JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Distribution of Cone Orientations as an Explanation of the Stiles-Crawford Effect*tARAN SAFIRI AND LYON HYAMS§

Mount Sinai School of Medicine, Fifth Avenue at 100th Street, New York, New York 10029(Received 16 August 1968)

The retinal directional effect of several subjects was measured. Flicker brightness matches were madeusing monochromatic lights. The experiment was performed in two parts. The first measured the Stiles-Crawford effect across only the central 6 mm of the pupil. Objective statistical methods were used to derivethe best-fitting parabolas by least-squares criteria. Variances of the data were calculated from the replicationsand indices of goodness of fit were derived. On these grounds, the parabola was found to be unacceptable.The second part of the experiment included replicate matches made at points across the entire width of thedilated pupil. Similar methods were again used to derive the best-fitting gaussian and parabolic functions.The gaussian proved to be statistically acceptable as a description of the change of brightness with positionof pupil entry while the parabola did not. The consequences of accepting the gaussian function are far-reaching. It is argued that the shape of the Stiles-Crawford-effect curve as measured across the pupil is theresult of the normal distribution of cone angulations, each cone having inherent directional sensitivity. Itfollows from this that the directional sensitivity of individual cones cannot be as broad as the observedStiles-Crawford-effect curve. It is probable that the individual cone will accept energy incident upon itwithin only a very small angle from its long axis.INDEX HEADINGS: Vision; Photometry.

In 1933, Stiles and Crawford noted the remarkabledecrease of brightness of light as its position of entryinto the pupil of the eye became more peripheral. Theirfindings were soon corroborated; and the effect, whichbears the name of its co-discoverers, was shown to bedue, not to absorption in the transparent ocular media,but rather to an inherent directional sensitivity of theretina.

Many problems concerning the Stiles-Crawfordeffect (SCE) remain unsolved today in spite of con-siderable study. There is no single preeminent theory ofthe origin of the SCE. Nor is there agreement concern-ing the mathematical function that best fits the experi-mental data. Several equations have been suggested.The parabola has been offered1 as an empirical de-scription of the central portion of the curve of bright-ness as a function of position of pupil entry. To dealwith the full extent of the observed curve, a fourth-order polynomial, 2 a gaussian, 3 and a trigonometricfunction4 have been suggested.

There are two main reasons for this diversity of view-points: the difficulties of the physical experiment, andthe lack of suitable mathematical analysis. While it iscomparatively easy to demonstrate that the SCE is

* Paper presented at Washington meeting of Optical SocietyMarch, 1968 [J. Opt. Soc. Am. 58, 727A (1968)].

t This research was supported by Grant No. NB-05895,National Institutes of Health, United States Public HealthService, National Institute of Neurological Diseases and Blindness.

$ Supported in part by Special Fellowship BT-869, NationalInstitutes of Health, United States Public Health Service, Na-tional Institute of Neurological Diseases and Blindness.

§ Present address: Dept. of Statistics and Community Medicine,Rutgers Medical School, Rutgers, The State University, NewBrunswick, New Jersey 08903.

'W. S. Stiles, Proc. Roy. Soc. (London) B123, 90 (1937).2 P. Moon and D. E. Spencer, J. Opt. Soc. Am. 34, 319 (1944).3 B. H. Crawford, Proc. Roy. Soc. (London) B124, 81 (1937).1 J. M. Enoch, Suminated Response of the Retina to Light Entering

Different Parts of the Pupil, doctoral dissertation, Ohio StateUniv. (1956). University Microfilms 18, 789, Ann Arbor, Michigan.

present and to measure it roughly, it is very difficultto measure it with precision and reliability. Even withgood data, previous studies have not used quantitativetechniques of determining how well their proposedmathematical functions fitted the observed data points.

A powerful tool for the production of new theory andexperiments results from knowing the mathematicalfunction that describes the data in accordance withquantitative criteria. We set up statistical criteria forthe acceptance or rejection of hypothesized mathemati-cal relations; and we designed and performed experi-ments that gave us sufficient information to test ourhypotheses. The results of the tests show that thegaussian function acceptably describes the SCE. Onthe basis of this evidence, we advance a new theory thatsignificantly differs from current concepts of retinalcone function.

I. PREVIOUS MATHEMATICAL AND PHYSICALMODELS OF THE SCE

Stiles and Crawford' made the original observationsusing white light. They proposed the possibility that theloss of brightness was due to absorption in pigmentepithelial cells intruding between the outer segments ofthe photoreceptors. Wright and Nelson6 soon confirmedthe effect, implied that it might have different magni-tudes at different wavelengths, and, in a few words,proposed several ideas of far-reaching importance. Theysuggested that: (a) the retinal cone itself is directionallysensitive, requiring that the light enter it directly, (b)the irregular orientation of the cones might then ex-plain the curves, (c) the light might be totally reflectedinside the walls of the cone.

