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Visual brightness: Some applications of a model
Yision Res. Vol. 12, pp. 1409-1423. Pergamon Press 1972. Printed
VISUAL
BRIGHTNESS:
in Great Brimin.
SOME APPLICATIONS MODEL’
OF A
LAWRENCE E. MARK.? John B. Pierce Foundation Laboratory and Yale University, New Haven, Connecticut 06519, U.S.A. (Received 3 September 1971; in revisedform 9 December 1971)
paper aims to explore some applications of a relatively simple model (analogous to an electrical filter with feedback) that attempts to account for visual responses to flashes of light. More specifically, the model gives a good description of the way brightness varies with the intensity of flashes presented to the dark-adapted eye. Already established is the capacity of filter models to deal with effects of stimulus duration. An extension of the model enables it to deal also with the parametric effects of other stimulus variables, namely the retinal locus of stimulation and stimulus size. The last part of the paper will look at possible correspondences between elements and operations of the model and anatomy and physiology of the visual system. THE PRESENT
THE
BASIC
MODEL
The
essence of the model is the postulation of a cascade of filter-like stages; the relation between the output from the filter system and input to it will be seen to parallel the brightness response of the visual system to light. Filter systems have been proposed frequently to explain a number of other visual phenomena, such as flicker detection (LEVINSON, 1968; MATIN, 1968; SPERLING and SONDHI, 1968), differential sensitivity and increment detection (MATIN, 1968; SPERLING and SONDHI, 1968), contrast detection (SPERLING, 1970), and temporal gap detection (URAL and HIERONYMUS, 1970). A general input-output equation for any one stage i of the filter is
where J+ is the output from stage i, a1reflects the sensitivity to input of the stage, and y, _ 1 is the input to the stage (i.e. the output from the previous stage). A filter such as that described by equation 1 is analogous to a simple electrical (RC) stage, i.e. a linear integrator with a time constant equal to TV. There is evidence that an RC-stage can provide a representation of a neuron that receives both excitatory and inhibitory inputs (see, for example, KATZ, 1966). Sperling based his model of luminance and flicker detection on an analogy between neuron and RC-stage (SPERLING, 1970; SPERLING and SONDHI, 1968). On the other hand, FUORTES and ’ This study was supported by Grant No. AFOSR 70-1950 from the U.S. Air Force Office of Scientific Research. ’ Address: John B. Pierce Fomdation Laboratory, 290 Congress Avenue, New Haven, Connecticut 06519. “lmm
12/s-G
1409
1410
LAWRENCE E. MARKS
HODGKIN(1964) were unable to find anatomical basis for the large number (average about ten) of filter stages whose postulation was necessary in order to describe electrical responses to light in Limulus ommatidia. The present author tends to treat the filter stages primarily as elements of a functional model whose elements and activities provide a logical and operational role. It is not nceessary to ascribe structures of functions in the model to particular anatomical units. The filter system is a model for psychophysical phenomena, not necessarily a model for neurophysiological data. It is rare indeed to discover a model that is able to make quantitatively accurate predictions both of perceptual and of neural responses. Thus the terminology to be used in the paper, e.g. “excitatory”, “inhibitory”, “feedback”, etc., is aimed primarily at “functional” denotation. On the other hand, in certain places it will be difficult to avoid reference to clear anatomical or physiological correlations. The question of anatomical and physiological correspondence will be looked at specifically at the end of the paper. The critical assumption of the model is that the output of each filter stage is modulated by shunting feedback. To be more precise, we assume that the output from the final stage modifies the output from every prior stage by decreasing the gain and time-constant of the stage (and, therefore, of the system as a whole). Thus each stage may be considered to receive output-dependent, inhibitory (feedback) input as well as excitatory input. This assumption is not original to the present paper. FUORTE.Sand HODGKIN(1964) postulated a similar gain-control mechanism in order to describe electrical responses induced in Limulus. MATIN (1968) and SPERLINGand SONDHI(1968) incorporated such a feedback filter into more complex models that attempt to account for incremental intensity discrimination and flicker detection; SPERLING(1970) later extended his model to take into account spatial summatory and inhibitory influences. These papers utilized filter models to explain light adaptation. The present paper is an extension of the use of a feedback filter, not as a model of light adaptation, but to describe brightness responses to flashes of light presented to the dark-adapted eye. By incorporating shunting feedback into the parameter T[ in equation 1, the general equation for any filter stage becomes
$ Y,
=
QfYf-1 -
hY, (1 + ClY”).
The constant b, reflects the sensitivity of stage i to feedback and is inversely proportional to the time-constant uninfluenced by feedback; c1 gives the fraction of the final output y,, that is fed back to stage i. The overall time-constant of stage i equals [b,(l + c&)1-I. A visual representation of a two-stage filter is shown in Fig. 1. If the input to such a system is a pulse of constant intensity y, = I, then the equations for the two stages are
&Yl=a,/$Y2
= &Yl
-
by,
(1 + qy,)
bzY*
(1 +
C2Y2)
(4)
The primary concern here is the steady-state behavior of the system. In the steady state we obtain for the output from the last stage a31 y2
=
(1
+
ClY2) (1 + CzY2)
(5)
Visual Brightness: Some Applications of a Model
1411
FIG. I. Diagram of a two-stage feedback filter. Each stage receives both excitatory input (+) from the prior stage and inhibitory (feedback) input (-) from the last stage.
where a3 = (u~~~~~(~~~~),In order to simplify for exposition, we let cl = cz = c and set y = cy,. Then (6)
y (1 + yj2 = d where CI= as/c. For the values of y large in comparison to 1 y N (az)“3
(7)
In general, as SPERLING and SONDHI pointed out, a feedback filter of n stages approximates power law behavior (exponent equal to I/[n + 11) in the steady state. Figure 2 shows the input-output function for the steady-state behavior of a two-stage filter (equation 6). At high intensities the function is linear in double-logarithmic coordinates, with a slope
P-
3’ -
IO-
I
I
I
I
0 20 40 60 00 relative intensity (1)in decibels FIG. 2. Input-output function in the steady state for a two-stage filter: y(1 + r)* = I. The dashed curve shows the increase in curvature that obtains at the low end of the function if a -20
threshold parameter (lo = 094) is added.
LAWRENCE E. MARKS
1412
(exponent) equal to l/3. At low levels of intensity the function steepens, and, when y is small in comparison to 1, equation 6 yields y E al.
(8)
The model therefore displays small-signal linearity at low inputs, a property of considerable use and importance in accounting for threshold phenomena such as spatial summation. PSYCHOPHYSICAL
FUNCTION
FOR
BRIGHTNESS
A large number of psychophysical experiments have demonstrated power law behavior in the visual system. In particular, when subjects judge (by means of direct, ratio-scaling procedures, such as magnitude estimation) the brightness of flashes of light presented to the dark-adapted eye, the typical input-output relation is a cube-root function, both for group averages (e.g. J. C. STEVENSand SEVENS, 1963 ; S. S. STFVENSand STEVENS,1960) and for individual subjects (MARKS and STEVENS, 1966). Figure 3 gives an example of
I-
, I :d :
_
o while o red
: 05-
-
:’ -:
o-3
-i 0
I
t
I
IO
20
30
relative
intensity
40
in decibels
FIG. 3. Magnitude estimation of the brightness of red (squares) and white (circlea~ fovea1 stimuli. The solid line has a slope of l/3. The dashed linea indicate the steepen@ ofthe psychophysical brightness function near threshold. Data from Mnn~s (1971).
average numerical estimates of brightness versus intensity for 0.5~see flashes. Note that at low (near-threshold)intensities the simple power relation breaks down, and the brightness function steepens in double-logarithmic coordinates, somewhat like the lower portion of the function in Fig. 2. The filter model can be seen to give a good description of the relation between brightness judgments and stimulus intent&y, both at high intensities and near threshold. The brightness of a light changea as stimulus duration is varied. It is convenient to divide the time-course of brightness into three stages. The first is a transient stage over
VisualBrightness:Some Applicationsof a Model
1413
which temporal summation operates (stimulus durations up to about 0.1 set, depending on intensity). Application of the filter model to this stage will be made later in the paper. The second stage is a quasi-steady state (stimulus durations from a few tenths of a second to a few seconds). Brightness changes little if at all over this range of durations. The cube-root relation between brightness and intensity applies to this stage. In the third stage, brightness declines to a lower level. The extent of the decline varies with intensity. Thus, if stimulus duration is extended in order to permit the visual system to reach “equihbrium”, the cuberoot relation breaks down (J. C. STEVENSand STEVENS,1963). It is important, therefore, to keep in mind that the “steady-state” responses of the filter system provide an analogy to brightness responses at the second stage. Additional mechanisms would have to be postulated in order to account for the nature of the “equilibrium” brightness function. It is notable that the model does not contain a threshold parameter, i.e. the predicted function passes through y = 0, I = 0. Addition of a threshold parameter would increase the degree of curvature at low intensities (as, for example, is shown by the dashed curve of Fig. 2). There has been considerable concern with regard to the mathematical form of psychophysical functjons near threshold (for a review, see MARKSand STEVENS,1968), in particular whether threshold parameters for the psychophysical power equation should modify the input (stimulus) or the output (response). The present model and EKMAN and GUSTAF~WN’S (1968) analysis of some psychophysical data both suggest that the psychophysical function for brightness is approximately linear near threshold; such an outcome would dissolve the controversy between stimulus- and response-corrected versions of the power function, since the two alternatives would be mathematically identical. It is perfectly possible, of course, that some rather complex modification of the model will be necessary in order to predict accurately the variation of brightness with intensity at low levels. In any case, the precise nature of the brightness function near threshold is not of concern here. The major concern is with brightness well above threshold, over the enormous stimulus range to which the visual system responds.
