-cone-isolating stimuli as measuredby the 16 cone fundamentals. Note the different scaling of the y axes for the targeted (left-column panels) versus the two nontargeted(middle- and right-column panels) cone mosaics. L-, M-, and S-cone contrasts are plotted in light gray, black, and dark gray, respec-tively.


Table 1. Retinal Cone Contrasts Visible to the Targeted Mosaics and Relative Contaminations of theNontargeted Mosaics for Spatially Uniform L-, M-, and S-Cone-Isolating Stimulia

Cone-Fundamental Characteristics 20% L-Isolating 24% M-Isolating 84% S-Isolating

Identi-fication Source Value CL(%) eM(%) eS(%) CM(%) eL(%) eS(%) CS(%) eL(%) eM(%)

a S–P15 ODMP 3 2 21.6 13.6 21.3 24.1 26.0 13.8 82.0 22.0 24.6b S–P15 ODMP 3 1 20.0 0.0 0.0 24.0 0.0 0.0 84.0 0.0 0.0c S–P15 ODMP 3 1/2 19.0 22.1 10.6 23.7 13.4 21.5 84.8 11.5 13.1d S–P15 ODMP 5 0 17.9 24.5 11.1 23.3 17.3 22.8 85.4 13.4 16.9

a8 Macaque (from S–P15) ODMP 3 2 19.6 21.1 20.4 24.9 13.3 11.0 83.5 21.4 22.6b8 Macaque (from S–P15) ODMP 3 1 18.3 23.9 10.8 24.4 17.5 22.2 85.2 10.8 11.9c8 Macaque (from S–P15) ODMP 3 1/2 17.6 25.5 11.4 24.0 19.9 23.5 85.9 12.3 14.9d8 Macaque (from S–P15) ODMP 5 0 16.7 27.2 11.9 23.4 112.5 24.7 86.5 14.2 18.3

e S–P15 ODLP 3 2 23.8 17.0 22.0 22.6 219.4 18.0 79.7 20.6 22.4e8 Macaque (from S–P15) ODLP 3 2 20.9 11.6 20.9 23.6 24.7 13.1 82.2 0.0 20.2f S–P15 ODLP 3 1/2 18.1 24.1 11.6 24.5 18.5 24.2 86.4 10.8 11.7f 8 Macaque (from S–P15) ODLP 3 1/2 17.0 27.0 12.2 24.6 113.0 25.2 86.9 11.6 13.4

g Stockman et al.49 2 deg 20.6 12.2 11.2 22.9 25.0 23.6 85.9 11.6 12.4h Stockman and Sharpe50 2 deg 20.5 11.6 11.1 23.0 24.2 23.6 85.9 11.6 12.3g* Stockman et al.49 10 deg 18.7 22.9 12.0 23.1 13.2 25.4 86.8 13.3 16.3h* Stockman and Sharpe50 10 deg 17.9 25.2 12.2 23.4 16.7 25.7 87.0 13.8 17.0

a Rows show estimates for the 16 examined sets of cone fundamentals. ODMP stands for macular-pigment optical density, and ODLP stands for lens-pigment optical density.

mainly from a 11.8% L-cone-contrast artifact), and theS-cone-isolating stimulus yielding the smallest contami-

nation (a total of 2.7%). It follows that when presentedto foveal regions of the macaque visual system, spatiallyuniform, S–P-based L- and S-cone-isolating stimuli mayachieve strong isolation of the corresponding mosaic,whereas the M-cone-isolating stimulus may additionallyactivate, in phase and to a very modest level, the L-conemosaic.

Variations in the optical density of the macular pig-ment (cone fundamentals a –d and a8–d8) result in thefollowing changes in L- and M-cone-isolating stimuli: (i)the retinal contrast available to the targeted mosaic is at-tenuated (especially in L-cone-isolating stimuli) as macu-lar pigment density decreases; (ii) the residual M-cone-contrast artifact in L-cone-isolating stimuli remainsrelatively small (<0.75% for human, <1.2% for macaque);(iii) the residual L-cone contrast in M-cone-isolatingstimuli does not increase above 1.7% (human) and 2.9%(macaque); and (iv) artifacts visible to the S-cone mosaicare negligible (<1.1%). On the other hand, the contrastavailable to the targeted mosaic in S-cone-isolatingstimuli increases as the optical density of the macularpigment decreases. The residual L- and M-cone-contrastartifacts also increase, reaching 2.9% (L) and 5.8% (M) inhumans, and 3.6% (L) and 7.2% (M) in the macaque. Itmust be noted that the retinal artifacts are maximized(across all stimuli) in the no-macular-pigment condition,i.e., for stimuli presented at eccentricities >10 deg. Onemust also note that, although the absolute magnitudes ofartifactual contrasts are strongest for the high-contrast

S-cone-isolating stimuli, it is the M-cone-isolating stimulithat result in the strongest relative contaminations(Table 1).

Variations in the optical density of the lens pigment(cone fundamentals e –f and e8–f 8) affect the retinal conecontrast of the targeted mosaics more than comparablevariations in the optical density of the macular pigment,especially for L- and S-cone-isolating stimuli. Also, therelative contaminations in L- and M-cone-isolatingstimuli induced by variations in lens density are strongerthan those induced by comparable variations in macularpigment density. Note, however, that the residual conecontrasts induced by these lens density variations do notexceed 1.7% for L-, 4.4% for M-, and 3.0% for S-cone-isolating stimuli.