I W. S. Stiles and B. H. Crawford, Proc. Roy. Soc. (London)B112, 428 (1933).

6 W. D. Wright and J. H. Nelson, Proc. Phys. Soc. (London) 48,401 (1936).

757

VOLUME 59, NUMBER 6 JUNE 1969

ARAN SAFIR AND LYON HYAMVIS

Crawford,' in comparing the SCE obtained fromphotometric matches with that obtained bv brightness-threshold measurements, adopted suggestion (b) fromWright and Nelson. He plotted q, the relative luminousefficiency, against the position of pupil entry. Heassigned values to the coefficients in the gaussian ex-pression and suggested that the orientation of cone axesfollows some law of random distribution about a mean.His characterization of the curves as gaussian was notadopted by other investigators.

A proposal which met with wider acceptance was thatmade by Stiles' at nearly the same time. Using mono-chromatic lights, he showed that the SCE has slightlydifferent magnitudes at different wavelengths. Re-stricting the data points to the central 6 mm of thecurve, he found it useful to relate pupil entry and bright-ness by the parabolic expression logio -q=ar2 , where 7

relative pupil efficiency, r= displacement in mm of theperipheral beam from the pupil brightness center, anda= a constant representing the degree of curvature ofthe SCE curve. This form of the parabolic expressionomits the two constants which translate the curve alongthe X and Y coordinates.

Moon and Spencer2 pooled the data from severalexperiments and proposed two functions which they feltwere equally good,

a; = 1-0.0850x2 +0.0020x 4 (x = mm)

X= 0.379+0.621 cosO.515x (x= rad).

Enoch4 adopted a trigonometric function similar tothe one put forth by Moon and Spencer

nr =0.25(1+cos 9.50)2.

In all of these attempts to find the proper mathe-matical function, the published results were curvesmade of either single observations or of the arithmeticmeans of groups of observations not separately re-ported. There were no reports of the tests of the hy-potheses that the proposed functions acceptably fittedthe data.

II. MATERIALS AND METHODS

A. Physical Methods (see Fig. 1)

Light from a 28-V quartz-halogen lamp, S, operatedat 34-V dc is passed through a double grating mono-chromator, a collimating lens, L2, and a beam-splittingcube, Cl. The beam reflected by C, toward the first-surface mirror M, is referred to as the fixed beam, be-cause its position relative to the subject's pupil can bealtered onlv by moving the subject's head. The beamtransmitted by C1 is called the movable beam, be-cause its position relative to the subject's pupil can bechanged without changing the subject's head position.The fixed beam passes through beam-splitting cube C2,the ten-diopter maxwellian-view lens L3 and is con-

S -?an

E:= L 2

SD, AC

- w

~W2

FFS, Ad

C2z RPH" RPV

FS2

C3z

Ef

02

M

TM

FIG. 1. Drawing of the experimental apparatus.

verged to a focus in the plane of the subject's entrancepupil at E. The movable beam is reflected through 900by mirror IM2, is then partially reflected at C2, and thereflected fraction is also imaged in the pupil plane. Theimages of the exit slit formed by the two beams areidentical in size, 0.5 mm wide and 1.0 mm high.

The movable beam passes through two sets ofcounter-rotating prisms, one which can deviate ithorizontally, RPH, and another vertically, RPV. Thesedeviations cause the movable beam to strike the par-tiallv reflecting hypotenuse of C2 at various points andtherefore to be imaged by L3 at various positions in thepupil plane. All prisms, cubes, and lenses are low-reflec-tion coated; the lenses are achromats.

The fixed and movable beams are presented to thesubject alternately as the result of interruption by thesector disks SD, and SD2, which are driven by a servo-controlled dc motor whose speed is under the subject'scontrol.

The subject's field of view is limited by field stopsFSt and FS2, whose apertures in the experiments de-scribed below subtended 10 at the retina. FS, has aniatte white finish on the side fLacing the subject. Thissurface is illuminated by light passed through suitableKodak Wratten filters. The face of FS, forms a surroundof approximately 60.

Vol. 59758

EXPLANATION OF STILES-CRAWFORD EFFECT

A portion of the fixed and movable beams strikes thehypotenuse of beam divider C3, is directed away fromthe subject's eye, and is brought to a focus by L3 atplane G. There, two images are formed that lie in aplane equivalent to the plane of the subject's entrancepupil. These images are formed on a ruled screen onwhich each small box is 0.1 mm square. The illuminationof the grid and of the subject's iris are each under thecontrol of the experimenter. A combination telescopemicroscope is placed at TM. The objective lens, T, of10-cm focal length and 30-mm diam is placed 30 cmfrom the plane of G. An image of G is formed by T inthe plane of the stage of a compound microscope, M.This image is viewed by the experimenter through eitherhigh- or low-power objectives, 01 and 02, and ocularOC. The telescope objective, T, is immovably mountedon the optical bench that holds C3 and G.

