field, the increment-threshold functions have two distinct branches. It was shown,

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1 J. Phyaiol. (1979), 294, pp With 6 text-ftgurem Printed in Great Britain INITIAL-IMAGE AND AFTERIMAGE DISCRIMINATION IN THE HUMAN ROD AND CONE SYSTEMS BY WILSON S. GEISLER From the Department of Psychology, University of Texas, Austin, Texas 78712, US.A. (Received 28 June 1978) SUMMARY 1. The rod-isolation technique of Aguilar & Stiles (1954) was used to obtain scotopic increment-threshold functions in the dark-adapted eye. Incrementthreshold functions were obtained for background durations of 50 to 500 msec, but the onset of the background and increment fields was always simultaneous. In all conditions the duration of the increment field was 50 msec. 2. The pattern of results obtained is the same as that reported earlier for the cone system (Geisler, 1978). For background durations greater than that of the increment field, the increment-threshold functions have two distinct branches. It was shown, by measuring action spectra, that both branches reflect the sensitivity of the rod system. 3. When the increment thresholds are plotted as a function of background retinal illuminance, all the lower branches superimpose. This implies that those thresholds are dependent only on the number of background quanta absorbed during presentation of the increment field. On the other hand, when the increment thresholds are plotted as a function of background energy, all the upper branches superimpose, implying that those thresholds are determined by the total number of background quanta absorbed. 4. For the thresholds falling on the lower branches, observers reported that the increment field was detected in the initial image of the background and increment fields when they were flashed. For the upper branches, the increment field was detected in a short-term afterimage that appeared after the background was extinguished. The higher the background intensity the longer was the latency until the increment appeared in the afterimage. 5. All of the above findings appear to be consistent with the known properties of the electrical responses of vertebrate photoreceptors. A model based on Penn & Hagins' (1972) model for the photocurrent in rat rods predicts, fairly accurately, the rod and cone increment-threshold results. The parameters estimated by fitting the model support the hypothesis that the short-term rod and cone afterimages are due to the relatively slow decay of internal transmitter, but they suggest that postreceptor mechanisms are responsible for the threshold saturation observed with flashed backgrounds /79/ $ The Physiological Society

2 166 W. S. GEISLER INTRODUCTION In a previous paper I reported increment-threshold functions that were obtained in the dark-adapted fovea for backgrounds of various durations (Geisler, 1978; some of these data are shown in Fig. 6A). Briefly, when the background and increment fields were of equal duration, the increment-threshold curves were approximately described by the generalized Weber's law. However, when the duration of the background was lengthened, keeping the onset of the two fields simultaneous, the functions displayed two distinct branches. The lower branches followed a continuously accelerating curve, and they were superimposed for all background durations. On the other hand, the upper branches were approximately described by Weber's law, but with Weber fractions that increased with background duration. The two branches of each increment-threshold function were shown to reflect two different modes of intensity discrimination. For the thresholds described by the lower branch, the increment field was detected in the initial image of the background and increment fields when they were flashed, but for the upper branch, the increment field was detected in a short-term afterimage that appeared after the background field was extinguished. The purpose of the present paper is (1) to report a set of similar increment-threshold experiments for the rod system, and (2) to describe a quantitative model of initial-image and afterimage discrimination in the human cone and rod systems that is based on recent models of vertebrate receptor responses. METHODS The stimulus configuration, which was presented in Maxwellian view, is shown at the top of Fig. 1. The display comprised a 20 increment field that was centred in a 90 background field. Subjects viewed the display with the right eye, and fixated a dim red light that fell 60 from the centre of the increment field. The image of the tungsten filament at the pupil, which was dilated with 1 % Mydriacyl, was less than 2 mm in diameter. The rod system was isolated using the two-color technique of Aguilar & Stiles (1954). In particular, the background field was red (Kodak Wratten filter no. 26) and the increment field was a 521-nm green obtained with a Baird-Atomic interference filter. In order to achieve even better rod isolation, it is possible to make use of the Stiles-Crawford effect by having the background beam enter at the centre of the pupil and the increment beam enter at the edge of the pupil (Aguilar & Stiles, 1954). However, this latter technique was not used since, as will be shown below, sufficient rod isolation could be obtained without it. The advantage of having both beams enter at the centre of the pupil is that it reduces the possibility of errors due to slight misalignments of the subject's eye. The stimulus presentation sequences used in the main experiment are given in the first four lines of Fig. 1. The duration of the background field was varied from 50 to 500 msec. The duration of the increment field was always 50 msec, and its onset was simultaneous with that of the background. Each experimental session was preceded by at least 30 min of dark adaptation. The background and increment fields were flashed once every 10 sec. Control conditions, described below, showed that this presentation rate maintained the eye in a sufficiently dark-adapted state. The background and increment fields were turned on and off with rise and decay times of less than 1 msec. Stimulus presentation and response collection were under the control of a PDP-1 1/20 computer. All filters and wedges were periodically calibrated by reading an RCA phototube placed in the position normally occupied by the subject's pupil. The phototube was also used to set the maximum intensities of the background and increment beams before each experimental session,