PARAMETRIC EFFECTS ON BRIGHTNESS: RETINAL LOCUS OF STIMULATION If a two-stage feedback filter is to prove an adequate model of brightness perception, then we would expect that effects of parametric variations in stimulation should be explicable in terms of variables in the model or by simple extensions of the model. Two types of parametric effect need to be distinguished: level-dependent and level-independent effects. In the case of level-inde~ndent variation, curves of relative sensitivity are the same regardless of the level of output at which they are measured. An example is the approximate constancy of the photopic luminosity function; photopic luminosity is practically the same near photopic threshold as at high levels of brightness (see, for example, GRAHAM,1965). Variations in sensitivity that are independent of output level are generally easily incorporated into models; in the present model, for example, variations in luminosity can be subsumed in the parameter a, (sensitivity to input of the first stage). Variations in certain stimulus parameters, however, produce effects on sensitivity that are strongly level-dependent. A striking example is the variation in sensitivity across the dark-adapted retinal surface. Flashes of broad-spectrum, constant-intensity light arouse greater brightness in the peripheral retina than in the fovea (JAMESON,1965; MARKS, 1966, 1971). But the ratio of peripheral to fovea1 brightness varies with intensity level: at low
LAWRRNCEE. MARKS
1414
fovea A---A 12* .a---020" -
10
0 relative
intensity
in decibels
eccentrmly
20
in degrees
FIG. 4. (a) Magnitude estimation of the brightness of white light. The stimuli (O-2”dim.)were presented either to the fovea (circles) or to peripheral loci 12” (triangles) or 20” (squares) temporally displaced from the fovea. Data from MARKS(1971). (b) Relative intenshies requiredat d&rent retinal loci to produce the same degreeof brightness. The lowest contour corresponds to relatiwz sensitivity across the retina for near&m&old brightness. Upper contours correspond to measures of relative sensitivity at higher and higher levels of brightness. The points were calculated from Fig. 4a.
intensities the ratio is much larger than at higher intensities (see Fig. 4a). Alternatively stated, the peripheral retina requires a small fraction of the light intensity required by the fovea to produce the same low level of brightness; as the brightness level is increased, however, the difference in relative sensitivity decreases (see Fig. 4b). These results have been interpreted in terms of (1) the assumption that the brightness of a Bash of light presented peripherally is mediated primarily by retinal rods and their neural system, whereas the brightness of fovea1 &&es can be mediated only by the cone system ; (2) differences in neural connectivity between the rod and cone systems (connections of cones to bipolar cells are largely one-to-one, whereas many rods tend to converge on single bipolar@ ; and (3) local retinal density of receptors (MARKS, 1968, 1971). It is possible to elaborate the two-stage filter model in order to account for the effects on brightness of variations in retinal locus of stimulation. Elaboration consists of convergence of inputs in the filter system, as shown by the schematic representations of two alternatives in Fig. 5. That is to say, the converging stages produce both facilitation (at the +ond stage) and increased feedback inhibition (either at the first [Fig. Sa] or second Fig. 5bf ~tage).~ It turns out that the input-output relation is the same for both of the alternatives shown in Fig. 5. Let us consider, therefore, just the second version (Fig. 5b). The constants a, and ct in equations 3 and 4 give the sensitivities corresponding to excitation and inhibition; so if a aI* = ral and c2* = rcl, r L. 1, describe the simultaneous facilitory and inhibitory effects of convergence, then (substituting y,* for yz):
y2* (1 -t
cl
y,*) (1 + czv2*) = ad
(9)
3 SPEKLING’S (1970) model also incorporates spatial summation and inhibition. In that model, lateral interactions occur in a f&forward, not a f&back. stage. There is also some similarity batween the present formulation and RUSSTW’S(1965) hypothesis of a summation pool invohrad in adaptation. The prc6ent model does not attempt to account for light adaptation, however, but limits itself to the responsh+enessof the dark-adapted eye.
1415
Visual Brightness: Some Applications of a Model
b
FIG. 5. Expanded two-stage feedback filter. Excitatory inputs converge at the second stage. Inhibitory inputs may converge either at the first (5a) or second stage (Sb). Again simplifying with cl = c 2 = c, and setting a = a& and y* = y*(l Compare
equations
+ y*) (r-l
cy,*
+ y*) = al.
10 and 6. At low intensity levels, where y* is small compared
(10)
to 1,
equation 10 yields y* 2: arl
(11)
y* 21 (aZ)1’3
(12)
and y*/y _” r. If y* is relatively large
identical to equation 7, and y*/y N 1. At low intensities, therefore, the ratio of y* to y will be relatively large, but as intensity increases the ratio decreases, asymptotically approaching unity (see Fig. 6). Permitting r to vary between one and some large value (e.g. 100) enables the brightness function to vary in the desired manner in order to account for differences observed at various retinal loci. (Note, however, that it is only the relative variation in r that is important-that is, the ratio of maximum to minimum value of r). We have made no assumption concerning differences between excitation sensitivities of elements in the peripheral and fovea1 (rod and cone) systems. It is not as yet clear whether fovea1 and peripheral brightness functions for broadspectrum lights converge at high intensities or whether the functions maintain small constant ratio displacements relative to one another. In the latter case it would be necessary to assume in addition small local variations in sensitivity. There is some evidence (e.g. ARDEN and WEALE, 1954) that differences traditionally found between peripheral and fovea1 thresholds in the dark-adapted eye are the outcome primarily of differences in neural connectivity, rather than of differences in sensitivity of retinal elements. It is of considerable interest to compare brightness functions for the peripheral retina when the eye is stimulated with very long wavelength light. The results of several investigations (e.g. WALD, 1945; WALTERSand WRIGHT, 1943) suggest that cones are on an absolute basis more sensitive than are rods to long wavelengths. It turns out that long wavelength light of constant intensity appears brighter in the fovea than in the periphery (corresponding to the higher density of cones). Furthermore, there does not seem to be any dependence of
1416
-20
-40 rtlotive
0 intensity
20
40
fIf in decibels
FIG. 6. Input-output functions in the steady state, given a smaii (r = I) or a
degree of cower-.
0
IO
20
largeft = loo)
30
relative intensity in decibels FKI. 7. Mqnitwie &&nations of the brightnew of nod light. The s&n&i (1” dia.) were preioci 12” (tiWgk8) or W @&wW sented e&tlkerto the fovea (ciIck$) or to temporaiiy displaced fram the fovea. Bata from MAR= (1971).
Visual Brightness: Some Applications of a Model
1417
the brightness-locus relation on level (MARKS, 1971). See Fig. 7. The brightness functions are ail parallel. notwithstanding the possibility that the rod system contributes to the periphery’s brightness responses, it is nevertheless the case that differences in brightness, presumably reflecting differences in density of retinal cones at different loci, can be accounted for simply in terms of the variation in the parameter a in equation 6 or 10. TEMPORAL
SUMMATION
Thus far we have considered only steady-state responses of the filter. It has long been
known (BROCAand SULZER,1902; MCDOUGALL,1904; KLEITMANand PI~RON, 1925) that the brightness response of the visual system strongly depends on stimulus duration. At very brief durations (up to 0.02-0.1 second), duration and intensity can be traded equally one for the other to maintain constant brightness (AIBA and STEVENS,1964; BRINDLEY,1952; RAAB,1962). The complete reciprocity between intensity and duration is known as BLOCH’S (1885) law. Alternatively stated, given a constant stimulus intensity, brightness increases over these brief durations. At the longer durations (such as those we have considered thus far in this paper) brightness reaches a relatively steady value.4 See WALLACE(1937). Between the stages of growth and constancy, brightness reaches a maximum, often termed the Broca-Sulzer effect (BROCAand SULZER,1902). Of special interest is the fact that the critical duration for temporal brightness summation is level-dependent: the temporal limit on Bloch’s law and the temporal locus of the BrocaSulzer peak both decrease as intensity increases. There is mounting evidence that the critical duration is inversely proportional to the cube-root of intensity, or, equivalently, is inversely proportional to the “steady-state” level of brightness (ANGLIN and MANSFIELD,1968; MANSFIELD,1970; J. C. STEVENS and HALL, 1966; S. S. STEVENS, 1966). From these relations it follows that the brightness of a fiash of very brief, constant duration varies as the squareroot of intensity (see, for example, ANGLINand MANSFIELD,1968; MANSFIELD,1970). These temporal properties of brightness vision are accounted for quite satisfactorily by a feedback filter. The inhibitory feedback serves to decrease the time-constant of the filter system in inverse proportion to the magnitude of the output (FUORTESand HODGKIN, 1964; SPERLINGand SONDHI,1968). Sperling and Sondhi point out that the response to an intense impulse (extremely brief flash) will be proportional to the l/n power of the input energy, where n is the number of feedback stages. Thus a two-stage filter will predict a cube-root relation between brightness and intensity for long flashes, but a square-root relation for very brief flashes. Sperling and Sondhi further point out that the output, given a step input of constant intensity, reaches a maximum before declining to the steadystate value. It appears that a feedback filter provides very good first-order quantitative agreement with the brightness responses of the visual system to flashes of different durations. SPATIAL
SUMMATION
The final application of the filter model to be considered here is to some of the spatial summative properties of the visual system. Although summation at absolute threshold is readily observed both in the cone (GRAHAMand BARTLETT,1939; GRAHAM,BROWNand 4 In actual fact, the steady state cannot last very much more than one or two seconds; with further increases in duration, brightness diminishes with a time-constant of about 15 sec.