When the experimenter looks through the telemicro-scope he can see the grid superimposed on the anteriorsegment of the eye. When the subject is in proper po-sition, the pupillary border of his iris is seen clearlyagainst the grid, allowing precise measurements of pupilsize and position. The positions of entry into the pupilof the fixed and movable beams are measured by notingtheir positions on G. The position of each of the 2 beamsrelative to G can be set to the nearest 0.025 mm.

In the path of the fixed beam there is a neutral densityfilter F. In the path of the movable beam there is a 15-cm circular neutral density wedge, W1 , (Kodak) whoseoptical density progresses linearly from 0.00 to 2.00density units in 3450 of rotation. The movable beam alsopasses through the compensating wedge, W2, which hasthe same density gradient as W1 .

Wedge W, is fixed on a shaft and its position is underthe control of the subject. Mounted on the same shaft,in an unvarying relation to the wedge, is a precisionpotentiometer across which is placed a mercury batteryand adjustable dropping resistors arranged so that thevoltage drop across 3450 of potentiometer is 2.000 Vas measured on a digital voltmeter (Hewlett-Packard).The reading of the digital voltmeter affords a continuousindex of wedge position.

The fixed filters and the circular wedge have beencalibrated in a spectrophotometer (Unicam) and a tableof correction factors has been drawn up so that thetrue density at each wavelength can be calculated.

The optical elements are mounted on a stone tabletop, beneath which is attached the subject head-po-sitioning apparatus. There are micrometric screw ad-justments in the X (horizontal), Y (vertical), and Z(anterior-posterior) coordinates of the pupil plane.

On a brass bite rest, coated with dental-impressioncompound, an impression of both upper canine teethis made, supplying two points for determining theplane of the face. The third point is determined by thenose rest, which is a brass post with a molded dental-compound impression of the nasal bone at the bridge of

the nose where there is very little soft tissue betweenskin and bone. We have found this more stable than aforehead rest.

After the facial plane has been established, restraintsare adjusted to limit the subject's ability to twist aboutthis plane. Two threaded bars that project toward thehead, perpendicular to the parietal bones, are advanceduntil they rest firmly against the skull.

Positioning the subject at the beginning of an experi-mental run is begun by first bringing the grid G intofocus as seen with the telemicroscope, TM. The sub-ject's pupil is fully dilated and its margin is brought tolie in the plane of focus of the telemicroscope. The dis-tance of the subject's entrance pupil from the max-wellian-view lens is thus fixed. The sector discs arestopped and the lights so arranged that only the fixedbeam at it brightest strikes the subject's eye. The ex-perimenter turns the micrometer screws, moving thesubject's head until he sees, through the telescope, thesmall spot of light which is the virtual image of thedistant light source formed by the first corneal surfaceacting as a convex mirror.7 A small adjustment of themicroscope part of the telemicroscope brings this intofocus as a spot of monochromatic light approximately0.05-mm diam. The subject's head is moved carefullyleft and right and up and down until the spot has itsmaximum brightness. This is a very delicate adjust-ment. A lateral displacement of the head (and eye). of0.05 mm causes a very noticeable diminution in bright-ness of the image. The brightness of the spot is much lesssensitive to rotational movements of the eye than totranslational movements. With a telescope objective ofsmall aperture, and a field stop subtending a small angleat the cornea, the corneal mirror must occupy with greatprecision that unique position at which the incident raysof the fixed beam return upon themselves so that theserays may enter the telescope. Very small deviations fromthis condition of perpendicularity to the corneal vertexcause some or all of the returning rays to be deviatedaway from the telescope, resulting in dimming or disap-pearance of the corneal reflection. This system of ad-justing for maximum brightness is a very precise methodof getting the subject's eye back into the same position,time after time.

Compensation is made for the spherical aberration ofthe eye. As the movable beam is traversed across thepupil, the field stop FS2 is moved by micrometer screwsso that its image continues to occupy the same retinallocation. For the subject, there is a strong sense ofwobble when the two images are not superimposed. Theattainment of correct image position is attended by asudden sense of locking in, due to loss of wobble.

The pupils were dilated with the mydriatic, phenyl-ephrine 10%, and either of the cycloplegics, tropicamide

This method of positioning the subject was mentioned in apaper by W. S. Stiles, Proc. Royal Soc. (London) 27, 74 (1939). Hecredits the suggestion to Professor Hartridge.

