JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Variability of Accommodation during Steady Fixation at VariousLevels of Illuminance*

MATHEW ALPERNDepartment of Ophthalmology and the Vision Research Laboratories, The University of Michigan, Ann Arbor, Michigan

(Received October 25, 1957)

Using the stigmatoscope, the variability of the dioptric power of the eye during steady fixation wasmeasured for four observers over a range of intensities from 2.5 to 3.0 log trolands in one-half logarithmicsteps. Two different viewing conditions, (constant and variable size test letters), were studied but nosignificant difference between them was obtained. There was almost a fourfold decrease in the variabilityof the settings as the intensity was increased over this range. There was a sharp transition at one troland,suggestive of a discontinuity between scotopic and photopic vision. These changes could be eliminated bycycloplegia of the fixating eye. Essentially the same results were obtained when the refraction of one eye

was measured objectively (with a concidence optometer), while the other eye fixated the chart. This lattertechnique is, however, less valid at the lower intensities because of the tendency to fixate the measuringlight of the optometer. The data can be quantitatively described by a theory which postulates that the

accommodation is continually fluctuating and that the limits of fluctuation are proportional to thresholdAl/I.

INTRODUCTION

W HEN the refractive state of the normal eye ismeasured a number of times during steady

fixation of an object, the individual measurements willdiffer from one another by a small amount.',2 Obviousas this might appear on the surface, little is yet knownof the factors which influence this variability, or whatdetermines the limits of the range of variation. Thefollowing very simple experiments were carried out inorder to obtain some rudimentary information aboutthese matters.

METHOD

All that was involved in the present experiment wasthe measurement of variability of the refractive statefor a number of different illuminance levels of the testchart. Two different viewing conditions of the chartwere provided. In the first of these, the same chartwas used at all illuminance levels. It consisted of asingle row of ten 20/15 black Snellen letters immediatelyabove a row of ten 20/10 black Snellen letters. Thesewere seen against a rectangular white background26.7 cm wide and 11.4 cm high. In the second viewingcondition the background remained the same but thesize of the letters was varied (up to 20/200) so that theywere always barely suprathreshold.

The observer viewed the chart in an otherwisedarkened room through a 2 mm artificial pupil mountedjust before his eye (the other eye was occluded) andat a distance of 345 cm from the test chart (which wasilluminated by an argus 300 watt, 120 v, tungstenfilament projector). Wratten neutral density filterswere used to vary the intensity of illuminance on thechart. The room was carefully screened with baffles andvelvet cloth so that no stray light passed into the

* Supported by a grant from the Institute of Industrial Health,University of Michigan.

' Arnulf, Dupuy, and Flamant, Ann. opt. ocul. 3, 109-115(1955).

2 G. Westheimer, J. Opt. Soc. Am. 47, 714 (1957).

observer's eyes. Looking through a half-silvered mirrorthe observer viewed a stigmatoscope (which has beendescribed in detail elsewhere3 ) imaged by a Badaloptometer arrangement. A small point of light wasflashed one second in every two. It was seen as justclearly visible on the test chart. The observer wasinstructed to set this stigma at its sharpest focusbut always to keep fixation on the chart and not thestigma. He was told to make settings only when hewas certain that he was looking at the chart. Theintensity of the stigma could be varied by varyingthe current through the lamp with a variable resistancetransformer. (The resulting changes in color tempera-ture of the stigma were too small to be observed overthe range of intensities used in the present experiments.)The focus of the stigma was varied by moving the smallpoint source of light back and forth on an optical benchbefore a 122 mm focal length lens. A chin cup andhead rest helped to stabilize the position of the eyebehind the artificial pupil. Where a correction wasrequired it was placed immediately in front of theartificial pupil.

After dark adaptation for thirty minutes, the observermade ten stigmatoscope settings at the lowest level ofchart illuminance. He was given as much time asrequired. Ten settings were usually completed in about15 mintues. After adaptation to the next highestintensity level the process was repeated. The entireprocess was repeated in this way going from the verylowest to the very highest intensity. The two differentviewing conditions were studied on two different days.

The standard deviation of the ten optometer settingswas the dependent variable studied throughout.Measurements were made on four practiced observers.

RESULTS

The mean data for all four observers and the twodifferent viewing conditions are plotted in Fig. 1A.

3 M. Alpern, Am. Med. Assoc. Arch. Ophthalmol. 54,907 (1955).

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MARCH, 1958VOLUME 48, NUMBER 3

MATHEW ALPERN

.6 A

A \

MEAN DATAX .2 4 OBSERVERS

STIGMATOSCOPE

-Z of 0 1 2 3

LOG RETINAL ILLUMINANCE (TROLANDS)

FIG. 1. The means of the standard deviations of ten optometersettings under each of two viewing conditions as a function ofthe log of the retinal illuminance in trolands, during steadyfixation of the test chart. The points are empirical, the curvestheoretical [Eq. (2)]. A. Stigmatoscope measurements on thefixating eye of four observers. B. Coincidence optometer measure-ments on the nonfixating (right) eye while the other (left) eyefixated the chart. Two essentially emmetropic observers wereused. The photopic curves in both figures have constants: b=0.26K=2.0. The scotopic curves both have the constant K=900.In the scotopic curve 1A. b=0.76; the scotopic curve in B.b=0.58.