1418
LAWRENCE
E. MARKS
MOTE, 1939; HILLMAN, 1958 ; WILLMER, 1950) and in the rod systems (GRAHAMand BARTLETT,1939; GRAHAMet al., 1939; GRAHAM and MARGARIA, 1935), the evidence is somewhat contradictory and inconclusive concerning suprathreshold summation. WFLLMER (1954) provided some evidence for complete area-intensity reciprocity [RICCCYS (1877) law] in the fovea at brightness levels very near threshold. OGAWA,KOZAKI, TAKAE~O and OKAYAMA (1966) also found summation, extending beyond the fovea, at low brightness levels; however, the degree of summation (the relative efficacy of increases in area) was inversely related to brightness level. On the other hand, DIAMOND(1962) found little or no evidence for any summation at levels well above the fovea1 threshold. HANEY(1951) found increasing stimulus size to increase brightness at low intensity levels, but to decrease brightness at high levels. It has been reported that brightness of a very small (“point”) stimulus increases as the square-root, rather than cube-root, of intensity (S. S. STEVENS and STEVENS,1960). A larger exponent for very small stimuli would also be consistent with Hane’s results, but not, however, with Diamond’s. Furthermore, extensive study by MANSFIELD(1970) of the temporal locus of the Broca-Sulzer maximum suggests that spatial summation may display properties in certain respects quite similar to properties of temporal summation. MansBeld$ results imply that the brightness of a “steady-state point” stimulus varies as the square-root of intensity, but that the brightness of a very brief “point” stimulus varies proportionally to intensity. Extension of the feedback filter to account for the difference between peripheral and fovea1 (rod and cone) brightness responses (Fig. 5) may have a direct analogue in extension of the filter to deal with the problem of spatial summation. DLAMO~ (1962) and HANES (1951) argued for interaction of summation (facilitation) and inhibition to account for the change from relatively extensive summation at threshold to much less summation well above threshold. If we assume that area A is directly analogous to the term r in equation 10, then y* (1 + y*) (1 + Ay*) = a&.
(13)
Figure 8 shows input-output functions of the filter (equation 13) for stimuli varying in area over a 1000 : 1 range. Near threshold, when A is not too large and Ay* is much less than 1 y* N aAZ.
(14)
The small-signal linearity of the filter system thus provides a basis for complete threshold summation (R&o’s law) over a considerable area1 range. For somewhat larger areas, the degree of spatial summation decreases. See Fig. 9, It is possible, however, that the transition from complete reciprocity to independence of area that is seen in threshold data (e.g. GRAHAMet al., 1939) also reflects a decrease in effectiveness or magnitude of input from the more distant stimulated elements. It is evident from Fig. 9 and from equation 13 that at high levels of input and output the functions for different area1 extents converge. That is, when y* and Ay* are large Y * N (aZ)li3.
(15)
For intense and relatively large stimuli, therefore, area should have little or no effect on brightness. Spatial summation may be limited both by limitations on the spatial extent of facilitory convergence and also by the inhibitory effects of “lateral” feedback. There can
Visual Brightness: Some Applications
of a Model
IO“ -20 relative FIG.
8. Input-output
functions
20
40
60
(I) in decibels
in the steady state, given stimuli varying in area over a 1000 : 1 range.
I
,”
0 intensity
1
1
1
I-
:1-L-
-10 -
lo-*
10-I relative
IO0
IO’
102
area (A)
FIG. 9. Area-intensity relationships at three levels of output predicted by the filter model. The relative magnitude and spatial extent of area1 summation both decrease as the level of output increases.
1419
1420
LAWRENCE E. MARKS
be a small range of intensities for which y* is relatively large, but A is small enough SO that Ay* is small. Then, very approximately ?‘* 2 (aAZ)“2.
(16)
Equation 16 is satisfactory over a range of intensities of at most about three orders of magnitude (30 dB). It is possible, as MANSFIELD’S(1970) results suggest, that the square-root relation between brightness and intensity operates over a large portion if not all of the intensity range for very small stimuli. In that case, significant modification of the present model would be required. For example, sensitive “units” might be arranged as receptive fields, with small centers receiving spatially summating, excitatory input from the entire “receptive field” and producing one-stage of inhibitory feedback, plus an inhibitory, spatially summating surround. Stimulation of only the central region would parallel the activity of a one-stage feedback filter exhibiting spatial summation; stimulation of much greater extent would parallel the activity of a two-stage filter whose output is independent of area. Such a model could yield equation 16 for small stimulus sizes at all intensities not near threshold. With brief &ashes and small areas, the model would predict a linear relation between brightness and intensity throughout the range of stimulation. This same outcome is also predicted by MANSFIELD’s (1970) results. This prediction makes it specially interesting and important, therefore, to determine with precision the nature of the brightness function for very brief and very small stimuli. It is important to point out that the model as applied to spatial summation would require scaling of the area factor A according to retinal locus of stimulation. Since the limits of complete threshold summation are about 6-8’ diameter of arc in the fovea (GRAHAMet al., 1939 ; HUMAN, 1958 ; WILLMER,1950), but more in the region of 1” in the peripheral retina (GRAHAMand BARTLEIT, 1939; GRAHAMet al., 1939; GRAHAMand MARGAIUA,1935), the spatial scales for the model as applied to peripheral and foveal responses must be adjusted in order to make the summative properties for different retinal loci agree with the experimental data. It follows that in order to show a significant degree of suprathreshold summation in the fovea (or, equivalently, to show a clear change in the exponent of the brightness function), it is expected that a very small stimuli (less than 6’) would have to be used. A significant degree of suprathreshold summation should be observable, however, with larger stimuli presented to the peripheral retina. It is certainly not surprising, therefore, that for stimuli between O-2 and l*O”,size has little effect on the relation between brightness and retinal locus (MARKS, 1971). With very small stimuli, however, brightness should not show the same variation with locus that is evident with larger stimuli. Level-dependent variations should disappear, leaving at most only level-independent (e.g. luminosity-related) variations. SOME The
ANATOMICAL
AND
PHYSIOLOGICAL
SPECULATIONS
author having been unsuccessful in his attempt to avoid seeking correspondences between anatomy and physiology of the visual system and elements and operations in the filter model, we may examine some implications. SPERLING and SONDHI (1968) suggested receptor and bipolar cells as the locus for a two-stage feedback filter that was incorporated into a model for predicting luminance discrimination and flicker detection. The lateral interactions that are poatufated in the present formulation suggest that the locus extends to the level of ganglion cells. Known convergence of receptors onto bipolars and bipolars
Visual Brightness: Some Applications of a Model
1421
cells could parallel the second stage of the filter, leaving the first stage to the Feedback signals might be carried by amacrine cells and horizontal cells (DOWLING, 1967; DOWLING and BOYCO’IT, 1966). Consider, however, some implications of such anatomical assignments. In accordance with equation 6, the output y from the second state varies as the cube-root of intensity at high levels, but proportionally to intensity at low levels. Assuming a threshold corresponding to y 21 0.1, from Fig. 7 it is clear that for large. stimuli the magnitude of y would have to vary over a range of IO3 : 1 (30 de&logs) in order to follow a stimulus range of IO* : I (80 dB). Whether a range of 30 decilogs is too large to expect from a single channel is probably open to debate. On the other hand, it is not unreasonable to expect that the psychophysical brightness function for the peripheral retina would show regular increase in brightness with increasing intensity over a stimulus range of at least 120 dB (e.g. from a threshold of about lO-6 cd/m2 to at least lo6 cd/m2). Such a stimulus range would imply a response range for y of more than 40 decilogs. Clearly, such a response range would necessitate a shift in channels, with overlapping stimulus-response ranges. If these considerations do not create di~culties enough, consider the input~utput function for the first stage, which was ascribed to retinal receptors. The steady-state output y, can be derived from equations 3 and 5 as onto ganglion
receptors.
Yl
[I + (1 + bY*Y21 = Ml
(17)
where b = 4azc/b2, d = a,,%,, and cl = c2 = c. Thus the first stage of the filter also shows small-signal linearity y, ‘v dl (18) but at higher intensives the output increases in approximate proportion to the Z/3 power of intensity y, 1 (kZ)2’3 (19) where k = (a12b2/ca2b12)1/3. The degree of ‘“range compression” exerted at the first stage would, therefore, be much less than that found at the output from the second stage. A stimulus range of 80 dB would be compressed only to about 60 decilogs at the first stage. It is interesting to note, however, the results of some measurements made by BOYNTONand WHITTEN (1970) of late receptor potentials from the cones of macaque monkeys. Flashes of O-15 set duration (intensity range about 60 dB) were presented to the dark-adapted eye; the amplitude (P) of the late receptor potential was related to intensity by the equation
At high intensities equation 20 implies that the potential saturates; at low intensities, where I” is much less than K, the relation approximates a simple power function with exponent n. Boynton and Whitten report that most of their data could be fitted satisfactorily with n = 0.73, a value not very different from 2/3. REFERENCES T. S. and STEVENS, S. S. (1964). Relation of brightness to duration and luminance under light- and dark-adaptation. Vision Rex 4, 391-W. ANGLIN, J. W. and MANSFIELD, R. J. W. (1968). On the brightness of short and long flashes. Percept. PsychoAIBA,
phys. 4, 161-162.
