Model of visual adaptation and contrast detection

GEORGE SPERLINGl BELL TELEPHONE LABORATOR1ES

A three-component model of spatial other hand, shunting models of lateral vision is proposed, consisting of (1) a interaction intrinsically are models of feedback stage, (2) a feedforward stage, adaptation. (3) a threshold detector. The components :7' correspond to physiological processes; in I'~ Boundary Detection particular, the feedforward control signal o----l When viewing the boundary between I : corresponds to the "surround's" signal in I two adjacent, uniform fields of slightly the receptive fields ofretinal ganglion cells. FEEDFOAWAAD 1~ FEEDBACK different luminances, Os see an illusory The model makes appropriate qualitative ---_/ '---- light band near the boundary on the light predictions of" (1) a square-root law side and an illusory dark band near the ( ~Q a: QY2) for detection at low luminances, Fig. 1. Electronic analogue of shunting boundary on the darker side (Fig. 2). These (2) a Weber law (t:.Q a: Q) at high synaptic inhibition. The input current is H; illusory light-dark bands are called Mach luminances, (3) additivity of threshold the output voltage is G; the triangle bands because of their similarity to the masking effects at high background indicates an isolating transconductance. light-dark bands observed by Mach at luminances, (4) receptive fields that, in the The shunt path is R, which is controlled by discontinuities in luminance gradients dark, consist only of an excitatory center inhibition, I. Inhibition may arrive either (Ratliff, 1965). and that, in the light, also contain via feedforward or feedback paths; I Because of Mach bands, the apparent inhibitory surrounds, (5) the variation of increases the conductance (diminishes R) contrast between two fields ofnearly equal spatial characteristics of receptive fields of the shunt path proportionally to its own luminance is greatest at the boundary depending on the temporal characteristic strength. between them. Thus, in searching for a of the test stimulus used to measure them, liminal test field superimposed on a (6) the subjective appearance of Mach Eccles, and Fatt (1955) first established masking field, it usually is the boundary of bands, (7) sine-wave contrast-threshold the basic mechanism of neural inhibition. the test that the 0 detects (Lamar et al, transfer functions, (8) the frequent failure This mechanism may be called shunting 1947). For a boundary to be detected, it is of disk-detection experiments to inhibition (Furman, 1965) because assumed that the excitation in the light demonstrate inhibitory surrounds, and inhibitory signals cause a portion of the band (plus-zone, z+) must differ by a (9) various second-order threshold effects, excitatory signals to be diverted, or criterion amount from the excitation in the such as reduced spatial integration for shunted. An electronic analog of shunting z+ long-duration stimuli, reduced temporal inhibition is an RC·stage in which the value ------/--­ integration for large-area stimuli, and the of R is decreased by inhibition (Fig. 1). At ~~ increased effect of background luminance high levels of inhibitory input, the net Q on the detection of large-area stimuli. effect is analagous to arithmetic division of z Predictions areimproved by assuming there excitation by inhibition. exist various sizes of receptive fields that .r:": determine thresholds jointly. Lateral Interaction -----~/// All visual systems of more than one Q I. INTRODUCTION receptor exhibit phenomena of lateral l- This article describes a shunting interaction; namely, the outputs of the feedback-plus-feedforward model of visual adjacent receptors combine at various contrast detection." Shunting networks levels in the nervous system. A priori, at a derive from neurophysiology, where they particular level, lateral interaction may be describe a kind of synaptic inhibitory characterized as feedback or feedforward, process. Though nonlinear, the and as shunting or subtractive (Furman, mathematical analysis of the steady-state 1965), although these categories are not response of the shunting networks is exhaustive. For example, the lateral basically simple. On the other hand, human interaction in the eye of Limulus has been contrast detection is basically complex, characterized as feedback and subtractive probably involving several different (Hartline & Ratliff, 1958).3 Fig. 2. Mach bands at a boundary (a), mechanisms. Whereas a complete model for Various subtractive models of lateral and at the ends of gradients (b) and (c). contrast would be correspondingly interaction have been proposed for vision The anaular curves indicate the stimulus complex, the model proposed here is (e.g., Schade, 1956; von Bekesy, 1960; retinal illuminance £(x) as a function of x; neither complex nor exact. The Lowry & DePalma, 1961; Rodieck, 1965; the short horizontal lines indicate Q(x)=O. justification for proposing it is that, for a Nachmias, 1968). A desirable feature of The smooth curves are G(x), the model of its simplicity, it has remarkable these models is that, because they are predictions by the model of the retinal correspondences to vision. linear, the full power of linear analysis response to £(x). G(x) corresponds applies. Unfortunately, linear models do approximately to the subjective appearance Shunting Inhibition not handle the problem oflight adaptation, of the stimulus. In (a), the light band z+ Fatt and Katz (1953) and Coombs, except by ad hoc mechanisms. On the and the dark band z_ are indicated.

Perception &Psychophysics, 1970, Vol. 8 (3) 143 jDIAM(WH Fig. 3. Example of an excitatory >1 weightinS function WJI(r) and of an r- --, inhibitory weighting functiOR WI[r]. Both I I functions are two-dimensional normal I distributions-; 3. These functions I or/ou = are used in all numerical examples of this article; The indicated effective diameters of wH and WI are 2OUv'2 aad 201..fi, respectively.

------DIAM (WI> ------Equation 3 arises from the circuit of Fig. 1. which describes shunting inhibition. For H dark band (minus-zone, z_). An expression are not interested in these details may skip and I constant in time, Eq.3 reduces to for the difference in excitation between z+ to Section 111. G = R• H/(I +kl), For convenience, the and z_ is derived from the feedforward parameter R is absorbed into G so that-for shunting model. lIa. Basic Assumptions steady inputs-Eq. 3 reduces to the form (1) Excitation: At each point in space II. FEEDFORWARD MODEL x,y and at each instant in time t, excitation H (3a) Outline H is given by the weighted sum of retinal G=I+k1' Let ~(x,y), the illuminance distribution illuminance ~ in a spatial and temporal on the retina, be the input to the model. region around x,y,t, namely It is interesting to note that a similar The input undergoes a transformation equation for lateral interaction originally (feedforward neural field) so that H(x,y,t) = ~ (x',,/,t') wH was proposed by Mach one hundred years corresponding to each point x,y there f {[ ago (Mach, 1868; Ratliff, 1965).4 ultimately is an output G(x,y). The output (4) Prior adaptation. The adaptation G is composed of an excitation term H and (x-x',y-y'}vH (t-t') dx' dy' dt'. mechanism proposed by Fuortes and an inhibition term I. The excitation term H Hodgkin (1964) consists of a cascade of n is given by a weighted sum ofilluminances (l) RC stages in each of which the R is falling on the excitatory area around x,y; controlled by shunting feedback from the inhibition is given by the weighted sum of Here WH represents the excitatory final stage. This n-stage feedback system is illuminance falling on a larger, concentric area-weighting function, VH represents the described by equations of the form inhibitory area(Fig. 3). Inhibition interacts excitatory temporal-weighting function, with excitation by feedforward shunting to and R represents integration over the area produce the net output G. A detector scans of the retina. G(x,y) to find the point z+ where G(z) is a By defining the symbol * to denote the maximum and the point z_ where G(z) is a operation of convolution, Eq. 1 can be j=I,···,n minimum. When .lG, rewritten as where X = t/RC and Yo = ~ (Sperling & .lG = max G(z) - min G(z) H(x,y,t)= ~x,y,t) * wH(x,y) * vH(t). Sondhi, 1968). For constant input i', the output Y is defined by = G(z+) - G(z_), (Ia) (4) exceeds a threshold criterion e, detection is The definition of H by an integral is an signaled. approximation to a sum that involves an In the case of prior adaptation, the ~ of the receptor elements; because these are Eqs, 1 and 2 is replaced by Y of Eq, 4. Adaptation numerous, the approximation is good. (5) Boundary detection. The detector As a complication, the input to the (2) Inhibition. Inhibition I(x,y,t) is given locates the point z; with maximum output feedforward neural field may not be ~(x,y) by in the light band near the boundary and directly, but some transformation of ~. the point z_ with minimum output in the There are good reasons for supposing that I(x,y,t)= ~(x,y,t) ... wI(X,y) * vIet) (2) dark band of the boundary. It forms the the H and I terms of the feedforward field difference .lG = G(z+) - G(z_), adds a are not composed simply of weighted sums where WI and VI are the inhibitory random variable N to this difference, and of ~, but that a feedback transformation area-weighting and temporal-weighting produces a detection response if (adaptation) precedes the feedforward functions. Figure 3 illustrates the wH and .lG + N;;;' e. The criterion E may vary from transformation (Sperling' & Sondhi, 1968). WI that are used in the examples of this experiment to experiment but is assumed If the initial adaptation is taken to be the article. to remain fixed within an experiment. shunting feedback transformation proposed by Fuortes and Hodgkin (1964) (3) Output. The output G(x,y,t) is lib. Simplifying Assumptions; Corollaries and by Sperling and Sondhi (1968), then defined by The aim of this article is to illustrate results obtained with pure feedforward general properties of the model that obtain shunting carry over-with only slight d 1+ kl(x,y,t) over wide ranges of parameters. We, modification-to the case of feedback plus dt G(x,y,t) + RC G(x,y,t) therefore, make the following simplifying feedforward. assumptions. (I) Homotropic. The weighting The next section presents a formal I statement of the assumptions. Readers who = CH(x,y,t). (3) functions WH(X,y), WI(X,y) are assumed

