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A technique for estimating the contribution of photomechanical responses to visual adaptation
A TECHNIQUE FOR ESTIMATING THE CONTRIBUTION OF PHOTOMECHANICAL RESPONSES TO VISUAL ADAPTATION MANDYAM V. SRINIVASAN* and GARYD. BERNARD Department of Ophthaimology and Visual Science. Yale University School of Medicine, 333 Cedar Street. New Haven, Connecticut 06510. U.S.A. (Receired I3 Ju~~.l979) Abstract-This paper presents a technique for isolating and quantifying the contribution of photomechanical responses to visual adaptation. The technique is developed in the context of the pupillary mechanism that is active in the primary visual cells of the insect eye. Using a feedback-control model to represent the combination of retinular cell and pupil, it is shown that the effect of pupillary movements on retinal illumination can be inferred by analysing an intensity-response function of the pupil. When the technique is applied to the pupils of the butterfly and the fly, the results indicate that, in each case. the pupil decreases retinal illumination by approximately 0.7 log units when the ambient light level is increased from pupillary threshold to a level 2.5 log units higher. The validity of the technique is examined by applying it to the human pupil. The results predict changes of retinal illumination which are in close agreement with those expected on the basis of changes in iris diameter, including the Stiles-Crawford effect. The procedure presented here is simple and can, in principle, be applied to many forms of photom~hani~l adaptation.
INTRODUcT1ON
The eye employs several mechanisms to adapt to changes in the ambient intensity of light. The peripheral mechanisms are (i) photomechanical adap tation, including pupillary responses, (ii) adaptation of photoreceptor membrane, (iii) photochemical adaptation and (iv) range fractionation, as in duplex retinae. These adaptive mechanisms, acting in combination with those of higher-order neurons, enabte vision at Ii~t-levels that vary over a range of ten decades (rev. Barlow, 1972). Over the years, considerable interest and effort has been directed toward unravel@ the contribution of each mechanism to visual adaptation. Progress has been restricted, because of the difficulty of measuring the effect of just one mechanism in isolation from others. This paper describes a technique which isolates and quantifies photomechanical contributions to visual adaptation. Photomechanical responses contribute to visual adaptation by regulating illumination of the visual pigment. The optical effects of some forms of photomechanical response can be modelled in a relatively straightfo~rd manner, and their contribution to visual adaptation can be estimated on a theoretical basis. For example, the pupil of the human eye regulates retinal illumination just as the variable aperture of a camera regulates illumination of the photographic film. Thus, the contribution of the human
* Present address: Departments of Neurobiology and Applied Mathematics, Australian National University, Canberra. A.C.T. ZYXIO, Australia
pupillary response to adaptation can be easily estimated from the change in diameter of the iris. However, there are many forms of photomechanical response which are not as amenable to optical modelling as the human pupil. For example, in the pupils of some insect eyes, regulation of light is mediated by movement of granules of pigment within the retinular cells (Kirschfeld and Franceschini, 1969; Menzel and Lange, 1971; Brunnert and Wehner, 1973; Stavenga; 1975a; Franceschini, 1975). The effect of these movements on retinal illumination is governed by a complex opti~l-waveguide phenomenon for which important parameters such as refractive index and absorbance are not accurateiy known (see, for example, Snyder and Menzel, 1975). As another example, photoreceptors of many visual systems undergo changes in size, shape, or position when illuminated by light (e.g. Walcott, 1969; White and Lord, 1975; Blest and. Day, 1978). It is difficult to theoretically estimate the optical effects of these morphological changes. Here we present a general technique for estimating the contribution of photomechanical responses to visual adaptation. The theory behind the technique is developed in the context of the pupil of insect eyes, but the technique itself is applicable to many visual systems, and to many forms of photomechanical response. In most diurnal insects, the pupillary function is mediated by pigment granules that migrate within the retinular cells. When the eye is thoroughly darkadapted, the granules are dispersed throughout much of the cytoplasmic volume. When the eye is illuminated, the granules move intraceilularly and congregate
next to the rhabdom, at its distal end. By doing so. they reduce the fraction of light available for absorption by the photopigment, which is located within the rhabdomere formed by the retinular-cell membrane. The pupillary function is a complex optical phenomenon, not completely understood at the present time. The current concept is that the pupiliary granules reduce the light available to the photopigment by bteeding light from the rhabdom, thereby impairing its effectiveness as an optical waveguide. It is believed that the granules accomplish this by (a) raising the average refractive index of the medium surrounding the rhabdom and (b) absorbing and scattering the evanescent light which propagates along the outside of the rhabdom (Kirschfeld and Franceschini. 1969: Snyder and Horridge, 1972; Stavenga er a!., 1973). The pupil responds quite rapidly to changes of illumination: in flies and butterflies, the response has a time constant of only a few seconds (Franceschini, 1972; Bernard, 1973; Stavenga et al.. 1977). It is known that pupiliary movements are associated with some parameter of the lint-evoked response of the retinular cell (Franceschini. 1972; Stavenga. 1975b; Bernard and Stavenga, 1977), although it is not yet clear as to which particular parameter, or combination of parameters (e.g. release of calcium ions, change of membrane conductance, sodium-ion influx. electrical depolarization, etc.) is actually driving the
pupillary response. When the intensity of light that illuminates the eye is increased, the depolarization of the retinular cell increases. followed by a pupillary movement which partially counteracts the increased retinal illumination. This action is the analo_eue in the insect eye of the pupillary reflex in humans. THEORY
The combination of retinular cell and pupil can be viewed as a feedback control system (Franceschini. 1975; Stavenga, l97Sa). as shown in Fig. I. We characterize the optical effect of the pupil by a filter of variable optical density. interposed between the retinular cell and the incident illumination. The change in stimulation of the retinular cell that is produced by a given pupillary movement is represented by an equivalent change in the optical density of this filter. This representation can be applied to any type of pupillary m~hanism-longitudina1 (as in many insects), iris (as in the human), or otherwise. It provides a unifted way of characterizing the effect of any form of photomechanical response on retinal sensitivity. Defniriort of‘ pupillary respotue The response of the pupil is monitored
by a vari-
Opttcal attenuation Meowed jxpillary response
producedbyp~pll Stimulus intensily 5l
1
I
i AU
/
Optical/
AR
stimrdotion of rertnulor cetl (ktimal Illuminatiori~
Time -
Fig. I. (a) Fe~back-control model of retinular cell and pupil. (b) Schematic illustration of time course of pupifiary response(R) and retinal illumination (L’) evoked by a sudden increase of stimulus intensity (I).
Contribution ol photomechanical able which i2dicates the state of the pUpii, i.e. its degree of closure. For example, in the case of the buttertly eye, we view the deep pseudopupil and monitor the intensity of eyeshine that is produced by a deep-red. subthreshold light-beam (as in Stavenga et al., 1977). In the case of the hoverfly, we view the deep pseudopupil and monitor the light that is backscattered from the distal tips of the retinular cells (as in Franceschini, 1975). The pupillary response is defined as the change in the level of the monitoring variable from that when the pupil is in the fully dark-adapted state. Experimental details are described under “Methods”. We assume that the variable used to monitor the pupillary response has the following properties:
(i) The variable is a monotonic function of pupillary density (ii) Over the range of measurement, changes of the variable are associated with pupillary movements that produce measurable changes of retinal sensitivity. In other words, the combination of retinular cell and pupil functions as a closed-loop system.
responses to adaptation
513
lens and rhabdom.) Within a few tens of milliseconds, the increased retinal illumination causes increased depolarization of the retinular cell. This is followed by closure of the pupil, which settles to a new steadystate value of R + AR, within a few seconds thereafter. As the pupil closes; it reduces the increase of retinal illumination from the initial value kAi to a steady-state value given by k. Ai- TfARf, where T(AR) is defined as the ratio of the steady-state value of the increase of retinal illumination to the initial value of that increase. In general, T(AR) wilI be a nonlinear function with a shape that depends upon both the nature of the pupillary mechanism and the variable that is selected to measure the pupillary response. However, T(AR) is a monotonically decreasing function of AR (by assumption), and T(0) = I. Therefore, when AR is small T(AR) can be approximated by its first-order Taylor-series expansion: T(AR) 2: I - PAR and the increase of retinal illumination steady-state value AU given by AU z kAl(1 - PAR)
Analysis of intensityresponse
function
Assume that we have experimentally obtained a curve which describes the steady-state relationship between the intensity of light incident on the eye, and the response of the pupil. In this section we develop a scheme for analyzing this intensity-response (i-R) curve to quantify the changes of optical density that are associated with pupillary movements. The analysis is developed in two stages. First, we develop a procedure for analyzing a small segment of the I-R curve to determine the vaiue of a parameter P which characterizes changes of pupillary density within this segment. Next, we show how the values of this parameter, obtained by analysing different segments, can be used to compute the relationship between the intensity of light that stimulates the eye, and the optical stimulation of the retinular cell. Using this relationship, it is possible to calculate the changes of density that are associated with pupillary movements. Let I denote the intensity of light that illuminates the eye, R denote the steady-state response of the pupil, and U denote the stimulation received by the retinular cell, which we shall call “retinal illumination”. Consider an experimental situation in which the stimulus intensity has been constant at the level I for a long time, so that the pupillary response has stabilized at a level R. Now, suppose that the stimulus intensity is suddenly increased by a small amount Al. This produces a sudden increase in retinal illumination, AU, proportional to AI. Thus, after. the onset of the stimulus and before the pupil has had time to respond, we have AU = kAi (Fig. I b). (Here k, the constant of pro~rtionality. depends upon the pupillary density at the initiaf steady-state, and upon nonpupillary factors such as the optical properties of the
(1) will have a (2)
The parameter P is equal to (-dT)/[d(AR)], and has dimensions (AR)-‘. The larger the value of P, the greater the decrease in transmission (or increase in optical attenuation) produced by a given pupillary closure AR (P = 0 implies that pupillary closure has no effect on transmission, and P-m implies that a change in transmission has no effect on the response variable.) Equation (2) specifies the relationship between a small change of stimulus intensity and the corr~~nding steady-state change of retinal illumination. ‘Now, consider the relationship between retinal illumination and pupillary response. In general, this relationship will be a nonlinear one. However, a small increment AU of retinal illumination is linearly related to the corresponding increment AR of the pupillary response, provided that AU is sufficiently small. Thus, we express the relationship between U and’R in piecewise-linear fashion by writing AR = bAU
(3)
where the parameter b depends upon U. That is, the value of b depends upon such variables as photochemical state, state of adaptation of the photoreceptor membrane, etc., when the retinal illumination is equal to U. Eliminating AU between (2) and (3), we obtain the relation EAR = k~1(1 - PAR) or AR = where m = WI.