Estimations of retinal cone contrast via the 2- and 10-deg Stockman et al.49 and Stockman and Sharpe50 conefundamentals ( g, h, g* , h* ) show that the artifactualcone contrasts that are visible to the nontargeted mosaicsare small for L- and M-cone-isolating stimuli and remainso even for the 10-deg versions: <0.93% (L-isolating)and <1.57% (M-isolating). The S-cone-isolating stimulihave somewhat stronger residual cone-contrast artifacts:<2.1% (2-deg) and <6.1% (10-deg).

Overall, these data can be summarized as follows. Un-der all conditions, the residual contrast visible to theS-cone mosaic for both L- and M-cone-isolating stimuli isnegligible, as is the residual M-cone contrast for L-cone-isolating stimulation. The residual L-cone contrast forM-cone-isolating stimulation is also relatively small, ex-cept at macaque retinal locations with little or no macularpigment (c8, d8), in macaque eyes with a low-optical-


density lens ( f 8), and in human eyes with a high-optical-density lens (e), in which cases, it is moderate. Finally,the artifactual L-cone and, especially, M-cone contraststhat occur during S-cone-isolating stimulation are gener-ally the strongest of all. However, this is mostly a resultof the high contrast of the S-cone-isolating stimulus used(compared with that of L- and M-cone-isolating stimuli).In terms of the sums of the absolute values of relativecontaminations, the S-cone-isolating stimulus almost al-ways results in smaller artifacts than the M-cone- (butnot the L-cone-) isolating stimulus.

2. Spatially Structured Stimuli: Sinusoidal GratingsBefore being imaged on the retina, spatiochromaticstimuli are further subjected to axial chromatic aberra-tion, which causes additional spectrally selective filteringaccording to the stimulus spatial-frequency content. Themodel spatio-spectro OTF used in the present study isshown in Fig. 5. As expected, radiant energy of zero-spatial-frequency image components is passed unattenu-ated across all wavelengths. For image components witha spatial-frequency content greater than 0 cycles per de-gree (c/deg), i.e., for spatially patterned stimuli, radiantenergy at short wavelengths is attenuated most stronglyand radiant energy near the in-focus wavelength is at-tenuated least strongly. Long-wavelength energy is alsoattenuated but to a lesser degree than short-wavelengthenergy.

In the case of extended sinusoidal gratings, the differ-ential retinal cone excitations can be estimated readily bymultiplying the stimulus differential SPD by the OTFvalue at the spatial frequency of the sinusoid, f, wave-length by wavelength, and then summing over all wave-lengths. The total cone excitation is given by the sum-mation of the mean cone excitation and the differentialcone activation,

Econestim,f i

~ f ! 5 Econef i

1 K(j

$@SPDstim~l j!

2 SPDbkgnd~l j!#OTF~ f, l j!%fconei ~l j!Dl,

(20)

Fig. 5. Surface plot of the spatio-spectro optical transfer func-tion of the eye focused at 580 nm.

and the corresponding retinal cone contrast is

Cconef i

~ f ! 5Econe

stim,f i~ f ! 2 Econe

f i

Econef i

. (21)

Figures 6(a), 6(b), and 6(c) show estimates of retinal conecontrasts as a function of spatial frequency for a 20% L-, a24% M-, and a 84% S-cone-isolating sinusoidal grating,respectively. The format of this figure is similar to thatof Fig. 4, with the results obtained from the sixteen conefundamentals displayed within the four adjacent subpan-els of each main panel. The first quadruplet of curves inthe leftmost subpanel show retinal cone-contrast esti-mates obtained by varying the optical density of macularpigment in the human eye (fundamentals a –d). Thesecond quadruplet of curves in the second-from-the-leftsubpanel show retinal cone-contrast estimates obtainedby the same variations in the optical density of macularpigment in the macaque eye (fundamentals a8–d8). Thethird quadruplet of curves in the third-from-the-left sub-panel show retinal cone-contrast estimates obtained byvarying the optical density of the lens in human (funda-mentals e –f ) and macaque (fundamentals e8–f 8) eyes.Finally, the fourth quadruplet of curves in the fourth-from-the-left subpanel show retinal cone-contrast esti-mates obtained from the 2- and 10-deg versions of theStockman et al.49 (fundamentals g –g* ) and Stockmanand Sharpe50 (fundamentals h –h* ). Retinal contrastsvisible to the targeted mosaics are shown in the left col-umn, and the residual contrasts visible to the nontargetedmosaics are shown in the middle and right columns.

A number of observations are notable. As spatial fre-quency increases from 0 c/deg to 40 c/deg (maximum spa-tial frequency tested), the retinal cone contrast visible tothe targeted mosaics decreases monotonically (except forthe small nonmonotonic changes near 10 c/deg in S-cone-isolating stimuli), as a result of chromatic aberration.Conversely, as spatial frequency increases, the retinalcone contrast visible to the nontargeted mosaics does notalways change monotonically. Instead, it may change itssign and even increase (in absolute magnitude) from itsvalue at 0 c/deg, depending on the targeted and nontar-geted cone mosaic and on the preretinal absorption condi-tion tested. L-cone-isolating stimuli are least affected bythe combined effects of chromatic aberration and varia-tions in the preretinal absorption conditions, with re-sidual artifactual cone contrasts never exceeding 1.7%(M) and 0.5% (S), and maximal contamination values(Fig. 7) of 26.3% (M) and 3.6% (S), across all conditions.M-cone-isolating stimuli result in stronger residual arti-facts, up to 4.4% (L) and 1.82% (S), and maximal contami-nation across all conditions of 77.4% (L) and 14.9% (S).S-cone-isolating stimuli have the highest residual artifac-tual contrasts, up to 3.6% (L) and 7.2% (M), and contami-nations that approach infinity beyond 7 c/deg. It mustalso be noted that for all stimuli, the relative contamina-tions remain constant up to spatial frequencies in therange of 4–10 c/deg (depending on the targeted and non-targeted cone mosaics), after which they begin increasingin absolute magnitude (Fig. 7). Also note that there ismore variation (with preretinal absorption condition) in


Fig. 6. Combined effects of preretinal absorption and axial chromatic aberration on the retinal contrasts of (a) 20% L-, (b) 24% M-, and(c) 84% S-cone-isolating sinusoidal gratings as a function of spatial frequency and as measured by the 16 cone fundamentals. Note thedifferent scaling of the y axes for the targeted (left-column panels) versus the two nontargeted (middle- and right-column panels) conemosaics. Figure format is similar to that of Fig. 4.

the relative contamination associated with M-cone con-trast than there is with L- and S-cone contrasts.