144 • Perception & Psychophysics, 1970, Vol. 8 (3) Fig. 4. Typical spatial and temporal relations between background (B), masking (M), and test (T) fields. Spatial relations only are illustrated at left; spatial and temporal relations at right. On the right, the Q·axis represents retinal illuminance Q, the t-axis represents time and the r-axis represents the diameter of the stimuli '"-_------, £B (after Sperling, 1965).

is based on the difference in output between two points, z+,z_, rather than between two areas, e.g., appropriately weighted neighborhoods of z.; and z_. Comparison of neighborhoods rather than points is equivalent to spatial filtering in the detector, that is, to a convolution integral of G(x,y) with the neighborhood not to vary within the domain being denotes a time-varying field QM(X,y,t). weighting function of the detector. Since, studied. Further, they are assumed to be Typical relations between T, M, and B are in the model, the effects of spatial filtering radially ~mmetric, that is, functions only illustrated in Fig. 4. at different levels are difficult to v'x~ of r = The weighting functions The contribution of a field j of retinal discriminate, all spatial filtering is assumed WH,WI are defined so that (a) the illuminance Qj(x,y,t) to H at t m is given by to occur in the feedforward neural field. maximum value of each is I, and (b) WI Hj = Qj * wH * VH. The units of Hj are total extends beyond wH' Assumption a is illuminance energy, i.e., retinal illuminance IIc. Derivation of Predicted Contrast arbitrary for convenience; Assumption b X area X time; it is useful to think of Hi as Thresholds for Incremental Tests upon is amply supported at the retinal level by the total energy contributed by Field J to Uniform Backgrounds physiological evidence. H at the moment of detection. To calculate Variation of incremental threshold (2) Additivity of luminance Hj it usually is not necessary to know luminance .:lQ with background luminance components. When the net retinal WH(X,y) and VH(t) exactly. For example, Q. Approximate solution. The critical illuminance Q(x,y,t) is composed of several when QB(X,y,t) = QB, i.e., a constant, it difference .:lG for detection is assumed to component illuminances, such as a only is necessary to know be background illuminance QB(x,y,t), a masking illuminance QM(X,y,t), and a test QT(X,y,t), then the contributions of each of these illuminance components to H and to I can be computed separately and the the effective area of WH , and (6) results added together. For example,

H = [~Qj(x,y,t)] * wH(x,y) * vH(t) J the effective duration of VH (cf. Fig. 3). The energy contributed by B to H then is = ~Q/x,y,t) * wH(x,y) * vH(t) (5) HB = QB • WH • VH· J When a field does ndt cover w or v entirely, estimates can be made of the where Qj denotes a component illuminance. intersection between the field and the Equation 5 follows from the superposition weighting function, i.e., of their fractional (linear) property of convolution. overlap (Fig. 5). In the case of area (3) Effective areas and durations. In weighting functions, good estimates can be order to determine whether or not a made for the cases of interest (see below boundary is detected, it is not necessary to and Fig. 5). For the case of temporal r;;)\-,.@ find .:lG as a function of time; it is only weighting functions, two cases are by far V'- 0 necessary to find its maximum value. the most frequent: (a) Qj(x,y,t) = Suppose .:lG attains its maximum at time Qj(x,y) 5 (t) and (b) Qj(x,y,t) = Q(x,y). In (e) (d) tm . By assuming the interaction to be of Case I, the temporal weight is 1.0; and in the form of Eq. 3a, the problem of Case 2, only the effective areas (VH, VI) Fig. S. Location z+ of maximum calculating .:lG reduces to calculating H matter. By confining our treatment to T response and z_ of minimum response for and I at t m. fields, which either are impulses or steady, two stimuli: (a) and (b) a light-dark The calculations of H and I usually can to M fields, which are impulses coincident boundary; (c) and (d) a small test spot. The be simplified by considering Q to be with T or steady, and to B fields, which are effective areas of WH and of wI are drawn composed of several fields; a field to be steady, only one parameter, VI/VH, need around z+ and z_. Receptive fields are detected-the test, T, a background field, be known to describe the temporal illustrated for (a) z+ inside light areas; B, and/or a masking field, M. B and M weighting functions. (b) z_ in dark area; (c) z+ centered around impair the detectability of T over the (4) Comparison-detection ofneighboring stimulating spot; (d) z_ located with test time and areas of interest. Usually M areas. It may seem artificial that detection spot at fringe of WH but within WI.

Perception & Psychophysics, 1970, Vol. 8 (3) 145 Defining ~H = H'( z ,") - H(z_), of Eq. 9 vanishes. At threshold, t.G is remains valid. For test stimuli oflarge area, t.1 = I(z+) - I(z_),substituting into Eq. 6, constant so Eq, 9 reduces to the correction factor becomes about 2 at and algebraically reducing gives high luminance and, therefore, is (10) significant. Details of the definition and calculation of the area factors are given where the positive constants c, and C2 below. depend on the overlap of the B, M, and T The effect of test stimulus area on ~Q vs Rearranging terms of Eq. 7 gives fields with the weighting functions, on the Q. (1) Point-stimulus. In the expansion of threshold criterion ~G, and on the relative Eq. 8 into Eq. 11, a term containing the amounts of QM and QB. This elementary ratio of inhibition to excitation occurred equation describes the response of the repeatedly, and it was convenient to Equation 8 can be rewritten as feedforward system to the usual contrast .designate it by the symbol a. In fact, a is detection stimulus, i.e., an incremental test the ratio of two pure numbers t.H = t.G + [kt.G] I(z _) + [kG(z+)] t.I. field T of threshold luminance l2T = ~Q [~I] ~H added onto a larger background field B of r r' ~) (9) luminance l2. Variations in procedure cause a = U(z_ll • LH(z_lJ = /3/0.. predictable changes in the constants c, and To simplify Eq.9 further, the following C2' In words, a is the ratio of the proportional assumptions are made: Variation of ~l2 with 12.. Complete increase in inhibition /3 to the proportional (1) M and B entirely cover the area of solution. The exact solution of Eq. 6 is increase in excitation a.. It reflects the interest so that only T contributes to much more complicated than the amount by which T increases inhibition differences between z., and z_ in the approximate solution (Eq. 10), but it is relative to the amount by which T amount of H and I received. basically quite similar to Eq. 10. To derive increases excitation. For the case of M and (2) ~H can be expressed in terms of this solution, we may start with Eq. 8, B fields that uniformly cover the entire ~I, 2 fractional overlaps as follows. Let expand I(z+) into I(z_) + G(z_) into area of interest, a reduces to a- ~l/~H, H(z_) and I(z_), and make the following t.Q = max l2T(X,y). From the definition of the form used in Eq. II. Finding the effect substitutions, which will be explained later. H(x,y) by Eq. 1, it follows that HT(Z+) can of test area on threshold ~Q involves first For H(z_) substitute l(z_)/a2; for ~I be expressed as some constant calculating a. The first case to be substitute aa 2 ~H; for ka 2 ~G write E. This [p(z+,T) • WH] times ~l2, where p(z+,T) considered is: T = a point. ultimately yields represents the fractional overlap ofthe area A point stimulus at (0,0) is defined as QT(x,y) = 5(0,0)l2T' The factor ~H is WH with T at the point z+. Similarly, ~G[1 ~l2. + kI(z_)] HT(Z_) = p(z_ ,T) • WH • To calculate ~H = • (11) t.H, the additivity property (Eq. 5) is 1 - eo: - a/[I + l/kl(z_)] ~H = QT(x,y) * wH(x+-x,y+-y) convenient: For a = 0, the denominator of Eq. 11 is - QT(x,y) * wH(x_ -x,y _ -y) t.H = H(z+) - H(z_) 1.0; the numerator of Eq. 11 is identical to the first two terms of the right-hand side of (15) t.H = [HT(z+) + HM+B(z+)] Eq. 9. Therefore, for a = 0, Eq. 11 is equivalent to Eq. 10, which used only the - [HT(z_) + HM+B(z_)]. first two right-side terms of Eq. 9. More Maximum excitation occurs at z, = 0,0; generally, Eq. 11 may be written in terms minimum excitation occurs ~x away, at Since it was assumed in (1) that M and B ofEq. 10 as z_ = (~x,O). Writing WH as a function of uniformly covered the entire area of distance ~x and letting ET denote the total interest, HM+B(Z+)= HM+B(Z_) giving energy in the test gives

