JOURNAL OF THE OPTICAL SOCIETY OF AMERICA V

The Two-Quanta Hypothesis as a General Explanation for the Behavior of ThresholdValues and Visual Acuity for the Several Receptors of the Human Eye

M. A. BOUMAN AND H. A. VAN DER VELDEN

Physical Institute of the University of Utrecht, Utrecht, Netherlands

(Received January 20, 1948)

With the aid of the three methods for the determinationof the number of absorbed quanta necessary for lightperception described in previous papers, it is found fromextensive threshold measurements with flashes for fovealand peripheral vision that a "red cone" in the peripheryprobably gives rise to a light impression when two quantaare absorbed in it within a time r.

For the foveal cone systems a light impression is causedfor every wave-length by the absorption of two quantawithin a time and within an angular distance of 2-4minutes. The different kinds of receptors proved to be

capable of reacting in mutual dependence on each other,and the conclusion is drawn that all receptors send a nerveimpulse to the nerve connection when one quantum isabsorbed.

A light impression will occur when a second quantum isabsorbed after the first absorption within sec. within areceptor within a distance D of the first receptor.

Experiments on the visual acuity demonstrate that forall wave-lengths, for foveal as well as for peripheral vision,the dependence of the visual acuity on the intensity agreeswith the two-quanta theory.

I. INTRODUCTION

IT has turned out that for the peripheral rodvision the fluctuating results in determining

threshold values, the dependence of these valueson time and area, and the visual acuity as afunction of intensity could all of them be de-scribecl by the two-quanta explanation of van derVelden1 -3 ; two quanta have to be effectivelyabsorbed in the rhoclopsin of two rods situatedwithin a distance corresponding with a visualangle D of each other and within a time inorder to result in a light perception. Each of theabsorbed quanta gives rise to a nerve impulsein the rod in which it is absorbed, and thesetwo stimuli will cooperate in producing a lightperception only when the conditions just men-tioned are satisfied.

These conclusions were drawn from three in-dependent ways of approach for the determina-tion of the number of quanta k necessary forthe light perception. For none of these threemethods the losses of light by reflection andabsorption in the various parts of the eye haveto be known. One of them consists in measuringthe dependence of the chance of observation of alight flash on the average number of quanta perflash. From the slope of this curve the upper limitof the number k can be deduced with the aid of

I H. A. van der Velden, hysica 11, 179 (1944).2 H. A. van der Velden, Opthalmologica 111, 321 (1946).3 M. A. Botuman and H. A. van der Velden, J. Opt. Soc.

.\m. 37. 908 (1947).

the well-known Poisson formula. When k provedto be not equal to one, the exact determination isnot possible with this method as the chance fora light perception will also depend on the casualtime between the two absorbed quanta in theflash and the casual distance of the rods in whichthey are absorbed: When the time between thetwo absorptions increases the chance for a lightperception will not suddenly fall from a constantvalue to zero, and the same is true with regardto the distance. For these reasons the slope ofthe curves according to Poisson's formula willbecome steeper so that the real number of quantak cannot be deduced from the slope of the ex-perimental curves and only an upper limit for kis obtained. It is evident that the agreement withthe theoretical curve for the number k will beclosest when the time of the flash as well as thevisual angle are as small as possible.

In a previous paper3 it was shown that thetwo other methods for the determination of kare free from these complications. The thresholdvalues N60 prcent (which we defined as the averagenumber of quanta of the flash for which thechance of observation is 60 percent) as a func-tion of the flash time t for small visual angles forthe case k = 2 is proportional to the root squareof t when t>r and independent of t when tr.

Foor te general case No perceit is proportionalto tk-i/k when t is large. From the experimentsaccording to this second method for rod vision

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VOLUME 38. NUMBER JULY. 1948

BEHAVIOR OF THRESHOLD VALUES

70 nasal from the fovea for 5100A, k proved tobe 2 and about 0.03 sec.

The threshold values N60 percent for shortflashes as a function of the visual angle d is pro-portional to d2(k-1)/k when d>>D and almostindependent of d when d<D. Again k=2 wasfound with this third method and D was about12'.

The slope of the probability curve yielded bythe first method for the smallest visual anglesand shortest flash times agreed, moreover, fairlywell with k = 2.

In accordance with these facts the visualacuity has to be proportional to the reciprocalvalue of the intensity, so long as the details ofthe -figure--that 4nust be recognized are greaterthan the distance within which the stimuli ofthe absorbed quanta can cooperate to a singlelight perception. For the threshold of recognitionwe chose the threshold for which the chance ofrecognition was again 60 percent. We found thatthe visual acuity for the peripheral rod visionobeys indeed the two-quanta case.