A final observation concerns the polarity of the residualcontrast artifacts visible to the nontargeted mosaics. InL- and M-cone-isolating stimuli, the artifactual cone con-trasts reverse their polarity as spatial frequency is in-creased, and for low spatial frequencies they remain inantiphase across almost all preretinal absorption condi-tions. Therefore, in neurons with nonopponent cone in-puts, the residual contrast artifacts are likely to evoke an-tagonistic contributions that would tend to cancel eachother. On the other hand, the residual L- and M-cone-contrast artifacts in S-cone-isolating stimuli are almostalways in phase with each other, across the spatial-frequency range and across the different preretinal ab-sorption conditions. This creates an additional problemfor S-cone-isolating stimuli, since the consistentlymatched polarities of the residual L- and M-cone-contrastartifacts will always evoke synergistic contributions,which would reinforce each other, in neurons with nonop-ponent cone inputs. This becomes especially problematicin the case of reduced or no macular pigment. In theseconditions, the S-cone contrast is in phase with the re-sidual L- and M-cone-contrast artifacts, and the summedL1M-cone-contrast artifact can be as high as 8.8% in hu-mans and 10.7% in macaques at 0 c/deg. This result in-dicates that during stimulation with low- and moderate-

spatial-frequency S-cone-isolating gratings, peripheral(.10 deg of eccentricity) neurons would sense a strongcombined L1M-cone-contrast artifact whose polaritywould be the same as that of the S-cone contrast. Thisartifact would be especially problematic if sensed by neu-rons in the magnocellular pathway that have a strongcontrast gain control, and it could potentially lead to theincorrect interpretation that a neuron with no real S-coneinput receives significant S-cone inputs that are in phasewith its L- and M-cone inputs. A method to guardagainst this incorrect interpretation is presented below.

3. Spatially Structured Stimuli: Rectangular PulsesRectangular-pulse stimuli are commonly used in physi-ological studies, especially in spatiochromatic receptive-field mapping studies.17,18,20 The spatial distribution ofretinal L-, M-, and S-cone contrasts evoked by rectangu-lar stimuli can be computed by first multiplying theirspatial-frequency spectra at each frequency, f, with thecorresponding Ccone

f i( f ) (see previous subsection) and then

by inverse Fourier transforming the resulting spatial-frequency spectra.

Examples of the resulting spatial distribution of retinalcone contrasts for a 20% L-, 24% M-, and an 84% S-coneisolating pulse with a width of 0.3 deg are shown in Figs.8(a), 8(b), and 8(c), respectively. The format of this figure


is similar to the format of Fig. 6, with results obtainedfrom the sixteen cone fundamentals grouped every fourwithin the four adjacent subpanels of a main panel. Notethat the width of the retinal pulse visible to the targetedmosaic is closer to the intended width for L- and M- thanfor S-cone-isolated pulses. This is expected from theasymmetric shape of the eye’s spatio-spectro OTF (Fig. 5),which results in increased spread and attenuation of reti-nal S-cone contrast, especially for narrow stimuli. Alsonote that the spatial profiles and the polarities of the re-sidual cone-contrast artifacts visible to the nontargetedmosaics vary with the preretinal condition tested andwith the targeted and nontargeted cone mosaics.

Figure 9 summarizes the dependence of retinal L-, M-,and S-cone contrasts on the width of rectangular pulsestimuli for the 16 preretinal absorption conditions. Here,the average retinal cone contrasts that fall within therectangular pulse and in an equal area surrounding thepulse are plotted separately against pulse width. As ex-pected, within the region of intended pulse width, theretinal contrast visible to the targeted cone mosaic ap-proaches the intended contrast as pulse width increasesand starts to asymptote beyond 1 or 2 deg. However, re-sidual contrast artifacts visible to the nontargeted conemosaics also increase as pulse width increases and as-ymptote together with the contrast visible to the targeted

Fig. 7. Contaminations in L-, M-, and S-cone-isolating sinu-soidal gratings due to the combined effects of preretinal absorp-tion and axial chromatic aberration as a function of spatial fre-quency.

cone mosaic. This occurs because as pulse width in-creases, chromatic aberration decreases and the artifactsgenerated due to non-S–P-like preretinal absorption con-ditions are imaged unattenuated on the retina. On theother hand, for small pulse widths, chromatic aberrationattenuates the imaged cone contrast, including artifactsgenerated as a result of non-S–P-like preretinal absorp-tion conditions. It must be noted that the targeted andnontargeted cone contrasts contained within the intendedpulse width tend to increase together for pulse widthsabove 0.1 deg, yielding rather constant contaminationvalues (Fig. 10). Outside the region of intended pulsewidth, contrast artifacts visible to the targeted and non-targeted cone mosaics disappear as pulse width is in-creased beyond 1 deg. Note that for L- and M-cone-isolating pulses, the reduction in retinal contrasts outsidethe region of intended width is more or less monotonic,whereas for S-cone-isolating pulses, retinal contrasts ini-tially increase up to pulse widths of ;0.07–0.09 deg andthen gradually decline toward zero as pulse width in-creases further. This occurs because for very smallpulses, axial chromatic aberration attenuates all retinalS-cone contrast within or without the rectangular pulse.As pulse width is increased, retinal S-cone contrast is in-creased, but it is strongly defocused, resulting in extendedspatial spreading outside the intended boundaries. Fur-ther increase in pulse width decreases the defocusing ofretinal S-cone contrast, thus reducing leakage outside theintended boundaries.