mAi 1 + mPAI
(4)
Equation 1-I)describes the relation between AI and AR over a segment of the I-R CWFC. It shows that the shape of a given segment of the I-R curve depends on the local value of tn. and the local value of P. The values of m and P for a given segment are obtained by analyzing the segment as follows. If equation (-I) is rewritten as AR z-=1
tn + tnPA1
(5)
which shows that the value of the parameter tn is given by the initial slope of the segmental curve. If equation (4) is rewritten as mSIAR
then P is determined from equation (6) by inserting the value of M and the co-ordinates of any point (Af,. AR,) on the segmental curve.* Note that equation (4) is valid only over a se_ment of the f-R curve which is sufhciently small that equations (2) and (3) are valid. ‘Therefore we require k. b and P to be approximately constant over the segment. Thus, we analyze the l-R curve by dividing it into
small segments. and we use the above scheme to determine the value of m and P for each segment. Details of this procedure are described in the “analytical” section under “Methods”. Using the P values, it is possible to compute the steady-state relationship between stimulus intensity (I) and retinal illumination (V). This relationship is derived as follows. Assume that the f-R curve is divided into n segments. Referring to Fig. 2, let the computed value of P be P, in the first segment (0 < I < Ii ; 0 < R ,< R,), P2 in the second segment (Ii < I < I,; R, 4 R < R2) and so on. Consider a situation in which the stimulus intensity is equal to zero for a long time, after which it is suddenly increased to I,. Before the pupil responds, the retinal illumination would be a,Ji, where a0 is an unknown constant of proportionality. After the pupil has settIed down to its new steady-state R,. the reduced retinal illumination is expressed as
U, is the steady-state retinal illumination
(7)
when the
+ ao(lz - [,)[I
do exhibit this property.
For each segment of the I-R curve. the parameter I)I measures the initial slope, and the parameter P is a measure of the curvature of the segment.
R2. the
L2 = a&,[1 - P,Rt] - [,)[I
-
P,R,ICI- PAR, - RI)1 (8)
L,m2 is the steady-state retinal illumination when the stimulus intensity is fz. The process can be repeated for each succeeding segment. The steady-state retinal illumination L’, corresponding to the highest stimulus intensity I,, is expressed as lj” = No i
(f 8 - [i-l)
fI
,= I
[l - Pj(Rj - Rj-L)](91
where I, = 0 and R. = 0. Equations (7) through (9) define the steady-state relationship between L’ and I at the points of junction between successive segments along the intensit>.response curve. We use the same reasoning to derive the relationship between G’ and i at a point that lies within a segment. Consider a point with co-ordinates (I, R). lying within the ith se_gment (ii_, < I < Ii: Ri_ 1 < R c Ri). The relationship between C and I at this point is expressed as Li = Ui-i + UO(f- li) X fi [I - Pj(R, - Rj-I)] j= I
(10)
Thus, given any stimulus intensity-. we use the appropriate equations (7 through IO) to compute the corresponding steady-state retinal iIIumination. We can now calculate the increase of pupillary density when the pupillary response increases from zero (fully open) to any given level R, according to the following procedure: first. use the I-R curve to determine the steady-state stimulus intensity I corresponding to this pupillary response. Nest, use equations (7) through (IO) to calculate the steady-state retinal illumination U corresponding to this stimulus intensity. Finally, use the computed values of L’ and I to calculate the increase of optical density D(I), produced by the pupillary response when the stimulus intensity is increased from 0 to i. This quantity is given by D(I) = log,*
*The parameter P should be positive. to be consistent with our assumption that T(AR) is a monotonically decreasing function of AR (see equation I and the paragraph preceding it). This implies that the ratio of AR to Al should continuously decrease as A/ increases (see equation 5). We find that experimentally measured 1-R curw for the pupil
- P,R,]
After the pupil attains its new steady-state reduced retinal illumination is expressed as
i=l
mAI - AR
li, = a&,[1 - P,R!]
- P,R,]
c’= uJ,[I
+ a,(!,
then
P=
stimulus intensity is 1,. Now. consider a sudden increase of stimulus intensity from I, to IL. Before the pupil responds. the retinal illumination would be
a; !
i
(II)
In equation (I I), a,,1 is the initial retinal illumination when the stimulus intensity is suddenly increased from 0 to 1. 1/ is the retinal illumination when the pupiliary response has attained the steady-state level R corresponding to this intensity. Therefore the Iogarithm of the ratio (a&/V represents the effective increase in optical density due to closure of the pupil.
Contribution of photomechanical responses to adaptation
515
defined as the percentage change in reflectance from the dark-adapted value. The dark period between stimulus flashes is longer than the time required for the reflectance to return to the value for the darkadapted eye. Duty cycles for the butterfly are 4Osec flashes every 4 min. and for the fly are 16 set flashes every I.5 min. The intensity-response function of the pupil is obtained by plotting the steady-state value of the pupillary response as a function of stimulus intensity.
Stmulus lntenslty ( linear scale)
Fig. 2. Illustration of scheme for analyzing the intensityresponse curve. The curve is divided into segments, and the value of m and P for each segment is computed as described in the text.
,METHODS
The pupillary response of the insect is experimentally measured as follows: The intact insect is immobilized with wax and mounted on the universal stage of a Leitz OrtholuxMPV microspectrophotometer. A 10X/0.18 objective is focussed on the deep pseudopupil (Franceschini, 1975). collecting light from about 40 ommatidia for the butterfly and about IO ommatidia for the fly. Two beams axially illuminate the eye through this microscope objective. The first is a measuring beam, to monitor the reflectance of the deep pseudopupil. This beam is from a 45 W tungsten source covered by a 3 mm Schott KG3 heat filter plus either an RG665 (for the butterfly) or an RG630 (for the fly) cutoff filter plus neutral filters of sufficient density that the measuring beam elicits no pupillary response from the fully dark-adapted eye. The second beam provides stimulating flashes of monochromatic light, 5 nm in bandwidth, of adjustable intensity, and of wavelength 540 nm (for the butterfly) or 500 nm (for the fly). The photomultiplier of the MPV is covered with an absorption filter that is transparent to the measuring beam but opaque to the stimulating flashes. The procedure for measuring pupillary responses is to first dark-adapt the eye for at least 1 hr, then turn on the measuring beam, checking to be sure that the reflectance of the deep pseudopupil remains stable at its initial value; this beam remains on during the remainder of the experiment. Responses to stimulating flashes are obtained by adjusting the neutral filter in the stimulating beam, opening the shutter, and waiting until the reflectance stabilizes. The response is
The step-by-step procedure for analyzing the experimentally measured intensity-response function is as follows. Refer to Fig. 2. (I) Plot the I-R curve on a linear-linear scale. Center the origin of the co-ordinate system at the lowest-intensity point on the I-R curve. (2) Measure the slope of the tangent to the curve at the origin of the co-ordinate system. This slope gives M,, the value of the ,n parameter for the first segment. (3) The length of the first segment is determined as follows: Pick a nearby point S, on the curve. and measure its co-ordinates (AI,, AR,). Insert the COordinates and the value of m, into equation (6) to obtain PI. Test this value of P, by inserting m, and P, into equation (4) and checking whether the equation accurately describes the I-R curve between 0 and S,. If the fit is not accurate, then pick a point on the curve that is closer to the origin of the co-ordinate system. and repeat step (3). If the fit is accurate over this range, then determine the maximum intensity 1, (beyond S,) for which the I-R curve can be approximated by equation (4). with m = m, and P = PI. The range (0 < I < I, ; 0 < R < R,) defines the length of the first segment. (4) Shift the origin of the (AI, AR) co-ordinate system to the end point of the first segment. This marks the starting point of the second segment. Determine PZ and the length of the second segment by repeating steps (2) and (3). (5) Proceed along the I-R curve, calculating a P value for each segment, until the high-intensity end of the curve is reached. (6) To calculate D(f), use the procedure described in the paragraph that contains equation (I I). RESULTS Intensity-response curves were measured for the pupil of a butterfly and a hoverfly, and the pupillary density function, D(I). calculated for each case. Figure 3a shows the experimentally measured f-R curve for the pupil of the butterfly, Nynphalis (Aglais) t&cue. Figure 3b shows the steady-state relation between stimulus intensity and pupillary density, computed by analyzing the I-R curve of Fig. 3a. This computation was carried out by drawing a smooth curve through the data points (by eye) and dividing the I-R curve into seven segments. The length of each
M. V. SRINIVASAXand Ci. D. BERNARD
(bl
( a)
a
Log inlensity
Log fntemty
Fig. 3. (a) Experimentally measured intensity-response curve for the butterfly pupil. (b) Steady-state relation between stimulus intensity and pupillary density. computed by analyzing curve of (a).
segment was determined according to the procedure described in the “Analytical” section under “Methods”. Figtire 4a shows the experimentally measured I-R curve for the pupil of the hoverfly, Sq’rpious sp. (2). Figure 4b shows the steady-state relation between stimulus intensity and pupillary density, computed as in the case of the butterfly, except that the I-R curve was divided into eight segments. At each stimulus intensity in Figs 3a and 4a, the standard deviation of the measured response is less than 2%. The largest pupillary densities plotted in Figs 3b and 4b are about 0.85 log units (butterfly) and 0.7 log units (fly). The stimulus intensity in each case is about 2.5 log units above pupillary threshold, where threshold is arbitrarily defined as a 4y.i change of reflectance from the dark-adapted level (butterfiy) or a lo/ change from the dark-adapted level (fly). -‘D Figure 5a shows an intensity-response function for the‘ human pupil. plotted according to an empirical relation published by DeGroot and Gebhard (1952): log,,d = 0.8558 - o.OOO401(log,,l + 8.1)’ where I represents stimulus intensity (mL), and d represents pupillary diameter (mm). In Fig. Sa we have replotted a portion of this empirical relation by defining pupillary response as the decrease in diameter of so
ments over the range l-IOmL (log I = O-1). nine IOmL segments over the range IO-1OOmL (log f = i-2), and nine IOOmL segments over the range
~~~~rnL {fog I = 3-3). The results were checked by repeating the calculation using segments that were smaller by a factor of IO in each range. The second calculation yielded a pupillary density curve which differed from the first curve by less than 3%. Figure 5b also shows, for comparison, two theoretically predicted pupillary-density functions. The broken curve depicts the expected pupillary density if one assumes that the pupillary response is driven pri10
40 if
to)
t
Fig. 4.
the iris, starting from a baseline diameter of 4.4mm corresponding to a stimulus intensity of 1 mL. A steady-state relation between stimulus intensity and pupil~ary density was inferred by app!ying our technique to the f-R curve of Fig. 5a. In this case the computation was not done graphicafly. but rather by a computer program. The program divided the f-R curve into a specified number of segments, calculated the ~ltand P values for each segment by analyzing ths empirical relation, and, finally, computed pupillary density versus stimulus intensity according to our procedure. The result of this computation is shown by the solid curve of Fig. 5b. In computing this curve. the I-R curve was divided into nine 1mL seg-
(a) Ex~imentaliy
* 0 0
0
(bf 08
i
t
measured intensity-response curve for hoverfly pupil. (b) Steady-state relation be!ween stimulus intensity and pupiliary density, compute by analyzing curve of (a].