In this paper we present the experiments aboutthe threshold values and visual acuity for variousother wave-lengths for peripheral as well as forfoveal vision, both for cones and rods.

II. THE EXPERIMENTAL ARRANGEMENT

In Fig. 1 we give the experimental arrange-ment. The tungsten ribbon filament lamp L isfocused on the slit of a Hilger monochromatorwith a prism of constant deviation. The secondslit is focused by the lens a on the artificialpupil p of 2-mm diameter. The visual anglecould be changed by diaphragms b in front ofthe lens a. The time of observation could beadjusted by a disk c driven by a synchronousmotor. The range of wave-lengths let through bythe monochromator never exceeded 150A. Forthe peripheral as well as for the foveal measure-ments the right eye of M.A.B. was used. Thiseye was fixed peripherally by means of a redfixation light observed with the fovea of the lefteye, in such a way, that the spot of the retinawith which the flashes were seen was 70 nasalfrom the fovea of the right eye. For fovealmeasurements a light source fixed by the leftor right eye is of no use as the flashes would thenappear on the same place of the visual field as

the fixation light. We used for the foveal meas-urements a ring concentric with the place of theflashes and provided with two arrows, as shownin Fig. 1. This figure was illuminated with whitelight as weak as possible so that it was justvisible to the same eye, with which the measure-ments were performed. The diameter of the ringwas about 7 degrees so that its image activatedthe peripheral rods. The observer tried to fixas well as possible the center of this figure. Inthis center the flashes were seen. As regards thefoveal measurements the fixation figure and theflashes were seen in this way with the same eyeso as to prevent deviations in fixation by thenon-parallelism of the two eyes. The intensityof the flashes was varied by means of the currentof the tungsten ribbon filament lamp, which wascalibrated in the way developed and used byOrnstein. 4 The energy at the pupil p was meas-ured with a thermopile calibrated for absolutevalue. The eye of the observer was always com-pletely dark-adapted before the experiment wasstarted.

III. THE PERIPHERAL THRESHOLDMEASUREMENTS

In all previous papers the experimental cir-cumstances were such that only the peripheralrods were stimulated. By varying the wave-length from the extreme blue to the extreme redwe pass from rod vision to cone vision, so thata region of wave-lengths will exist in which thesensitivity of the two systems is of the sameorder of magnitude. Outside this region the

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4 L. S. Ornstein, W. J. H. Moll, and H. C. Burger, Ob-jective Spektral Photometrie Sammlung Vieweg, Braun-schweig, 1932; L. S. Ornstein, J. Opt. Soc. Am. 20, 573(1930); L. S. Ornstein, Mlle, J. G. Eymers and M. D.Vermeulen, Revue d'optique 12, 385 (1933).

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slope of the probability curve giving the chanceof observation as a function of the averagenumber of quanta per flash will on the blue siderefer to the rod system and on the red side tothe cone system. As already pointed out in theintroduction and in reference 3, we can onlydeduce from this slope the upper limit of thenumber of quanta k necessary for the two sys-tems. It is possible that the slopes of the proba-bility curves do not agree, even when the twonumbers k are equal. Indeed, the time and dis-tance within which the k quanta must be ab-sorbed may differ for the two systems so thatthe deviations from the theoretical slope ac-cording to Poisson may differ for the two systemsfor reasons mentioned in the introduction.

A question of great importance is whether ornot the several systems react independently. Inother words, will the absorption of N1 quanta bythe first system and N2 quanta by the secondone result in a light perception or not when bothN1 and N2 are smaller than the numbers k re-quired by each system separately?

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FIG. 2. The chance of ob-servation W(V,t,d) as a functionof N for various wave-lengthsand for different combinations ofd (12' and 85') and t (0.025, 0.2,0.85 seconds). The theoreticalslopes for the 1, 2, 3, and 4quanta cases according to Pois-son's law are submitted (dottedlines). The level of energy foreach wave-length is indicatedalong the abscissa.

The slope of the probability curve in the regionwithin which the sensitivity of the two systemsare of the same order will depend on these curvesfor the two systems separately. In the case ofindependence, so that the stimuli of the systemscannot cooperate to produce a light impressionin part of this region, the slope of the resultingprobability curve will be steeper than those ofthe two curves of the separate systems. In thecase of dependence, so that the stimuli of thetwo systems can cooperate in producing a lightimpression, the resulting slope will be inter-mediate between the slopes of the individualsystems.

Figure 2 shows the probability curves forvarious combinations of flash time and visualarea, related to four wave-lengths, the dark-adapted right eye, 7 nasal from the fovea.Besides the slight increase of the slope with in-creasing time or visual angle which was to beexpected,3 it appears fromn Fig. 2 that the curvesare steeper in the red end of the spectrum.