LAWRENCE E. MARKS
1422 ARDEN, G.
B. and WEALE,R. A. (1954). Nervous mechanisms and dark-adaptation. .I. Physioi., Lot&. 125 417-426. BWCH, A.-M. (1885). Exp&iences sur la vision. C.r. S&UK.Sot. Biol. 37,493-495. BonurON, R. M. and W-N, D. N. (1970). Visual adaptation in monkey cones: Recordings of late receptor potentials. Science, N. Y. 170, 1423-1426. BRINDLEY, G. S. (1952). The Bunsen-Roscoe Law for the human eye at very short durations. J. Physiol., Lend. 118, 135-139. BR~CA,A. and SULZER,D. (1902). La sensation lumineuse en fonction du temps. C.r. hebd. S&nc. Acad. Sci., Paris 134,831-834. DIAMOND, A. L. (1%2). Brightness’of a field as a function of its six.. J. opt. Sot. Am. 52,700-706. DoWLING, J. E. (1967). The site of visual adaptation. Science, N. Y. 155,273-279. DoWLING, J. E. and BOYCOT& B. B. (1966). Organization of the primate retina: Electron microscopy. Proc. R. Sot. Lond. B. 166,80-l 11. EKMAN,G. and G-T-N, U. (1968). Threshold values and the psychophysical function in brightness vision. Vision Res. 8, 747-758. FUORTES,M. G. F. and H-KIN, A. L. (1964). Changes in time scale and sensitivity in the ommatidia of Limtdtts. I. Physiol., Land. 172,239-263. GRAHAM,C. H. (l%S). Discriminations that depend on wavelength. In Vision and Visual Perception (edited by C. H. Gw), pp. 350-369. Wiley, New York.
GRAHAM,C. H. and BARTLETT, N. R. (1939). The relation of size of stimulus and intensity in the human eye. II. Intensity thrahokfs for red and violet light. J. exp. Psychol. 24, 574-587. GRAJUM,C. H., BROWN,R. H. and MOTE, F. A., JR. (1939). The relation of size of stimulus and intensity in the human eye. I. Intensity thresholds for white light. J. exp. Psychol. 24,555-573. GRAHAM,C. H. and MAROARIA, R. (1935). Area and the intensity-time relation in the peripheral retina. Am. J. Physiol. 113,299-305. HANES, R.
H. (1951). Suprathreshold area brightness relationships. J. opt. Sot. Am. 41, 28-31. HILLMAN,B. M. (1958). Relationship between stimulus size and threshold intensity in the fovea measured at four exposure times. J. opt. Sot. Am. 48,422-428. JAMES~N,D. (1%5). Threshold and supra-threshold relations in vision. In Proceedings of the International Colour Meeting: Lucerne, Vol. I, pp. 128-136. Musterschmidt-Verlag, Gbttingen. KATZ, B. (1966). Newe, Muscle and Synapse, McGraw-Hill, New York. KLEITMAN, N. and P&RON, H. (1925). Recherches sur I’btablissement de la sensation lumineuse. A&e Psychol. 25,34-84. LEVINSON,J. Z. (1968). Flicker fusion phenomena. Science, iV. Y. 160,21-28. R. J. W. (1970). Intensity relations in vision: Analysis and synthesis in a non-linear sensory system. Unpublished doctoral dissertation, Harvard University.
MANSPIUD,
MARW, L. E. (1966). B&htness as a function of retinal locus. Percept. Psychephys. 1,335-341. Mm, L. E. (1968). Bri&tnass as a function of retinal locus in the light-adapted eye. yisfgn Res. 452.5-535. MAIIKS,L. E. (1971). Brightness and retinal locus: Effects of target size and spectral compo&ion. Percept. Psychophys. 9,26-H). MARKS, L. E. and Sm. J. C. (1966). Individual brightness functions. Percept. Psycliopys. 1, 17-24. MARKS,L. E. and Srave~@,J. C. (1968). The form of the psychophysical function near thrcdtod. Perwpt. Psychophyss. 4315-318. MAP, L. (1368). Critical duration, the differential luminance threshold, critical flicker fmuency and visual awtion: A &ore&al treatment. J. opt. Sot. Am. S&404-415. Mcmwrsu, W. (19Q4). The variation of the intensity of visual sensation with the duration of the stimulus. Br. J. Psychol. 1,151-189. &AWA, T., K.IXZAKI, T., TAKANO, Y. and OKAYMU, K. (1966). Effect of area on apparent brightness. Report from the Psychological Laboratory, Keio University. RUB, D. H. (1962). Magnitude estimation of the brightness of brief fovea1 stimuli. Science, N. Y. 135.42-44. RI&, A. (1877). B&&one fra il n&&no angolo visuals e I’inW&& luminopa. Am. Ottalmol..6,373-479. RUSHTON,W. A. H. (1965). Visual adaptation. Proc. R. SUC.Land., B. 1% 20-46. spwq G. (W&j). &&I of visual adaptation and contraat detection. Pe?e$pt. Pey&phys. % 142fS7. spE#UtrX3, G. pnd s
(1966) Duration luminance and the brightness exponent. Percept. Psycho&s. 1,96-100. S;sTpNs, J. d. (1960). Tl;e dynamics of visual brightness. -1 Pro&et Report
%-EVENS;S: S: d
PPR-246, Harvard University.
Visual Brightness: Some Applications
UTTAL,
W. and
HIERONYMUS,R.
(1970). Spatio-temporal
of a Model
1423
effects in visual gap detection. Percept. Psychophys.
8, 321-326. WALD, G. (1945). Human vision and the spectrum. Science, N. I’. WALLACE, S. R., JR. (1937). Studies in binocular interdependence:
101, 653-658. I. Binocular relations in macular adapta-
tion. J. gen. Psychol. 17, 307-322. H. V. and WRIGHT, W. D. (1943). The spectral sensitivity of the fovea and the extrafovea in the Purkinie range. Proc. R. Sot. Land.. B 131. 340-361.
WALTERS,
WILI.MER,E. N. 71950). The relationshid betwe& area and threshold in the central fovea for lights of different colours. Q. .I. exp. Psycho/. 2, 53-59. WILLMER, E. N. (1954). Subjective brightness and size of field in the central fovea. J. Physiol., Land. 123,
315-323.
Abstract-Several models have postulated networks analogous to electrical filters in order to account for neurophysiological and psychophysical behavior of the visual system. A filter system that contains inhibitory feedback, such as those postulated by Fuortes and Hodgkin and by Sperling and Sondhi, can be used to account for the cube-root relation that typically obtains between judged brightness and intensity of light flashes. Elaboration of the model in terms of lateral facilitation and lateral inhibitory feedback enables it to account for differences among
brightness-intensity
elaboration
functions
measured at different retinal loci. The same type of
may also help to account for the way brightness depends on stimulus area.
R&urn&Pour expliquer le comportement neurophysiologique et psychophysique du systeme visuel, divers modeles supposent des rtseaux analogues B des filtres blectriques. Un systbme de filtres comprenant un feedback avec inhibition, comme I’ont suppos6 Fuortes et Hodgkin ainsi que Sperling et Sondhi, peut rendre compte de la relation en racine cubique typique entre luminosit6 appr&%e et intensitC des &lairs lumineux. L’Blaboration de ce modele & partir de la facilitation la&ale et de I’inhibition latbale par feedback permet d’expliquer les diffkrences entre fonctions IuminositGntensite obtenues pour diverses localisations rbtiniennes. Le meme genre d%laboration
aide aussi B rendre compte de la variation de IuminositC avec I’aire
du stimulus.
Zusammenfassung-In verschiedenen Modellen wurden Netzwerke entsprechend elektrischen Filtem vorgeschlagen, die das neurophysiologische und psychophysische Verhalten des visuellen Systems deuten sollen. Ein solches System mit inhibitorischer Riickkopplung, wie es von Fuortes und Hodgkin sowie von Sperling und Sondhi vorgeschlagen wurde, kann man benutzen, urn die kubische Abhangigkeit, wie sie zwischen geschatzter Helligkeit und Intensitlt von Lichtblitzen besteht, zu erliiutem. Wenn man das Mode11 mit den GrBBen der lateralen Verstindigung und inhibitorischen Rfickkopplung beschreibt, kann man damit Unterschiede in der Helligkeit-Intensitlt-Funktion in Abhangigkeitvom Netzhautort beschreiben. Das gleiche gilt fiir die Abhlngigkeit der Helligkeit von der Reizfliiche.