When the data for the two different viewing conditionswere compared (using the nonparametric pairedreplicate statistic described by Wilcoxon4 ) no significantdifference was found; accordingly the results from bothviewing conditions have been grouped together. Thereis a marked decrease in the variability of the settingsas the illuminance is increased until at high intensitiesthe curve appears to level off. The mean of the standarddeviations (representing 80 observations at eachintensity) undergoes almost a fourfold change as theintensity is changed by a factor of 32 000. The possibil-ityt that such a change could have occurred by chancewhen there is no relation between illuminance and thevariability of the settings is very small (X2 = 18;9 d.f.; p=0.036). At one troland of retinal illuminancethere is an abrupt discontinuity. Presumably thisrepresents the usual transition between scotopic andphotopic vision.5

4 F. Wilcoxon, Somse Rapid Approxhinale Slatisticul Procedures(American Cyanamid Company, Stamford, Connecticut, 1949).

t When statistical statements are made without furtherqualification they are based on the nonparametric chi squaretest also described by Wilcoxon.4

5 S. Hecht, Physiol. Rev. 17, 239 (1937).

Do these changes in variability of the optometersettings reflect actual changes in the range of fluctuationof the lens at each intensity level or are they due tosome artifact in the measuring device or perhaps tosome other optical or sensory variability within theeye? In order to answer one of these questions theexperiment was repeated for one observer with thevariable size letter viewing condition when his ciliarymuscle was paralyzed by applying two drops of onepercent homatropine into his conjunctival sac. Theresults of the experiment (Table I) were clear andunequivocal. After the onset of cycloplegia the varianceof the refractive state was significantly (p=0.01,using the paired replicate test) smaller than was thecase without cycloplegia. The variance found at eachintensity with cycloplegia was about the same. Theaverage value for the standard deviation for all inten-sities under the influence of homatropine was 0.25 dwhich was slightly smaller than the average value at thehighest intensity in Fig. 1A. These results can onlymean that the change in variability illustrated inFig. 1A requires a normal unparalyzed ciliary muscle.

Are the results illustrated in Fig. 1A merely anartifact of the method of measurements of the refractivestate? In order to answer this question the experimentswere repeated using an objective measurement of therefractive state. The technique employed, the objectivecoincidence optometer described, by Fincham,6 hadcertain obvious disadvantages for its present purpose.It was selected, however, because of the availablemethods it was the one which was the most differentfrom the stigmatoscope. Both adapting disks wereremoved and their source of light occluded. The opaladapting disk for the left eye was replaced by a mirrorwhich reflected the image of the test chart along theline of sight of the observer's left eye while the opticalaxis of the optometer coincided with the line of sight ofhis right eye. Two observers were selected who wereessentially emmetropic in the fixating eye. The physicalarrangement was such that the 2-mm diam artificial

TABLE I. Standard deviations of ten optometer settings withvariable size letters on chart when fixating eye was paralyzedvith two drops 1% homatropine. One observer.

Log I Standard deviation

3.0 0.262.5 0.152.0 0.371.5 0.271.0 0.160.5 0.230.0 0.151.5 0.241.0 0.302.5 0.33

MAN 0.25

E. F. Fincham, Proc. Phys. Soc. (London) 49, 456 (1937).

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March1958 ACCOMMODATION AT VARIOUS LEVELS OF ILLUMINANCE

pupil before the fixating (left) eye was only 225 cmfrom the test chart. This produced a somewhat largerangular substense for the chart which was the same asthat used for the stigmatoscope measurements. Meas-urements were made on one meridian of the right eyewhile the other eye fixated the chart. The sequence ofadaptation, testing, and the viewing conditions other-wise were the same as those followed during thestigmatoscope measurements.

The results of this experiment are illustrated inFig. 1B. The obvious disadvantage of the optometer forthe present purpose was the fact that at low levels ofchart illuminance, there was an almost irresistibletendency for the observer to fixate the bright mono-chromatic line images of the optometer target ratherthan the letters, despite specific instructions to thecontrary. This fact undoubtedly accounts for themajor discrepancies (especially at the lower lightlevels) between these data and those in Fig. 1A.Aside from this, there is remarkable agreement betweenthe two kinds of measurements in view of the differencesin testing conditions and observers involved. Thevariation of the standard deviation with intensityis again highly significant (X2= 26.2; 10 d.f.; p= 0.003).Too much importance should not be given to these data,except to point out the similarity in the trends to thosein Fig. 1A. However, it is of some interest, that thesmooth curves for the higher intensities in both sets ofdata are identical. Comparison of the measurements bythe two methods for the one observer common to bothexperiments failed to show a significant differencebetween the objective and subjective measurements(X2 = 1.0; 1 d.f.; p=0.32).