In one of the graphs of Fig. 2 we plotted also

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BEHAVIOR OF THRESHOLD VALUES

FIGs. 3a and 3b. The averagenumber of quanta, necessary fora chance of observation of 60percent as a function of t and d,respectively, for various wave-lengths. The theoretical slopesfor the 1, 2, 3, and 4 quantacases are submitted for d>>Dand t>>r (dotted lines).

(a) (b)

the theoretical slopes according to Poisson's law.We can deduce along the lines given in the intro-duction and in reference three that for rod visionthe upper limit of the number of effectively ab-sorbed light quanta k necessary for the lightperception is 2. From the previous papers weknow that k is 2. For cone vision this upper limitis obviously higher, namely, about 3. As we canexpect that at about 6000A the rods and conesare almost equally sensitive, the slopes of thecurves in this region can give information aboutthe dependence of the different systems. It isnoticed that the slopes are not steeper comparedwith the slopes for 7000A and 5100A. If theywere independent, the slope should be steepercompared with the slope for 7000A as well as for5100A. Although the reliability of the measure-ments is rather poor for drawing this conclusion,it is likely that the systems are not independentat the threshold of vision.

In Section V we shall use a much more appro-priate method by which this conclusion is con-firmed.

For the exact determination of k we mustmake use of the two other methods indicated. 3

Figure 3 shows the threshold values N60 percent

for various combinations of flash time andvisual angle. Regarding the curves as a functionof time of Fig. 3a we see that the shape is almostthe same for all wave-lengths. The slope for tvery large agrees for all wave-lengths with k = 2,as for this case N60 percent has to be propor-

tional to t'. The slope certainly doesn't agreewith k=3, as proportionality with t would berequired.

The curves of Fig. 3a do not show the existenceof the two separate systems because the differ-ences in the time value at which the N60 percent

from being independent of t alters into a functionof I, proportional to t0, is very small and mightbe due to the unavoidably limited accuracy ofmeasurements of this kind.

We can conclude that for the rod and conesystem the number k is 2 and these two quantamust be absorbed in a time, which is the samefor the two systems, and is equal to about0.04 sec.'-3

The slope of the threshold curves N6 0 percent asa function of the visual angle d of the flashes ofFig. 3b when d is large agrees again for all wave-lengths with k = 2, since N60 percent is proportionalto d. Using the method of the probability curves(see Fig. 2) we found for 6000A and 7000A anupper limit for k equal to 3 and for smallerwave-lengths 2. This confirms the fact that in theregion of the longer wave-lengths another kindof receptor is stimulated than in the green andblue part of the spectrum. This is further con-firmed by the appearance of a sensation of colorin our threshold measurements in the red regionof the spectrum. For all wave-lengths largerthan 6000A we experience a reddish sensation,for the other wave-lengths a colorless rod sensa-tion. In the behavior of the threshold curves as

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a function of the visual angle we observe a welldefined difference between the rod and the conesystem. The visual angle value at which N60 percent

from being nearly independent of d becomes aquantity proportional to d is almost equal tothe distance D within which the two quanta khave to be absorbed in order to result in a lightimpression.'- It is seen that for the wave-lengths 6000A and 7000A this value becomesabout 3 to 4 minutes, whereas for the other wave-lengths D is about 12 minutes. Such a depend-ence on the wave-length of the distance D withinwhich the two quanta have to be absorbed isonly possible when there are more systems thanone. In the region of wave-lengths within whichthe sensitivity of the two systems is of the sameorder of magnitude the threshold curve as afunction of d will change with wave-length fromthe curve belonging to the rods to the curve ofthe cones. The way in which the threshold curveis changed from being independent of d to beingproportional to d depends not only on this ratio,but also on any possible interaction between the

FIG. 4. The chance of ob-servation W(N,t,d) as a functionof N for various wave-lengthsand for different combinationsof d (12' and 48') and t (0.025,0.2, 0.85 seconds). The theoreti-cal slopes for the 1, 2, 3, and 4quanta case according to Pois-son's law are submitted (dottedlines). The level of energy foreach wave-length is indicatedalong the abscissa.

5 10r'

two systems and, moreover, on the facts bywhich the method of the measuring of the proba-bility curve is not suitable for the exact de-termination of the number k mentioned in theintroduction as these facts can differ for the twosystems. For these reasons it cannot be ascer-tained from the way in which the thresholdcurve is changed from D - 12' to D - 3' whetheror not the two systems can cooperate, nor can theratio between their sensitivities as a function ofwave-length be determined out of this change.