June 1969 75S9

ARAN SAFIR AND LYON HYASoIS

1% or cyclopentolate 1%. Drops were instilled fre-

quently to keep the pupil at maximum size.Flicker brightness matches were made. The subject

set the flicker rate so that at the wedge position yieldingequal brightness in the two beams a point was reachedof minimum flicker, just bordering on no flicker. At thisend point, the subject pressed a switch which signalleda small digital computer (Mathatron 8-48, with PTP& APS) to accept the reading from the digital voltmeter.The computer was programmed to multiply each read-ing by the wedge-correction factor for the specific wave-length and to print out the reading as well as the arith-metic mean at the end of a series of readings at one

point of pupil entry. This reading Dm, corresponds tothe density of the wedge, Wi.

The density of the fixed filter was sufficient to allow

the fixed beam to match the movable beam at the mostextreme positions of pupil entry measured. The circularwedge attenuates the movable beam, a function in-creasingly taken over by the intrinsic directionalism ofthe retina as pupil entry becomes more peripheral. Atfar peripheral points it is not desirable to allow the sub-ject to make matches near the 0.0 density end of thewedge, where the sudden transition from 0.0 to 2.0 gives

a very visible cue. Therefore, the fixed filter was in-creased by about 0.2D to push the matching point fur-ther up on the wedge.

B. Mathematical Considerations

The dependent variable, Y, used for the calculationswas the wedge density through which the movablebeam passed. It can be shown that this reading Dmn, isequivalent to loglo v plus a constant KI.

In curve fitting to the gaussian hypothesis, we use theexpression

Y=logo1i-K,=K 2 +Ae=B(X-C)2 (1)

in which C is the centering constant, K2 is the hori-zontal asymptote, K, is an arbitrary constant dependentupon the optics of the apparatus, and A and B areparameters of spread. This expression is a general form

of the gaussian function. It neither assumes an asymp-tote of zero nor does it normalize the area under thecurve to unity.

C. Statistical Considerations

(1) Design

The experiments were originally undertaken to ex-amine the spectral variation of the SCE. We derivedour technique largely from Stiles.' Accordingly, the firstgroup of experiments was composed of measurements ofthe SCE at many wavelengths with pupil positions re-stricted to the central 6 mm. At each position of pupilentry, brightness matches were made throughout thespectrum.

We calculated best-fitting parabolas according toleast-squares criteria, in order to derive the coefficientsfor comparing the curves obtained at different wave-

lengths. Before making these comparisons, we calcu-lated the goodness of fit of the data to the parabolicfunction. In nearly all of our experiments, we were com-pelled to reject the parabola. The central problem thenbecame to determine the function properly describingthe data points. We therefore designed an experimentto do this. At a single wavelength, we made multiplereplications of brightness matches at many points acrossthe entire diameter of the dilated pupil.

Since the SCE is more marked at the short wave-lengths, we decided to use a wavelength as close to 400nm as possible. We wished to keep the brightness levelhigh enough so that hues were clearly sensed, in orderto bring out any effects of hue shift on the variances ofmatches. Since the amount of stimulus available atshorter wavelengths was insufficient, we performed thesesingle runs at 480 nm.

The single-wavelength runs were repeated 6 times,one run in the morning and one in the afternoon of 3consecutive days. Each run consisted of 10 successivereplications of brightness match, yi, at k positions ofpupil entry, xj. The independent variable, xj, repre-senting positions of the peripheral beam, was chosen sothat in averaging the 6 runs the effects of any trends inthe order of selection of x; were balanced. Pupil positionswere chosen at 0.50-mm increments. When the startingposition was an integer, the values were as follows:xj= 0.00, 0.50, 1.00, 1.50-. At other times, the sameincrements were taken between the previous locations,e.g., 0.25, 0.75, 1.25---. The dependent variable is yij,where the subscript i gives the replication number ofthe brightness match at pupil-entry position j. Therange of the y observations and their mean yj weredetermined and plotted. The variances of yij were com-puted and evaluated for homogeneity. Finally, thethe means of the means, (9j), were determined.

If our aim had been only to test the goodness of fitof a particular function, this design would have beenunnecessarily complex. But, because of the scarcity ofstatistical data, we wanted also to examine the related-ness of y observations when taken consecutively, trendsof data related to fatigue or peripheral entry, and thevariations of the parameters measured from one ex-perimental trial to another.

(2) Analysis

In fitting a mathematical function to the observeddata points, we have two main objectives: (1) to obtaina close spatial correspondence of the proposed functionto the points and, (2) to be able to derive a usefulphysiological interpretation from the function.

Achieving the first objective alone is not enough.Many functions can yield acceptable fits to the observedSCE. We think that it is unwise to choose one or another

on this ground alone.Partly for historical reasons, we originally considered

four functions for testing: a parabola, a gaussian, atrigonometric function, and a 4th-degree polynomial.

Vol. 59760

EXPLANATION OF STILES-CRAWFORD EFFECT

The higher-order polynomial was eliminated from con-sideration since it does not readily offer us a physiologi-cal interpretation, regardless of how well it fits. Further-more, it is always possible to achieve a good fit to anyset of data with a polynomial of high-enough degree.We also eliminated trigonometric functions because: ifsimple they are periodic (unlikely for the SCE), and ifcomplex enough to eliminate peripheral oscillations,they make biological interpretations very difficult.