3 ROD AND CONE THRESHOLDS 167 and to check their intensities after the sessions. Absolute luminance measurements were obtained by Rushton's method (Westheimer, 1966). All thresholds were obtained by the method of adjustment. At moderate background intensities, the threshold judgments were often rather difficult. Variability and uncertainty were greatly reduced by allowing the subjects to view the background flash alone whenever they wished. A highly practised observer (the author) and a less experienced observer (a male in his early 20's) served as subjects. X - Fixation light II~ O lincrement field 1-90 I Increment -l 10 sec: - BackgroundOte4 (1) (2).g1.FHo. sec 4. - Background field (3) El (4) 4-O*5 sec-* (5) C2 mi;nov J n T2 (6) J_ qyl * 05 sec Fig. 1. Stimulus display and stimulus presentation sequences for rod threshold experiments. RESULTS The increment-threshold functions obtained for subject W. G. are shown in Fig. 2. Each symbol represents the average of at least three threshold settings (the x 's are the results of a control experiment that will be described below). The results obtained for the other subject are very similar to those shown in Fig. 2. The pattern of results in Fig. 2 is strikingly similar to those obtained for the cone system (compare with Fig. 6A). The increment-threshold function obtained with the 50-msec background is approximately described by the generalized Weber's law, and each of the increment-threshold functions obtained with the longer duration backgrounds clearly shows two branches. The lower branches are roughly superimposed

4 168 W. S. GEISLER for all background durations, which implies that those thresholds are only dependent on the background quanta absorbed while the increment field is being presented, and not on the quanta that are absorbed later. Although the dynamic range of the curve describing the lower branches is about the same as it is for the cones (see Fig. 6A), the shape of the curve is slightly different. In the rod curve, there is a greater range of background intensities over which Weber's law holds. 3- W.G Z 0, 0-1 -j Log background (scotopic td) Fig. 2. Upper curves: scotopic increment-threshold functions obtained for flashed backgrounds of various duration. For all the thresholds that fall along the curved line the increment field was detected in the initial image, and for all the thresholds that fall along the straight, parallel lines it was detected in the short-term afterimage. Lower curve: a control condition in which absolute threshold was measured 500 msec prior to onset of the 50-msec background, as a function of background intensity. Background duration (msec): *-*, 50; Q-Q, 100; U-f, 200; A-A, 500; x -x, 50 (control). The background intensity at which the upper branch begins is an increasing function of background duration. However, the upper branches nearly satisfy Weber's law, their slopes appearing to be slightly less than 1-0. Another systematic aspect of the data is revealed in Fig. 3. In Fig. 3, the data shown in Fig. 2 are plotted as a function of background energy instead of retinal illuminance. As can be seen, the lower branches are now separated, but the upper branches are roughly superimposed (notice that the thresholds for the 500-msec background fall slightly above the others). This suggests that to a first approximation the thresholds on the upper branches are determined by the total energy of the background, independent of how it is distributed temporally, at least for background durations up to around msec. The phenomenology of the present experiment is also very similar to that observed in the cone experiment. For the lower branches, the observers reported that the increment field is visible at the moment the background and increment fields are flashed, but for the upper branches, the increment field is not visible until after the