Accepting the results of Qsterberg,5 there areabout 100 rod-like receptors in an area with avisual angle of 10 minutes 7 nasal from thefovea, within which the two quanta must beabsorbed. As the chance is negligible that atthe threshold for rod vision the two quanta areabsorbed in one of this number, a rod reacts onthe absorption of one quantum. The rod willsend a nerve impulse to its nerve connection.Two impulses of two rods within the distance D

5 G. Osterberg, Acta Ophthalm. 13 (Supplement) (1935).

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BEHAVIOR OF THRESHOLD VALUES

mentioned will cooperate and give rise to alight perception.

In a region of 3 to 4 minutes within which thetwo quanta must be absorbed for cone vision thenumber of cone-like receptors is very small at70 nasal from the fovea, namely, about two, ac-cording to Osterberg. 5

So when we identify the rod-like and cone-likereceptors of sterberg with our rod and conesystems, it may be that for the cone system 70nasal since the wave-length is larger than about6500A the two quanta have to be absorbed withinthe self-same receptor. Perhaps by this reasonthe probability curves of Fig. 2 are steeper in thered compared with the slopes in the blue part ofthe spectrum, in agreement with the remarks inthe introduction. Of course, it might be that thenerve fibers of the separate cones are not longenough to reach the nerve-connection of thenext cone as the number of cones is so few in thisregion of the retina. It is clear that in this caseimpulses from two cones cannot cooperate. Thetime within which the two quanta must be ab-sorbed does not differ for the two systems. As forthe rod system, each of the two quanta gives riseto a nerve impulse which mutually cooperatesin the nerve system; it is very probable that eachof the absorbed quanta in a cone for 6500A orlarger also causes a nerve impulse and that thesetwo impulses cooperate in a nerve element ofthe retina similar to the element in which the

FIGS. 5a and 5b. The averagenumber of quanta, necessary fora chance of observations of 60percent as a function of t and d,respectively, for various wave-lengths. The theoretical slopesfor the 1, 2, 3, and 4 quantacases are submitted for d>>Dand tr (dtted lines).

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rod cooperation occurs. If the two quanta ab-sorbed in a cone would give rise to one singlenerve impulse, it is rather strange that the timewithin which the two quanta must be absorbedin the cone agrees with the time mentioned forthe rods, which is of nervous origin. From theexperiments of Section V it is confirmed thatindeed each absorbed quantum arouses a nerveimpulse.

IV. THE CENTRAL FOVEAL THRESHOLDMEASUREMENTS

Figure 4 shows probability curves for variouscombinations of visual angle.and flash time forthe central fovea of the dark adapted right eye.Neglecting the possibility that for one or moreof the chosen wave-lengths the sensitivities oftwo or more receptor systems are of the sameorder of magnitude and, moreover, react inde-pendently, we can deduce from these curvesfor the smallest t and d that for all wave-lengthsthe upper limit of the number k is 3. Most ofthe curves are steeper than the Poisson formulafor k=2 requires and less steep than for k=3.In paragraph 5 we shall give the experiments asto whether or not an interdependence betweenvarious cone systems of the fovea exists accord-ing to the method mentioned in the previousparagraph.

For the exact determination of the number kwe again used the two other methods. In Fig. 5

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H. A. VAN DER VELDEN

we give the threshold values N60 percent for variouswave-lengths as a function of time and visualangle. The shape of the curves as a function oftime is again almost the same for all wave-lengths. The slope with time when t is largeagrees with the case k = 2. From these curves theexistence of more systems is not apparent, ashere too the difference in the time value, at whichthe N60 percent shift from being independent of tto the region in which N6 0 percent is proportionalto t0, is very small.

The slope of the threshold curves N60 percent

as a function of the visual angle d of the flasheswhen d is large agrees again for all wave-lengthswith k = 2. Till now we have from our measure-ments no evidence that for the foveal visionmore than one receptor system exists. The slopesof the probability curves as well as the thresholdcurves as a function of time or visual angle donot give any clear indication of the actual exis-tence of more than one system. For the measure-ments of the dependence of N6 0 percent on thevisual angle it is of the greatest importance toapply with the utmost care the necessary cor-rections for the chromatic aberration of the eye.Without these a great dependence on the wave-length of the region of the visual angle d inwhich N6 0 percent is almost independent of d isfound, as without a good correction the illu-minated area of the retina will not decreasebelow a definite value, dependent on the wave-length, when the visual angle of the flash isdecreased. In our experimental arrangement itwas necessary for the right eye of the observerto apply the above correction with a lens of-0.75 diopter for 7000A and with -2.25 for4200A in front of the artificial pupil.