t.H = HT(z+) - HT(z_) ~H = ET[wH(O) - wH(~x)]

where C3(l2) = Ea + akl(z )/ [I + kl(z )]. =ET[1-wH(~x)], = [p(z+,T) - p(z_,T)]WH • t.Q For extreme values of l2, the limits of c-;(l2) are ~p. ~l2 = WH • and Iimc3(l2)=a(I+E) (13a) where the constant ~p is defined as l2--+oo p(z+,T) - p(z_ ,T). Thus, ~H, the left side ~l = ET [wI(O) - wI(~x)] of Eq. 9, reduces to a constant (t.p • WH) lim C3 (l2) =ae . (13b) times ~l2. lI--+0 = ET [1 - wI(t.x)]. (3) The time-relations and geometry of This means that ~l2's computed from M, T, and B are assumed to remain fixed Eq. 10 (the simplified solution of Eq.6) The total energy contributed to H at z., and either (a) QM and QB are varied must be increased by a correction factor of (and at z_) by a uniform masking field is together (Q = cQM + c'l2B) or (b) only one (1 - ue )-1 at low background luminances simply the effective area WH of WH times of QM and l2 B (denoted l2 is varied). By this and a correction factor of l2 H = l2M WH. If WI is of the same form definition of l2, I(z_) becomes c" -t--c"'l2, M: [1 - a(l + E)] - 1 at high luminances to as WH, but larger by a factor of where the c's depend on the time relations satisfy the exact solution. These factors a[wI(x) = wH(ax)] then the effective area and geometry of M, T, and B. 2WH, depend on the area of T. For test stimuli WI of WI is a and 1= a 2l2MWH. (4) For the moment, ~I will be assumed that are points, the correction factors Putting these terms together in Eq. 14 gives to be negligibly small so that the last term differ insignificantly from 1.0, and Eq. 10 (for a point stimulus):

146 Perception & Psychophysics, 1970, Vol. 8 (3) 2 Using the same normal functions for WH gives: J1G(threshold) = E/(ka ), or 2 and wj in the case of the half-field as in the E = ka J1G. This is the definition of E given point test gives, for the half-field: earlier. It also is presumed that E is small, p(z+) "'" 0.94, p(zJ"" 0.06, q(z+) "" 0.70, i.e., E';;; 0.1. q(z_) "" 0.30, so that a = J1q/J1p = Equation 18 indicates another reason for (0.40)/(0.88) "" 0.45. This value of a defining the threshold E as above. Together cannot be ignored in Eqs. II and 12, which with J1p, E gives the main factor in the shall now be reconsidered. slope of J1Q vs Q, dllT/d£B, the Weber Ta .ing Effect of test stimulus area on J1Q vs Q. fraction. When J1p is 1.0 (T overlaps 2 (3) General form. According to Eqs. 13a perfectly with WH at z., and not at all with wH(X) e _x /2, wI(X) e-X 2 /(2(72), = = and 13b, the area correction is smallest at WH at z_) E is simply the fractional low retinal illuminances, when it is I- ae. amount by which QT must increase the flux a = 3 and J1x "'" 2.80 The threshold criterion E is assumed to be being received by WH at z+ in order to small, that is, E .;;; 0.1. Using the values of a bring the increase to threshold. When 5 gives that were calculated above, we see that J1p < 1, the Weber fraction is larger than E ce < 0.045 even for large-area tests; the because T stimulates WH at z, imperfectly I area correction always is negligible at zero [p(z+) I] and/or because it stimulates a = 9" (0.351)/(0.980) = 0.038. < background illuminance. At high wH at z [p(z ) > 0]. The effect of T on illuminances, the correction factor is inhibition (wr) is smaller, and it is The effect of test stimulus area on J1Q vs [1 - a(J + 1:)]-1. For point tests subsumed in the area term a of the Q. (2) Half-field stimulus. For a half-field (a "" 0.038), the area correction differs denominator of Eq. 18. QT(X,y) = U(X)QT, where u(x) is a unit step insignificantly from 1.0. For large-area Critical area. When Q is large, Eq. 18 at the point zero [u(x) =0, x < 0; u(x) = I, tests, a = 0.45; assuming E = 0.1 gives reduces to a simpler form, x > OJ. When QT is very small, the [1 a(1 + 1:)]-1 = (0.505)-1 "" 2.0, i.e., maximum and minimum of G are an area correction of about 2 for J111 vs Q. lim J1Q = (E/J1p)Q =--:------:,.,-EQ_ symmetrical around the boundary, For the case of an incremental test Q-->oo I- a(1 + 1:) J1p (J + E)J1q z, =-z . Choosing WH and WI as above stimulus of illuminance QT on a large givesf z , =J1x/2 "" 1.57. Substituting uniform background 11 it is possible to (19) QT(X,y) in Eq. 15 gives: write an almost-explicit exact form of Eq. 11 to define the threshold value J1l1 of From Eq. 19, it is possible to predict the QT. Using J1p as defined above, ignoring observed "critical area" of the feed forward time effects, assuming large, uniform M field; i.e., the intersection of two and B fields gives asymptotes in a graph of the energy of a threshold test stimulus vs the area of the stimulus. The equation of the small-area asymptote is A • J1Q = Eo (Ricco's law), J1Q = where A is the area of the test stimulus, and Eo is the minimum energy for detection. The equation of the large-area The calculation of J11 is the same as Eq. 16 asymptote is Ac • J1Q = Eo, where Ac is the with WI substrtuted for WH· (18) critical area. To solve for critical area, these To simplify Eq. 16, it is convenient to two quantities are set equal, namely define the fractional overlap of T with WH The area parameter a Illj.Ist be calculated at z , as p(z+) QT(X,y) * separately for each test stimulus WH (x, -x.y+-y)/(QTWH)' When QT(X,y) configuration. It must also be calculated covers all of WH at z., then p(z+) = 1. The giving range of p is 0';;; P .;;; I. By defining p(z ) separately for each background illuminance similarly (with z substituted for z+), and Q because a depends on z_ and z+, and z defining J1p = p(z+) p(z_), J1H may be and z, vary somewhat with the values of Q written as liTWHJ1p. Analagous fractional and of J1Q. Fortunately, using the same The subscript 0 denotes a small test field overlaps of liT with WI may be defined as value of a for all liT ,liB' introduces only a minor error into Eq. 18. (i.e., a point) and the subscript 1 denotes a q(z+), q(z ), and J1q. Substituting these large test field (i.e., a half-field). To into Eq. 14 gives The parameter J1p, which describes the difference in overlap of T with WH at z, calculate Ac , we substitute Eq. 19 into and at z _' reaches values near 1.0 for large Eq. 19a, giving 2 a = a WH • J1~\( QMWH ) =~q. test fields (0.884 of a possible 1.0 for the t'-T' Ell , 2 WH and WI assumed in the examples) and A -A \ 11 M ' a WH /\lITWH J1P p J1p goes to zero as the area of T goes to c- IJ1po[1-ao(I+E)] (I 7) zero. This means that the main effect of test area on threshold occurs in the EQ ,- 1 Equation 17 was derived without any calculation of J1p; the effect of test area via '/J1Pdl -al(1 +E)] specific assumptions about geometry and, a is only of second-order importance. therefore, holds for all test stimuli. Note The meaning of E. For a large uniform AJ1pI [1- al(1 + E)] that the area-factor a2 appears wherever field B, the asymptotic value of G as A = (19b) 2 c J1Po[1 - a o(1 + E)] WI appears, i.e., in both the numerator and liB ..... 00 is G =H/kl = I/(ka ). Assuming denominator of J1I(I, and therefore a2 is that the threshold of detection J1G is some Equation 19b is valid for large values of Q. cancelled in Eq, 17. fractional value E of the limiting value ofG To reduce Eq. 19b further, it is necessary