The results with the objective optometer emphasizethe fact that the increase in he variability of themeasurements of the refraction state as the illuminanceis decreased, illustrated in Fig. 1A is not merely anartifact of stigmatoscopy. One might suspect that asthe chart illuminance level was lowered there would bemore and more tendency for the observer to accom-modate on the stigma. To obviate this, the stigma wasflashed (instead of being continuously exposed), itsintensity was reduced along with that of the backgroundso that it was always equally suprathreshold regardlessof the intensity of the background, and specific instruc-tions were given for the subject to make settings onlywhen he was confident that he was looking at the chartand not the stigma. If, despite all this, that artifactpersisted it might tend to make the stigmatoscopesettings show a higher variability at lower levels.But such an effect would operate in exactly the oppositeway with the objective optometer. In this instrument,the exit pupil is so small that the settings of theoptometer position can be varied through a wide rangewithout any perceptible blur of the target. Consequentlydifferent settings of the optometer do not providestimuli for changing accommodation. If the decrease inchart illuminance was only associated with an ever

i.6I-

z .2

I..1

-20 -IS -10 -5 o 5 to 15 20

DISTANCE ALONG RETINA (ARBITRARY UNITS)

FIG. 2. Distribution of illuminance in the retinal image of theborder between a white and a black field according to Fry.The solid curve line represents optimum focus while the dottedcurved line represents the effect of a small amount of defocusproduced by either over or under accommodation.

increasing tendency to fixate the target of the optometerinstead of the letters on the chart, then it would beexpected that the decrease in chart illuminance wouldbe associated with a decrease in variability of thesettings. This would occur as the assommodation of theeye became more and more stabilized on the highintensity, high contrast, sharply defined optometertarget. The opposite result illustrated in Fig. 1Bstrongly suggests that other factors must be responsiblefor the change in variability obtained.

DISCUSSION

The most obvious result of the experiments justdescribed was the fact that the variability of themeasurement of the dioptric power of the eye dependedin a rather striking way upon the retinal illuminance ofthe fixating eye.

How is this best to be explained? Fry's recentanalysis7 suggests that the distribution of intensity inthe retinal image of a border between a black and whitefield may be represented by a curve similar to the solidline in Fig. 2, which is a plot of retinal illuminance as afunction of distance along the retina from the chief rayimage of the border. Suppose that this distributionrepresents the case when the eye is in optimum focus.Fry shows that the slope of the intensity distributionfunction at its midpoint decreases progressively as theeye becomes defocused. As a first approximation, itwill be assumed that the slope of the curve at itsmidpoint is linearly related to the change in focus.

The results of a number of recent experiments suggestthat the dioptric power of the human eye is changingcontinuously even during the steadiest of fixations.These fluctuations in accommodation will at a givenmoment produce a distribution of intensity slightlydefocused from the optimum distribution. This isrepresented in Fig. 2 by the dotted line. Whether the

7 G. A. Fry, Tle Blur of the Retinal Image (The Ohio StateUniversity Press, Columbus, 1955).

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MATHEW ALPERN

I

TIME

FIG. 3. A possible explanation for the response of the accom-modation of the eye to the addition and removal of a minus lens.For simplicity the normal oscillations of accommodation havebeen assumed to follow a simple harmonic motion. The lowercurve shows the variation in intensity of retinal illuminance atthe critical point x.

eye is defocused positively or negatively the fluctuationsof accommodation will constantly change the distribu-tion of intensity in the retinal image between the oneextreme of the solid line and some other distribution(with less slope at its midpoint than the solid line)at the other extreme. How is this other extremeestablished?

Assume that the limitation on the fluctuation ofaccommodation is made by a receptor, (or a group ofreceptors), at a fixed but unspecified distance, x, fromthe chief ray image on the bright side of the retinalimage of the border. Let the intensity of retinal illu-minance at this point at the moment of optimum focusbe Ii. Let the intensity of retinal illuminance at thispoint at another moment when the eye is slightlydefocused be equal to 12. The basic theoretical assump-tion is that as long as the value (II-12)/Il is so smallthat the observer is unable to detect a difference inthe intensity between It and 12 then the accommodationwill fluctuate freely. If however, the accommodationis such that (-12)/Il exceeds a critical value ofAl/I 1 (in which Al is the intensity difference threshold)then this intensity difference will be detected and ashift of the accommodation will occur back withinthe range of imperceptible departure from optimumintensity. According to this scheme the limiting casefor the variation of the accommodation will be whenthe difference between Ii and 2 equals the value forAl at that particular value of I, and the range ofvariation in accommodation is directly proportionalto the critical value of Al/I. This latter varies withthe value of 1, in a manner which has been adequatelydescribed$ for the human eye,8