It is desirable to characterize the mathematical re-lationship by the simplest common statement fulfillingthe objectives mentioned above. Both the gaussian andparabolic functions are simple, biologically occurringrelationships. The parabola often relates to surfacearea, the gaussian to population distribution.

Another question influenced our choice of analysis.In what domain of X would we test our hypotheses: inthe central part only or throughout the entire range ofthe phenomenon? If the SCE is one phenomenon, themathematical expression describing it must hold forperipheral as well as central entry of rays. It seemed de-sirable therefore to test over the complete, observablerange of X. Using some arbitrary segment of this rangemay introduce error. For example, it is possible to choosea range of operation such that an acceptable fit may begiven by a straight line. Nevertheless, since the centralportion of the curve had been used in the past, we de-cided to test our functions in that range as well.

Two null hypotheses regarding the form of the func-tion were tested in each of the two domains of X, thecomplete and the restricted central part. The datapoints used for testing were the mean values of theindividual runs, gj. These means were handled as singledata points since they are independent between trials.Least-squares estimators of the parabolic and gaussianparameters were determined, the latter by iterativetechniques. Each proposed function was tested sepa-rately by the analysis-of-variance F ratio, at a signifi-cance level of 5%.

It was also necessary to compare the variances of yijat various values of xi in order to determine the mostappropriate testing model. The variances were found tobe homogeneous. The two null and alternate hypothesesand the algebraic working rules are given below.

Ho,: Y= f(X)'= a+bX+cX 2 = parabolaHA1: YsoparabolaHo2: Y =f(X)I = K2+ A e-B (XC) 2

= gaussianHA 2 : Ywzgaussian

In each case, we determine the statistic,

k

j=1F(0 bs)

k V.E E[Yx-(y)]2 (N-k)j=1 i=1

where p= number of parameters.

Rule: if F(ObS)>0.950, k-p, N-k reject the hy-pothesis, otherwise accept it.

III. RESULTS

A. Preliminary Experiments-Spectral runsIn Figs. 2 and 3, the means of the observations are

plotted for 2 different observers at various wavelengths.Each value is the mean of 4 matches. The data, wheninspected visually, appear to be similar in form andquality to those of Stiles and others. Those curvesmarked with an asterisk were the only ones found tobe acceptable as parabolas by the criteria describedabove.

B. Principal Experiments-Fixed Wavelength

Single trials: Table I presents the numerical data fortwo typical trials. The means and extremes of repeatedmeasurements at fixed pupil positions are given in Fig.4 for one typical trial. Figure 5 shows the mean valuesfrom another trial and the best-fitting gaussian. Severalfeatures can be seen. The data are tight in two senses:the spread of replications is narrow, and the means ap-pear to depict a smoothly changing continuous func-tion. The SCE of this subject is eccentric, allowing usto view the shouldering-off effect, a phenomenonsimilarly seen in all six experimental trials. In addition,it is clear that the gaussian function corresponds closelyto the observations (Fig. 5).

Subject AS.. 480

.9- 0

.7 .

.5 0

.3 0

.1 .-9 520 I.2.3

.7A .

.5.

Is.3 .

0 -560

.7..

.5

.9600 ...9.

.7 *

.5

-3-2-1 0 12 3NASAL TEMPORAL

500 . .

540 **'.

* 0

580 ..

*620 *.* 0

-3-2-1 0 1 2 3NASAL TEMPORAL

FIG. 2. Results of a spectral run restricted to 6 mm of pupilentry, right eye. The subject is a protanomalous 40-year-old male,4.00 diopters myopic, with normal visual acuity. The points for620 nm are acceptably fitted with a parabola. The others are not.

June 1969 761

ARAN SAFIR AND LYON HYAMS

Subject TO.- 420

.7-

.5 .

.3.

.7 * 6.5 .

*.3.

.7 5400.5 . *

: 7

2-3-2- 501 23

L .3 T O.7 620 ...... Z

.5.

.3

.7 60..

.5 .- *

.3

-3-2-1 1 '23NASAL TEMPORAL

*4401.0

0.9

0.8

0.7480 ..

520 *,

560

600,.

*640

680 ....

-3-2-1 0 I 2 3NASAL TEMPORAL

FIG. 3. Results of a spectral run restricted to 6 mm of pupilentry, right eye. The subject is a color-normal 21 year old female,1.00 diopter myopic, with normal visual acuity. The points for440, 460, 640, and 660 nm are acceptably fitted with parabolas.The others are not.

-

0.3 -

0.2 1-

0.1

-4.0 -3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0

Nasal Temporal

-0.,

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

-1.0

Pupil position (mm.)