5 ROD AND CONE THRESHOLDS 169 background is extinguished. This dichotomy is particularly evident with the 500-msec background, since there is such a long interval between presentation of the increment field and the offset of the background field. Another important aspect of the phenomenology, which has also been observed by Adelson (1977), is that the latency until the increment field is detected in the afterimage is an increasing function of background intensity. For example, if the 50-msec background is at 03 log scotopic td, 3 - W.G. - 0 ~ ~ ~ ~ ~~~~~ ~0 C j -2 -oo Log background (scotopic td sec) Fig. 3. The increment thresholds in Fig. 2 replotted as a function of background energy (intensity x time). Approximate superimposition of all the afterimage branches suggests that! the magnitude of the afterimage signal depends on the total energy of the stimulus not on how it is distributed temporally. Background duration (msec): * *, 50; 0-0, 100; *-*, 200; A-A, 500. the increment is seen immediately after the background flash; in fact, it is difficult to decide whether it was or was not visible in the initial image. On the other hand, at 3-8 log scotopic td the increment field is not seen until 1-2 sec after the background flash. In accordance with these observations, I will refer to the lower branches of the increment-threshold curves as initial-image branches and the upper branches as afterimage branches. Controls Spectral-sensitivity curves. In order to verify that only the rod system was being measured, increment-threshold functions were obtained with the 500-msec background for several wave-lengths of the increment field. The functions obtained for increment fields of 521, 500, and 467 nm are shown in Fig. 4A. The threshold intensities are expressed in scotopic td; thus if only rod responses were mediating threshold, the three functions should be superimposed. On the other hand, from the curves reported by Stiles (1978) it can be shown that if any one of the cone mechanisms were involved, the thresholds would be spread out over a range of at least 0-6 log units. It seems reasonable, therefore, to conclude that the thresholds shown in Fig. 2 reflect the sensitivity of the rod system alone. Hallett (1969) was also able to obtain adequate

6 170 W. S. GEISLER rod isolation under conditions similar to those used in the present experiment. In addition, by varying the wave-length of the background field Hallett showed that there were no rod-cone interactions under his stimulus conditions. Replication of the Aguilar & Stiles experiment. In a classic study, Aguilar & Stiles (1954) obtained rod increment thresholds against steady backgrounds. Using a A W.G. Continuous 0 ~~~~~~~~~~~background msec background 0 0~ -J B 10 sec 40sec W,,a40 secj -oo Log background (scotopic td) Fig. 4. Results of control experiments. A, the filled circles are the increment thresholds obtained when the background is presented continuously (see line 5 of Fig. 1). The open symbols are the increment thresholds obtained with the 500-msec background, for several wave-lengths of the increment field. Increment wavelength (nm): A-A, 521; []-El, 500; 0-0, 467; *-*, 521. B, absolute thresholds measured 500 msec before onset of the 500-msec background when the background was presented once every 10 see (open squares) and once every 40 see (filled squares). Increment thresholds measured when the 500-msec background was presented once every 10 see (open circles) and once every 40 sec (filled circles). increment field, they found that rod thresholds approximately followed the generalized Weber's law up to background intensities of about 2-0 log scotopic td, but above this value threshold increased sharply, indicating that the rod system was fully saturated around 3-0 log scotopic td. As a check on the stimulus calibration and procedure used in the present experiment, increment thresholds were also measured against steady backgrounds. The stimulus configuration was the same as in the main experiment. The increment field was flashed for 50 msec once every 10 sec, and the subject adapted to each background intensity for at least 2 min before adjusting the increment flash to threshold (see line 5 of Fig. 1). The increment-threshold function that was obtained is also shown in Fig. 4A (filled circles). As in the Aguilar & Stiles results, the increment-threshold function has two branches; a lower rod branch and an upper cone branch (at the two highest background intensities the increment flash appeared violet). The shape of the lower branch is very similar to that obtained by Aguilar & Stiles, and as expected, it appears to saturate around 3 0 log scotopic td. In fact, above log scotopic td the

7 ROD AND CONE THRESHOLDS filled circles shown in Fig. 4A lie about 0-2 log units above the average curve reported by Aguilar & Stiles. Below - I'0 log scotopic td they lie further above the Aguilar & Stiles curve (e.g. 0-6 log units above at absolute threshold). However, this discrepancy is probably due largely to spatial summation, since Piper's law correctly predicts the difference between the absolute thresholds. In conclusion, the incrementthreshold function shown in Fig. 4A appears to be in reasonable agreement with the results reported by Aguilar & Stiles (1954). State of adaptation. The interval between stimulus presentations was kept at 10 sec for all the flashed-background conditions in order to maintain the eye in a relatively dark-adapted state. Was this interval sufficient? This was tested by measuring absolute thresholds 500 msec before onset of the 50-msec background field. The stimuluspresentation sequence is shown in line 6 of Fig. 1. In order to allow the adaptation effects to build up, the subject adapted to each intensity level of the flashed background for a couple of minutes before adjusting the increment field to threshold. The thresholds that were obtained are given by the x 's in Fig. 2. These results indicate that the eye remained essentially dark adapted for 50-msec backgrounds of up to around 3 0 log scotopic td (1.7 log scotopic td sec). Thus, at least the initial-image branches and part of the afterimage branches reflect the response of the darkadapted eye. The question remains, however, as to whether the observed departures from the dark-adapted state actually affected the increment thresholds. This was tested in a second experiment. In the first condition, absolute thresholds were measured 500 msec before onset of the 500-msec background, which was presented at intervals of either 10 or 40 sec. In the second condition, increment-thresholds were measured for the same inter-flash intervals. As can be seen in Fig. 4B, increasing the inter-flash interval from 10 to 40 sec produced a substantial decrease in absolute threshold (open and filled squares), but no decrease in increment threshold (open and filled circles). Thus, the increment-threshold functions in Fig. 2 are apparently identical to those that would be obtained in a completely dark-adapted eye. 171 DISCUSSION Alpern, Rushton & Torri (1970), King-Smith and Webb (1974), Shevell (1977), and Hood, Ilves, Maurer, Wandell & Buckingham (1978) have all reported cone incrementthreshold functions that appear similar to the initial-image branches in Figs. 2 and 6A, but do not have afterimage branches. However, in all these cases, the stimulus conditions were such that one would not expect to see afterimage branches. For more discussion of this point see Geisler (1978). Several studies have measured intensity discrimination in afterimages (Brindley, 1959; Standing & Dodwell, 1972; Geisler, 1978), but Sakitt (1976) was apparently the first to obtain measurements for the isolated rod system. The measurements were obtained on a rod monochromat. Sakitt first found the intensity of the continuous background at which threshold saturation occurred. In agreement with the lower curve in Fig. 4A, this was found to occur around 3'0 log scotopic td. Discrimination was then measured in the afterimage at background intensities above 3-0 log scotopic td, by having the subject close her eyes immediately following the 'click' of the