To summarize, we found that for foveal visionagain two quanta must be effectively absorbedwithin a distance of about 2 to 4 minutes andwithin a time of about 0.04 sec. This time agreesonce more with the time found for the peripheralcone and rod system.

Of course it is necessary to measure this timefor more places of the retina, but it seemed thatthe nerve element in which the cooperation of thetwo impulses of the two absorbed quanta occursis for all receptor systems for the whole retinaof the same kind.

It is not possible to conclude from our foveal

measurements that for foveal vision the twoquanta must be absorbed in the same receptor.In a region of about 2 to 4 minutes there are,according to Qsterberg, 20 to 90 receptors inthe central fovea. The chance that the twoquanta are absorbed in two different receptorsis not negligible, even down to the smallestareas dealt with. The area of the retina coveredby the flash will be enlarged by the size of theAiry diffraction disk. For the experimental ar-rangement described in Section II we must addto the size of the flashes 2.3 to 1.3 minutescaused by diffraction. In order to test whetherthis influences the region of d in which N60 percent

is almost independent of d, we made an arrange-ment for which the diffraction was smaller, andonly about 0.8 to 0.4 minutes had to be addedto the size of the flashes computed purely geo-metrically. We found no influence on the regionof d within which N6 0 percent is almost independ-ent. When the various kinds of receptors are notevenly distributed over the fovea, as suggestedby Hartridge, 6 but form clusters of cones of thesame kind, the distance D seemed indeed to belarger. When the area of the retina covered by theflash is smaller than such a cluster, a part of theflashes for a certain wave-length can fall on re-ceptors which are insensitive to the wave-lengthemployed. For this reason for the smallestangles N60 percent would become higher than agreeswith a homogeneous distribution.

There might be another reason by which theD of the threshold measurements is influenced,namely, the possibly small involuntary move-ments of the eye which always occur even whenthe observer tries to fix his eye as well as possible.By these movements the -absorption of twoquanta, within a certain area, is impeded andincreases the threshold when the visual angle ofthe flash is of the same order of magnitude asthe size of such an area. If we accept the resultsof sterberg, for peripheral cone vision in thered region the two quanta must probably beabsorbed in one cone. So it might be that inthe fovea too this condition must be satisfiedfor these red wave-lengths. In the previous para-graph we also suggested that each quantum ab-sorbed in a cone in the red region may give rise to

6 H. Hartridge, Nature 158, 303 (1946).

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BEHAVIOR OF THRESHOLD VALIUES

a nerve impulse. These two nerve impulses mustbe transmitted by the nerve connection of thiscone within a time of about 0.04 sec. for the lightperception. From the electrophysiology of nervefibers it is known that the impulses can only fol-low each other with intervals greater than thelatency period. These periods are about 10-4 sec-ond. It would seem, therefore, that with decreas-ing flash time the No0 percent will increase for t

values of 10- second. As far as we know, such aneffect has never been found. In our opinion, thisis not an objection to the suggestion that undercertain circumstances (for instance, in the red re-tion of foveal and peripheral vision) two quantamight be necessarily absorbed in one cone for thelight perception and that the two quanta eachcause a nerve impulse in the nerve connection ofthe cone. The way in which the absorption ofa quantum gives rise to a nerve impulse of thecone is practically unknown. In any case, thetime between the absorption of a quantum andthe start of the impulse in the nerve connectionwill not be infinitely small and differ time aftertime by the inevitable natural fluctuations in thecourse of the process transmitting the energy ofthe quantum to the starting of a nerve impulse.If this fluctuation in time is longer than thelatency time of the nerve connection, two quantaabsorbed at the same moment will give rise totwo separate nerve impulses in the nerve fiber.

At the Physiological Conference in Oxford andthe International Conference on Colour Vision inJuly 1947 Hecht dealt with, among other things,his experiments concerning the determination ofthe number k for the absolute and differentialthreshold for cone vision. His conclusion that forthe absolute, as well as for the differential thresh-old, the absorption of four quanta in a cone of thetest spot is necessary, is found on probabilitycurves obtained by the first method described inSection I. As we pointed out at the time, thismethod is not suitable for the exact determina-tion of k, moreover, the case k=4 can certainlynot explain the dependence of the threshold withtime or visual angle. As we found k =2 for theabsolute threshold we also have serious objec-tions to the conclusion as regards the differentialthreshold (intensity discrimination). It is hardto understand how for all intensity levels of thedifferential threshold the behavior will be in

TABLE I. Chance of observation of flashes with visualangle of 25' and time 0.025 second 7 nasal from the foveafor various wave-lengths and combinations of wave-lengths.