Pe3loMe--nOCTyJIIfpOBaHb1 HeCKOJIbKO MOneJIeii CeTek aHaJIOr&PIHblXSJIeKTpWYeCKHM f&IiJIbTpaMC UcJIrOo6wIcHeHHR HetipO&i3HOJIOrFIeCKOrOH IICHXO&f3Ii%CKOrO IIOBelleHAR 3pHTeJIbHOP CWCTeMbI. CHCTeMa @HJIbTpOB, KOTOpaB COAepEHT TOpMOJHyIO 06paTHyIo cBR3b, TaKym KaK 3~0 nocrynepym~ Fourtes H Hodkin, a TaKXe Sperling u Soundhi, MOXCeT 6bITb KCIIOJIb30BaHa MB O6MICHeHHBTHmPIeCKOrOOTHOIlIeHHII MemjQ’ CymneHHeM 0 Cy6beKTHBHOfiSIPKOCTH H WHTeHCHBHOCTllMH CBeTOBblX BClIbIIIIeK (OTHOlJIeHHe KOPHFI Ky6H’Iecroro). Paspa6oTaHa Monenb, C y%TOM naTepanbHor0 o6nerueHHn H nalepanbHoro TOPMOXCKeHUII C 06parHofi CBIIJbiO, KOTOPU lI03BOJIleT 06bSCHHTb pl3SIHWR B CY~WKTHBHO BOCllPAHHMil‘ZMOti IlpKOCTkiB JaBHCHMOCTH OT RPKOCTK CTIiMyJIa, IIpH pa3HOt nOKanU3auHH er0 Ha CeT’laTKe. TOT me CaMbIBTHIIpa3pa60TKH MOxeT IIOMOYbH B IIOHHMaHHU MeXaHH3Ma, onpenennmuero 3aBHcHMocTb BocnpHnTHn Bprt~cTH 0~ rmouantf CruMyna.
VISUAL
BRIGHTNESS:
in Great Brimin.
SOME APPLICATIONS MODEL’
OF A
LAWRENCE E. MARK.? John B. Pierce Foundation Laboratory and Yale University, New Haven, Connecticut 06519, U.S.A. (Received 3 September 1971; in revisedform 9 December 1971)
paper aims to explore some applications of a relatively simple model (analogous to an electrical filter with feedback) that attempts to account for visual responses to flashes of light. More specifically, the model gives a good description of the way brightness varies with the intensity of flashes presented to the dark-adapted eye. Already established is the capacity of filter models to deal with effects of stimulus duration. An extension of the model enables it to deal also with the parametric effects of other stimulus variables, namely the retinal locus of stimulation and stimulus size. The last part of the paper will look at possible correspondences between elements and operations of the model and anatomy and physiology of the visual system. THE PRESENT
THE
BASIC
MODEL
The
essence of the model is the postulation of a cascade of filter-like stages; the relation between the output from the filter system and input to it will be seen to parallel the brightness response of the visual system to light. Filter systems have been proposed frequently to explain a number of other visual phenomena, such as flicker detection (LEVINSON, 1968; MATIN, 1968; SPERLING and SONDHI, 1968), differential sensitivity and increment detection (MATIN, 1968; SPERLING and SONDHI, 1968), contrast detection (SPERLING, 1970), and temporal gap detection (URAL and HIERONYMUS, 1970). A general input-output equation for any one stage i of the filter is
where J+ is the output from stage i, a1reflects the sensitivity to input of the stage, and y, _ 1 is the input to the stage (i.e. the output from the previous stage). A filter such as that described by equation 1 is analogous to a simple electrical (RC) stage, i.e. a linear integrator with a time constant equal to TV. There is evidence that an RC-stage can provide a representation of a neuron that receives both excitatory and inhibitory inputs (see, for example, KATZ, 1966). Sperling based his model of luminance and flicker detection on an analogy between neuron and RC-stage (SPERLING, 1970; SPERLING and SONDHI, 1968). On the other hand, FUORTES and ’ This study was supported by Grant No. AFOSR 70-1950 from the U.S. Air Force Office of Scientific Research. ’ Address: John B. Pierce Fomdation Laboratory, 290 Congress Avenue, New Haven, Connecticut 06519. “lmm
12/s-G
1409
1410
LAWRENCE E. MARKS
HODGKIN(1964) were unable to find anatomical basis for the large number (average about ten) of filter stages whose postulation was necessary in order to describe electrical responses to light in Limulus ommatidia. The present author tends to treat the filter stages primarily as elements of a functional model whose elements and activities provide a logical and operational role. It is not nceessary to ascribe structures of functions in the model to particular anatomical units. The filter system is a model for psychophysical phenomena, not necessarily a model for neurophysiological data. It is rare indeed to discover a model that is able to make quantitatively accurate predictions both of perceptual and of neural responses. Thus the terminology to be used in the paper, e.g. “excitatory”, “inhibitory”, “feedback”, etc., is aimed primarily at “functional” denotation. On the other hand, in certain places it will be difficult to avoid reference to clear anatomical or physiological correlations. The question of anatomical and physiological correspondence will be looked at specifically at the end of the paper. The critical assumption of the model is that the output of each filter stage is modulated by shunting feedback. To be more precise, we assume that the output from the final stage modifies the output from every prior stage by decreasing the gain and time-constant of the stage (and, therefore, of the system as a whole). Thus each stage may be considered to receive output-dependent, inhibitory (feedback) input as well as excitatory input. This assumption is not original to the present paper. FUORTE.Sand HODGKIN(1964) postulated a similar gain-control mechanism in order to describe electrical responses induced in Limulus. MATIN (1968) and SPERLINGand SONDHI(1968) incorporated such a feedback filter into more complex models that attempt to account for incremental intensity discrimination and flicker detection; SPERLING(1970) later extended his model to take into account spatial summatory and inhibitory influences. These papers utilized filter models to explain light adaptation. The present paper is an extension of the use of a feedback filter, not as a model of light adaptation, but to describe brightness responses to flashes of light presented to the dark-adapted eye. By incorporating shunting feedback into the parameter T[ in equation 1, the general equation for any filter stage becomes
$ Y,
=
QfYf-1 -
hY, (1 + ClY”).
The constant b, reflects the sensitivity of stage i to feedback and is inversely proportional to the time-constant uninfluenced by feedback; c1 gives the fraction of the final output y,, that is fed back to stage i. The overall time-constant of stage i equals [b,(l + c&)1-I. A visual representation of a two-stage filter is shown in Fig. 1. If the input to such a system is a pulse of constant intensity y, = I, then the equations for the two stages are
&Yl=a,/$Y2
= &Yl
-
by,
(1 + qy,)
bzY*
(1 +
C2Y2)
(4)
The primary concern here is the steady-state behavior of the system. In the steady state we obtain for the output from the last stage a31 y2
=
(1
+
ClY2) (1 + CzY2)
(5)
Visual Brightness: Some Applications of a Model
1411
FIG. I. Diagram of a two-stage feedback filter. Each stage receives both excitatory input (+) from the prior stage and inhibitory (feedback) input (-) from the last stage.
where a3 = (u~~~~~(~~~~),In order to simplify for exposition, we let cl = cz = c and set y = cy,. Then (6)
y (1 + yj2 = d where CI= as/c. For the values of y large in comparison to 1 y N (az)“3
(7)
In general, as SPERLING and SONDHI pointed out, a feedback filter of n stages approximates power law behavior (exponent equal to I/[n + 11) in the steady state. Figure 2 shows the input-output function for the steady-state behavior of a two-stage filter (equation 6). At high intensities the function is linear in double-logarithmic coordinates, with a slope
P-
3’ -
IO-
I
I
I
I
0 20 40 60 00 relative intensity (1)in decibels FIG. 2. Input-output function in the steady state for a two-stage filter: y(1 + r)* = I. The dashed curve shows the increase in curvature that obtains at the low end of the function if a -20
threshold parameter (lo = 094) is added.
LAWRENCE E. MARKS
1412
(exponent) equal to l/3. At low levels of intensity the function steepens, and, when y is small in comparison to 1, equation 6 yields y E al.
(8)
The model therefore displays small-signal linearity at low inputs, a property of considerable use and importance in accounting for threshold phenomena such as spatial summation. PSYCHOPHYSICAL
FUNCTION
FOR
BRIGHTNESS
A large number of psychophysical experiments have demonstrated power law behavior in the visual system. In particular, when subjects judge (by means of direct, ratio-scaling procedures, such as magnitude estimation) the brightness of flashes of light presented to the dark-adapted eye, the typical input-output relation is a cube-root function, both for group averages (e.g. J. C. STEVENSand SEVENS, 1963 ; S. S. STFVENSand STEVENS,1960) and for individual subjects (MARKS and STEVENS, 1966). Figure 3 gives an example of
I-
, I :d :
_
o while o red
: 05-
-
:’ -:
o-3
-i 0
I
t
I
IO
20
30
relative
intensity
40
in decibels
FIG. 3. Magnitude estimation of the brightness of red (squares) and white (circlea~ fovea1 stimuli. The solid line has a slope of l/3. The dashed linea indicate the steepen@ ofthe psychophysical brightness function near threshold. Data from Mnn~s (1971).
average numerical estimates of brightness versus intensity for 0.5~see flashes. Note that at low (near-threshold)intensities the simple power relation breaks down, and the brightness function steepens in double-logarithmic coordinates, somewhat like the lower portion of the function in Fig. 2. The filter model can be seen to give a good description of the relation between brightness judgments and stimulus intent&y, both at high intensities and near threshold. The brightness of a light changea as stimulus duration is varied. It is convenient to divide the time-course of brightness into three stages. The first is a transient stage over
VisualBrightness:Some Applicationsof a Model
1413
which temporal summation operates (stimulus durations up to about 0.1 set, depending on intensity). Application of the filter model to this stage will be made later in the paper. The second stage is a quasi-steady state (stimulus durations from a few tenths of a second to a few seconds). Brightness changes little if at all over this range of durations. The cube-root relation between brightness and intensity applies to this stage. In the third stage, brightness declines to a lower level. The extent of the decline varies with intensity. Thus, if stimulus duration is extended in order to permit the visual system to reach “equihbrium”, the cuberoot relation breaks down (J. C. STEVENSand STEVENS,1963). It is important, therefore, to keep in mind that the “steady-state” responses of the filter system provide an analogy to brightness responses at the second stage. Additional mechanisms would have to be postulated in order to account for the nature of the “equilibrium” brightness function. It is notable that the model does not contain a threshold parameter, i.e. the predicted function passes through y = 0, I = 0. Addition of a threshold parameter would increase the degree of curvature at low intensities (as, for example, is shown by the dashed curve of Fig. 2). There has been considerable concern with regard to the mathematical form of psychophysical functjons near threshold (for a review, see MARKSand STEVENS,1968), in particular whether threshold parameters for the psychophysical power equation should modify the input (stimulus) or the output (response). The present model and EKMAN and GUSTAF~WN’S (1968) analysis of some psychophysical data both suggest that the psychophysical function for brightness is approximately linear near threshold; such an outcome would dissolve the controversy between stimulus- and response-corrected versions of the power function, since the two alternatives would be mathematically identical. It is perfectly possible, of course, that some rather complex modification of the model will be necessary in order to predict accurately the variation of brightness with intensity at low levels. In any case, the precise nature of the brightness function near threshold is not of concern here. The major concern is with brightness well above threshold, over the enormous stimulus range to which the visual system responds.