Contribution
of photomechanical
517
responses to adaptation
07 , i o=t
3. _
la)
Ib)
2-
Lag htmslty
Log nrensity
Fig. 5. (a) Experimental intensity-response curve for human pupil, derived from data compiled ^. .Iby DeGroot and Gebhard (1952). Pupillary response is defined as the decrease tn diameter 01 the ITIS. starting from a baseline diameter of 4.4 mm corresponding to a stimulus intensity of I ml. (b) Solid curve depicts steady-state relation between stimulus intensity and pupiIlary density, computed by analyzing curve of (a) according to the technique described in this paper. Broken curve depicts theoretically predicted steady-state relation between stimulus intensity and pupillary density, assuming that the pupillary response is dominated by rods. Dotted curve depicts theoretically predicted steady-state relation between stimulus intensity and pupillary density, assuming that the pupillary response is dominated by cones. Arrow indicates approximate intensity at which the rod system saturates (1WOcd/m2: Pirenne, 1962). Details in text and Appendix A. marily by rods, and neglects the Stiles-Crawford effect.‘The dotted curve depicts the expected pupillary density if one assumes that the ~pupillary response is
dominated by cones, and takes the Stiles-Crawford effect into account. The derivation of these curves is explained in Appendix A. The solid curve, obtairied by applying the technique described in this paper to the I-R function of Fig. Sa, fails between the theoretical extremes represented by the “rod” and “cone” curves. DlSCUsslON In general, the problem of isolating and measuring of photom~hani~i responses to visual adaptation is difficult. It varies in complexity, depending upon the particular mechanism and upon the particular visual system in which it is studied. In the human eye, for example, changes of pupillary density can be readily estimated from the changes in diameter of the pupil. In the case of the insect pupil, however, the problem is far from trivial. The difficulty stems partly from the optically complex nature of the pupil, which confounds theoretical attempts at estimating pupillary density. Furthermore, the time course of the pupillary response is not very different from those of other, co-existing mechanisms of visual adaptation. This complicates e~ectrophysiolog~cal approaches to the problem. For example, changes of pupillary density cannot be directly related to changes in sensitivity of retinular or higher-order cells, because sensitivity depends on additionaf variables such as neuronal adaptation and photochemical state. the contribution
This paper presents a technique to determine the adaptive contribution of photomechanical responses, and applies the technique to the insect pupil and the human pupil. In developing the technique. we treat the retinular cell-pupil system as a nonlinear feedback-control system with unknown parameters. and show how the parameter of interest, P, can be estimated by analyzing an experimentally measured input-output relationship. The essence of the scheme lies in expressing the light-attenuating effect of the pupil as a multiplicative interaction between output (pupillary response) and input (stimulus intensity), as shown in equation (2). Such a formulation makes it possible to mathematically isolate and measure just that com~nent of the overail non~inea~ty which is attributable to the effect of the pupil. Other nonlinearities, such as those that are involved in the photoreceptor’s transduction process and in the process that mediates pupillary movements, are confined to the forward path of the retinular cell-pupil system. In our formulation, these nonlinearities appear as variations in the estimated value of the parameter b, as we proceed from one segment of the intensityresponse curve to the next. These other nonlinearities will not contaminate the estimated values of P. provided that the P values are determined by analyzing sufficiently small segments of the intensity-response curve. In Appendix B we demonstrate the validity of the analysis developed in the paper by applying it to a hypothetical, mathematically defined pupil model. This exercise also illustrates the magnitude of the errors that can be expected to arise from the graphical procedures required by the analysis. Our technique fails when the actual value of P is
very small (P- 0) or very large (P- G ). The former case represents a situation in which pupillary responses are accompanied by insignificant changes of pupillary density. when the system is, in effect. openloop. The latter case represents a situation in which changes of pupillary density are not accompanied by significant pupiilary responses, as when an inappropriate variable is chosen to measure the pupillary response. In applying the technique. therefore, one has to use some judgement in choosing the monitoring variable and in estimating the range of stimulus intensities over which accurate results can be expected. In our experiments, we used the reflectance of the deep-pseudopupil to monitor the pupillary response. In the case of the butterfly, this reflectance is caused primarily by light which has travelled all the way down the rhabdom, and is reflected by the tapetum (Miller and Bernard, 1968). Since this light makes two traverses through the entire length of the rhabdom, changes in its intensity must necessarily be accompanied by changes of pupillary density. However, saturation of the observed reflectance does not necessarily mean that the pupil has attained the fully-closed state. If the microscope photometer collects too much scattered light from the surrounding pigment cells, it could lead to a P-cc condition in which the pupillary density continues to increase, but is accompanied by unmeasurable decreases in reflectance. We avoided the latter situation by careful optical adjustments and by avoiding extreme intensities. Therefore, we believe that our butterRy results are reliable for stimulus intensities up to 2.5 log units above pupillary threshold. In the case of the fly, changes in the reflectance of the pseudopupil are caused primarily by light which is backscattered by the pupillary granu~~(~rance~hini, 1975). This could conceivably lead to a P--, ~0 condition at low intensities, wherein changes of pupillary density are not accompanied by measurable changes in the intensity of the backscattered light. In addition, a P--+0 condition is possible at high stimulus intensities wherein, for example, the granules are packed so closely around the rhabdom that the pupiiIa~y density has attained its maximum value, and yet the reflectance continues to change with stimulus intensity because granules continue to accumulate in regions outside those which mediate a pupillary effect. We believe that both conditions have been avoided in our experiment. The human pupil affords a good test of the accuracy of our technique because, in this system, the op-
* A stimulus wavelength of 500 nm can, in theory, stimulate the central cell RB in addition to the six peripheral ceils Ri+ (Hardie, 1979). Under our experimental conditions. however, the measured pupillary response was dominated by the contributions from the peripheral cells. A separate contribution, if any, from the two central cells R, and R8 was not detectable.