Perception & Psychophysics,1970, Vol. 8 (3) 147 to calculate the l1p's and the a's. This has shape of l1Q vs Q, which cannot be an exact solution is derived to relate all the already been done for the case of point T's compensated by linear transformations of factors. (l1po ,ao) and for half-field T's (l1PI ,al ). the coordinates. Because H(z+), H(z_), When a sine stimulus Q(x,y) is the input For T = a point, l1po I(z+), I(z_) occur in C3(Q) only in terms of to the feedforward system, the output is: [A/WH)[I - WH(l1X)] "'" 0.98 A/WH, and Q directly and in terms of the area factor a, ao = 0.038. For T = a half-field, PI "'"0.88, the problem reduces to that of finding the Q(x,y) * wH(x) independent of A, and al = 0.4,. effect of temporal stimulus variations upon G(x,y) = I + kQ(x,y) * Wt(x) Substituting the algebraic quantities into a. Eq. 19b yields In Eq. 14, a was defined as the ratio of (3 = l11/I(z_) to 0: = l1H/H(z_). Visual responsiveness to transients means that test (I + kQ) + rnkQ cos * WI stimulus illuminance QT contributes more effectively to l1H and to l11 than (20) (l9c) background illuminance QB contributes to H(z_) and I(z_). Let 1/H be the factor To find the threshold it is necessary only The last expression of Eq. 19c is a expressing the increased effectiveness of a to determine AG = max G(x) - min G(x). convenient, rough approximation. transient input to l1H relative to that of a The algebra is simplified by disposing early Substituting the numerical values obtained steady-state input; i.e., we use 1/HQT to of unneeded terms. Equation 20 is of the for the case where wH, WI are normal calculate l1H rather than QT. Let 1/1 be the form (CI + VI)/(C2 + V2) where c and v distributions, and al/aH = 3, gives fac tor expressing the increased denote constant and variable terms (as a effectiveness of a transient inpu t to l11 function of x). Furthermore, the maximum relative to that of a steady-state input. Let lim Ac/WH "'" 0.47. of G occurs when x = 0, the minimum Q.-o 1/= 11I/1/H· By substituting 1/H and 111 into occurs at x = 1T/W, so that VI (x+) = Eq. 14, the revised area factor a' is seen to -VI (x _) and V2 (x+) = -V2 (x_). Thus l1G A similar calculation of the critical area be a' =na. is of the form of the dark-adapted model shows it to The effect of temporal variations on the differ negligibly from WH. form of l1Q vs Q is thus given by a single 1 + a J Il-aj a-b l1G=c[-- -c -- =2c-- General solution. The values of H, I, and parameter, 1/. When T and B have the same 1+ b 1- b 1- b2 G are defined by Eqs. I, 2, and 3a, even temporal waveshape [i.e., QT(t) = cQB(t)], when a closed-form solution does not exist. then by definition 1/= 1.0. When T is a step (21) By using summations to approximate function, superimposed on a steady integrals, G may be calculated for any background B, then the effective value of1/ Before substituting for a, b, and c in Eq. 21 explicitly defined QT, QM, QB. The author is will vary with time. Suppose inhibition it is desirable to change scale factors indebted to Mrs. J. T. Budiansky for a grows more slowly than excitation [for slightl~ and to define w'H = computer program to calculate G(x,y) for example, VI(t) = vH(mt), m > 1]. Then 1/ (21T)-V2 a H - l w H and W'H = the special case of T = a disk stimulus of becomes a function 1/{t) of the time t after (21T)- \l,al- I WI, so that W'H(X) and W'I(X) radius r, B = a uniform background of onset of the step. Initially 1/(t) < I, finally are probability distribution functions. By illuminance Q, and the weighting functions 1/(t) = I, and for reasonable assumptions defining AG' = (21T)-'12aH- IAG and WH, WI proportional to normal about VH(t), VI(t), 1/(t) will overshoot the k' = (21Tt\l,alk the primed system becomes distributions (Fig. 3). The results of these value 1.0 for some intermediate value of 7 equivalent to the unprimed system. In computations are illustrated later, in t. Eq. 20, and in all equations derived from Section III. The variation of 1/ with time means that it, changing from unprimed to primed WH , Transient effects. In vision, transient the spatial response characteristics vary WI, l1G, and k preserves the equalities. responses generally are greater than with the duration of a test stimulus. In Considering Eq. 20 to be of the form steady-state responses, and test stimuli are considering psychophysical data, a c( I + a)/O + b) gives the following detected by their transient response. In the simplifying assumption will be made; identifications of a, b, c: model, transient effects are subsumed namely, at a moment of detection td, the model can be represented by the currently under the temporal weighting functions, a s m cos e w'q =mexp [-~n2a~J; VI! , VI, which are not known exactly. In effective value of1/,1/(td). this section, some conclusions are derived b = my cos * w'l = m exp [-~ n2ail; from very general considerations about lid. Sine-WaveContrast Thresholds A sine-wave contrast stimulus is defined VH, VI· c=""(/k'; In considering temporal effects, the by Q(x,y), concern here is primarily with the form of ""( = kQ/(I + kQ). the l1Q vs Q function rather than with the Q(x,y)= Q(l + m cos wx). particular values of l1Q. E.g., the basic form Substituting these values into Eq. 22 gives of the l1Q vs Q function is l1Q = c I + C2 Q. Q(x,y) is a one-dimensional stimulus in the The temporal weighting functions affect CI sense that retinal illuminance varies only in l1G' = m(2""(/k) exp [ -~ n2a~] and C2' On a graph of l1Q vs Q, changes in the x-dimension. For a given threshold CI and C2 merely shift the curve vertically criterion AG of the feedforward model, [I-""(exp [-~(af -a~)J] and/or change the scale of the horizontal and for a given average illuminance Q, the coordinate; they do not change the form of modulation threshold m is a function ofw. /[I-m2""(2exp(-n2ai)].(23) the curve. When area effects are Significant, The function depends on Q and on AG. By assuming, as before, that l1Q is given by Eq. 12, which contains a EQuation 23 is of the form factor of [1 - C3(Q)] -I. The factor C3(Q) m depends on Q, producing changes in the A = (24) 1- m2 B'

148 Perception & Psychophysics, 1970, Vol. 8 (3) 6.------., Fig. 6. Theoretical log AQ vs log Q (test threshold vs adapting bact.ground} functions. (a) Pure feedforward forward 5 model; (b-I) models with feedback plus feedforward (see Fig. 7). The ratio r of feedback to feedforward for each of the curves is (a) r = 0, (b) r = I, (c) r = 10, 4 6 (d)r= 102,(e)r= 104,(f)r= 10 • ~