PARAMETRIC EFFECTS ON BRIGHTNESS: RETINAL LOCUS OF STIMULATION If a two-stage feedback filter is to prove an adequate model of brightness perception, then we would expect that effects of parametric variations in stimulation should be explicable in terms of variables in the model or by simple extensions of the model. Two types of parametric effect need to be distinguished: level-dependent and level-independent effects. In the case of level-inde~ndent variation, curves of relative sensitivity are the same regardless of the level of output at which they are measured. An example is the approximate constancy of the photopic luminosity function; photopic luminosity is practically the same near photopic threshold as at high levels of brightness (see, for example, GRAHAM,1965). Variations in sensitivity that are independent of output level are generally easily incorporated into models; in the present model, for example, variations in luminosity can be subsumed in the parameter a, (sensitivity to input of the first stage). Variations in certain stimulus parameters, however, produce effects on sensitivity that are strongly level-dependent. A striking example is the variation in sensitivity across the dark-adapted retinal surface. Flashes of broad-spectrum, constant-intensity light arouse greater brightness in the peripheral retina than in the fovea (JAMESON,1965; MARKS, 1966, 1971). But the ratio of peripheral to fovea1 brightness varies with intensity level: at low
LAWRRNCEE. MARKS
1414
fovea A---A 12* .a---020" -
10
0 relative
intensity
in decibels
eccentrmly
20
in degrees
FIG. 4. (a) Magnitude estimation of the brightness of white light. The stimuli (O-2”dim.)were presented either to the fovea (circles) or to peripheral loci 12” (triangles) or 20” (squares) temporally displaced from the fovea. Data from MARKS(1971). (b) Relative intenshies requiredat d&rent retinal loci to produce the same degreeof brightness. The lowest contour corresponds to relatiwz sensitivity across the retina for near&m&old brightness. Upper contours correspond to measures of relative sensitivity at higher and higher levels of brightness. The points were calculated from Fig. 4a.
intensities the ratio is much larger than at higher intensities (see Fig. 4a). Alternatively stated, the peripheral retina requires a small fraction of the light intensity required by the fovea to produce the same low level of brightness; as the brightness level is increased, however, the difference in relative sensitivity decreases (see Fig. 4b). These results have been interpreted in terms of (1) the assumption that the brightness of a Bash of light presented peripherally is mediated primarily by retinal rods and their neural system, whereas the brightness of fovea1 &&es can be mediated only by the cone system ; (2) differences in neural connectivity between the rod and cone systems (connections of cones to bipolar cells are largely one-to-one, whereas many rods tend to converge on single bipolar@ ; and (3) local retinal density of receptors (MARKS, 1968, 1971). It is possible to elaborate the two-stage filter model in order to account for the effects on brightness of variations in retinal locus of stimulation. Elaboration consists of convergence of inputs in the filter system, as shown by the schematic representations of two alternatives in Fig. 5. That is to say, the converging stages produce both facilitation (at the +ond stage) and increased feedback inhibition (either at the first [Fig. Sa] or second Fig. 5bf ~tage).~ It turns out that the input-output relation is the same for both of the alternatives shown in Fig. 5. Let us consider, therefore, just the second version (Fig. 5b). The constants a, and ct in equations 3 and 4 give the sensitivities corresponding to excitation and inhibition; so if a aI* = ral and c2* = rcl, r L. 1, describe the simultaneous facilitory and inhibitory effects of convergence, then (substituting y,* for yz):
y2* (1 -t
cl
y,*) (1 + czv2*) = ad
(9)
3 SPEKLING’S (1970) model also incorporates spatial summation and inhibition. In that model, lateral interactions occur in a f&forward, not a f&back. stage. There is also some similarity batween the present formulation and RUSSTW’S(1965) hypothesis of a summation pool invohrad in adaptation. The prc6ent model does not attempt to account for light adaptation, however, but limits itself to the responsh+enessof the dark-adapted eye.
1415
Visual Brightness: Some Applications of a Model
b
FIG. 5. Expanded two-stage feedback filter. Excitatory inputs converge at the second stage. Inhibitory inputs may converge either at the first (5a) or second stage (Sb). Again simplifying with cl = c 2 = c, and setting a = a& and y* = y*(l Compare
equations
+ y*) (r-l
cy,*
+ y*) = al.
10 and 6. At low intensity levels, where y* is small compared
(10)
to 1,
equation 10 yields y* 2: arl
(11)
y* 21 (aZ)1’3
(12)
and y*/y _” r. If y* is relatively large
identical to equation 7, and y*/y N 1. At low intensities, therefore, the ratio of y* to y will be relatively large, but as intensity increases the ratio decreases, asymptotically approaching unity (see Fig. 6). Permitting r to vary between one and some large value (e.g. 100) enables the brightness function to vary in the desired manner in order to account for differences observed at various retinal loci. (Note, however, that it is only the relative variation in r that is important-that is, the ratio of maximum to minimum value of r). We have made no assumption concerning differences between excitation sensitivities of elements in the peripheral and fovea1 (rod and cone) systems. It is not as yet clear whether fovea1 and peripheral brightness functions for broadspectrum lights converge at high intensities or whether the functions maintain small constant ratio displacements relative to one another. In the latter case it would be necessary to assume in addition small local variations in sensitivity. There is some evidence (e.g. ARDEN and WEALE, 1954) that differences traditionally found between peripheral and fovea1 thresholds in the dark-adapted eye are the outcome primarily of differences in neural connectivity, rather than of differences in sensitivity of retinal elements. It is of considerable interest to compare brightness functions for the peripheral retina when the eye is stimulated with very long wavelength light. The results of several investigations (e.g. WALD, 1945; WALTERSand WRIGHT, 1943) suggest that cones are on an absolute basis more sensitive than are rods to long wavelengths. It turns out that long wavelength light of constant intensity appears brighter in the fovea than in the periphery (corresponding to the higher density of cones). Furthermore, there does not seem to be any dependence of
1416
-20
-40 rtlotive
0 intensity
20
40
fIf in decibels
FIG. 6. Input-output functions in the steady state, given a smaii (r = I) or a
degree of cower-.
0
IO
20
largeft = loo)
30
relative intensity in decibels FKI. 7. Mqnitwie &&nations of the brightnew of nod light. The s&n&i (1” dia.) were preioci 12” (tiWgk8) or W @&wW sented e&tlkerto the fovea (ciIck$) or to temporaiiy displaced fram the fovea. Bata from MAR= (1971).
Visual Brightness: Some Applications of a Model
1417
the brightness-locus relation on level (MARKS, 1971). See Fig. 7. The brightness functions are ail parallel. notwithstanding the possibility that the rod system contributes to the periphery’s brightness responses, it is nevertheless the case that differences in brightness, presumably reflecting differences in density of retinal cones at different loci, can be accounted for simply in terms of the variation in the parameter a in equation 6 or 10. TEMPORAL
SUMMATION
Thus far we have considered only steady-state responses of the filter. It has long been
known (BROCAand SULZER,1902; MCDOUGALL,1904; KLEITMANand PI~RON, 1925) that the brightness response of the visual system strongly depends on stimulus duration. At very brief durations (up to 0.02-0.1 second), duration and intensity can be traded equally one for the other to maintain constant brightness (AIBA and STEVENS,1964; BRINDLEY,1952; RAAB,1962). The complete reciprocity between intensity and duration is known as BLOCH’S (1885) law. Alternatively stated, given a constant stimulus intensity, brightness increases over these brief durations. At the longer durations (such as those we have considered thus far in this paper) brightness reaches a relatively steady value.4 See WALLACE(1937). Between the stages of growth and constancy, brightness reaches a maximum, often termed the Broca-Sulzer effect (BROCAand SULZER,1902). Of special interest is the fact that the critical duration for temporal brightness summation is level-dependent: the temporal limit on Bloch’s law and the temporal locus of the BrocaSulzer peak both decrease as intensity increases. There is mounting evidence that the critical duration is inversely proportional to the cube-root of intensity, or, equivalently, is inversely proportional to the “steady-state” level of brightness (ANGLIN and MANSFIELD,1968; MANSFIELD,1970; J. C. STEVENS and HALL, 1966; S. S. STEVENS, 1966). From these relations it follows that the brightness of a fiash of very brief, constant duration varies as the squareroot of intensity (see, for example, ANGLINand MANSFIELD,1968; MANSFIELD,1970). These temporal properties of brightness vision are accounted for quite satisfactorily by a feedback filter. The inhibitory feedback serves to decrease the time-constant of the filter system in inverse proportion to the magnitude of the output (FUORTESand HODGKIN, 1964; SPERLINGand SONDHI,1968). Sperling and Sondhi point out that the response to an intense impulse (extremely brief flash) will be proportional to the l/n power of the input energy, where n is the number of feedback stages. Thus a two-stage filter will predict a cube-root relation between brightness and intensity for long flashes, but a square-root relation for very brief flashes. Sperling and Sondhi further point out that the output, given a step input of constant intensity, reaches a maximum before declining to the steadystate value. It appears that a feedback filter provides very good first-order quantitative agreement with the brightness responses of the visual system to flashes of different durations. SPATIAL
SUMMATION
The final application of the filter model to be considered here is to some of the spatial summative properties of the visual system. Although summation at absolute threshold is readily observed both in the cone (GRAHAMand BARTLETT,1939; GRAHAM,BROWNand 4 In actual fact, the steady state cannot last very much more than one or two seconds; with further increases in duration, brightness diminishes with a time-constant of about 15 sec.