W(N) W(N2) W(N,,N2) W(N1N,2) W(N1,N2)experimental theoretical theoretical

independent dependent

6900A, 27% 5320A, 43% 85% 58% 72%6900A, 27% 5320A 30% 80% 49% 62%6900A, 27% 4800A, 25% 64%/, 450 60%6900A, 27% 4620A, 25% 67% 45% 60%6900A, 27% 4400A, 25% 67% 45% 60%6900A, 27% 4200A, 30% 65% 49% 62%6900A, 27% 4700A, 25% 72% 45% 60%6900A, 27% 5060A, 27% 75% 47% 72%6900A, 27% 3900A, 30% 77% 49% 62%6900A, 27% 4500A, 22% 64% 43% 57%6900A, 27% 4500A, 40% 80% 56% 70%

agreement with k =4 except when the surround-ing intensity is zero. Our preliminary experi-ments on the intensity discrimination are inagreement with the suggestion of de Vries,7 ex-cepting for the lowest intensities.

According to de Vries the intensity-dis-crimination is proportional to the reciprocal ofthe square root of the intensity.

V. THE MUTUAL DEPENDENCE OF THERECEPTORS OF DIFFERENT KINDS

IN THE RETINA

Pirenne5 used a method for investigating anypossible mutual influence at the threshold ofvision between the two eyes. We used thismethod in a somewhat different way for in-vestigating whether a mutual dependence existsbetween the receptors of different kinds in thesame eye.

Suppose the chance of observation of a flashpresented separately with an average number ofquanta N1 is W(N1 ) and of another flash W(N 2 ).When these flashes are presented simultaneouslyto the observer the resulting chance of ob-

TABLE 1I. Chance of observation of flashes with visualangle of 25' and time 0.025 second from foveal measure-ments for various wave-lengths and combinations ofwave-lengths.

W(Ni) W(N2) W(N,N2) W(N1N2) W(N,N2)experimental theoretical theoretical

independent dependent

6900A, 32% 4200A, 29% 73% 52% 65%6900A, 38% 4200A, 25% 71% 54% 66%6900A, 38% 5250A, 28% 90% 55% 70%6900A 45% 4000A, 30% 88% 62% 74%6900A, 45% 5100A, 30% 85% 62% 74%6900A, 45% 6000A, 25% 85% 59% 72%

7 H. de Vries, Physica 10, 553 (1943).8 M. H. Pirenne, Nature 152, 698 (1943).

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FIGS. 6a, 6b, and 6c. The average number of quanta from the field between the two black stripes as a function of for different values of the visual angle a for a chance of recognition of 60 percent for some wave-lengths as well for fovealas for peripheral vision.

servation will be W(V1,N2) = 1 - [1 - W(Ni)]X [1-W(2V 2)] for the case that the receptorsystem stimulated by the two separate flashesreact completely independently. As soon as aslight dependence exists, the resulting chancewill be greater.

For the dark adapted eye at 7 nasal from thefovea at the threshold of vision only the conesare stimulated by light of wave-length 7000Aand only the rods by quanta from about 6000Adownward. In the experimental arrangement ofFig. , a glass plate was inserted behind the sec-ond slit of the monochromator. By means of thisplate the light of a second tungsten ribbonfilament lamp L is focused by lens a on theartificial pupil p. The light of L first passedthrough a Shott RG9 filter so that only quanta of6900A and greater are used. Now we first de-termined the current value of L for which thechance of observation of the flash is about 30percent. The visual angle was always 25' and thetime 0.025 second. We next determined the cur-rent values of L for which the chance of observa-tion of flashes of various wave-lengths was againabout 30 percent. By the flashes of L only thecones are stimulated. We now expose the eye tothe flash of L together with the flashes via themonochromator.

The result is that on the same spot of theretina, and at the same time, quanta of twokinds are absorbed, namely, of 6900A and ofanother wave-length, depending on the adjust-ment of the monochronmator.Il Table I we givethe results of these experiments. In the firsttwo columns the chances are given for the com-

ponents of the mixture separately, in the thirdthe measured resulting chance. From the firstand second column the chance is computed forthe case of complete independence, and thesevalues are given in column four. In the lastcolumn the resulting chances are given for thecase where the behavior would have been as ofone system and on the assumption that the slopeof the probability curve agrees with the Poissonformula for the two quanta case. Table I showsthat the resulting chances are always even greaterthan the chances of the last column. As theexperimental resulting chances are greater thanthe values of a complete independence we canalready conclude that there exists mutual de-pendence. We were forced to choose a ratherlarge visual angle, namely, 25'. It is difficult tocheck whether for peripheral vision the quantaof the different wave-lengths fall indeed on thesame spot of the retina when the flashes aresmaller. When for foveal vision the two flashescoincide as regards place, the quanta of the twowave-lengths, if they differ much, are not fo-cused in the same way on the retina when theeye is turned to 7 nasal, as the chromatic devia-tions of the eye are rather large. In order tomake sure that the two flashes coincide, thevisual angle of the flashes is chosen rather largeand the slopes of the probability curves for theseparate flashes will be steeper than agrees withthe smallest visual angle (see introduction and ref-erence 3). The slopes will certainly be steeper thanthe slope of the purely theoretical Poisson curve,so that in the case that there exists mutual de-pendence of the systems the resulting chances