FIG. 5. Subject A.S. Data from day 2 of 3-day experiment.Solid dots represent the means of 10 brightness matches. The solidline is the best-fitting gaussian.

Combined data: Figure 6 is a plot in which each pointis the combined mean of three separate means, eachcomposed of ten observations. Figure 7 superimposes onthe points of Figure 6 the curves representing the gaus-

1.0

0.9 t- 0-0

0 -

0S

SS

1.0

0.9

0.8

0.7

0.6

0.7 h

0.4 h

0.3 l

0.2 F-

0.1 F-

.

0.5 -S

0.4 -S

0.3 -

-00 - 0.2 I-

S

0.1 I-

0

-4.0 -3.0 -2.0

Nasal-1.0 0 1.0 2.0 3.0 4.0

TemporalPupil position (mm.)

FIG. 4. Subject A.S. (same subject as in Fig. 2). Solid dot-represent means of 10 brightness matches. Horizontal lines repressent extremes. Data taken on day 1 of 3-day experiment, right eye.

-4.0 -3.0 -2.0

Nasal

00

go

0o...0 0 a

-1.0 0 1.0 2.0 3.0 4.0

TemooralPupil position (mm.)

FIG. 6. Subject A.S. Combined data from AM and PM of all3 days of 3-day experiment. Each solid dot represents the meanof 30 observations.

762 Vol. 59

0.8

0.6

0.50

S

0S

S

See.

I I I .--- I I I I Io a

l) | r x s

I I I I I

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IetaTI

EXPLANATION OF STILES-CRAWFORD

TABLE 1. Brightness matches of two typical trials.

Pupil Wedge Readings y = log n + K1 Mean VarianceDate Posit ion Yij

i 2 3 4 5 6 7 8 9 10 -

10/11/67 -3.75 .1887 .1785 .2121 .1683 .2172 .1499 .2274 .1162 .1346 .1326 .1725 .00151P.M. -3.25 .229S .2539 .2264 .2274 .2499 .2101 .1275 .1968 .2335 .1683 .2123 .00152

-2.75 .2407 .2580 .2172 .2274 .2713 .2845 .3345 .2713 .2529 .2050 .2562 .00140

-2.25 .3876 .3529 .3641 .3804 .3600 .3763 .3315 .3784 .3957 .3937 .3720 .000401-1.75 .4957 .5263 .4549 .4702 .4998 .5120 .5151 .4498 .5222 .4712 .4917 .000795-1.25 .6232 .5926 .6528 .6721 .6507 .6191 .6i46 .6048 .6613 .6650 .6386 .000735- .75 .7619 .6966 .7221 .7078 .7272 .7099 .6936 .7089 .7160 .7282 .7172 .000381- .25 .7843 .7548 .7813 .7915 .7721 .8425 .8007 .8058 .7945 .8109 .7938 .000566.25 .8221 .8098 .8109 .8445 .7864 .8670 .8394 .8180 .8343 .8945 .8326 .000961.75 .8649 .9241 .8547 .8364 .8139 .9078 .9180 .8965 .9516 .8619 .8829 .001881.25 .9047 .8690 .8415 .9190 .8955 .8598 .8445 .8527 .9057 .8965 .8788 .0008121.75 .8527 .8537 .8221 .8506 o9241 o8608 .8629 .8425 .8200 .8823 .8571 .0008952.25 .8741 .87:0 .8486 .7976 .7752 .8415 .8863 .7976 .7935 .7425 .8227 .00231

2.75 .8384 .8323 .8047 .8119 .8068 .8017 .7884 .7456 .7854 .8068 .8022 .000670.-

10/12/67 -4.00 .2244 .1866 .1397 .1387 .1560 .1132 .1703 .1111 .1836 .1509 .1574 .00122

A.M -3.50 .2213 .2519 .2142 .2193 .2009 .2172 .2397 .2692 .1927 .2682 .2294 .000712-3.00 .2172 .2468 .3111 .2927 .3080 .2050 .2223 .2264 .2305 .2376 .2497 .00154-2.50 .2845 .2856 .2754 .2601 .2621 .3325 .2794 .2825 .3121 .2580 .2832 .000551-2.00 .3988 .3661 .3957 .4182 .3957 .3386 .3661 .3621 .3682 .3284 .3737 .000793

-1.50 .4814 .4977 .4926 .4834 .4508 .4671 .5191 .4804 .5110 15]71 .4900 .000484-1.00 .5732 .5569 .6058 .5875 .6130 .6181 .5854 .5630 .5946 .5589 .5856 S000500- .50 .7211 .7293 .7231 .7017 .7119 .7109 .7089 .7191 .7303 .7160 .7172 .0000833.00 .7956 .7752 .7456 .7905 .7568 .7854 .7915 .7599 .7731 .7639 .7737 .000286