tical effects of the pupil are sufficiently wellunderstood that one can independently predict, on a theoretical basis, the effect of pupillary movements on retinal i~iumination. This prediction can then be compared with results that are obtained by applying our technique. The solid curve of Fig. 5b shows the resuf: of applying our analysis to the intensityresponse curve of the human pupil (Fig. Sa). In order to make an independent prediction of this pupillary density curve, one has to take into account the fact that pupillary movements have different effects depending upon whether vision is photopic or scotopic. Cone-mediated vision exhibits a StilesCrawford effect. That is. light which enters the pupil through its center stimulates the retina much more effectively than fight which enters the pupil near its periphery. In the case of rod-mediated vision, this effect is very weak; retinal sensitivity to light entering the eye via different regions of the pupil is approximately constant (rev. Crawford, 1972). Therefore, at scotopic luminosities. it should be possible to predict changes of pupillary density in terms of changes in the area of the iris. This prediction is illustrated by the broken curve of Fig. 5b. At photopic luminosities, on the other hand. changes of pupillary density must be estimated in terms of changes of iris area that are weighted by the Stiles-Crawford directionalsensitivity function. This prediction is illustrated by the dotted curve of Fig. 5b. OetaiIs of the calculation are described in Appendix A. The range of intensities displayed in Fig. 5b (I-IOOOmL) lies approximately within the mesopic range. The threshold of the cone system is known to be at about 0.03 mL, or -0.5 log td (Rodieck. 1973, p. 455): this level corresponds to - I.5 log units on the intensity scale of Fig. 5b. The rod system is known to saturate at about 3OOmL or 1OOOcd:m’ (Pirenne. 1962. p. 1653: this level corresponds to +2.5 log units on the intensity scale of Fig. 5b. and is indicated by the arrow. Thus we should expect the actual curve of pupillary density to approximate the theoretical “rod” curve at the lowintensity end of Fig. 5b, and to run parallel to the theoretical “cone” curve at the high-intensity end. Our result (solid curve) substantiates this expectation. thereby reinforcing our confidence in the validty of the technique presented in this paper. and demonstrating its potential for applications elsewhere. In the experiments on the insect pupils, we have chosen the stimulating wavelengths so as to selectively stimulate only one spectral type of retinular ceil-R I _b in the hoverRy.* and the green type in the butterfiy (Bernard, 1976). In the case of the hoverfly. the stimulating wavelength (500 nm) is probably close to that which the pupil absorbs most effectively. The estimated relative-absorbance spectrum of the pupil of the blowffy Calliphora peaks at about 480 nm (Stavenga et ai., 1973). The relative-absorbance spectrum of the butterfly pupil has yet to be accurately measured, but preliminary evidence suggests that it is
Contribution
of photomechanical
broad, with maximum absorption at wavelengths greater than 550nm (Stavenga, 197%. Fig. 3). Our results for the fly and the butterfly indicate that pupillary density increases by approximately one log unit when the pupil moves from the dark-adapted state to the light-adapted state. Since ambient light intensities can vary over ten decades, the lightregulating capability of the pupil is only modest, when reckoned over this range. However, the results demonstrate that the pupil is quite effective in regulating small changes of stimulus intensity, especially when these changes are superimposed upon a large background intensity. For example, the computed pupillary-density curve for the fly (Fig. 4b) shows that a 0.5 log unit increase in stimulus intensity, starting from a level of 2.0 log units, produces a 0.25 log unit increase in pupillary density, resulting in. a net increase in retinal illumination of only 0.25 log units. The pupillary response of individual retinular cells may therefore be a useful mechanism by which different regions of the com~und eye adapt to the spatial variations of intensity that exist within a given visual scene. These intensity variations seldom exceed a range of about one log unit (Wyszecki and Stiles, 1967, p. 185). Another pupillary function could be helping prepare the eye for a return to lower intensities. That is, pupillary closure might reduce the severity of lightadaptation, and, consequently, speed up subsequent dark-adaptation, as has been suggested for the human pupil (Woodhouse and Campbell, 1975). Such a function could be especially beneficial to those insect species which often Ily to and fro between sunny and shady areas. Sup~rting this idea is the observation that such species usually have faster pupils than species that prefer open fields. For example, the pupils of satyrid buttertlies such as Cercionis peg& or nymphalids such as Poiygonia inrerragarionis have time constants of about 1 set, while those of sulfur butterflies such as Colias phiIodice have time constants of at least 5 set (unpublished measurements). Yet another role of the pupillary response may be to modify the visual fields of retinular cells. In many species, it has been demonstrated that closure of the pupil causes a sharpening of the angular-sensitivity functions of retinular cells (e.g. Butler and Horridge, 1973; Hardie, 1979; Beersma, 1979). In conclusion, this paper presents a technique for isolating and quantifying the photomechanical contribution to visual adaptation. The procedure is experimentally simple-requiring measurement of only the intensity-response function-and can, in principle, be applied to many forms of photomechanical response. Acknowledgemenrs-We thank Dr Doekele G. Stavenga for many illuminating discussions during the course of this work, and for a critical reading of the manuscript. This work was supported by Research Grant EYOt140. and Center Grant EY00785.from the National Eye Institute, U.S.P.H.S., and by the Connecticut Lions Eve Research Foundation, Inc. _
responses to adaptation
519
REFERENCES Barlow H. B. (1972) Dark and light adaptation: Psychophysics. In Handbook of Sensory Physiology. Vol. VII/4 (Edited by Jameson D. and Hurvich L. M.). p. I-28. Springer. Berlin. Beersma D. (1979) Spatial characteristics of the visual field of flies. Ph.D. Thesis, Rijksuniversiteit te Grijningen, Netherlands. Bernard G. D. (1973) Rapid nhotom~~ni~1 adaptation in the butterfly eye. Ab&. .&n. Meet. ARVO, p. 47. Bernard G. D. (1976) Snectral sensitivitv of the darkadapted butte&y pupil.’ Abstr. Ann. Mebt. AR VO, p. 17. Bernard G. D. and Stavenga D. G. (1977) The pupillary response of flies as an optical probe for determining spectral sensitivities of retinular cells in completely intact animals. Biol. Bull. 153. 415. Blest A. D. and Day W. A. (1978) The rhabdomere organization of some nocturnal Pisaurid spiders in lighi and darkness. Phil. Trans. R. Sot. 283. l-23. Brunnert A. and Wehner R. (1973) Fine structure of lightand dark-adapted eyes of desert ants, Cataglyphis bicolor, (Formicidae, Hymenoptera). J. Morph. 140, 15-30. Butler B. and Horridge G. A. (1973) The electrophysiology of the retina of Peripfunera americana L.-I. Changes in receptor acuity upon Iight/dark adaptation. J. eomp. Physjol. 83, 263-278.
Crawford B. H. (1972) The Stiles-Crawford effects and their significance in vision. In Handbook of Sensory Physiology, (Edited by Jameson D. and Hurvich L. M.), Vol. VIU4. DD.470-483. Snrinaer. Berlin. DeGroot S. G: and Gebhaid J: W. (1952) Pupil size as determined by adapting luminance. J. opt. Sot. Am. 42, 492-495. Franceschini N. (1972) Pupil and pseudopupil in the compound eye of Drosophila. In Information Processing in the Visual Systems of Arthropods (Edited by Wehner R.), pp. 75-82. Springer, Berlin. Franceschini N. (1975) Sampling of the visual environment by the compound eye of the fly: Fundamental and applications. In P~otorecepror Optics (Edited by Snyder A. W. and Menrel R.), pp. 98-125. Springer, Berlin. Hardie R. C. (1979) El~trophysioIogica~ analysis of By retina. I: comparative properties of Rt_s and R, and Rs. J. camp, Physiol. 129, 19-33. Jacobs D. H. (1944) The Stiles-Crawford effect and the design of telescopes. J. opt. Sot. Am. 34,694. Kirschfeld K. and Franceschini N. (1969) Ein Mechanismus zur Steuerung da Lichtflusses in den Rhabdomeren des Komplexauges von Musca. Kybernetik 6, 13-22. Menzel R. and Lange G. (1971) Anderungen der Feinstruktur im Komplexauge von Formica polyetena bei der HelIadaptation. 2. Naru$26b, 357-359. Miller W. H. and Bernard G. D. (1968) Butterfly glow. J. ~~tr~tr~~t. Res. 24, 286-294.
Pirenne M. H. (1962) In The Eye, Vol. 2. (Edited by Davson H.), chaps i-10. Academic Press, New York. _ Snyder A. W. and Horridae G. A. (1972) The ontical function of changes in the-medium‘surrbunding the cockroach rhabdom. J. camp. Physiol. 81, 1-8. Snyder A. W. and Menzel R. (1975) Photoreceptor Optics. Springer, Berlin. Stavenga D. G. (197Sa) Optical qualities of the fly eye-an approach from the side of geometrical, physical and waveguide optics. In Photoreceptor Optics (Edited by Snyder A. W. and Menzei R.), pp. 126-144. Springer, Berlin. Stavenga D. G. (1975b) Photopigment conversions expressed in pupil mechanism of blowfly visual sense ceils. Narure 253, 740-742. Stavenga D. G. (197%~) Visual adaptation in butterflies. Narure 254,435-437.
M. V. SRIVWASAN
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Stavenga D. G.. Zantema A. and Kuiper J. W. (1973) Rhodopsin processes and the function of the pupil mechanism in flies. In Biochemisrry and Phplology o/‘ Visual Pigments (Edited by Langer H.). Springer, Berlin. Stavenga D. G., Numan J. A. J.. Tinbergen J. and Kuiper J. W. (1977) Insect pupil mechanisms. II. Pigment migration in retinula cells of butterflies. J. camp. PIz+oi. 113. 73-93. Walcott B. (1969) Movement of retinula cells in insect eyes in light adaptation. IVVarure223, 971-972. White R. H. and Lord E. (19753 Dimunition and enlargement of the mosquito rhabdom in light and darkness. J. gen. Physiol. 65, 593-598. Woodhouse J. M. and Campbell F. W. (1975) The role of the pupil light reflex in aiding adaptation to the dark. Vision Res. 15, 649-653. Wyszecki G. and Stiles W. S. (1967) Color Science, p, 185, Fig. i-58. Wiley, New York.
APPEiYDtX
A
and G. D.
BERVARD
portional to the area of the iris, This assumption is valid ii the pupillary response is dominated by rods (Crawford. 1972). Thus. if (i(l) is the steady-state pupillary diameter corresponding to stimulus intensity I. then the rod stimulation is given by L‘ll) = i f (d(l)]:
(A21
Inserting this relation into equation (Al). we obtain the theoretical increase of pupillary density as (A31 The broken curve depicts this function. The dotted curve of Fig. Sb depicts the expected pupillary density if one assumes that the pupillary response is dominated by cones, and takes into account the StilesCrawford effect. Thus. cone stimulation is computed by summing elementary areas of the iris, appropriatel! weighted by the Stiles-Crawford function (Crawford, 1972). Jacobs (1944) has performed this calculation, the result oi which is:
We have derived the theoretical pupiilary density curves for the human eye (broken and dotted curves of Fig. 5b) as follows. Let G(I) denote the steady-state retinal stimulation corresponding to stimulus intensity 1. The increase of pupillary density associated with an increase of stimulus intensity from 1, to I is given by
The broken curve of Fig. 5b depicts the expected pupillary density if one assumes that retinal stimulation is pra-
The quantity in braces represents the net efficiency of cone stimulation; this quantity has a value of 1 for rods. The dotted curve of Fig. 5b depicts the theoretical pupillary density that is obtained by inserting the above relation into equation (A I).
005 lb)
I
2
3
s:: ( cl
0
05
15
Lq&
Fig. 6. (a) Hypothetical pupil modei. used to demonstrate the validity of the graphical analysis developed in the paper. (b) Graphical analysis of I-R function of the model. (cl Solid curve depicts the theoretical relationship between stimulus intensity and pupillary density, calculated as described in the text. Circles indicate the relationship between stimulus intensity and pupillary density. inferred by the graphical analysis.