C. Applications to Color Physiology: ImprovingSpatial Receptive-Field-Map Estimates of Cone Inputs toMacaque Monkey NeuronsQuantitative methods of spatio-temporo-chromaticreceptive-field (STCRF) mapping use the reverse-correlation technique51 combined with binary, white-noisemodulation of the L-, M-, or S-cone contrast of spatiallycontiguous ensembles of rectangular pulses.17,18 Thesemethods estimate a neuron’s linear STCRF by first com-puting the spike-triggered average contrast (STAC) thatprecedes spiking by some time delay, t, as follows:

STAC(cone, x, t) 51

N (i51

N

Ccone~x, ti 2 t!, (22)

where N is the total number of spikes fired in response tothe stimulus Ccone(x, t) and ti is the time of the ith spike.The STCRF is obtained from the STAC after normaliza-tion with respect to stimulus contrast power:

STCRF(cone, x, t) 5STAC(cone, x, t)

Ccone2

. (23)

Figure 11(a) shows slices at the time of peak response,tmax , through the STCRF for a magnocellular lateral gen-iculate nucleus (LGN) neuron recorded at an eccentricityof 15 deg. Because the STCRF is computed with respectto the stimulus at the corneal plane, it may be corruptedby responses from the nontargeted mosaics, a result of theresidual cone-contrast artifacts that we quantified in theprevious sections. Here we describe a method to compen-sate for these effects.


Fig. 8. Spatial distributions of retinal L-, M-, and S-cone contrasts for (a) a 20% L-, (b) a 24% M-, and (c) a 84% S-cone-isolating rect-angular pulse with a width of 18 arc min (0.3 deg). The specified pulse is depicted by dotted curves in the left column. The width ofthe specified pulse is shown as dotted lines in the middle- and right-column plots. Note the different scaling of the y axes for the tar-geted (left-column panels) versus the two nontargeted (middle- and right-column panels) cone mosaics.

The spatial profiles of the corneal L-, M-, and S-cone-isolating stimuli that are most likely to evoke a responsefrom this neuron tmax ms after their presentation can becomputed as

CL~x ! 5STAC~L, x, tmax!

max@STAC~L, x, tmax!#3 20%, (24)

CM~x ! 5STAC~M, x, tmax!

max@STAC~M, x, tmax!#3 24%, (25)

CS~x ! 5STAC~S, x, tmax!

max@STAC~S, x, tmax!#3 84%, (26)

and are shown in Fig. 11(b). Following the procedure de-scribed in Subsection 3.B.3, we can compute for each ofthese optimal corneal profiles the resulting spatial pro-files of retinal L-, M-, S-cone contrasts, [e.g., CL

L,f i(x),

CML,f i

(x), CSL,f i

(x) for the CL(x) corneal profile] that mostlikely preceded this neuron’s firing by tmax ms. Sincethis neuron was recorded at an eccentricity .10 deg, webased our calculations on the derived macaque no-macular-pigment fundamentals (f i 5 f d8). The resultsof this computation are shown in Fig. 11(c). Note that forM- and S-cone-isolating stimulation there were signifi-cant residual retinal contrast artifacts that may have con-tributed a significant component to the neuron’s response.

To improve our estimates of the spatiochromatic RF, weneed to (i) estimate the artifactual response contributionfrom all cone mosaics separately for L-, M-, and S-cone-isolating stimulation and (ii) compensate for their effects.

1. Neurons with Linear Contrast-ResponseRelationshipsFor neurons with a linear contrast-response relationship,an estimate of the response contributions from the differ-ent cone mosaics can be obtained as follows:

RLstim 5 (

xCL

stim,f i~x !STCRF~L, x, tmax!, (27)

RMstim 5 (

xCM

stim,f i~x !STCRF~M, x, tmax!, (28)

RSstim 5 (

xCS

stim,f i~x !STCRF~S, x, tmax!. (29)

Note that these response contributions depend on themeasured STCRF, which is based on contaminated re-sponses, and therefore are not very accurate. However,they can be used as a starting point to generate a first cor-rection in the RF, STCRF8, as follows:


STCRF8~L, x, tmax! 5 STCRF~L, x, tmax!

3 S 1 2RM

L 1 RSL

uRLLu 1 uRM

L u 1 uRSLuD ,

(30)

STCRF8~M, x, tmax! 5 STCRF~M, x, tmax!

3 S 1 2RL

M 1 RSM

uRLMu 1 uRM

Mu 1 uRSMu

D ,

(31)

Fig. 9. Combined effects of preretinal absorption and axial chromatic aberration on the retinal contrasts of (a) 20% L-, (b) 24% M-, and(c) 84% S-cone-isolating rectangular pulses as a function of pulse width and as measured by the 16 cone fundamentals. Contrastswithin and outside the spatial extent of the rectangular pulse are plotted separately. Note the different scaling of the y axes for thetargeted (left-column panels) versus the two nontargeted (middle- and right-column panels) cone mosaics.


STCRF8~S, x, tmax! 5 STCRF~S, x, tmax!