.50 .9261 .8761 .9445 .8659 .8690 .9118 .9322 .8914 .8690 .8792 .8965 .000875

10/12/67 1.00 .9496 .9180 .9424 .9914 .9526 .9608 .9027 .9598 .9690 .9659 .9512 .000650

A.M. 1.50 .9659 .9486 .9506 .9445 .9730 .9628 .9741 .9282 .9506 .9180 o9516 .0003362.00 .9965 .9822 .9832 .9792 .9333 .9832 .9894 1.0067 .9292 o8996 .9682 .001212.50 .9812 1.0138 1.0016 .9945 .9608 .8802 .8751 .8323 1.0026 .8721 .9414 .004693o00 .9302 .8761 .8629 o8353 .8364 .8517 .8782 .9435 .9098 .8619 .8786 .00141

sian and parabola, as'determined by least-squarestechnique.

The parabolic hypothesis was tested within the com-plete range of pupil entry" (-3.50 to +3.00) and re-jected (F24,54 =6.27, P<0.001). The gaussian hypothe-sis was similarly tested and was accepted (F 2 3 ,54=0.937,P>0.5). The SCE in the complete range ofobservation is therefore satisfactorily described by agaussian but not by a parabolic function.

In testing these hypotheses within the central domainof X(-2.0 to +3.0), we found both functions to fitequally well (F1 7,42 (g) = 0.59, P> 0.5, F,8,42 (p) = 0.51,P>0.5), Fig. 8. Thus, there are at least two functions,and probably more, that describe the relationship in anarbitrary x interval. Of those functions which we con-sider reasonable, only the gaussian fulfills the criteriaover the complete operational domain. Figure 9 pre-sents this function in relation to the data.

I Since the computer program we used for these calculationsallowed us to use only points with equal numbers of replications,we were forced to omit two peripheral points.

IV. DISCUSSION

The knowledge that the SCE in at least one person isproperly described by a gaussian function stimulated asearch for its meaning. We have been unable to findany biological event conforming to the gaussian functionin which neither of the variables is related to a frequencyof occurrence. It is a long step from the observation thatthe SCE is gaussian to the inference that the SCE, beinggaussian, is the result of a population distribution. Still,we know of no alternative except the unlikely suggestionthat the SCE is unique as compared to other knownbiological phenomena. We therefore make the crucialassumption: the SCE is a function of a population dis-tribution. Assuming that every cone will accept energyincident upon it within only a very small angle from itsaxis, and further that the cones respond equally uponstimulation, and if the angulations of the axes of thepopulation of cones are normally distributed, the SCEwill be gaussian. We have observed the gaussian andinferred the population distribution.

EFFECT 763June 1969

ARAN SAFIR AND LYON HYAMS Vol. 59

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in the area subtending 1 deg of visual angle at the tovea,some 20 000 cells.9 If this group of cones exhibits anover-all sensitivity to light as shown in Fig. 6, may wenot properly ask, "In what way do the responses ofthe individual members of the population contributeto. the overall SCE as measured across the pupil"? Weare aware of no evidence proving that the brightnessresponse of an individual cone is described by the overallSCE.

Our results are consistent with the argument that thepattern of directionalism of single cones is not describedby the well-known overall SCE, that individual cones aremore highly directional than indicated by the over-allSCE, and that they may indeed be very much morenarrowly directional than the overall SCE.

What justification is there for the construction ofsingle-cone models that achieve the directionalism ofthe overall SCE when the overall SCE does not de-scribe the directionalism of a single cone? Very little, wethink. By analogy, it would be an error to fly over afootball stadium and, hearing the crowd roar, observe,"I hear the sound of a human voice." We have beentraversing the pupil and "hearing the voices" ofthousands of cones in concert.

There are several possible sources of error in theseexperiments, physical and statistical. The apparatusdescribed enables us to monitor the position of the

9 S. Polyak, The Retina (University of Chicago Press, Chicago,1941).

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FIG. 8. Subject A. S. Combined datafrom3 days (same as Fig. 6)Functions fitted using only points data included between solidvertical lines. Solid line (-) best-fitting gaussian, dashed line(- - -) best-fitting parabola.

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corneal reflection during the brightness-matching pro-cedure when the subject is making his best efforts to

- >N hold his head and eye as still as possible in the correctposition. We have tested the stability of a number ofsubjects. A good subject can repeatedly re-positionhimself on the bite and headrests with an error of less

A/ than 0.10 mm. The same subject will unpredictablydevelop periods of small wandering movements of upto 0.25 mm in random directions from the true loca-tion. These could increase the variances of the matchesbut would have little effect on the means of the readings.