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BEHAVIOR OF THRESHOLD VALUES

can even be greater than the chances of the lastcolumn. From these considerations we can con-clude that the rods and cones react almost com-pletely mutually dependent, as the measuredresulting chances are indeed greater than thechances of the last column. The differences ofcolumns 3 and 5 are in fair agreement with thedifferences between the slopes of the experi-mental probability curves (see Fig. 2) and thePoisson curve for the two-quanta case.

From the preceding paragraphs and papers itis clear that a rod reacts on the absorption ofone quantum and that two rod impulses resultin a light perception when the two quanta con-ditions are satisfied. For peripheral vision in thered region of the spectrum it is probable that twoquanta must be absorbed within one cone.

From the interdependence of the rod and conesystems in the periphery we must conclude thatif one quantum is absorbed in a cone which issensitive in the red region of the spectrum andone other quantum is absorbed in a rod situatedwithin a distance D' of the cone a light impres-sion will result.

In Table II we give the results for the fovealmeasurements. The measured resulting chancesshow that the receptor system sensitive for acertain wave-length reacts in any case in com-plete interdependence with the systems sensitivefor another wave-length. From what is knownconcerning the fundamental response curves,reviewed by Wright, 9 the dependence foundcannot be explained by a possible overlapping of

7 t- I sec. 7 nasal

2 t 17000 A -_

.2__

CS

52

the sensitivity curves of the three systems. Whenthe two systems are completely independentlyreacting at the threshold of vision, the resultingchances measured by us should have been smallerfor most of the chosen combinations of wave-lengths, even for the curves of Hecht. If ourmeasurements were to be explained by a com-plete independence, the three response curvesmust lie closer to each other than Hecht's curves.It turned out to be possible to analyze fromfoveal threshold measurements the curve repre-senting the sensitivity as a function of wave-length in three components lying very close tothe curves of Pitt. We will report on this matterin a later paper. Our components are even some-what smaller than Pitt's curves. At any rate wemust conclude that an almost complete de-pendence exists between the different conesystems, so that also each absorbed quantumarises a nerve impulse.

We summarize here the information concern-ing rod and cone vision obtained from our thresh-old measurements and the investigation abouttheir dependence.

A rod reacts with a nerve impulse in its nerveconnection after the effective absorption of onequantum. A light impression is caused by theabsorption of a second quantum in a rod withina distance D of the first and within a time .

A "red cone" in the periphery gives rise toa light impression when within a time T twoquanta are absorbed in it. The mutual depend-ence of the peripheral rod and "red cone" sys-

ic

4/

I~ t..lWsec. foveal -5 cc 5000A..

2 - -~~[ I-

l � SO I I I 10 I- I' I'- --'162' -

10- 2 S 2 5 i10i6 2 5 10" 02 O 2 5

(a) (b) (c)

FIGS. 7a, 7b, and 7c. The reciprocal of the visual angle a (visual acuity) as a function of 2, the brightness expressed inthe average number of quanta per second per square area (located on the retina) with a visual angle of 10-2 radian. Theshape of the theoretical curves are given for foveal vision, using the shape of the threshold values as a function of thevisual angle from the two-quanta theory.

I W. D. Wright, Researches on Normal and Defective Colour Vision (Kimpson, London, 1946), p. 371.

79

ic

50 M. A. O IT AN AND T1. A. VA N DER VELD EN

tems can only be explained by each absorbedquantum in a cone giving rise to a nerve impulsein the nerve connection of the cone and by onerod stimulus which can cooperate with one conestimulus to produce a light impression. It is veryprobable that also for this kind of cooperationthe quanta must be absorbed within the time T

and the distance D.For the foveal cone systems for every wave-

length a light impression is caused by the ab-sorption of two quanta within a time T and adistance D1 of 2-4 minutes.

The foveal cone systems sensitive for a certainwave-length react at the threshold of vision incomplete mutual dependence with the systemssensitive for another wave-length.