3 S 1 2RL

S 1 RMS

uRLSu 1 uRM

S u 1 uRSSuD .

(32)

Now STCRF8 can be used to compute an improved esti-mate of the response contributions from the different conemosaics as follows:

RLstim 5 (

xCL

stim,f i~x !STCRF8~L, x, tmax!, (33)

RMstim 5 (

xCM

stim,f i~x !STCRF8~M, x, tmax!, (34)

RSstim 5 (

xCS

stim,f i~x !STCRF8~S, x, tmax!. (35)

By plugging these response contributions into Eqs. (30)–(32) we get a second-order improvement in the STCRF.

Figure 12(a) illustrates how the estimates of responsecontributions from the different cone mosaics change dur-ing the first three iterations. The uncorrected spatio-chromatic RF is shown with the first two corrected RF es-timates in Fig. 12(b). Note that the method

Fig. 10. Contaminations within L-, M-, and S-cone-isolatingrectangular pulses due to the combined effects of preretinal ab-sorption and axial chromatic aberration as a function of pulsewidth.

progressively attenuates the S-cone RF to discount thecontaminating response component that occurs duringS-cone stimulation as a result of this neuron’s L- andM-cone sensitivity to the residual L- and M-cone con-trasts. The convergence of this corrective method isshown in Fig. 12(c), which plots the relative L-, M-, andS-cone weight estimates as a function of iteration number.The S-cone relative weight is 10.5% before any correctionsand rapidly (within 4 or 5 iterations) converges to a valueof 6.0%.

2. Neurons with Nonlinear Contrast-ResponseRelationshipsThis iterative procedure can be easily modified for neu-rons with nonlinear contrast-response relationships. As-suming that this relationship can be described by an ana-lytical relationship, such as N(c) 5 kc/(c 1 c50) , we cansimply substitute N(Ccone

stim,f i(x)) in the place of Ccone

stim,f i(x)

in Eqs. (27)–(29) and (33)–(35). Figure 13 shows the re-sults for the case where c50 5 15%. The neuron’s L-, M-,and S-cone contrast-response functions are shown in Fig.13(a). Figure 13(b) shows how the estimates of response

Fig. 11. Estimating artifacts in spatiochromatic RF mapping.(a) Slices at time of peak magnitude through the spatio-temporo-chromatic RF of a magnocellular LGN neuron as measured withreverse correlation and with no corrections for the combined ef-fects of preretinal absorption and axial chromatic aberration.(b) Most probable spatial distribution of corneal cone contrasts.(c) Most probable spatial distribution of retinal L- (light-gray),M- (black), and S- (dark-gray) cone contrasts computed with useof the macaque no-macular-pigment cone fundamentals.


contributions from different cone mosaics change duringthe first three iterations. Note that relative to theS-cone-mosaic response contribution, the contributionsfrom the L- and the M-cone mosaics are stronger thanthose in the linear contrast-response situation, becausethe retinal contrast of the S-cone isolating stimulus usedis well into the neuron’s saturating regime. Therefore inthis case the uncorrected S-cone RF estimate is exagger-ated more than if the neuron had a linear contrast-response relationship. The uncorrected spatio-chromaticRF and the first two improved RF estimates are shown inFig. 13(c). Note that the S-cone RF is attenuated morethan in the linear contrast-response relationship case.The convergence of this corrective method is illustrated inFig. 13(d), which plots the relative L-, M-, and S-coneweight estimates as a function of iteration number. TheS-cone relative weight is 10.5% before any corrections andrapidly converges to a value of 4.8%.

It is interesting to see what semisaturation valuewould be necessary for these high-contrast S-cone-isolating stimuli to generate measurable S-cone RFs inneurons with no S-cone inputs. Our simulations indicatethat in such a neuron, the corrected relative S-cone

Fig. 12. Demonstration of the iterative procedure for improvingspatiochromatic RF estimates: neuron with a linear contrast-response function. (a) The first three estimates of contributionsoriginating in the L-, M-, and S-cone mosaics in response to themost probable L-, M-, and S-cone isolating stimulus. (b) Uncor-rected and first two corrected estimates of spatio-chromatic RF.(c) Convergence of the iterative corrective procedure demon-strated by plotting the relative L-, M-, and S-cone weights as afunction of iteration number.

weight drops to 3% for c50 5 5%, and to 1% for c505 1.4%. Note that values of c50 less than 5% are ratherrare for LGN magnocellular neurons,52,53 although theyare not so rare for cortical neurons (personal observa-tions). Therefore one can safely conclude that this mag-nocellular neuron has a valid S-cone RF and that its rela-tive S-cone weight is ;6.0% (c50 for this neuron was 15%).

D. Retinal Artifacts in Isoluminant StimuliThe shape of the luminous-efficiency function, Vl , de-pends not only on the preretinal absorption conditions

Fig. 13. Demonstration of the iterative procedure for improvingspatiochromatic RF estimate: neuron with a nonlinearcontrast-response function. (a) Contrast-response functions.(b) The first three estimates of contributions originating in theL-, M-, and S-cone mosaics in response to the most probable L-,M-, and S-cone-isolating stimulus. (c) Uncorrected and first twocorrected estimates of spatiochromatic RF. (d) Convergence ofthe iterative corrective procedure demonstrated by plotting therelative L-, M-, and S-cone weights as a function of iterationnumber.