1418
LAWRENCE
E. MARKS
MOTE, 1939; HILLMAN, 1958 ; WILLMER, 1950) and in the rod systems (GRAHAMand BARTLETT,1939; GRAHAMet al., 1939; GRAHAM and MARGARIA, 1935), the evidence is somewhat contradictory and inconclusive concerning suprathreshold summation. WFLLMER (1954) provided some evidence for complete area-intensity reciprocity [RICCCYS (1877) law] in the fovea at brightness levels very near threshold. OGAWA,KOZAKI, TAKAE~O and OKAYAMA (1966) also found summation, extending beyond the fovea, at low brightness levels; however, the degree of summation (the relative efficacy of increases in area) was inversely related to brightness level. On the other hand, DIAMOND(1962) found little or no evidence for any summation at levels well above the fovea1 threshold. HANEY(1951) found increasing stimulus size to increase brightness at low intensity levels, but to decrease brightness at high levels. It has been reported that brightness of a very small (“point”) stimulus increases as the square-root, rather than cube-root, of intensity (S. S. STEVENS and STEVENS,1960). A larger exponent for very small stimuli would also be consistent with Hane’s results, but not, however, with Diamond’s. Furthermore, extensive study by MANSFIELD(1970) of the temporal locus of the Broca-Sulzer maximum suggests that spatial summation may display properties in certain respects quite similar to properties of temporal summation. MansBeld$ results imply that the brightness of a “steady-state point” stimulus varies as the square-root of intensity, but that the brightness of a very brief “point” stimulus varies proportionally to intensity. Extension of the feedback filter to account for the difference between peripheral and fovea1 (rod and cone) brightness responses (Fig. 5) may have a direct analogue in extension of the filter to deal with the problem of spatial summation. DLAMO~ (1962) and HANES (1951) argued for interaction of summation (facilitation) and inhibition to account for the change from relatively extensive summation at threshold to much less summation well above threshold. If we assume that area A is directly analogous to the term r in equation 10, then y* (1 + y*) (1 + Ay*) = a&.
(13)
Figure 8 shows input-output functions of the filter (equation 13) for stimuli varying in area over a 1000 : 1 range. Near threshold, when A is not too large and Ay* is much less than 1 y* N aAZ.
(14)
The small-signal linearity of the filter system thus provides a basis for complete threshold summation (R&o’s law) over a considerable area1 range. For somewhat larger areas, the degree of spatial summation decreases. See Fig. 9, It is possible, however, that the transition from complete reciprocity to independence of area that is seen in threshold data (e.g. GRAHAMet al., 1939) also reflects a decrease in effectiveness or magnitude of input from the more distant stimulated elements. It is evident from Fig. 9 and from equation 13 that at high levels of input and output the functions for different area1 extents converge. That is, when y* and Ay* are large Y * N (aZ)li3.
(15)
For intense and relatively large stimuli, therefore, area should have little or no effect on brightness. Spatial summation may be limited both by limitations on the spatial extent of facilitory convergence and also by the inhibitory effects of “lateral” feedback. There can
Visual Brightness: Some Applications
of a Model
IO“ -20 relative FIG.
8. Input-output
functions
20
40
60
(I) in decibels
in the steady state, given stimuli varying in area over a 1000 : 1 range.
I
,”
0 intensity
1
1
1
I-
:1-L-
-10 -
lo-*
10-I relative
IO0
IO’
102
area (A)
FIG. 9. Area-intensity relationships at three levels of output predicted by the filter model. The relative magnitude and spatial extent of area1 summation both decrease as the level of output increases.
1419
1420
LAWRENCE E. MARKS
be a small range of intensities for which y* is relatively large, but A is small enough SO that Ay* is small. Then, very approximately ?‘* 2 (aAZ)“2.
(16)
Equation 16 is satisfactory over a range of intensities of at most about three orders of magnitude (30 dB). It is possible, as MANSFIELD’S(1970) results suggest, that the square-root relation between brightness and intensity operates over a large portion if not all of the intensity range for very small stimuli. In that case, significant modification of the present model would be required. For example, sensitive “units” might be arranged as receptive fields, with small centers receiving spatially summating, excitatory input from the entire “receptive field” and producing one-stage of inhibitory feedback, plus an inhibitory, spatially summating surround. Stimulation of only the central region would parallel the activity of a one-stage feedback filter exhibiting spatial summation; stimulation of much greater extent would parallel the activity of a two-stage filter whose output is independent of area. Such a model could yield equation 16 for small stimulus sizes at all intensities not near threshold. With brief &ashes and small areas, the model would predict a linear relation between brightness and intensity throughout the range of stimulation. This same outcome is also predicted by MANSFIELD’s (1970) results. This prediction makes it specially interesting and important, therefore, to determine with precision the nature of the brightness function for very brief and very small stimuli. It is important to point out that the model as applied to spatial summation would require scaling of the area factor A according to retinal locus of stimulation. Since the limits of complete threshold summation are about 6-8’ diameter of arc in the fovea (GRAHAMet al., 1939 ; HUMAN, 1958 ; WILLMER,1950), but more in the region of 1” in the peripheral retina (GRAHAMand BARTLEIT, 1939; GRAHAMet al., 1939; GRAHAMand MARGAIUA,1935), the spatial scales for the model as applied to peripheral and foveal responses must be adjusted in order to make the summative properties for different retinal loci agree with the experimental data. It follows that in order to show a significant degree of suprathreshold summation in the fovea (or, equivalently, to show a clear change in the exponent of the brightness function), it is expected that a very small stimuli (less than 6’) would have to be used. A significant degree of suprathreshold summation should be observable, however, with larger stimuli presented to the peripheral retina. It is certainly not surprising, therefore, that for stimuli between O-2 and l*O”,size has little effect on the relation between brightness and retinal locus (MARKS, 1971). With very small stimuli, however, brightness should not show the same variation with locus that is evident with larger stimuli. Level-dependent variations should disappear, leaving at most only level-independent (e.g. luminosity-related) variations. SOME The
ANATOMICAL
AND
PHYSIOLOGICAL
SPECULATIONS
author having been unsuccessful in his attempt to avoid seeking correspondences between anatomy and physiology of the visual system and elements and operations in the filter model, we may examine some implications. SPERLING and SONDHI (1968) suggested receptor and bipolar cells as the locus for a two-stage feedback filter that was incorporated into a model for predicting luminance discrimination and flicker detection. The lateral interactions that are poatufated in the present formulation suggest that the locus extends to the level of ganglion cells. Known convergence of receptors onto bipolars and bipolars
Visual Brightness: Some Applications of a Model
1421
cells could parallel the second stage of the filter, leaving the first stage to the Feedback signals might be carried by amacrine cells and horizontal cells (DOWLING, 1967; DOWLING and BOYCO’IT, 1966). Consider, however, some implications of such anatomical assignments. In accordance with equation 6, the output y from the second state varies as the cube-root of intensity at high levels, but proportionally to intensity at low levels. Assuming a threshold corresponding to y 21 0.1, from Fig. 7 it is clear that for large. stimuli the magnitude of y would have to vary over a range of IO3 : 1 (30 de&logs) in order to follow a stimulus range of IO* : I (80 dB). Whether a range of 30 decilogs is too large to expect from a single channel is probably open to debate. On the other hand, it is not unreasonable to expect that the psychophysical brightness function for the peripheral retina would show regular increase in brightness with increasing intensity over a stimulus range of at least 120 dB (e.g. from a threshold of about lO-6 cd/m2 to at least lo6 cd/m2). Such a stimulus range would imply a response range for y of more than 40 decilogs. Clearly, such a response range would necessitate a shift in channels, with overlapping stimulus-response ranges. If these considerations do not create di~culties enough, consider the input~utput function for the first stage, which was ascribed to retinal receptors. The steady-state output y, can be derived from equations 3 and 5 as onto ganglion
receptors.