Perception & Psychophysics, 1970, Vol. 8 (3) 149 ratio of feedback to feedforward signal signals than equally detectable ~l/ signals each compornent l/i must already be in the may be, the same asymptotes ultimately on low-l/ backgrounds. To utilize the Weber-law range, and the ~l/'s must be are achieved! stimulus information efficiently, the a taken in the same direction (e.g., all Figure 6 illustrates how the nature of would have to adjust his detection criterion increments or all decrements). the transition from dark-adapted behavior delicately to each background level l/. In The right-hand equality has not been (horizontal line) to light-adapted behavior fact, human as are notoriously unable to widely tested. Sperling (1965) found it to (line of slope 1) can be determined by the make accurate judgments of background hold over a wide range of masking flashes ratio of feedback to feedforward signals. retinal illuminance, which poses the superimposed on backgrounds of varying For example, a line of slope ~ over a problem of how as could adjust a criterion luminances (cf. Fig. 4), but his range of horizontal range of about 3 logs (Curve C) to depend on something (l/B) that they backgrounds was limited. 2. occurs when r = 10 The threshold cannot estimate. On the other hand, Many theorists have proposed that the relation of ~l/ ex l/v, is a square-root law. It humans are able to make stable detection visual system behaves linearly for small is commonly observed at low levels of l/, judgments of ~l/, even with little practice. signals. Small-signal linearity is a trivial particularly in peripheral viewing, i.e., in These difficulties are resolved naturally by consequence of any realizable continuous rod vision. According to the model, the the shunting feedback component. system (i.e., also having a continuous difference between threshold functions Shunting feedback produces an output derivative). The problem with small-signal observed in different regions of the retina that is asymptotically proportional to Qv,. linearity is that the linear range may be is explained by a regional variation in the Thus, intensity discrimination would be very small. The significance of Eq.28 is parameter r. impaired because of the reduced dynamic that it predicts superposition to hold for In terms of neural mechanism, a large range of the output. On the other hand, large signals; the larger the background value of r (in the peripheral retina) requires small threshold increments of 0' • Qv, would input signals, the better it holds. that feedback signals be mediated by produce output increments proportional to Considering how nonlinear the elements of special cells-possibly horizontal cells. This 0' an d therefore would be equally the model are, it is remarkable that is because the feedback is effective even detectable. The shunting feedback system, additivity should ever obtain, all the more when only one quantum per hundreds of which reduces the input by a factor of l/v" remarkable that the model should become rods is being absorbed from l/, so that the is the ideal match to a stimulus in which a linear system with large masking signals. probability of a quantum from ~l/ striking RMS noise increases by a factor of l/Y'. In a rod that recently received a quantum the output of a shunting feedback system, Receptive Fields from l/ is small. Adaptation of a rod is the RMS value of quantal noise in l/ Consider the output of the model G(x,y) determined by its environment (e.g., the remains constant, independent of Q. Thus, at a particular point x,y. Let the input be a rod pool, Rushton, 1963). Although the as Q varies, increments ~l/ of constant point of light ~l/(x',y') superimposed on a same also may be true for foveal vision statistical significance produce constant steady uniform background l/. It was (Le., cones), it cannot be proved on a priori incremental outputs, independent of l/. assumed (Section II) that the excitatory grounds, because cones respond only at With one stage of shunting feedback, the WH and inhibitory WJ weighting functions higher levels of retinal illumination (e.g., detection rules can be independent of l/-an were radially symmetrical, so that G is a several quanta/cone; Brindley, 1960, enormous simplification of the detection function only of ~r =../(X_X')1 + (y_y')2, p. 187ft) and therefore the feedback loop problem. the distance of spot ~l/ from x,y. When theoretically could be within a single The second remarkable property of y' =y, then ~r = ~x. receptor. shunting feedback is that it does not alter To calculate G(r), specific distributions Shunting feedback: An evolutionary the desirable properties, at high input must be given for wH(r) and wJ(r). In the adaptation to quantum noise. At low levels luminances, of a subsequent feedforward examples of this section, it is assumed that of illumination, visual discrimination component. The Weber-law characteristics WH and WJ are proportional to approaches the limits set by photochemical of the system result from the feedforward two-dimensional normal distributions, with absorption and the quantum variability of component; these characteristics are left the standard deviation aJ of WJ equal to the stimulus (Hecht et al, 1942; DeVries, entirely intact by shunting feedback. 3aH (Fig. 3). In terms of area, the effective 1943; Pirenne, 1962; Barlow, 1962b). That Additivity of illuminance components. area WJ of WJ is nine times the effective is, the performance of the eye approaches Consider a stimulus composed of several area Wjj OfwH: WJ!WH =(aJ/aH)2, the performance of a perfect detector components; for example, of a brief The shape and magnitude of the which absorbs the same number of quanta masking flash and of a steady background response G(r) to ~l/ depend on both Qand as does the eye (Barlow, 1962a). At high (Fig. 4). The threshold luminance ~l/. Three values of l/ are considered: Q= 0, 3 retinal illuminances, the eye is not nearly measured in the presence of each 1/18, and 10 , representing darkness, a so efficient. component stimulus, individually, is ~l/h dim background, and an The measure of quantal ~l/2, ••• , ~Qn, etc., and the threshold in asymptotically-intense background. The (Poisson-distributed) stimulus noise is the the presence of all components together is input value of ~l/ is chosen arbitrarily so noise's standard deviation, i.e., its ~l/1 + 2 ••• + n- Then the model (Eq. 10) that-in the absence of self-inhibition-it root-mean-square (RMS) value. If the limit predicts that would produce a maximum response ~G of of visual discrimination were determined magnitude equal to the asymptotic by quantum noise of the stimulus, then the steady-state value of G in the light (e.g., so detection threshold ~l/ would increase as that 2 l/1/ , because the RMS value of quantum (28) 2 2). noise increases as l/1/ • If there were no max ~G =lim G(l/B) = l/ka l/400 visual adaptation, test signals ~l/ of equal The left equality holds for low retinal detectability would then produce vastly illuminances (where the ~l/ vs l/ function is Because the self-inhibiting effect of ~l/ on different visual responses at the point of flat) and the right equality holds at high itself is neglected in choosing the value of detection; i.e., ~l/ signals from high-Q illuminances (where ~l/ is proportional to ~l/, the actual value of max ~G is 2 backgrounds would produce much larger l/). For the right-hand equality to hold, somewhat less than l/ka .

150 Perception & Psychophysics, 1970, Vol. 8 (3) POINT LINE HALF-FIELD Fig. 8. Output of the feedback component of the model at one point (x,y) as the location (x + Llx,y) of a test stimulus is varied. Horizontal scale markings indicate radial distances of 1.0 uH' Sections a, b, c x show responses to a point stimulus; Sections

;' 11 = 2; Phase 2 (b.e.h) transient rebound, / 11 = 0.5; steady-state (c,f,i) 11 = 1.0. The

,I three different curves in each section represent responses G(Llx,QAQ) calculated at three different levels of background Q: ------darkness (lowest curves), asymptotically intense Q (highest curves), and an RADIAL DISTANCE OF STIMULL!S IN"H J"'TS intermediate level of Q(middle curves). The dashed curves in Sections c, f, i indicate Consider first Fig. Be. It illustrates the Figures 8a and 8b illustrate the test-induced excitation ~H and inhibition steady-state response G(x,y) to a point LlQ point-response of the model in Phase I and LlI separately (-LlI is illustrated). The as a function of Llr, the distance of LlQ Phase 2, respectively, with these values of illuminance LlQ of the test stimuli was so from x,y. In darkness, the response G has .;vershoot and undershoot. The actual chosen that the maximum value of LlH is no undershoot. At high backgrounds, the amount by which transient responses the same for all 27 curves; within a section response G has a slight undershoot. The exceed steady-state responses is immaterial the entire curves for LlH and LlI are the curves illustrating the response G(x,y) to to the shape of the spatial response same at all three levels of background Q. LlQ(x + Llr,y) are analogous to the usual characteristics illustrated in Fig. 8. The See text for more details. physiological representation of neural various temporal effects on spatial response responses, e.g., of the firing rate of a retinal characteristics were shown in Section IIc to adaptation. These effects of light ganglion cell as a function of the retinal be determined by a single parameter 11 adaptation occur without any change of location of a stimulating spot of light. The which is a ratio of two ratios, B!«: (3 is the connectivity of the model; they simply reasons for identifying the feedforward ratio of effectiveness of transient (test) reflect the nonlinear, intensity-dependent stage with retinal ganglion cells are set inhibition to steady-state (background) mode of response of the feedforward forth by Sperling and Sondhi (1968, inhibition, and a is the ratio ofeffectiveness component. p. 1139). The WH function corresponds to of transient (test) excitation to steady-state Comparison of the Phase I and Phase 2 the center of the receptive field and the WI (background) excitation. For Phase I and responses shows that: for small values of function to the surround. Phase 2, 11 is assumed to be 0.5 and 2.0, Llr, both Phase I and Phase 2 responses are To represent visual processes more respectively; in the steady state, 11 is 1.0 by positive; for intermediate values of Llr, accurately, temporal factors need to be definition. Phase I responses are positive and Phase 2 taken into account. Using a similar model In the usual type of physiological responses are negative; for large values of (feedback plus feedforward), Sperling and recording, the maximum of response is Llr, both Phase I and Phase 2 responses are Sondhi (1968) showed that its temporal noted without regard for the precise time negative. The areas defined by these values response to an impulse of lightwas of occurrence of the maximum. Therefore, of ~r correspond to the on-center, on-off monophasic at low background luminances the appropriate responses to compare to surround, and pure-off surround of an and biphasic at high luminances. The physiological recordings are the maxima of on-center receptive field. These areas are surround portion contributed its response Phase I combined with the minima of similarly defined for off-center receptive more slowly than the center and thus Phase 2. However, the main features of the fields, except that the sign of the model's seemed to be delayed. Step inputs to the comparison are independent of these response must be reversed. feedforward system may be considered to details. First, they show that in the dark Impulse response (point-spread induce three recognizable phases. In the receptive field has a response that is function). The curves of Figs. 8a, 8b, and Phase I, the excitatory input H is at its characteristic only of the center. With 8c give the output G(x,y) at a single point maximum overshoot (e.g., two times the increasing light adaptation, the receptive x,y, as the location x + ~x,y of a steady-state value) and the inhibitory field shows, more and more, the effects of stimulating spot is varied. A little reflection component has acquired its steady-state an antagonistic surround. The light-adapted will convince the reader that when the spot value. In Phase 2, the excitatory receptive field contains areas that had is stationary at x,y and the output is component has subsided to its steady-state responded with the center characteristic in measured at different locations, x + Llx,y, value, but the inhibitory component is the dark, but which respond with the the resulting curves describing G(x + Llx,y) overshooting to two times its steady-state surround characteristic in the light. Thus are identical to those shown in Fig. 8. This value. Phase 3 is the steady-state, described the effective diameter of the center of the symmetry between input and output exists above. receptive field shrinks with light whenever the properties of the