Fig. 14. Luminance contrast artifacts in a nominally isoluminant color plane (top) are plotted as a function of chromatic angle, Q, forthe 16 examined Vl functions. Within each panel, data for 0, 0.08, 0.31, 0.86, 1.88, 4.23, 9.1, and 19.6 c/deg are shifted along the verticalaxis (bottom to top) by one vertical tick mark (3% luminance contrast). Artifact peaks (shown in circles) are connected by dotted lines.

and eccentricity (as cone fundamentals do) but also onother factors such as the chromatic adaptation point, spa-tial and temporal factors, the ratio of L to M cones, andeven the measurement criterion. Since Vl is dependenton the experimental paradigm and does not generalizeacross different conditions of chromatic adaptation, it is oflimited applicability. In the following simulations, wewill assume a constant and neutral (grayish) chromaticadaptation point and that Vl

f iis equal to a fixed weighted

sum of fLi and fM

i , the 16 L- and M-cone fundamentalsanalyzed. We will also examine the effects of variations

in the weighting of the L- and M-cone fundamentals tosimulate the effects of varying numbers of L and M cones,but we will ignore changes that would be caused by varia-tions in the chromatic adaptation point. On the basis ofthese alterations in the shape of Vl , we will quantify theretinal luminance artifacts in S–P-based nominally isolu-minant stimuli.

The nominally isoluminant color plane on whichstimuli are specified is shown in the top panel of Fig. 14.In our computations we consider only those stimuli alongthe ellipse whose major axes have the following differen-


tial excitation vectors (DELM , DES , DE lum)T : (3.20,0.0, 0.0)T and (0.0, 0.53, 0.0)T. The lengths of these vec-tors are the maximum possible for the chosen backgroundand CRT display. To ease representation, stimuli arespecified by their chromatic angle, Q. Within each of the16 panels in Fig. 14, the luminance contrast is plotted asa function of stimulus chromatic angle, from 0 to 360 deg,as estimated under the 16 different Vl

f ifunctions. Eight

different curves are shown corresponding to sinusoidalgrating stimuli with spatial frequency of (bottom to top)0.0, 0.08, 0.31, 0.86, 1.88, 4.23, 9.1, and 19.6 c/deg. Foreasier visualization, these curves are shifted along thevertical axis by one tick interval. Each vertical tickmarks the origin for each curve, and the separation be-tween adjacent ticks is 3% luminance contrast.

Several observations are notable. When the standardhuman luminosity function (Vl

b 5 fLb 1 fM

b ) is used tomeasure luminance, the luminance contrast remains zeroacross all chromatic angles for zero spatial frequency, asexpected. As spatial frequency is increased, a smallluminance contrast is generated, due purely to axialchromatic aberration. This contrast reaches its peakvalue (1.96%) at a spatial frequency of 10.7 c/deg and for

Q 5 290 deg. Note that the angle of peak contrast is notconstant but changes gradually as spatial frequencyincreases. When the Vl

b8 (macaque standard pigmenta-tion) is used to measure luminance, artifacts remain con-stant (;1.7%, attained at a rather constant chromaticangle, 145 deg) from 0 up to ;1 c/deg. As spatial fre-quency is increased beyond 1 c/deg, the luminance con-trast peak gradually shifts toward chromatic angles inthe range of 230–320 deg, with small fluctuations in mag-nitude.

When the Vla,a8 luminosity functions (double-density

macular pigment condition) are used to measure lumi-nance, artifacts are somewhat stronger, ;2.6% (human)and ;1.6% (macaque) and are remarkably stable in bothmagnitude and peak chromatic angle (Q 5 295 deg, hu-man; Q5250 deg, macaque) up to ;2 c/deg, after whichthe artifacts begin to decline. Compared with thedouble-density macular pigment condition, artifacts un-der the half-density macular pigment condition havesomewhat smaller magnitude in humans (;1.85%) andsomewhat larger magnitude in macaques (;3.3%) andpeak at chromatic angles 180 deg. away. Luminance ar-tifacts are maximized in the no-macular-pigment condi-

Fig. 15. Magnitude of peak luminance-contrast artifacts as a function of stimulus spatial frequency and L:M-cone ratio (see text), forthe 16 examined Vl functions. Solid black curves correspond to a Vl with a 2:1 L:M-cone ratio except for fundamentals g, h, g* , h* ,which have a 1.5:1 L:M-cone ratio. Dotted dark-gray curves correspond to a Vl with an L:M-cone ratio of 0.76:1, and dotted light-graycurves correspond to a Vl with an L:M-cone ratio of 3.62:1.


tion, 4.2% (humans) and 5.4% (macaques). Significantluminance contrast artifacts are also seen in the humanhalf-density lens pigment condition (3.4%) and in themacaque double-density lens pigment condition (3.0%).

Finally, when the 2-deg Stockman and Sharpe50 lumi-nosity function (Vl

g,h 5 1.5fLg,h 1 fM

g,h) is used to mea-sure luminance artifacts, peak contrasts are small,;1.7%. For the 10-deg version of the Stockman andSharpe50 luminosity function (Vl

g* ,h* 5 1.5fLg* ,h*

1 fMg* ,h* ), artifacts are large, ;4.0–4.6%. The

luminance-contrast peak magnitudes and the correspond-ing chromatic angles for the 10-deg Stockman et al.49 andStockman and Sharpe50 luminosity functions are on a parwith those computed from the derived human andmacaque, no-macular-pigment cone-fundamental-basedluminosity functions (Vl

d , Vld8).

The dependence of luminance-contrast magnitude onstimulus spatial frequency is summarized in Fig. 15 (solidcurves). Note that for Vl

d,d8 and Vlg* ,h* , artifacts in the

nominally isoluminant color plane are strongest at thelowest spatial frequencies, because high-spatial-frequency contrast is attenuated by axial chromatic aber-ration. It must be noted, however, that this is not alwaysthe case: For Vl

b,c,a8,b8,e8,f, g,h-based data, axial chromaticaberration has the opposite effect, causing artifacts topeak at high spatial frequencies.