Contribution APPESDIX
of photomechanical
B
Here we demonstrate the validitv of the analvsis which we have developed in the naner to’infer nunillarv density. We do this by applying the analysis to a hypothetical, mathematically defined pupil model, and comparing the inferred pupillary density with the density that is actually incorporated into the model. The pupil model (Fig 6a) is defined by the following arbitrarily chosen nonlinear relationships. The pupillary response R is related to retinal illumination U according to R=3fl
(Bf)
The retinal illumination is related to stimulus intensity according to rJ = le-O.LR (BZ) The I-R function for the model is derived by eliminating U between equations (Bl) and (BZ):
The increase of ouoillarv densitv. D(I). associated with an increase of stimuk intensity from 1, to I is given by
D(f) = log,, (t)
=
R’eO.lR 9
(B3)
This function is depicted in Fig. 6b. The relationship between U and I for the model is obtained by eliminating R between equations (BI) and (BZ): 1 = ue”-‘&
(B4)
- tog,, (G-j
(B6)
where I and U are related according to (84). Equations (B4) and (B5) together define the theoretical relationship between pupillary density and stimulus intensity for the model. The solid curve of Fig. 6c depicts this relationship for I, = 0.2. If our graphical analysis is valid, it should infer the same relationship. Figure 6b illustrates the application of our graphical analysis to the I-R curve. The curve has been divided into 3 segments. starting from a baseline intensitv In of 0.2. The mea&red values 07 m and the calculated ial& of P for each segment are: m, - 7_.50, P, = 0.24; m, = 1.17, P, = 0.16: m, = 0.79. P, = 0.14. The P values are inserted into equations (7) through (10) to infer pupillary density, as described in the text. The circles in Fig. 6c indicate the values of pupillary density inferred in this way. The results of the graphical analysis agree reasonably well with the theoretical curve. The accuracy of the graphical analysis is, of course, limited by the number of segments used, and by the accuracy with which the m values (slopes) are measured. a
1
521
responses to adaptation
2
s
INTRODUcT1ON
The eye employs several mechanisms to adapt to changes in the ambient intensity of light. The peripheral mechanisms are (i) photomechanical adap tation, including pupillary responses, (ii) adaptation of photoreceptor membrane, (iii) photochemical adaptation and (iv) range fractionation, as in duplex retinae. These adaptive mechanisms, acting in combination with those of higher-order neurons, enabte vision at Ii~t-levels that vary over a range of ten decades (rev. Barlow, 1972). Over the years, considerable interest and effort has been directed toward unravel@ the contribution of each mechanism to visual adaptation. Progress has been restricted, because of the difficulty of measuring the effect of just one mechanism in isolation from others. This paper describes a technique which isolates and quantifies photomechanical contributions to visual adaptation. Photomechanical responses contribute to visual adaptation by regulating illumination of the visual pigment. The optical effects of some forms of photomechanical response can be modelled in a relatively straightfo~rd manner, and their contribution to visual adaptation can be estimated on a theoretical basis. For example, the pupil of the human eye regulates retinal illumination just as the variable aperture of a camera regulates illumination of the photographic film. Thus, the contribution of the human
* Present address: Departments of Neurobiology and Applied Mathematics, Australian National University, Canberra. A.C.T. ZYXIO, Australia
pupillary response to adaptation can be easily estimated from the change in diameter of the iris. However, there are many forms of photomechanical response which are not as amenable to optical modelling as the human pupil. For example, in the pupils of some insect eyes, regulation of light is mediated by movement of granules of pigment within the retinular cells (Kirschfeld and Franceschini, 1969; Menzel and Lange, 1971; Brunnert and Wehner, 1973; Stavenga; 1975a; Franceschini, 1975). The effect of these movements on retinal illumination is governed by a complex opti~l-waveguide phenomenon for which important parameters such as refractive index and absorbance are not accurateiy known (see, for example, Snyder and Menzel, 1975). As another example, photoreceptors of many visual systems undergo changes in size, shape, or position when illuminated by light (e.g. Walcott, 1969; White and Lord, 1975; Blest and. Day, 1978). It is difficult to theoretically estimate the optical effects of these morphological changes. Here we present a general technique for estimating the contribution of photomechanical responses to visual adaptation. The theory behind the technique is developed in the context of the pupil of insect eyes, but the technique itself is applicable to many visual systems, and to many forms of photomechanical response. In most diurnal insects, the pupillary function is mediated by pigment granules that migrate within the retinular cells. When the eye is thoroughly darkadapted, the granules are dispersed throughout much of the cytoplasmic volume. When the eye is illuminated, the granules move intraceilularly and congregate
next to the rhabdom, at its distal end. By doing so. they reduce the fraction of light available for absorption by the photopigment, which is located within the rhabdomere formed by the retinular-cell membrane. The pupillary function is a complex optical phenomenon, not completely understood at the present time. The current concept is that the pupiliary granules reduce the light available to the photopigment by bteeding light from the rhabdom, thereby impairing its effectiveness as an optical waveguide. It is believed that the granules accomplish this by (a) raising the average refractive index of the medium surrounding the rhabdom and (b) absorbing and scattering the evanescent light which propagates along the outside of the rhabdom (Kirschfeld and Franceschini. 1969: Snyder and Horridge, 1972; Stavenga er a!., 1973). The pupil responds quite rapidly to changes of illumination: in flies and butterflies, the response has a time constant of only a few seconds (Franceschini, 1972; Bernard, 1973; Stavenga et al.. 1977). It is known that pupiliary movements are associated with some parameter of the lint-evoked response of the retinular cell (Franceschini. 1972; Stavenga. 1975b; Bernard and Stavenga, 1977), although it is not yet clear as to which particular parameter, or combination of parameters (e.g. release of calcium ions, change of membrane conductance, sodium-ion influx. electrical depolarization, etc.) is actually driving the
pupillary response. When the intensity of light that illuminates the eye is increased, the depolarization of the retinular cell increases. followed by a pupillary movement which partially counteracts the increased retinal illumination. This action is the analo_eue in the insect eye of the pupillary reflex in humans. THEORY
The combination of retinular cell and pupil can be viewed as a feedback control system (Franceschini. 1975; Stavenga, l97Sa). as shown in Fig. I. We characterize the optical effect of the pupil by a filter of variable optical density. interposed between the retinular cell and the incident illumination. The change in stimulation of the retinular cell that is produced by a given pupillary movement is represented by an equivalent change in the optical density of this filter. This representation can be applied to any type of pupillary m~hanism-longitudina1 (as in many insects), iris (as in the human), or otherwise. It provides a unifted way of characterizing the effect of any form of photomechanical response on retinal sensitivity. Defniriort of‘ pupillary respotue The response of the pupil is monitored
by a vari-
Opttcal attenuation Meowed jxpillary response
producedbyp~pll Stimulus intensily 5l
1
I
i AU
/
Optical/
AR
stimrdotion of rertnulor cetl (ktimal Illuminatiori~
Time -
Fig. I. (a) Fe~back-control model of retinular cell and pupil. (b) Schematic illustration of time course of pupifiary response(R) and retinal illumination (L’) evoked by a sudden increase of stimulus intensity (I).
Contribution ol photomechanical able which i2dicates the state of the pUpii, i.e. its degree of closure. For example, in the case of the buttertly eye, we view the deep pseudopupil and monitor the intensity of eyeshine that is produced by a deep-red. subthreshold light-beam (as in Stavenga et al., 1977). In the case of the hoverfly, we view the deep pseudopupil and monitor the light that is backscattered from the distal tips of the retinular cells (as in Franceschini, 1975). The pupillary response is defined as the change in the level of the monitoring variable from that when the pupil is in the fully dark-adapted state. Experimental details are described under “Methods”. We assume that the variable used to monitor the pupillary response has the following properties:
(i) The variable is a monotonic function of pupillary density (ii) Over the range of measurement, changes of the variable are associated with pupillary movements that produce measurable changes of retinal sensitivity. In other words, the combination of retinular cell and pupil functions as a closed-loop system.
responses to adaptation
513
lens and rhabdom.) Within a few tens of milliseconds, the increased retinal illumination causes increased depolarization of the retinular cell. This is followed by closure of the pupil, which settles to a new steadystate value of R + AR, within a few seconds thereafter. As the pupil closes; it reduces the increase of retinal illumination from the initial value kAi to a steady-state value given by k. Ai- TfARf, where T(AR) is defined as the ratio of the steady-state value of the increase of retinal illumination to the initial value of that increase. In general, T(AR) wilI be a nonlinear function with a shape that depends upon both the nature of the pupillary mechanism and the variable that is selected to measure the pupillary response. However, T(AR) is a monotonically decreasing function of AR (by assumption), and T(0) = I. Therefore, when AR is small T(AR) can be approximated by its first-order Taylor-series expansion: T(AR) 2: I - PAR and the increase of retinal illumination steady-state value AU given by AU z kAl(1 - PAR)
Analysis of intensityresponse
function
Assume that we have experimentally obtained a curve which describes the steady-state relationship between the intensity of light incident on the eye, and the response of the pupil. In this section we develop a scheme for analyzing this intensity-response (i-R) curve to quantify the changes of optical density that are associated with pupillary movements. The analysis is developed in two stages. First, we develop a procedure for analyzing a small segment of the I-R curve to determine the vaiue of a parameter P which characterizes changes of pupillary density within this segment. Next, we show how the values of this parameter, obtained by analysing different segments, can be used to compute the relationship between the intensity of light that stimulates the eye, and the optical stimulation of the retinular cell. Using this relationship, it is possible to calculate the changes of density that are associated with pupillary movements. Let I denote the intensity of light that illuminates the eye, R denote the steady-state response of the pupil, and U denote the stimulation received by the retinular cell, which we shall call “retinal illumination”. Consider an experimental situation in which the stimulus intensity has been constant at the level I for a long time, so that the pupillary response has stabilized at a level R. Now, suppose that the stimulus intensity is suddenly increased by a small amount Al. This produces a sudden increase in retinal illumination, AU, proportional to AI. Thus, after. the onset of the stimulus and before the pupil has had time to respond, we have AU = kAi (Fig. I b). (Here k, the constant of pro~rtionality. depends upon the pupillary density at the initiaf steady-state, and upon nonpupillary factors such as the optical properties of the
(1) will have a (2)
The parameter P is equal to (-dT)/[d(AR)], and has dimensions (AR)-‘. The larger the value of P, the greater the decrease in transmission (or increase in optical attenuation) produced by a given pupillary closure AR (P = 0 implies that pupillary closure has no effect on transmission, and P-m implies that a change in transmission has no effect on the response variable.) Equation (2) specifies the relationship between a small change of stimulus intensity and the corr~~nding steady-state change of retinal illumination. ‘Now, consider the relationship between retinal illumination and pupillary response. In general, this relationship will be a nonlinear one. However, a small increment AU of retinal illumination is linearly related to the corresponding increment AR of the pupillary response, provided that AU is sufficiently small. Thus, we express the relationship between U and’R in piecewise-linear fashion by writing AR = bAU
(3)
where the parameter b depends upon U. That is, the value of b depends upon such variables as photochemical state, state of adaptation of the photoreceptor membrane, etc., when the retinal illumination is equal to U. Eliminating AU between (2) and (3), we obtain the relation EAR = k~1(1 - PAR) or AR = where m = WI.
mAi 1 + mPAI
(4)
Equation 1-I)describes the relation between AI and AR over a segment of the I-R CWFC. It shows that the shape of a given segment of the I-R curve depends on the local value of tn. and the local value of P. The values of m and P for a given segment are obtained by analyzing the segment as follows. If equation (-I) is rewritten as AR z-=1
tn + tnPA1
(5)
which shows that the value of the parameter tn is given by the initial slope of the segmental curve. If equation (4) is rewritten as mSIAR
then P is determined from equation (6) by inserting the value of M and the co-ordinates of any point (Af,. AR,) on the segmental curve.* Note that equation (4) is valid only over a se_ment of the f-R curve which is sufhciently small that equations (2) and (3) are valid. ‘Therefore we require k. b and P to be approximately constant over the segment. Thus, we analyze the l-R curve by dividing it into
small segments. and we use the above scheme to determine the value of m and P for each segment. Details of this procedure are described in the “analytical” section under “Methods”. Using the P values, it is possible to compute the steady-state relationship between stimulus intensity (I) and retinal illumination (V). This relationship is derived as follows. Assume that the f-R curve is divided into n segments. Referring to Fig. 2, let the computed value of P be P, in the first segment (0 < I < Ii ; 0 < R ,< R,), P2 in the second segment (Ii < I < I,; R, 4 R < R2) and so on. Consider a situation in which the stimulus intensity is equal to zero for a long time, after which it is suddenly increased to I,. Before the pupil responds, the retinal illumination would be a,Ji, where a0 is an unknown constant of proportionality. After the pupil has settIed down to its new steady-state R,. the reduced retinal illumination is expressed as
U, is the steady-state retinal illumination
(7)
when the
+ ao(lz - [,)[I
do exhibit this property.