Yl
[I + (1 + bY*Y21 = Ml
(17)
where b = 4azc/b2, d = a,,%,, and cl = c2 = c. Thus the first stage of the filter also shows small-signal linearity y, ‘v dl (18) but at higher intensives the output increases in approximate proportion to the Z/3 power of intensity y, 1 (kZ)2’3 (19) where k = (a12b2/ca2b12)1/3. The degree of ‘“range compression” exerted at the first stage would, therefore, be much less than that found at the output from the second stage. A stimulus range of 80 dB would be compressed only to about 60 decilogs at the first stage. It is interesting to note, however, the results of some measurements made by BOYNTONand WHITTEN (1970) of late receptor potentials from the cones of macaque monkeys. Flashes of O-15 set duration (intensity range about 60 dB) were presented to the dark-adapted eye; the amplitude (P) of the late receptor potential was related to intensity by the equation
At high intensities equation 20 implies that the potential saturates; at low intensities, where I” is much less than K, the relation approximates a simple power function with exponent n. Boynton and Whitten report that most of their data could be fitted satisfactorily with n = 0.73, a value not very different from 2/3. REFERENCES T. S. and STEVENS, S. S. (1964). Relation of brightness to duration and luminance under light- and dark-adaptation. Vision Rex 4, 391-W. ANGLIN, J. W. and MANSFIELD, R. J. W. (1968). On the brightness of short and long flashes. Percept. PsychoAIBA,
phys. 4, 161-162.
LAWRENCE E. MARKS
1422 ARDEN, G.
B. and WEALE,R. A. (1954). Nervous mechanisms and dark-adaptation. .I. Physioi., Lot&. 125 417-426. BWCH, A.-M. (1885). Exp&iences sur la vision. C.r. S&UK.Sot. Biol. 37,493-495. BonurON, R. M. and W-N, D. N. (1970). Visual adaptation in monkey cones: Recordings of late receptor potentials. Science, N. Y. 170, 1423-1426. BRINDLEY, G. S. (1952). The Bunsen-Roscoe Law for the human eye at very short durations. J. Physiol., Lend. 118, 135-139. BR~CA,A. and SULZER,D. (1902). La sensation lumineuse en fonction du temps. C.r. hebd. S&nc. Acad. Sci., Paris 134,831-834. DIAMOND, A. L. (1%2). Brightness’of a field as a function of its six.. J. opt. Sot. Am. 52,700-706. DoWLING, J. E. (1967). The site of visual adaptation. Science, N. Y. 155,273-279. DoWLING, J. E. and BOYCOT& B. B. (1966). Organization of the primate retina: Electron microscopy. Proc. R. Sot. Lond. B. 166,80-l 11. EKMAN,G. and G-T-N, U. (1968). Threshold values and the psychophysical function in brightness vision. Vision Res. 8, 747-758. FUORTES,M. G. F. and H-KIN, A. L. (1964). Changes in time scale and sensitivity in the ommatidia of Limtdtts. I. Physiol., Land. 172,239-263. GRAHAM,C. H. (l%S). Discriminations that depend on wavelength. In Vision and Visual Perception (edited by C. H. Gw), pp. 350-369. Wiley, New York.
GRAHAM,C. H. and BARTLETT, N. R. (1939). The relation of size of stimulus and intensity in the human eye. II. Intensity thrahokfs for red and violet light. J. exp. Psychol. 24, 574-587. GRAJUM,C. H., BROWN,R. H. and MOTE, F. A., JR. (1939). The relation of size of stimulus and intensity in the human eye. I. Intensity thresholds for white light. J. exp. Psychol. 24,555-573. GRAHAM,C. H. and MAROARIA, R. (1935). Area and the intensity-time relation in the peripheral retina. Am. J. Physiol. 113,299-305. HANES, R.
H. (1951). Suprathreshold area brightness relationships. J. opt. Sot. Am. 41, 28-31. HILLMAN,B. M. (1958). Relationship between stimulus size and threshold intensity in the fovea measured at four exposure times. J. opt. Sot. Am. 48,422-428. JAMES~N,D. (1%5). Threshold and supra-threshold relations in vision. In Proceedings of the International Colour Meeting: Lucerne, Vol. I, pp. 128-136. Musterschmidt-Verlag, Gbttingen. KATZ, B. (1966). Newe, Muscle and Synapse, McGraw-Hill, New York. KLEITMAN, N. and P&RON, H. (1925). Recherches sur I’btablissement de la sensation lumineuse. A&e Psychol. 25,34-84. LEVINSON,J. Z. (1968). Flicker fusion phenomena. Science, iV. Y. 160,21-28. R. J. W. (1970). Intensity relations in vision: Analysis and synthesis in a non-linear sensory system. Unpublished doctoral dissertation, Harvard University.
MANSPIUD,
MARW, L. E. (1966). B&htness as a function of retinal locus. Percept. Psychephys. 1,335-341. Mm, L. E. (1968). Bri&tnass as a function of retinal locus in the light-adapted eye. yisfgn Res. 452.5-535. MAIIKS,L. E. (1971). Brightness and retinal locus: Effects of target size and spectral compo&ion. Percept. Psychophys. 9,26-H). MARKS, L. E. and Sm. J. C. (1966). Individual brightness functions. Percept. Psycliopys. 1, 17-24. MARKS,L. E. and Srave~@,J. C. (1968). The form of the psychophysical function near thrcdtod. Perwpt. Psychophyss. 4315-318. MAP, L. (1368). Critical duration, the differential luminance threshold, critical flicker fmuency and visual awtion: A &ore&al treatment. J. opt. Sot. Am. S&404-415. Mcmwrsu, W. (19Q4). The variation of the intensity of visual sensation with the duration of the stimulus. Br. J. Psychol. 1,151-189. &AWA, T., K.IXZAKI, T., TAKANO, Y. and OKAYMU, K. (1966). Effect of area on apparent brightness. Report from the Psychological Laboratory, Keio University. RUB, D. H. (1962). Magnitude estimation of the brightness of brief fovea1 stimuli. Science, N. Y. 135.42-44. RI&, A. (1877). B&&one fra il n&&no angolo visuals e I’inW&& luminopa. Am. Ottalmol..6,373-479. RUSHTON,W. A. H. (1965). Visual adaptation. Proc. R. SUC.Land., B. 1% 20-46. spwq G. (W&j). &&I of visual adaptation and contraat detection. Pe?e$pt. Pey&phys. % 142fS7. spE#UtrX3, G. pnd s
(1966) Duration luminance and the brightness exponent. Percept. Psycho&s. 1,96-100. S;sTpNs, J. d. (1960). Tl;e dynamics of visual brightness. -1 Pro&et Report
%-EVENS;S: S: d
PPR-246, Harvard University.
Visual Brightness: Some Applications
UTTAL,
W. and
HIERONYMUS,R.
(1970). Spatio-temporal
of a Model
1423
effects in visual gap detection. Percept. Psychophys.
8, 321-326. WALD, G. (1945). Human vision and the spectrum. Science, N. I’. WALLACE, S. R., JR. (1937). Studies in binocular interdependence:
101, 653-658. I. Binocular relations in macular adapta-
tion. J. gen. Psychol. 17, 307-322. H. V. and WRIGHT, W. D. (1943). The spectral sensitivity of the fovea and the extrafovea in the Purkinie range. Proc. R. Sot. Land.. B 131. 340-361.
WALTERS,
WILI.MER,E. N. 71950). The relationshid betwe& area and threshold in the central fovea for lights of different colours. Q. .I. exp. Psycho/. 2, 53-59. WILLMER, E. N. (1954). Subjective brightness and size of field in the central fovea. J. Physiol., Land. 123,
315-323.
Abstract-Several models have postulated networks analogous to electrical filters in order to account for neurophysiological and psychophysical behavior of the visual system. A filter system that contains inhibitory feedback, such as those postulated by Fuortes and Hodgkin and by Sperling and Sondhi, can be used to account for the cube-root relation that typically obtains between judged brightness and intensity of light flashes. Elaboration of the model in terms of lateral facilitation and lateral inhibitory feedback enables it to account for differences among
brightness-intensity
elaboration
functions
measured at different retinal loci. The same type of
may also help to account for the way brightness depends on stimulus area.
R&urn&Pour expliquer le comportement neurophysiologique et psychophysique du systeme visuel, divers modeles supposent des rtseaux analogues B des filtres blectriques. Un systbme de filtres comprenant un feedback avec inhibition, comme I’ont suppos6 Fuortes et Hodgkin ainsi que Sperling et Sondhi, peut rendre compte de la relation en racine cubique typique entre luminosit6 appr&%e et intensitC des &lairs lumineux. L’Blaboration de ce modele & partir de la facilitation la&ale et de I’inhibition latbale par feedback permet d’expliquer les diffkrences entre fonctions IuminositGntensite obtenues pour diverses localisations rbtiniennes. Le meme genre d%laboration
aide aussi B rendre compte de la variation de IuminositC avec I’aire
du stimulus.
Zusammenfassung-In verschiedenen Modellen wurden Netzwerke entsprechend elektrischen Filtem vorgeschlagen, die das neurophysiologische und psychophysische Verhalten des visuellen Systems deuten sollen. Ein solches System mit inhibitorischer Riickkopplung, wie es von Fuortes und Hodgkin sowie von Sperling und Sondhi vorgeschlagen wurde, kann man benutzen, urn die kubische Abhangigkeit, wie sie zwischen geschatzter Helligkeit und Intensitlt von Lichtblitzen besteht, zu erliiutem. Wenn man das Mode11 mit den GrBBen der lateralen Verstindigung und inhibitorischen Rfickkopplung beschreibt, kann man damit Unterschiede in der Helligkeit-Intensitlt-Funktion in Abhangigkeitvom Netzhautort beschreiben. Das gleiche gilt fiir die Abhlngigkeit der Helligkeit von der Reizfliiche.
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