E. Comparison of Luminance Artifacts Due toNonstandard Preretinal Absorption with Artifacts Dueto Varying L:M-Cone RatiosA question arises as to the significance of these luminancecontrast artifacts compared with those resulting from re-alistic departures from the 2:1 L:M-cone ratio that is as-sumed in the standard calculation of Vl . Such depar-tures are commonly seen among individuals and can beverified by a multitude of methods,54,55 including directoptical viewing of the cone mosaics.33 This informationis particularly relevant for neurophysiology experiments,since there is recent evidence by Dobkins et al.56 that inthe macaque visual system the L:M cone ratio may becloser to 1:1 than to 2:1. (Note, however, that the stimuliused by Dobkins et al. were presented against a yellowbackground, and therefore, their results may not applyduring modulation around chromatically neutral back-grounds, since Vl depends on chromatic adaptation.) Tocompare luminance artifacts that arise from variations inpreretinal pigmentation with those that arise from varia-tions in the L:M-cone ratio, we computed luminosity func-tions based on the same 16 L- and M-cone fundamentalsbut with two different L:M-cone weights: 2.1911.43(high) and 2.1921.43 (low). These values are the mean6 one standard deviation of the L:M-cone ratios for 38normal subjects in a recent report.55

The spatial-frequency dependence of the luminance-contrast artifacts estimated by these luminosity functionsis plotted as light-gray (high L:M ratio) and dark-gray(low L:M ratio) dotted lines in Fig. 15. Generally, the lowL:M-cone-ratio condition produces greater luminance con-trast artifacts than both the high and the standard L:M-cone-ratio conditions. However, note that the luminanceartifacts induced by these variations in the L:M-cone ratio

and those introduced by variations in the macular pig-ment density are of similar magnitudes. For 0 c/deg andfor Vl

b , a change in the L:M-cone ratio from normal to lowresults in a luminance contrast of 4.7% (humans) and5.9% (macaque), whereas a change in the macular pig-ment density from normal to zero with a 2:1 L:M-cone ra-tio results in a luminance contrast of 4.2% (humans) and5.4% (macaques). Variations in lens density from normalto double or half-density produce somewhat smaller arti-facts (3.4% or 1.8% in humans, 0.8% or 3.0% inmacaques). The human-to-macaque species change (2-deg fundamentals) results in the least amount ofluminance-contrast artifact (0.0% for Vb to 1.7% for Vb8 at0 c/deg). Finally, it must be noted that in most cases, de-viations from the 2:1 L:M-cone ratio compound the arti-facts generated by variations in preretinal absorptionconditions. For none of the conditions tested however,did the luminance-contrast artifact exceed 7.4% (humans)or 8.6% (macaques).

4. CONCLUSIONSAlthough some of the most recent physiological studieshave designed spatiochromatic stimuli based on theStockman et al.49 fundamentals (fMRI study of Wade andWandell29 and electrophysiological study of Chatterjeeand Callaway28) or the Stockman and Sharpe50 funda-mentals (electrophysiological study of Johnson et al.27),the vast majority of physiological studies have employedSmith–Pokorny15-based stimuli to examine the chromaticproperties of neurons in the LGN16,17,19,21 and cortical ar-eas V1,18,20,22,23,57 V2,24 V3,25 and MT.26 However, evenin the few studies that have employed the more recentlyderived cone fundamentals, stimuli have not been de-signed to account for the differences in preretinal absorp-tion conditions between humans and macaques, differ-ences in pigmentation among individual macaques, ordifferences in pigmentation as a function of a neuron’s ec-centricity. The reason for this is that customizing conefundamentals for each animal and for the range of eccen-tricities that may be examined would be very difficultwithout a trained behaving macaque monkey.

The present paper has two goals. The first goal is toexamine the magnitude and nature of retinal artifacts inSmith–Pokorny15-based spatiochromatic stimuli thatarise from a wide range of preretinal absorption condi-tions and axial chromatic aberration. This informationis crucial in providing confidence levels to physiologicalfindings, especially to measurements of small, but possi-bly significant, quantities such as the 10% S-cone inputrecently reported by Chatterjee and Callaway to magno-cellular neurons,28 by Cottaris et al. to V1 directionalneurons,57 and by Seidemann et al. to MT neurons.58

Our results demonstrate clearly that it is imperative totake into account the decline in macular pigment densitywith eccentricity. The second goal of this study is to sug-gest ways of discounting the effects of such artifacts inmeasurements obtained with S–P-based cone-isolatingstimuli. One method that compensates for the artifac-tual retinal cone contrasts in such cone-isolating stimuliis proposed and demonstrated for data from a magnocel-lular LGN neuron.


Since designing individualized cone fundamentals, es-pecially for nonfoveal regions of the macaque visual sys-tem, is technically difficult, investigators have developedalternative ways for providing controls. In the Chatter-jee and Callaway28 study for example, in which the originof an S-cone input to magnocellular neurons was investi-gated, a yellow adaptation light technique was used to se-lectively attenuate responses from L and M cones whileleaving S-cone responses largely unaffected. Our resultsand correction method can be used to further improvesuch controls by providing estimates of the reduction inthe S-cone response expected by selective adaptation of Land M cones. Also, since selective chromatic adaptationlikely involves nonlinear mechanisms, it may be more ap-propriate to use the method presented here for the quan-tification of a neuron’s true cone inputs and use selectiveadaptation only to confirm or dismiss the hypothesis thata neuron receives an S-cone input. Moreover, our resultsand correction technique can be used to correct spatio-chromatic receptive-field data that have been collectedwithout selective adaptation controls. The only require-ments for this corrective method are knowledge of a neu-ron’s contrast-response relationship, which can be easilyobtained before or after the receptive-field data are col-lected, and knowledge of the preretinal absorption char-acteristics at the neuron’s eccentricity. The latter re-quirement cannot be known for each monkey, but foreccentricities greater than 10 deg, one can safely assumea complete lack of macular pigment. In such a case, theonly unknown variable is the optical density of themacaque lens, which can be calculated as done by Bayloret al.40 At the very least however, one could assume a re-alistic range for both the macular pigment and the lenspigment and obtain a confidence range for the validity ofa measured cone-isolating response.