For each segment of the I-R curve. the parameter I)I measures the initial slope, and the parameter P is a measure of the curvature of the segment.
R2. the
L2 = a&,[1 - P,Rt] - [,)[I
-
P,R,ICI- PAR, - RI)1 (8)
L,m2 is the steady-state retinal illumination when the stimulus intensity is fz. The process can be repeated for each succeeding segment. The steady-state retinal illumination L’, corresponding to the highest stimulus intensity I,, is expressed as lj” = No i
(f 8 - [i-l)
fI
,= I
[l - Pj(Rj - Rj-L)](91
where I, = 0 and R. = 0. Equations (7) through (9) define the steady-state relationship between L’ and I at the points of junction between successive segments along the intensit>.response curve. We use the same reasoning to derive the relationship between G’ and i at a point that lies within a segment. Consider a point with co-ordinates (I, R). lying within the ith se_gment (ii_, < I < Ii: Ri_ 1 < R c Ri). The relationship between C and I at this point is expressed as Li = Ui-i + UO(f- li) X fi [I - Pj(R, - Rj-I)] j= I
(10)
Thus, given any stimulus intensity-. we use the appropriate equations (7 through IO) to compute the corresponding steady-state retinal iIIumination. We can now calculate the increase of pupillary density when the pupillary response increases from zero (fully open) to any given level R, according to the following procedure: first. use the I-R curve to determine the steady-state stimulus intensity I corresponding to this pupillary response. Nest, use equations (7) through (IO) to calculate the steady-state retinal illumination U corresponding to this stimulus intensity. Finally, use the computed values of L’ and I to calculate the increase of optical density D(I), produced by the pupillary response when the stimulus intensity is increased from 0 to i. This quantity is given by D(I) = log,*
*The parameter P should be positive. to be consistent with our assumption that T(AR) is a monotonically decreasing function of AR (see equation I and the paragraph preceding it). This implies that the ratio of AR to Al should continuously decrease as A/ increases (see equation 5). We find that experimentally measured 1-R curw for the pupil
- P,R,]
After the pupil attains its new steady-state reduced retinal illumination is expressed as
i=l
mAI - AR
li, = a&,[1 - P,R!]
- P,R,]
c’= uJ,[I
+ a,(!,
then
P=
stimulus intensity is 1,. Now. consider a sudden increase of stimulus intensity from I, to IL. Before the pupil responds. the retinal illumination would be
a; !
i
(II)
In equation (I I), a,,1 is the initial retinal illumination when the stimulus intensity is suddenly increased from 0 to 1. 1/ is the retinal illumination when the pupiliary response has attained the steady-state level R corresponding to this intensity. Therefore the Iogarithm of the ratio (a&/V represents the effective increase in optical density due to closure of the pupil.
Contribution of photomechanical responses to adaptation
515
defined as the percentage change in reflectance from the dark-adapted value. The dark period between stimulus flashes is longer than the time required for the reflectance to return to the value for the darkadapted eye. Duty cycles for the butterfly are 4Osec flashes every 4 min. and for the fly are 16 set flashes every I.5 min. The intensity-response function of the pupil is obtained by plotting the steady-state value of the pupillary response as a function of stimulus intensity.
Stmulus lntenslty ( linear scale)
Fig. 2. Illustration of scheme for analyzing the intensityresponse curve. The curve is divided into segments, and the value of m and P for each segment is computed as described in the text.
,METHODS
The pupillary response of the insect is experimentally measured as follows: The intact insect is immobilized with wax and mounted on the universal stage of a Leitz OrtholuxMPV microspectrophotometer. A 10X/0.18 objective is focussed on the deep pseudopupil (Franceschini, 1975). collecting light from about 40 ommatidia for the butterfly and about IO ommatidia for the fly. Two beams axially illuminate the eye through this microscope objective. The first is a measuring beam, to monitor the reflectance of the deep pseudopupil. This beam is from a 45 W tungsten source covered by a 3 mm Schott KG3 heat filter plus either an RG665 (for the butterfly) or an RG630 (for the fly) cutoff filter plus neutral filters of sufficient density that the measuring beam elicits no pupillary response from the fully dark-adapted eye. The second beam provides stimulating flashes of monochromatic light, 5 nm in bandwidth, of adjustable intensity, and of wavelength 540 nm (for the butterfly) or 500 nm (for the fly). The photomultiplier of the MPV is covered with an absorption filter that is transparent to the measuring beam but opaque to the stimulating flashes. The procedure for measuring pupillary responses is to first dark-adapt the eye for at least 1 hr, then turn on the measuring beam, checking to be sure that the reflectance of the deep pseudopupil remains stable at its initial value; this beam remains on during the remainder of the experiment. Responses to stimulating flashes are obtained by adjusting the neutral filter in the stimulating beam, opening the shutter, and waiting until the reflectance stabilizes. The response is
The step-by-step procedure for analyzing the experimentally measured intensity-response function is as follows. Refer to Fig. 2. (I) Plot the I-R curve on a linear-linear scale. Center the origin of the co-ordinate system at the lowest-intensity point on the I-R curve. (2) Measure the slope of the tangent to the curve at the origin of the co-ordinate system. This slope gives M,, the value of the ,n parameter for the first segment. (3) The length of the first segment is determined as follows: Pick a nearby point S, on the curve. and measure its co-ordinates (AI,, AR,). Insert the COordinates and the value of m, into equation (6) to obtain PI. Test this value of P, by inserting m, and P, into equation (4) and checking whether the equation accurately describes the I-R curve between 0 and S,. If the fit is not accurate, then pick a point on the curve that is closer to the origin of the co-ordinate system. and repeat step (3). If the fit is accurate over this range, then determine the maximum intensity 1, (beyond S,) for which the I-R curve can be approximated by equation (4). with m = m, and P = PI. The range (0 < I < I, ; 0 < R < R,) defines the length of the first segment. (4) Shift the origin of the (AI, AR) co-ordinate system to the end point of the first segment. This marks the starting point of the second segment. Determine PZ and the length of the second segment by repeating steps (2) and (3). (5) Proceed along the I-R curve, calculating a P value for each segment, until the high-intensity end of the curve is reached. (6) To calculate D(f), use the procedure described in the paragraph that contains equation (I I). RESULTS Intensity-response curves were measured for the pupil of a butterfly and a hoverfly, and the pupillary density function, D(I). calculated for each case. Figure 3a shows the experimentally measured f-R curve for the pupil of the butterfly, Nynphalis (Aglais) t&cue. Figure 3b shows the steady-state relation between stimulus intensity and pupillary density, computed by analyzing the I-R curve of Fig. 3a. This computation was carried out by drawing a smooth curve through the data points (by eye) and dividing the I-R curve into seven segments. The length of each
M. V. SRINIVASAXand Ci. D. BERNARD
(bl
( a)
a
Log inlensity
Log fntemty
Fig. 3. (a) Experimentally measured intensity-response curve for the butterfly pupil. (b) Steady-state relation between stimulus intensity and pupillary density. computed by analyzing curve of (a).
segment was determined according to the procedure described in the “Analytical” section under “Methods”. Figtire 4a shows the experimentally measured I-R curve for the pupil of the hoverfly, Sq’rpious sp. (2). Figure 4b shows the steady-state relation between stimulus intensity and pupillary density, computed as in the case of the butterfly, except that the I-R curve was divided into eight segments. At each stimulus intensity in Figs 3a and 4a, the standard deviation of the measured response is less than 2%. The largest pupillary densities plotted in Figs 3b and 4b are about 0.85 log units (butterfly) and 0.7 log units (fly). The stimulus intensity in each case is about 2.5 log units above pupillary threshold, where threshold is arbitrarily defined as a 4y.i change of reflectance from the dark-adapted level (butterfiy) or a lo/ change from the dark-adapted level (fly). -‘D Figure 5a shows an intensity-response function for the‘ human pupil. plotted according to an empirical relation published by DeGroot and Gebhard (1952): log,,d = 0.8558 - o.OOO401(log,,l + 8.1)’ where I represents stimulus intensity (mL), and d represents pupillary diameter (mm). In Fig. Sa we have replotted a portion of this empirical relation by defining pupillary response as the decrease in diameter of so
ments over the range l-IOmL (log I = O-1). nine IOmL segments over the range IO-1OOmL (log f = i-2), and nine IOOmL segments over the range
~~~~rnL {fog I = 3-3). The results were checked by repeating the calculation using segments that were smaller by a factor of IO in each range. The second calculation yielded a pupillary density curve which differed from the first curve by less than 3%. Figure 5b also shows, for comparison, two theoretically predicted pupillary-density functions. The broken curve depicts the expected pupillary density if one assumes that the pupillary response is driven pri10
40 if
to)
t
Fig. 4.
the iris, starting from a baseline diameter of 4.4mm corresponding to a stimulus intensity of 1 mL. A steady-state relation between stimulus intensity and pupil~ary density was inferred by app!ying our technique to the f-R curve of Fig. 5a. In this case the computation was not done graphicafly. but rather by a computer program. The program divided the f-R curve into a specified number of segments, calculated the ~ltand P values for each segment by analyzing ths empirical relation, and, finally, computed pupillary density versus stimulus intensity according to our procedure. The result of this computation is shown by the solid curve of Fig. 5b. In computing this curve. the I-R curve was divided into nine 1mL seg-
(a) Ex~imentaliy
* 0 0
0
(bf 08
i
t
measured intensity-response curve for hoverfly pupil. (b) Steady-state relation be!ween stimulus intensity and pupiliary density, compute by analyzing curve of (a].