AI/I = c[l+ 1(KI) JJ2. (1)

t The equation has been derived from a photochemical theoryof visual intensity discrimination but its use in the present paperis based on its ability to describe the data of intensity discrimina-tion for the human eye in a satisfactory way. It is not the purposeof the present work to enter into the relative merits of the various

Since the critical extent of variation of accommodationAF (in diopters) is directly proportional to critical valueof Al/I (that is AF= Y/A\/I), we may write

(2)

in which b= cb'. Equation (2) has been used in fact todraw the smooth curves in Fig. 1. The agreementbetween theory and data particularly in Fig. 1A (thedata for which must be considered more valid) isstriking. This in turn, suggests that the distance xbetween the chief ray image of the border and thelocus at which the hypothesized discrimination occursmust be reasonably small. For it is only within theregion that the intensity distribution function is astraight line, that the slope can be proportional to thechange in focus.

There is little doubt that Eq. (2) is an adequatedescription of the data in Fig. 1. The possibility existshowever, that this equation may be derived by makingassumptions different from those actually used so thatin no sense must this derivation be considered as a finalsolution. Nevertheless, the derivation serves to pointout the striking parallel between variation of measure-ments of the refractive state with intensity and bright-ness discrimination. The same type of analysis hasbeen made for visual acuity for a single black bar on awhite background, by Hecht and Mintz9 and forstereoscopic sensitivity by Mueller and Lloyd10 andrecently by Lit.1

The above speculations lead to a hypothetical answerto an important theoretical question. What is thestimulus to accommodation? Figure 3 illustrates apossible answer to this question. Assume that oscilla-tions of accommodation persist in normal everydayseeing. To simplify the illustration these oscillationshave been taken as a simple harmonic motion butthe same basic principles would apply irrespective ofthe type of oscillation involved. When the eye isfocused on an object, these oscillations show thateither an increase or a decrease of dioptric power overand above the optimum focus leads to a decrease inintensity of the image at the critical retinal point x.Suppose now that a minus lens is suddenly introducedbefore the eye. The continuing oscillations of accom-modation would soon make it apparent that now anincrease in accommodation would lead to an increaseof intensity at the point x, while a decrease in accom-modation would lead to a decrease in intensity at thispoint. Consequently the posture of the ciliary musclewould then change in the direction which would lead

theories of visual detection. The present experiments offer noevidence for this question. Equations derived from other theoret-ical formulations would be just as satisfactory for the presentpurposes provided they described the intensity discriminationdata just as well.

8 S. Hecht, J. Gen. Physiol. 18, 767 (1935).9 S. Hecht and E. U. Mintz, J. Gen. Physiol. 22, 593 (1939).10 C. G. Mueller and V. V. Lloyd, Proc. Natl. Acad. Sci. U. S.

34, 223 (1948).11 A. Lit (personal communication).

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JAFJ =bEl+ J/ (KI) i]2,

March1958 ACCOMMODATION AT VARIOUS LEVELS OF ILLUMINANCE

to an increase in intensity at x. The eye would continueto increase accommodation until a perceptible decreasein intensity of the retinal image at x occurred. At thispoint the dioptric power of the eye would be decreasedagain and the normal oscillations would resume aboutthe new position of optimum focus until some othercondition, such as the removal of the lens or the shiftof the gaze to some other object in space, wouldreactivate the entire process.

Such a formulation is an obvious simplification.Undoubtedly other factors (such as chromatic aberra-tion) play a part in the process at least for certainsubjects. 2 Nevertheless, it is a possible explanation forthe response in subjects for whom chromatic aberrationis not the only factor. It would explain why Finchamwas unable to produce a sudden shift in the posture ofaccommodation by the sudden interchange of a blurred

12 E. F. Fincham, Brit. J. Ophthalmol. 35, 381 (1951).

target in the field for a clear one. Under the foregoinghypothesis, none would be expected since the oscilla-tions of accommodation would continue to show thatdefocusing either in a positive or negative directionwould still produce a dimmer retinal image at thecritical point x.

SUMMARY

A decrease in the illuminance on a fixation chart wasassociated with a large increase in the variability ofthe successive settings of both an objective and asubjective optometer. On the assumption that theaccommodation is continuously fluctuating a theory isdeveloped which accounts for the limits of fluctuationas a special example of intensity discrimination. Thetheory is a reasonable agreement with the empiricaldata and suggests an additional answer to the question,"What is the stimulus to accommodation"?

197