Our results also show that two additional correctionsmay be necessary when one is mapping the spatiochro-matic receptive-field structure of a neuron’s cone inputs,especially its S-cone inputs. First, one must not use thestimulus corneal contrast power, CS

2 , to normalize theSTAC in Eq. (23) but instead normalize with respect toCSCS

fi, where CS

fiis the stimulus retinal S-cone contrast

computed by using the appropriate (f i) fundamental.Since the retinal S-cone contrast is greatly attenuated(attenuation is greater than 50% for pulse widths of 0.1deg), the true S-cone input may be greater than the oneestimated with respect to the corneal contrast power (butthis also depends on the neuron’s contrast-response func-tion). The second correction that may be needed is a spa-tial deconvolution of the measured receptive field to com-pensate for the retinal blurring of the S-cone stimulus.Both of these additional corrections become increasinglyimportant as the receptive-field-mapping stimulus getssmaller.

Another concern with most physiological studies of spa-tiochromatic receptive-field structure is that usually,S-cone stimulation is performed at high contrasts (>60%).One rare exception is the study of Johnson et al.,27 inwhich S-cone-isolating stimulation was performed at 24%contrast. We have shown that the high-contrast S-cone-isolating stimuli generate strong retinal L- and M-cone-contrast artifacts that are in phase with the retinal

S-cone contrast. The magnitudes of the synergistic L-and M-cone-contrast artifacts are well into the sensitivityrange of most V1 neurons and would drive neurons withmagnocellular inputs very effectively. This contamina-tion in S-cone-isolating stimulation would be further am-plified in neurons at eccentricities >10 deg because of thecombined effects of three factors: (i) artifacts are stron-gest in the no-macular-pigment condition, (ii) peripheralneurons have higher contrast gain control than more cen-trally located neurons,53 and (iii) peripheral neurons aretuned to low spatial frequencies, and artifacts due to non-standard preretinal absorption are maximized at low spa-tial frequencies. (At high spatial frequencies, axial chro-matic aberration eliminates all retinal contrasts; note,however, that for conditions of standard preretinal ab-sorption, artifacts are maximized at high spatial frequen-cies because of axial chromatic aberration.) One way toreduce the magnitude of artifacts when one is mappingthe S-cone inputs in neurons at eccentricities >10 deg isto employ S-cone-isolating stimuli with a contrast similarto or somewhat higher than the contrast of L- andM-cone-isolating stimuli, since we have shown that therelative contamination of S-cone-isolating stimuli is small(generally smaller than the relative contamination inM-cone-isolating stimuli).

The results of this study can also be utilized in the de-sign of cone-mixed corneal stimuli, which become coneisolating on the retina. For example, a 1CS 2 r(CL1 CM), r.0 Smith–Pokorny-based corneal stimuluswould become an S-cone-isolating stimulus on the retina.The exact value of r depends on the conditions of prereti-nal absorption and on the spatial characteristics of thestimulus. This information can be obtained from thedata shown in Figs. 6 and 9.

In addition, our results demonstrate that the 10-degStockman et al.49 and Stockman and Sharpe50 fundamen-tals provide measurements that are close to the derivedhuman and macaque fundamentals in the no-macular-pigment condition. Therefore, they would be acceptablealternatives for stimulating macaque neurons at eccen-tricities >10 deg, if one could not derive individualizedperipheral fundamentals for the macaque visual system.

Finally, we showed that for Smith–Pokorny15-basedisoluminant stimulation, significant luminance-contrastartifacts are introduced in the nominally isoluminantplane as a result of changes in the preretinal absorptionconditions with eccentricity (macular pigment) and thatthe magnitudes of these artifacts are as strong as thoseintroduced by expected deviations from the 2:1 L:M-coneratio. The luminance-contrast artifacts that are intro-duced by expected variations in the L:M-cone ratios aresomewhat stronger than artifacts due to variations in thelens density or due to differences in the cone fundamen-tals between humans and macaques. Our simulationsalso indicate that, in addition to luminance artifacts,small chromatic rotations are introduced in the isolumi-nant plane as a result of nonstandard preretinal absorp-tion and axial chromatic aberration. Because both theluminance and the chromatic artifacts shift with prereti-nal absorption conditions and stimulus spatial frequency,and since nominal human isoluminance may be differentfrom macaque isoluminance,56 stimulation of macaque


neurons with patterns that are nominally isoluminant forhumans is prone to more uncertainties than stimulationwith cone-isolating patterns.

ACKNOWLEDGMENTSI thank Sylvia D. Elfar, Russell L. De Valois, and GeneSwitkes for their suggestions and their support duringthe various phases of this work. I am grateful to DavidBrainard, two reviewers, and Chien-Chung Chen for theirvery helpful comments.

N. Cottaris can be reached by e-mail at [email protected].

Present address, Departments of Ophthalmology andPhysiology, Ligon Research Center of Vision, Kresge EyeInstitute, Wayne State University School of Medicine, De-troit, Michigan 48201.

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