Contribution
of photomechanical
517
responses to adaptation
07 , i o=t
3. _
la)
Ib)
2-
Lag htmslty
Log nrensity
Fig. 5. (a) Experimental intensity-response curve for human pupil, derived from data compiled ^. .Iby DeGroot and Gebhard (1952). Pupillary response is defined as the decrease tn diameter 01 the ITIS. starting from a baseline diameter of 4.4 mm corresponding to a stimulus intensity of I ml. (b) Solid curve depicts steady-state relation between stimulus intensity and pupiIlary density, computed by analyzing curve of (a) according to the technique described in this paper. Broken curve depicts theoretically predicted steady-state relation between stimulus intensity and pupillary density, assuming that the pupillary response is dominated by rods. Dotted curve depicts theoretically predicted steady-state relation between stimulus intensity and pupillary density, assuming that the pupillary response is dominated by cones. Arrow indicates approximate intensity at which the rod system saturates (1WOcd/m2: Pirenne, 1962). Details in text and Appendix A. marily by rods, and neglects the Stiles-Crawford effect.‘The dotted curve depicts the expected pupillary density if one assumes that the ~pupillary response is
dominated by cones, and takes the Stiles-Crawford effect into account. The derivation of these curves is explained in Appendix A. The solid curve, obtairied by applying the technique described in this paper to the I-R function of Fig. Sa, fails between the theoretical extremes represented by the “rod” and “cone” curves. DlSCUsslON In general, the problem of isolating and measuring of photom~hani~i responses to visual adaptation is difficult. It varies in complexity, depending upon the particular mechanism and upon the particular visual system in which it is studied. In the human eye, for example, changes of pupillary density can be readily estimated from the changes in diameter of the pupil. In the case of the insect pupil, however, the problem is far from trivial. The difficulty stems partly from the optically complex nature of the pupil, which confounds theoretical attempts at estimating pupillary density. Furthermore, the time course of the pupillary response is not very different from those of other, co-existing mechanisms of visual adaptation. This complicates e~ectrophysiolog~cal approaches to the problem. For example, changes of pupillary density cannot be directly related to changes in sensitivity of retinular or higher-order cells, because sensitivity depends on additionaf variables such as neuronal adaptation and photochemical state. the contribution
This paper presents a technique to determine the adaptive contribution of photomechanical responses, and applies the technique to the insect pupil and the human pupil. In developing the technique. we treat the retinular cell-pupil system as a nonlinear feedback-control system with unknown parameters. and show how the parameter of interest, P, can be estimated by analyzing an experimentally measured input-output relationship. The essence of the scheme lies in expressing the light-attenuating effect of the pupil as a multiplicative interaction between output (pupillary response) and input (stimulus intensity), as shown in equation (2). Such a formulation makes it possible to mathematically isolate and measure just that com~nent of the overail non~inea~ty which is attributable to the effect of the pupil. Other nonlinearities, such as those that are involved in the photoreceptor’s transduction process and in the process that mediates pupillary movements, are confined to the forward path of the retinular cell-pupil system. In our formulation, these nonlinearities appear as variations in the estimated value of the parameter b, as we proceed from one segment of the intensityresponse curve to the next. These other nonlinearities will not contaminate the estimated values of P. provided that the P values are determined by analyzing sufficiently small segments of the intensity-response curve. In Appendix B we demonstrate the validity of the analysis developed in the paper by applying it to a hypothetical, mathematically defined pupil model. This exercise also illustrates the magnitude of the errors that can be expected to arise from the graphical procedures required by the analysis. Our technique fails when the actual value of P is
very small (P- 0) or very large (P- G ). The former case represents a situation in which pupillary responses are accompanied by insignificant changes of pupillary density. when the system is, in effect. openloop. The latter case represents a situation in which changes of pupillary density are not accompanied by significant pupiilary responses, as when an inappropriate variable is chosen to measure the pupillary response. In applying the technique. therefore, one has to use some judgement in choosing the monitoring variable and in estimating the range of stimulus intensities over which accurate results can be expected. In our experiments, we used the reflectance of the deep-pseudopupil to monitor the pupillary response. In the case of the butterfly, this reflectance is caused primarily by light which has travelled all the way down the rhabdom, and is reflected by the tapetum (Miller and Bernard, 1968). Since this light makes two traverses through the entire length of the rhabdom, changes in its intensity must necessarily be accompanied by changes of pupillary density. However, saturation of the observed reflectance does not necessarily mean that the pupil has attained the fully-closed state. If the microscope photometer collects too much scattered light from the surrounding pigment cells, it could lead to a P-cc condition in which the pupillary density continues to increase, but is accompanied by unmeasurable decreases in reflectance. We avoided the latter situation by careful optical adjustments and by avoiding extreme intensities. Therefore, we believe that our butterRy results are reliable for stimulus intensities up to 2.5 log units above pupillary threshold. In the case of the fly, changes in the reflectance of the pseudopupil are caused primarily by light which is backscattered by the pupillary granu~~(~rance~hini, 1975). This could conceivably lead to a P--, ~0 condition at low intensities, wherein changes of pupillary density are not accompanied by measurable changes in the intensity of the backscattered light. In addition, a P--+0 condition is possible at high stimulus intensities wherein, for example, the granules are packed so closely around the rhabdom that the pupiiIa~y density has attained its maximum value, and yet the reflectance continues to change with stimulus intensity because granules continue to accumulate in regions outside those which mediate a pupillary effect. We believe that both conditions have been avoided in our experiment. The human pupil affords a good test of the accuracy of our technique because, in this system, the op-
* A stimulus wavelength of 500 nm can, in theory, stimulate the central cell RB in addition to the six peripheral ceils Ri+ (Hardie, 1979). Under our experimental conditions. however, the measured pupillary response was dominated by the contributions from the peripheral cells. A separate contribution, if any, from the two central cells R, and R8 was not detectable.
tical effects of the pupil are sufficiently wellunderstood that one can independently predict, on a theoretical basis, the effect of pupillary movements on retinal i~iumination. This prediction can then be compared with results that are obtained by applying our technique. The solid curve of Fig. 5b shows the resuf: of applying our analysis to the intensityresponse curve of the human pupil (Fig. Sa). In order to make an independent prediction of this pupillary density curve, one has to take into account the fact that pupillary movements have different effects depending upon whether vision is photopic or scotopic. Cone-mediated vision exhibits a StilesCrawford effect. That is. light which enters the pupil through its center stimulates the retina much more effectively than fight which enters the pupil near its periphery. In the case of rod-mediated vision, this effect is very weak; retinal sensitivity to light entering the eye via different regions of the pupil is approximately constant (rev. Crawford, 1972). Therefore, at scotopic luminosities. it should be possible to predict changes of pupillary density in terms of changes in the area of the iris. This prediction is illustrated by the broken curve of Fig. 5b. At photopic luminosities, on the other hand. changes of pupillary density must be estimated in terms of changes of iris area that are weighted by the Stiles-Crawford directionalsensitivity function. This prediction is illustrated by the dotted curve of Fig. 5b. OetaiIs of the calculation are described in Appendix A. The range of intensities displayed in Fig. 5b (I-IOOOmL) lies approximately within the mesopic range. The threshold of the cone system is known to be at about 0.03 mL, or -0.5 log td (Rodieck. 1973, p. 455): this level corresponds to - I.5 log units on the intensity scale of Fig. 5b. The rod system is known to saturate at about 3OOmL or 1OOOcd:m’ (Pirenne. 1962. p. 1653: this level corresponds to +2.5 log units on the intensity scale of Fig. 5b. and is indicated by the arrow. Thus we should expect the actual curve of pupillary density to approximate the theoretical “rod” curve at the lowintensity end of Fig. 5b, and to run parallel to the theoretical “cone” curve at the high-intensity end. Our result (solid curve) substantiates this expectation. thereby reinforcing our confidence in the validty of the technique presented in this paper. and demonstrating its potential for applications elsewhere. In the experiments on the insect pupils, we have chosen the stimulating wavelengths so as to selectively stimulate only one spectral type of retinular ceil-R I _b in the hoverRy.* and the green type in the butterfiy (Bernard, 1976). In the case of the hoverfly. the stimulating wavelength (500 nm) is probably close to that which the pupil absorbs most effectively. The estimated relative-absorbance spectrum of the pupil of the blowffy Calliphora peaks at about 480 nm (Stavenga et ai., 1973). The relative-absorbance spectrum of the butterfly pupil has yet to be accurately measured, but preliminary evidence suggests that it is
Contribution
of photomechanical
broad, with maximum absorption at wavelengths greater than 550nm (Stavenga, 197%. Fig. 3). Our results for the fly and the butterfly indicate that pupillary density increases by approximately one log unit when the pupil moves from the dark-adapted state to the light-adapted state. Since ambient light intensities can vary over ten decades, the lightregulating capability of the pupil is only modest, when reckoned over this range. However, the results demonstrate that the pupil is quite effective in regulating small changes of stimulus intensity, especially when these changes are superimposed upon a large background intensity. For example, the computed pupillary-density curve for the fly (Fig. 4b) shows that a 0.5 log unit increase in stimulus intensity, starting from a level of 2.0 log units, produces a 0.25 log unit increase in pupillary density, resulting in. a net increase in retinal illumination of only 0.25 log units. The pupillary response of individual retinular cells may therefore be a useful mechanism by which different regions of the com~und eye adapt to the spatial variations of intensity that exist within a given visual scene. These intensity variations seldom exceed a range of about one log unit (Wyszecki and Stiles, 1967, p. 185). Another pupillary function could be helping prepare the eye for a return to lower intensities. That is, pupillary closure might reduce the severity of lightadaptation, and, consequently, speed up subsequent dark-adaptation, as has been suggested for the human pupil (Woodhouse and Campbell, 1975). Such a function could be especially beneficial to those insect species which often Ily to and fro between sunny and shady areas. Sup~rting this idea is the observation that such species usually have faster pupils than species that prefer open fields. For example, the pupils of satyrid buttertlies such as Cercionis peg& or nymphalids such as Poiygonia inrerragarionis have time constants of about 1 set, while those of sulfur butterflies such as Colias phiIodice have time constants of at least 5 set (unpublished measurements). Yet another role of the pupillary response may be to modify the visual fields of retinular cells. In many species, it has been demonstrated that closure of the pupil causes a sharpening of the angular-sensitivity functions of retinular cells (e.g. Butler and Horridge, 1973; Hardie, 1979; Beersma, 1979). In conclusion, this paper presents a technique for isolating and quantifying the photomechanical contribution to visual adaptation. The procedure is experimentally simple-requiring measurement of only the intensity-response function-and can, in principle, be applied to many forms of photomechanical response. Acknowledgemenrs-We thank Dr Doekele G. Stavenga for many illuminating discussions during the course of this work, and for a critical reading of the manuscript. This work was supported by Research Grant EYOt140. and Center Grant EY00785.from the National Eye Institute, U.S.P.H.S., and by the Connecticut Lions Eve Research Foundation, Inc. _
responses to adaptation
519
REFERENCES Barlow H. B. (1972) Dark and light adaptation: Psychophysics. In Handbook of Sensory Physiology. Vol. VII/4 (Edited by Jameson D. and Hurvich L. M.). p. I-28. Springer. Berlin. Beersma D. (1979) Spatial characteristics of the visual field of flies. Ph.D. Thesis, Rijksuniversiteit te Grijningen, Netherlands. Bernard G. D. (1973) Rapid nhotom~~ni~1 adaptation in the butterfly eye. Ab&. .&n. Meet. ARVO, p. 47. Bernard G. D. (1976) Snectral sensitivitv of the darkadapted butte&y pupil.’ Abstr. Ann. Mebt. AR VO, p. 17. Bernard G. D. and Stavenga D. G. (1977) The pupillary response of flies as an optical probe for determining spectral sensitivities of retinular cells in completely intact animals. Biol. Bull. 153. 415. Blest A. D. and Day W. A. (1978) The rhabdomere organization of some nocturnal Pisaurid spiders in lighi and darkness. Phil. Trans. R. Sot. 283. l-23. Brunnert A. and Wehner R. (1973) Fine structure of lightand dark-adapted eyes of desert ants, Cataglyphis bicolor, (Formicidae, Hymenoptera). J. Morph. 140, 15-30. Butler B. and Horridge G. A. (1973) The electrophysiology of the retina of Peripfunera americana L.-I. Changes in receptor acuity upon Iight/dark adaptation. J. eomp. Physjol. 83, 263-278.