A cone of a certain kind reacts with a nerveimpulse to its nerve connection after the effectiveabsorption of one quantum. A light impression iscaused by the absorption of a second quantumwithin *r in the same or another cone, whichmust lie within a rather small distance from thefirst one. It might be possible that for somewave-lengths the actual distance within whichthe two quanta must be absorbed are smallerthan for other wave-lengths, as the distance Dof the measurements ranges from 2 to 4 minutes.For these wave-lengths it is possible that the twoquanta must be absorbed within one cone.

A rod stimulus can cooperate with any othersecond stimulus within a time 7 and a distance Dto produce a light impression.

It might be possible that the color sensationof a light perception of a "mixed stimulus" candiffer definitely from the sensations of the stimuliof the separate systems. In the region withinwhich the sensitivity of two of the systems is ofthe same order of magnitude and very smallareas of the.retina are illuminated at the thresh-old of vision, statistical fluctuations will occurin the color of the light perception as the colorwill depend on the kinds of receptors in whichthe two quanta are absorbed. We have startedexperiments on this subject together with G. tenDoesschate. It seemed likely that for certainwave-lengths short and small flashes can giverise to more than three definitely differing colorsensations. 0

A certain kind of flash, for instance, is some-times called yellow, sometimes blue or white or

green. A further investigation is in the course ofbeing carried out.

VI. THE VISUAL ACUITY

In a previous paper' we pointed out that thevariation of the visual acuity with the intensityfor peripheral rod vision could be described withthe two-quanta explanation so long as the de-tails of the figure presented for recognition tothe observer are larger than the areas withinwhich the separate stimuli can cooperate. Ac-cording to this the visual acuity in this regionwas proportional to the intensity. We investi-gated this also for a few other wave-lengths forperipheral as well as for foveal vision. The re-sults are shown in Figs. 6 and 7.

For the threshold of recognition of the figuregiven in Fig. 6 we chose again the averagenumber of quanta 16 pCn~t of the area situatedbetween the two black stripes for which thechance of recognition was 60 percent. In Fig. 6the measurements for various flash times aregiven; Fig. 7 gives the reciprocal values of thedistance between the two black stripes as afunction of the intensity at the threshold ofrecognition.

All curves considered as functions of time aswell as of intensity agree fairly well with thetwo-quanta case. For t>O, I sec. the slopes inFig. 6 are proportional to the square root of tand for a large, the visual acuity is proportionalto the intensity.

As soon as a is increased to values of about25', 7 nasal, and to 3' foveal,11 deviations fromthe two-quanta relations occur. In a previouspaper3 it was proved for the rod system thatwithin an area 0 all stimuli contribute to thebrightness impression once a first couple ofquanta is absorbed. This area proved to corre-spond indeed to a visual angle of about 25'.

For the rod system the distance of 25' is al-most twice as large as the distance D withinwhich two quanta must be absorbed for the lightperception. Now it turned out that for the pe-ripheral cone system too this distance is 25',whereas the distance D within which the twoquanta for the light perception must be ab-sorbed is very much smaller, namely, about 3

10 The data for 3' foveal are similar to those for 26' ofFig. 6a.

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BEHAVIOR OF THRESHOLD VALUES

minutes. An area with a visual angle of 25'acts, therefore, indeed as a recipient unit at 7°nasal from the fovea in the observer's eye. Inthe fovea this unit is obviously much smaller.

Pirenne" also performed some measurementson the visual acuity and determined what theintensity of a very large screen must be so thata round black spot in the middle of it is seen in50 percent of the cases.

So long as the visual angle of the black spotis large compared with the size of the recipientunit, we must expect that it is "seen" whenparts of the surrounding area are seen. It wouldbe stretching the point too much to deduce fromthe slope of the resulting curve for large visualangle the number k required for the light per-ception. In the experiments of Pirenne the centerof the screen was seen 20° out. The visual angleof the dark spot in the region of large anglesranges from 30° to 3°. It is quite certain that the

11 M. H. Pirenne, Proceedings of the Cambridge Philo-sophical Society 42, 78 (1946).

sensitivity of the retina differs very much oversuch large angular distances. Moreover, in ouropinion, realizing the presence of a black spot insuch large surroundings is not very suitable forinvestigating the number of quanta required forthe light perception. For that purpose one mustchose the most simple arrangement. Concerningthe break in the curve of the visual acuity, wedid not find this in our visual acuity measure-ments, but especially in the region of smallvisual angles are Pirennes data more complete.In some of our preliminary experiments onbrightness discrimination considered as a func-tion of intensity we found a break. It might bepossible that the cause for this break and forthe sudden bend in the curve of Pirenne are thesame. We are collecting further informationabout this matter. We have some reason tobelieve that breaks of this kind are partly due topsychological aspects of vision and partly toaspects mentioned by Pirenne in his discussionof the break.

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