Perception & Psychophysics, 1970, Vol. 8 (3) lSI transformation remain homotropic 05,------, generated from Eq. 25 by assuming a (independent of absolute location). It detection threshold of € = 0.01 and follows that the curves of Figs. 8a, 8b, and neglecting the effect of number of visible 8c describe the spatial impulse response of cycles. The predicted sine wave thresholds the model (i.e., its response to an impulse (m vs w curves for various values of Q) are in space, a point). In fact, Fig. 8 illustrates illustrated in Fig. 9. only half of the spatial impulse response; Figure 9 shows that as average the response is mirror-symmetrical around 20 luminance increases, so does the model's its maximum. The full spatial impulse acuity for sine wave gratings. At high response function also is called a 50 luminances, this trend reverses itself for point-spread function, i.e., a function that 100 low-frequency gratings, thereby producing describes the spread of excitation and a pronounced peak of sensitivity at about inhibition around a point input. 0.09 cycles/og , At the peak, a modulation -3 - 2 -I o Line-spread function. The curves in Figs. LOG IO (CYCLES PER CTH) of about 0.007 is detected. Beyond this 8d, 8e, and 8f describe the line-spread peak, the loss of acuity with frequency is functions of the model; they represent the Fig. 9. Spatial sine-wave modulation very rapid, so that the highest frequency spread of excitation and inhibition around thresholds of the feedforward section of that can be detected is about a thin line stimulus. The intensity AQ of the model as a function of spatial 0.4 cycles/og , the line stimulus is such that, when it is frequency. Average retinal illuminance is The modulation transfer functions of centered directly over the receptive field, it the same for aU points of a curve, and Fig. 9 resemble transfer functions provides exactly as much stimulation to differs by 0.5 log, 0 units for adjacent measured for the human eye under various the center of the field (WH) as did the curves. Dashed curves are theoretical conditions, e.g., by Schade (1956), corresponding point stimulus in Figs. 8a, calculations for modulations greater than Westheimer (1960), Campbell and Green 8b, and 8c. However, line inputs induce 100%. Maximum resolution occurs at (1965), Green and Campbell (1965), more inhibition relative to excitation (by a highest average illuminances for gratings of Robson (1966), Campbell et al (1966), and factor of ol/oH) than do point inputs. approximately 0.1 cyc1es/oH. Campbell and Robson (1968). To In the model, only the T/ ratio of determine the amount of quantitative inhibition to excitation matters; it matters agreement, 0H must be specified. Perhaps not whether it is determined by temporal corresponding point and line stimuli in the the most easily specified feature of the or spatial factors. Thus, elongating a point left and center of Fig. 8. At high values of human detection data is the peak in the stimulus into a line is equivalent to background Q, this value of AQ is such that transfer function which, at high luminance, increasing T/ for a point stimulus by a AQIQ = 0.25, a rather low contrast ratio. occurs at about 8 cycles per degree factor of oI!oH = 3, thereby greatly Spreading a line stimulus into a half-field (Campbell & Green, 1965; Campbell et aI, enhancing the manifestations of inhibition. greatly enhances the manifestations of 1966) and at 2 to 4 cycles per degree For example, the peak of the Phase 2 inhibition. For example, at high (Campbell & Robson, 1968). Setting the excitatory response to a point stimulus is background retinal illuminances, the peak of model's response to 4 cycles per six times greater than the peak of the Phase 2 response to a half-field stimulus is degree determines 0H = 1.47 min. The inhibitory response; the peak of the negative everywhere, corresponding to a predicted acuity limit is then about 4.4 Phase 2 excitatory response to a line negative afterimage. times the peak (18 cycles per degree) stimulus actually is smaller than the peak Half-field stimuli are among those that whereas the observed acuity limit is about of the inhibitory response. produce the Mach band iHusion. Mach 50 cycles per degree for stimuli viewed Lines are one-dimensional impulses in a bands in the model are evident in Phase 1, through the normal optics of the eye and two-dimensional space. The line spread Phase 2, and in the steady-state (Figs. 8g, about 60 cycles per degree for retinal functions are the one-dimensional impulse 8h, and 8i), although the vertical scale of stimuli of 100% contrast modulation. responses, and are the appropriate ones to the figures is too small to show them well. Typically, the observed acuity limit consider in analyzing one-dimensional The transient, Phase 1 Mach bands are less exceeds the observed sine-sensitivity peak stimuli, such as those used to study grating prominent than the steady-state bands. by a factor of about 4 to 20. acuity and Mach bands. The acuity target Figure 8h (high background Q) illustrates Generally, then, the model predicts a that has been used for the most thorough inverted Mach bands, such as would be too-small range of peak-to-cutoff in and analytical experimental studies is the seen in a blurred afterimage. Mach bands sine-wave modulation thresholds. This is spatial sine-wave grid; responses of the are considered in more detail later. not a critical error because it results from model to these grids are considered later. the steep high-frequency attenuation Halffield responses. A half-field test Responses to Spatial Sine Waves characteristic of the normal ·distribution stimulus at Location x, is defined as being When the eye is confronted with a sine assumed for wH(r). In fact, from the of retinal illuminance AQ everywhere to the wave acuity grating (see Section lid), observed h.igh-f'requency attenuation right of x and of illuminance zero to the detection of the grid depends on three characteristic, it is possible to calculate a left of x. The response of the model to a main variables: the fineness of the grid wH(r) (different from a normal half-field test is illustrated in Figs. 8g, 8h, (spatial frequency w in terms of number of distribution) that would reproduce the and. 8i. Because the response has no cycles per degree), the average luminance observed high-frequency attenuation significant symmetries, it is necessary to (Q), and the modulation amplitude of the characteristic exactly and thereby calculate the entire response, instead. of sine perturbation (m, a fraction of Q). eliminate the prediction error completely. merely half of it as for points and lines. Detection also depends on other factors, Multiple sizes of receptive fields. The The intensity AQ of the half-field stimulus such as the D's internal threshold criterion calculation of an optimally fitting wH(r) is is such that, when the light area completely and the number of cycles viewed. not made here because the probable reason covers the center of the receptive field, it Predictions of the model (considering only for the failure of the model to reproduce delivers as much light to WH as do the the feedforward component) were the sine-wave threshold data precisely is