Crawford B. H. (1972) The Stiles-Crawford effects and their significance in vision. In Handbook of Sensory Physiology, (Edited by Jameson D. and Hurvich L. M.), Vol. VIU4. DD.470-483. Snrinaer. Berlin. DeGroot S. G: and Gebhaid J: W. (1952) Pupil size as determined by adapting luminance. J. opt. Sot. Am. 42, 492-495. Franceschini N. (1972) Pupil and pseudopupil in the compound eye of Drosophila. In Information Processing in the Visual Systems of Arthropods (Edited by Wehner R.), pp. 75-82. Springer, Berlin. Franceschini N. (1975) Sampling of the visual environment by the compound eye of the fly: Fundamental and applications. In P~otorecepror Optics (Edited by Snyder A. W. and Menrel R.), pp. 98-125. Springer, Berlin. Hardie R. C. (1979) El~trophysioIogica~ analysis of By retina. I: comparative properties of Rt_s and R, and Rs. J. camp, Physiol. 129, 19-33. Jacobs D. H. (1944) The Stiles-Crawford effect and the design of telescopes. J. opt. Sot. Am. 34,694. Kirschfeld K. and Franceschini N. (1969) Ein Mechanismus zur Steuerung da Lichtflusses in den Rhabdomeren des Komplexauges von Musca. Kybernetik 6, 13-22. Menzel R. and Lange G. (1971) Anderungen der Feinstruktur im Komplexauge von Formica polyetena bei der HelIadaptation. 2. Naru$26b, 357-359. Miller W. H. and Bernard G. D. (1968) Butterfly glow. J. ~~tr~tr~~t. Res. 24, 286-294.
Pirenne M. H. (1962) In The Eye, Vol. 2. (Edited by Davson H.), chaps i-10. Academic Press, New York. _ Snyder A. W. and Horridae G. A. (1972) The ontical function of changes in the-medium‘surrbunding the cockroach rhabdom. J. camp. Physiol. 81, 1-8. Snyder A. W. and Menzel R. (1975) Photoreceptor Optics. Springer, Berlin. Stavenga D. G. (197Sa) Optical qualities of the fly eye-an approach from the side of geometrical, physical and waveguide optics. In Photoreceptor Optics (Edited by Snyder A. W. and Menzei R.), pp. 126-144. Springer, Berlin. Stavenga D. G. (1975b) Photopigment conversions expressed in pupil mechanism of blowfly visual sense ceils. Narure 253, 740-742. Stavenga D. G. (197%~) Visual adaptation in butterflies. Narure 254,435-437.
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Stavenga D. G.. Zantema A. and Kuiper J. W. (1973) Rhodopsin processes and the function of the pupil mechanism in flies. In Biochemisrry and Phplology o/‘ Visual Pigments (Edited by Langer H.). Springer, Berlin. Stavenga D. G., Numan J. A. J.. Tinbergen J. and Kuiper J. W. (1977) Insect pupil mechanisms. II. Pigment migration in retinula cells of butterflies. J. camp. PIz+oi. 113. 73-93. Walcott B. (1969) Movement of retinula cells in insect eyes in light adaptation. IVVarure223, 971-972. White R. H. and Lord E. (19753 Dimunition and enlargement of the mosquito rhabdom in light and darkness. J. gen. Physiol. 65, 593-598. Woodhouse J. M. and Campbell F. W. (1975) The role of the pupil light reflex in aiding adaptation to the dark. Vision Res. 15, 649-653. Wyszecki G. and Stiles W. S. (1967) Color Science, p, 185, Fig. i-58. Wiley, New York.
APPEiYDtX
A
and G. D.
BERVARD
portional to the area of the iris, This assumption is valid ii the pupillary response is dominated by rods (Crawford. 1972). Thus. if (i(l) is the steady-state pupillary diameter corresponding to stimulus intensity I. then the rod stimulation is given by L‘ll) = i f (d(l)]:
(A21
Inserting this relation into equation (Al). we obtain the theoretical increase of pupillary density as (A31 The broken curve depicts this function. The dotted curve of Fig. Sb depicts the expected pupillary density if one assumes that the pupillary response is dominated by cones, and takes into account the StilesCrawford effect. Thus. cone stimulation is computed by summing elementary areas of the iris, appropriatel! weighted by the Stiles-Crawford function (Crawford, 1972). Jacobs (1944) has performed this calculation, the result oi which is:
We have derived the theoretical pupiilary density curves for the human eye (broken and dotted curves of Fig. 5b) as follows. Let G(I) denote the steady-state retinal stimulation corresponding to stimulus intensity 1. The increase of pupillary density associated with an increase of stimulus intensity from 1, to I is given by
The broken curve of Fig. 5b depicts the expected pupillary density if one assumes that retinal stimulation is pra-
The quantity in braces represents the net efficiency of cone stimulation; this quantity has a value of 1 for rods. The dotted curve of Fig. 5b depicts the theoretical pupillary density that is obtained by inserting the above relation into equation (A I).
005 lb)
I
2
3
s:: ( cl
0
05
15
Lq&
Fig. 6. (a) Hypothetical pupil modei. used to demonstrate the validity of the graphical analysis developed in the paper. (b) Graphical analysis of I-R function of the model. (cl Solid curve depicts the theoretical relationship between stimulus intensity and pupillary density, calculated as described in the text. Circles indicate the relationship between stimulus intensity and pupillary density. inferred by the graphical analysis.
Contribution APPESDIX
of photomechanical
B
Here we demonstrate the validitv of the analvsis which we have developed in the naner to’infer nunillarv density. We do this by applying the analysis to a hypothetical, mathematically defined pupil model, and comparing the inferred pupillary density with the density that is actually incorporated into the model. The pupil model (Fig 6a) is defined by the following arbitrarily chosen nonlinear relationships. The pupillary response R is related to retinal illumination U according to R=3fl
(Bf)
The retinal illumination is related to stimulus intensity according to rJ = le-O.LR (BZ) The I-R function for the model is derived by eliminating U between equations (Bl) and (BZ):
The increase of ouoillarv densitv. D(I). associated with an increase of stimuk intensity from 1, to I is given by
D(f) = log,, (t)
=
R’eO.lR 9
(B3)
This function is depicted in Fig. 6b. The relationship between U and I for the model is obtained by eliminating R between equations (BI) and (BZ): 1 = ue”-‘&
(B4)
- tog,, (G-j
(B6)
where I and U are related according to (84). Equations (B4) and (B5) together define the theoretical relationship between pupillary density and stimulus intensity for the model. The solid curve of Fig. 6c depicts this relationship for I, = 0.2. If our graphical analysis is valid, it should infer the same relationship. Figure 6b illustrates the application of our graphical analysis to the I-R curve. The curve has been divided into 3 segments. starting from a baseline intensitv In of 0.2. The mea&red values 07 m and the calculated ial& of P for each segment are: m, - 7_.50, P, = 0.24; m, = 1.17, P, = 0.16: m, = 0.79. P, = 0.14. The P values are inserted into equations (7) through (10) to infer pupillary density, as described in the text. The circles in Fig. 6c indicate the values of pupillary density inferred in this way. The results of the graphical analysis agree reasonably well with the theoretical curve. The accuracy of the graphical analysis is, of course, limited by the number of segments used, and by the accuracy with which the m values (slopes) are measured. a
1
521
responses to adaptation
2
s