Even the best subject occasionally adopts a new po-,is sition, causing the reflection to move to a new location

/ P0.30 to 0.40 mm displaced. He has changed his criteria

l and now regards this as the zero position to which he

, returns with great precision. Such a shift of eye positionwill produce a large systematic bias in the subsequentmatches. We believe that this error accounts for the

, , ,groups of aberrant points seen in the curves in Figs. 2-4.0 -3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 and 3. The data for these curves were taken at a time

Nasal Temporal when we monitored corneal reflection but did not alterPupil position /mm. )

the subject's head position to correct for changes, once'.Subject A.S. Combined data from all 3 days of 3-day theu ha strtd!nt (same as Figure 6). Solid line (-) best-fitting the run a started.

dashed line (-- -) best-fitting parabola. Functions Movements of the corneal reflection result from twoing all data points. types of eye movement, rotational and translational.

Translational movements are much more effective thanpopulation under study is the population of cones rotational ones in moving the reflection. The geometry

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FIG. 7experimegaussian,fitted usi

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EXPLANATION OF STILES-CRAWFORD EFFECT

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MUM iII IUVCIIILtrn 01 Lme corneai renection that could re-sult from rotating the eye to shift fixation from oneborder to the opposite border of the 1 deg matchingfield is 0.07 mm. Yet we observe movements of as muchas 0.40 mm. Evidently at least 0.33 of the 0.40 mm istranslational in origin and is due to a new head position.In our opinion this is by far the greatest source of errorin our experiments.

Our statistical approach has several weaknesses. Thestructure of the testing situation is particularly strongwhen rejecting a specific hypothesis, i.e., the chance oferring is a which is controlled at 5%. However, uponaccepting a function, we do not state the f error, i.e.,the probability of accepting falsely when compared tospecific alternate hypotheses. It is conceivable that otherfunctions such as the logistic," cosine,' 2 and Weibulli3

may be found to be equally acceptable. Of these, onlythe logistic is, we feel, worthy of further investigation,since it may have a physiological interpretation.

10 R. W. Ditchburn and B. L. Ginsborg, J. Physiol. (London)119, 1 (1953).

1 f(x) = Esech2 (x/d)] 2d,-X <x<oo2 f (x) = (+cosx) . 2 7 ,-7r<X<7r

= 0 , elsewhere3 f (x) = abxb-le-a.b , x>0

= 0 , elsewhere.

V. CONCLUSION

The Stiles-Crawford effect, measured at 480 nm inone subject, is acceptably described by the gaussianexpression

Y= K2+Ae-B(xc)2,

where A=0.9040, B=0.0670, C=1.4832, K2=0.0130.From this, we advance a physiological interpretation

that the SCE as seen across the pupil is a consequence ofthe gaussian distribution of cone orientations.

ACKNOWLEDGMENTS

Dr. Safir is grateful to Drs. F. W. Campbell and M.Alpern for their many helpful suggestions in the earlyphases of this work.

We wish to express our gratitude also to Drs. W. S.Stiles, B. H. Crawford, W. A. H. Rushton, M. Alpern,and E. H. MacNichol for their critical reading of thismanuscript.

We are indebted to Dr. Yonathan Bard, for makingit possible for us to use the computer program which heprepared.

We used a least-squares iterative solution run on anIBM-360/50 computer. Programs of this sort have well-known difficulties; e.g., they may stop iteration at a

* local minimum. We think our estimates of the param-eters do not suffer from such errors. Not only does theplot of expected values have good accordance to theobserved data but the parameters were independentlyestimated from the parabola that resulted from thenatural-logarithm transform of a gaussian with zeroasymptote. Since Y was found to have equal variance,lnY was fitted through a weighted least-squares solu-tion. The agreement between the two solutions wasexcellent.

It is apparent from our work and that of others thatprecise data can be obtained through careful experi-

* mentation. Yet the literature also offers examples of* data points so widely spread that interpretations and

inferences based on these results are presumptuous.While many investigators must have taken replicatereadings of brightness match in order to produce mean

,4a ,.Ii ,,values for the plots, none have reported their spreads in-4.0 -3. 0 -2. 0 -1.0 0 1.0 2.0 3.0 40 any form, i.e., range, variance, or standard deviation.

Nasal Temporal It is difficult for the reader to evaluate the significancePupil position (mm.) of those points and impossible to subject them to his

9. Subject A.S. Combined data from 3 days (same as Fig. 6). own analysis. In addition, the criteria used for theline (-) best-fitting gaussian derived using all Doints drawing of very specific continuous curves are never. A=0.9040, B=0.0670, C=1.4832, K=O.0130. given, and the techniques used to arrive at their param-

eters are not specified. While the practice of simply.1 known. Ditchburn'0 has calculated that a lateral drawing curves is common and often useful, it is notof 0.01 mm causes a deflection as great as an the best way to determine the true nature of the under-ar movement of 8 min of arc. lying mathematical function. This is better done bys reasonable to assume that fixation points outside using mathematical techniques, some of which are pre-Latching field are chosen only fleetingly. The maxi- sented in this paper.

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