152 Perception & Psychophysics, 1970. Vol. 8 (3) not its nonoptimal choice of wH(r) and makes good sense that a system for wl(r) but its attempt to fit all the data with detection of spatial detail should gather but a singlechoice of WH and WI. There are information from a small area (in order to now at least three lines of evidence to gain maximum spatial resolution), and for indicate that various sizes of 'receptive a long duration (in order to gain maximum fields operate jointly to determine statistical significance of its input). On the thresholds. other hand, a system for detecting Originally, the existence of two or more t e m poral variation must sample parallel detection systems with different information in relatively short periods of size constants was proposed by several time (in order to achieve temporal LOG I authors, based on their analyses of resolution) and, therefore, in order to detection thresholds (e.g., Hallett, 1963; obtain a statistically significant sample, it Fig. 10. Dependence of predicted test Campbell & Robson, 1968). would have to increase the size of the threshold All on the area of test stimulus. Unfortunately, various complex technical sampled area. Curve A is for a point stimulus; Curve B is factors, such as lack of perfect radial In most normal viewing situations, both for a half-field stimulus. For A and B, symmetry of grid detectors (e.g., Campbell the primarily temporal and the primarily detection is assumed to occur in steady et al, 1966), the effect of the number of spatial systems are likely to be stimulated. state. Curve C represents theoretical visible grid cycles (Coltman & Anderson, In certain conditions, however, even when thresholds of a half-field stimulus when 1960; Campbell & Robson, 1968; boundaries are available for spatial contrast detection is assumed to occur in Phase 2. Nachmias, 1968) and the fact that detection, they may not be used; that is, For ease of comparison, the curves were low-frequency sine waves extend over detection may be only of the temporal moved verticaUy so as to make the nonhomogeneous retinal regions, make it variation. This fact accounts for some thresholds superimpose at left (Le., in the difficult to prove or disprove the multiple obvious discrepancies between apparently dark). receptor hypothesis by simple sine-grid similar studies of detection (Sperling & detection experiments. Sondhi, 1968, p. 1143). Pulse duration. Barlow (1958) The strongest evidence for multiple demonstrated that pulses of a long receptor systems comes from Blakemore & Second-Order Threshold Effects duration imply a narrower spread function Campbell's (1968) study of selective Change ofarea of integration with light than do brief pulses. This effect is masking of sine-grids by sine-grids. adaptation. The spatial responses of the predicted by the model because 11 for long Blakemore observed that prolonged feedforward system to brief- and to pulses is larger than 11 for brief pulses, thus viewing of a particular frequency of long-duration pulses of light are indicated more shrinkage (cf. Eq. 19c, steady-state vs sine-wave grating selectively increases the by the Phase I and steady-state responses Phase I responses in Fig. 8). detection threshold for a grating of exactly in Fig. 8. To find the output G(x,y) for Pulse area. A third experimental the same spatial frequency. In Blakemore's small-increment test signals lIT(x,y) other observation is that large-area stimuli have analysis, there are at least as many sizes of than points, lIT(X,y) is convoluted (I.e., shorter periods of temporal integration neurons as there are frequencies that can weighted) by the point-response G from than do small-area stimuli. This effect is be selectively masked, and this appears to Fig. 8. This procedure uses the property of predicted by the temporal version of this be a very large number indeed. small-signal linearity in the model. model (Sperling & Sondhi, 1968) via the Indirect evidence for multiple detection Alternatively, G(x,y) for the test may be assumption that the feedforward systems. A third kind of evidence for the calculated directly. When the difference controlling signal comes from a larger area existence of at least two parallel detection between max G(x,y) and min G(x,y) (the surround area of the receptive field) systems, with different parameters, is exceeds the detection threshold E, than does the controlled signal (center area derived from studies that show that critical detection is signaled. of the receptive field). duration depends on the O's task; i.e., on To compare the model wiQ! the results A fourth observation (Barlow, 1957) is whether he detects acuity targets or of threshold experiments, it is more that thresholds of large-area stimuli brightness (Kahneman & Norman, 1964). convenient to reverse the procedure increase from their dark-adapted value at Additioral evidence comes from some outlined above, and to calculate the spatial lower luminances than do thresholds of A unpublished experiments by the author. impulse response G(x,y) implied by the small-area stimuli. This is predicted from Critical duration was measured under psychophysical threshold results, and then Eqs. 12 and 18. The model's prediction is conditions in which the boundaries of the to compare G(x,y) with G(x,y). illustrated in Fig. 10, which shows All vs II test field coincided with the boundaries of Calculations of this kind (Kincaid et al, threshold functions for a point test the masking field, and also in conditions in 1960; Blackwell, 1963)8 indicate that the stimulus and for a half-field test stimulus. which the test's boundaries lay within functional impulse response G becomes The parame ters for the curves are 11 = 1, those of a larger masking field. When the T narrower with light adaptation, much as E =0.1. and M boundaries coincide, detection is does G in Fig. 8, i.e., qualitative agreement Figure 10 also illustrates a speculative restricted to detection of a temporal with the model. The predicted shrinkage of All vs II function of a large test field for illuminance pulse. Under these conditions, the critical area of the spatial response is which it is assumed T/ =2, i.e., detection critical duration was found to be given by Eq. 19c. Shrinkage depends on occurs in Phase 2 of a long test flash. This substantially smaller than when test the area factor a. As the area factor a is function is particularly suggestive because boundaries were available for contrast altered by 11 to a' = al1, shrinkage also in a middle range of illuminances, All detection. The results imply that a system depends on the effective 11 value of the test increases more rapidly than lI, a mystifying with a long time-constant is operative when flash used to measure it. For 11 = 1, the but not uncommon phenomenon of a boundary is being detected and a system predicted shrinkage with light adaptation experimentally obtained All vs II curves. with a short time-constant is operative was a factor of ~, which is quantitatively The phenomenon occurs in the model when temporal pulses are being detected. somewhat less than typically was observed because of the combined changes in In terms of information processing, it by Blackwell (1963). se n sitivity and in spatial field

Perception & Psychophysics, 1970, Vol. 8 (3) 153 .27 1 3 Fig. II. Spatial response of the model to incremental disks of various diameters. Background retinal illuminance is very high. The responses are superimposed so that the left edge of all the disks coincides to with the center vertical line. Radii of the 100-...._::I::::::S~~!~:!:~~~~t-+-- -+-...,.:~:!!!lIp- ..."""'I~-t disks are indicated in aH units. The ...... maxima of the responses have been normalized to be equal. The horizontal scales are marked in units of one au . Responses are drawn only up to the center of the disk (except for r = 100 aH); each response is mirror-symmetrical to the right characteristics. The curve for 11 = 2 is From the locations and values of Gmin of its center. speculative because when 11 = 2, the model and Gmax it is possible to calculate test predicts that in the interior of an thresholds ill/. The results of these the (shifted) individual functions. The incremental test the response ilG is less calculations are illustrated in Fig. 12, overall function, perforce, has a shallower than it would have been in the absence of which shows how threshold depends on the minimum than any of the individual 2 the test. This condition describes a negative area (A =1Tr ) of a disk stimulus. Figure 12 functions. In fact, with certain afterimage, and frequently negative shows thresholds calculated at very high distributions of component receptive field afterimages of a test stimulus are more background retinal illuminances and also at sizes, the overall ill/-vs-radius function will detectable than the stimulus (Brindley, zero background retinal illuminance; the have no observable minimum. As a practical 1959; Sperling, 1960, 1965, p. 556). two sets of thresholds are scaled to matter, it would be impossible to However, the conditions under which ill/ coincide for point tests stimuli. discriminate psychophysically between a increases more rapidly than l/ probably are As in psychophysical experiments, for shallow minimum and no minimum at all. not limited to detection of afterimages. small areas of disk, the threshold depends Differences occur only with large-area The principle underlying the adaptation, only on the total energy (ill/ ex: r " 2 , disks; the gross nonhomogeneity of the area, and duration effects is that the more left-hand asymptote). For large disks, ill/ is retina makes such threshold data too inhibition (feedforward control signal) constant, independent of r. The difficult to interpret. there is in the system, the sharper is the intersections of the two asymptotes (2) Background retinal illuminance. If l/ spatial spread function and the shorter the determine the critical areas Ac of the disk is not asymptotically high, individual temporal function. In the model, these stimuli. In the dark Ac = WH (the effective ill/-vs-radius functions will be intermediate properties result because the inhibitory area of the excitatory center), between the extremes illustrated in Fig. 12; spatial weighting function WI is wider than corre sponding to a disk radius of the relative minimum will be WH, and because inhibitory control is r = aHV"2. In the light adapted model, proportionately shallower. slower than the controlled signal (Sperling A c = .47 WH. These graphically (3) Statistical fluctuation effects. There & Sondhi, 1968). Thus, large areas and extrapolated values are the same as those are various statistical factors that enter into long-duration stimuli favor inhibition and obtained by direct calculation from psychophysical thresholds. For example, produce the results characteristic of more Eq. 19c. detection may occur independently in inhibition. In fact, the various complex A shallow minimum in ill/ vs r occurs different parts of the test field. The second-order interactions of duration, area, when the radius of the test disk is 2.3 aH. number of opportunities for detection and background luminance are subsumed The minimum is about 0.2 log! 0 units in a single term [C3(l/), Eq. 12] , which can deep. The disks which contain the be interpreted as the amount of inhibition minimum threshold energy, however, have in the feedforward control pathway at the infinitesimally small areas (r ,."0). instant of detection. There are three basic reasons to expect, in psychophysical data, the depth of Thresholds of Disk-Shaped Test Stimuli observed relative minimum of ill/ vs r to be Although disk-shaped stimuli are the even less than the depth of the minimum most common psychophysical test predicted in Fig. 12: (1) multiple sizes of stimuli, their thresholds are extremely receptive fields, (2) failure to use high complicated to calculate in the model. background light levels, (3) statistical Threshold predictions were obtained under fluctuation effects. the f'ollow ing assumptions: (1) the (I) Multiple sizes ofreceptive fields. The " , LOG IltaOlUSI IITHI background retinal illuminance is large; evidence for multiple sizes of receptive (2) the threshold is small (e

154 Perception & Psychophysics, 1970, Vol. 8 (3) would be proportional to the producing Mach bands are precisely such same-because adaptation varies somewhat circumference of the test disk (i.e., to 2m) stimuli. Furthermore, the value of the across the field. for r > diam(wH)' Another factor is the illuminance difference between adjacent When predicting the effect of quantum fluctuation of disk energy, even areas of the stimulus determines ~G in the background retinal illuminance on when ~Q nominally is constant. Quantum output of the model. Thus, by using test threshold values of ~Q, FF and FB + FF fluctuations increase in proportion to the stimuli that are well above threshold, are quite different; here the difference was square root of the energy; the relative inhibitory effects are magnified in the explicitly considered (Fig. 6). fluctuation therefore decreases with the output. With such stimuli, the model The most important effect of FB is in its square root of energy. In a detection predicts quite prominent Mach bands; temporal control of adaptation. This was experiment, the decrease in relative these were illustrated in Fig. 2. subsumed in the parameter 1/. We consider fluctuation could yield an improvement in Other predictions about Mach bands by now a significant phenomenon of contrast detection proportional to Ay,. The slope at the model also are qualitatively in detection that, strangely, appears to have the right-hand side of the minimum of ~Q agreement with observations. The model been overlooked. In very brief flashes, Os (Fig. 12) is less than that of Ay, (slope = I) predicts that Mach bands are not observed cannot detect contrast differences of less so that if either of these statistical effects at extremely low retinal illuminances, that than about 5% across a boundary; 10% is a were operative in the region of the at moderately low illuminances the bright more common limit on performance minimum, the minimum would be Mach band is more prominent than the (Brindley, 1959; Sperling, 1965). In a long obscured entirely. In fact, in dark band, while at high illuminances the flash of 1.0 sec duration, Os easily detect a psychophysical experiments, threshold ~Q reverse is true. It also correctly predicts contrast of 2%. Os will fail to detect a usually continues to decrease with disk that Mach bands can be seen in brief low-contrast boundary in a brief flash and diameter, even for quite large disks, flashes, but that they are less prominent yet detect the same boundary in a long presumably for the reasons given above. than bands that appear in continuous flash that contains but a fraction of the Basically, the reason that the inhibitory viewing. energy. Better detection of long flashes field does not produce a large ~Q-minimum The main deficiency of the model is in (when energy is held constant) is the in the predicted disk-detection function is its predictions of apparent brightness of opposite of temporal integration. In terms that large disks are detected not in their large uniform areas. Once a uniform area of the model, it implies that the output center but at their boundaries. The reaches a moderate intensity, it no longer "saturates" when the input becomes large. presence of the ~Q-minimum has been a can induce significantly greater responses This kind of saturation, of course, is a traditional psychophysical criterion of in the model as its intensity is increased characteristic of the FB stage. However, to physiological inhibition. The small, shallow further; i.e., the model's response saturates predict this phenomenon quantitatively, minimum predicted by the model, even at quickly. Thus the model can predict indeed, to predict any of the vast number the highest background retinal illuminance, brightness phenomena that involve low and of complex temporal-spatial interactions in plus the additional facts that the moderate degrees of contrast but not vision, would require a much more multiplicity of receptive field sizes and brightness phenomena in general. In the complete dynamic specification of the FB statistical effects would tend to obscure it, case of Mach bands, this means that and FF stages than was necessary for the account for the difficulty of observing predictions of the model tend to emphasize eI emen t ary model of spatial vision evidence for inhibition in psychophysical the prominance of a band relative to the proposed here. detection experiments. brightness of adjacent uniform areas. It is When critical areas are calculated from quite probable that, in the visual system, IV. SUMMARY AND CONCLUSIONS the thresholds of disks viewed foveally in information about overall brightness is A three-component model for spatial absolute darkness, they typically have handled differently than information about vision is proposed. The first component is diameters of 4 to 5 min (Bouman & contours, i.e., about brightness differences. an RC stage controlled by shunting Walraven, 1957; Glezer, 1965).9 To fit the Insofar as this is true, the deficiency of a feedback. The second component is an RC model to these data, 0H is estimated from contrast-detection model to in predicting stage controlled by shunting fee dforward, 0H = diam (critical area in dark)/V8, giving overall brightness of uniform areas is not a with the controlling feedforward signal a 0H of about 1.4 to 1.7 min. The value of defect. originating from a larger area than the 0H estimated from disk detection controlled signal. Finally, a detector experiments thus is in agreement with the The Effect of Feedback in the Model compares the locations that produce the value of 1.47 estimated from the peak of Since spatial effects (as opposed to largest and the smallest outputs for a the observed sine-wave threshold function. adaptational effects) at the level of the particular signal and indicates detection feedback stage (FB) are very difficult to when the algebraic difference exceeds a Mach Bands distinguish from those at the level of the threshold criterion c. The shunting stages Figure 8 showed that the output of the feedforward stage (FF), in the foregoing are analogues of synaptic inhibitory model has a much shallower minimum for analyses, all spatial effects were assumed to processes. a point stimulus than it does for a line occur in the FF stage. We now consider the The model predicts that thresholds for stimulus, and the minimum for a line is conditions under which FB becomes increments ~Q on a background Q follow a shallower than for a half-field. If the height important. square-root law at low background of the uninhibited maximum is designated In predicting disk-thresholds (Figs. 11 luminance (~Q ex Q'!2) and a Weber law at as Ji, then the depth of the minimum is and 12), FF alone and FB + FF are high background luminance (~Q ex Q). The approximately proportional to h/o2 for a equivalent because the thresholds are model achieves its Weber-law response as a poin t, to h/u for a line and to h directly for determined at a constant adaptation level Q consequence of the feedforward a half-field. Since the minima of G are the and for infinitesimally small increments component and without having any signals signs of inhibition, to demonstrate ~Q. proportional to the logarithm of the inhibition one should use stimuli In predicting sine-wave contrast stimulus. In the model, square-root law containing relatively large uniform-or thresholds and Mach bands, FF alone and responses are a consequence of the nearly uniform-areas. The stimuli for FB + FF are similar-but not exactly the feedback component, not of quantum

Perception & Psychophysics, 1970, Vol. 8 (3) ISS noise directly; the square-root law is VISIon. Journal of Physiology (London), of America, 1960,50, 143-148. presumed to be an evolutionary adaptation 1962b, 160, 169-188. LAMAR, E. S., HECHT, S., SCHL<\:ER, S., & BLACKWELL, H. R. Neural theories of simple HENDLEY, C. D. Size, shape and contrast in (FB-component) to quantum noise. visual discriminations. Journal of the Optical detection of targets by daylight vision. I. Data With two stimuli A, B, both mask a test Society of America, 1963,53,129-160. . and analytical description. Journal of the ~Q, the model predicts that the combined BLAKEMORE, C., & CAMPBELL, F. W. Optical Society of America, 1947, 37, threshold equals the larger of ~QA and ~QB Adaptation to spatial stimuli. Journal of 531-545. at low luminances and equals ~QA + ~QB Physiology (London), 1968, 200, 11-13. LOWRY, E. M., & DePALMA, J. J. Sine-wave BOUMAN, M. A., & WALRAVEN, P. L. Some response of the visual system. I. 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156 Perception & Psychophysics, 1970, Vol. 8 (3) effects can be quite similar to the effects of H!(1 + kl) of Eq. 3a. However, Mach assumed implicitly that disk test .stimuli are detected at shunting interactions. The logarithmic subtractive that the output was not given directly by the their center. The procedure is invalid when test approach is unattractive because it requires two interaction term, but was proportional to input X fields are detected at boundaries. (See next complex operations (logarithms and subtraction) interaction; namely, H2/1 (Mach, 1868, p. 15). section.) to do what shunting processes do naturally in a 5. The minimum of G(x) in Fig. 8c indicates 9. These values are among the smallest critical single interaction. Even in Limulus, whose eye Ax. areas that have been reported. The small values represents the most successful case of the 6. See Fig. 8i; also, the zero of G(x) in Fig. 8f are used here because most experimental su-trr ,tive th.ory of interaction, the subtractive gives approximately ±Ax/2. artifacts, such as accommodation error (from theory is yielding to a shunting analysis (Purple & 7. See Sperling & Sondhi (1968) for a specific fixation lights of different wavelengths than the Do-tee, 1965), theory of slow inhibition. disks being detected) and inaccurate fixation ".- Mach computed a lateral interaction term of 8. The convolution procedure used by Kincaid tend to increase observed critical areas. the form HII, which is nearly equivalent to et aI (1960) and by Blackwell (1963) assumes (Accepted for publication November zo. 1969.)

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