8 172 W. S. GEISLER increment-field shutter. The shape of the afterimage function obtained by Sakitt is a bit difficult to determine since it only covers a 1 log unit range of background intensities; nevertheless, it appears to have a slope of around 1.5, which is greater than that of the afterimage branches in Fig. 2. The steeper slope obtained by Sakitt may be due to the adaptation produced by the continuous background, or to drift in the reaction time of the subject's eye blink. The similarity of the results in Figs. 2 and 6A suggests that the short-term afterimages in the rod and cone systems are generated by analogous mechanisms. Sakitt (1975, 1976) has suggested that the short-term rod afterimage arises from the photoreceptors, and Geisler (1975, 1978) has made a similar suggestion for the short-term cone afterimages. Three alternative sources for the afterimage signals within the receptors were considered. One possibility is that they arise from the final product of bleaching (the free opsin). Sakitt (1976) has argued rather convincingly that this hypothesis cannot account for the short-term rod afterimage. The argument is based on the considerable evidence that bleached rhodopsin (free opsin) generates the long-term afterimage that is often observed for many minutes following strong bleaches. From equivalent-background experiments it has been deduced that the equivalent intensity (IB) of the long-term afterimage associated with a given state of bleaching is approximately given by the following relationship: 'B = Io (1) where q is the proportion of pigment bleached, Io is the dark-light intensity in the dark-adapted eye (the eigengrau), and a is a constant of about 19 for the human rod system (see Barlow (1972) for a review). Using eqn. (1) and the results of Rushton's (1956) and Alpern's (1971) retinal densitometry measurements, Sakitt found that the long-term afterimages generated in her experiment were too weak to mediate detection of the increment field. Similar calculations for the stimulus conditions in the present experiments lead to the same conclusion. There is some evidence that eqn. (1) is also valid for the cone system (DuCroz & Rushton, 1965; Geisler, 1979); however, the value of a is about 3 0 (Rushton, 1963, 1965). Using eqn. (1) and the densitometry results for human cones reported by Rushton & Henry (1968), it can be shown, for the results in Fig. 6A, that the free opsin signal can be ruled out as the source of the short-term cone afterimage, except possibly for the two highest thresholds on the afterimage branch obtained with the 500-msec background. Another possibility is that the short-term afterimages arise from some intermediate stage of the bleaching sequence. In rods, they might arise from metarhodopsin II or Rushton & Powell's (1972) X-opsin, and in cones they might arise from analogous substances. On the basis of Rushton & Powell's results, Sakitt (1976) carried out an analysis like the one above for the free opsin signal, and concluded that discrimination in the short-term afterimage is also not mediated by X-opsin. However, the analysis seems much more tenuous in this case since the equivalentbackground hypothesis has not been adequately tested for X-opsin. Probably the most parsimonious alternative is that the short-term afterimage and the initial image are slightly different manifestations of the mechanism that produces the electrical responses that have been measured in vertebrate rods (Penn & Hagins,

9 ROD AND CONE THRESHOLDS , 1972) and in vertebrate cones (Baylor & Fuortes, 1970; Baylor & Hodgkin, 1973, 1974; Baylor, Hodgkin & Lamb, 1974). There is substantial evidence that absorbed quanta cause the release of an internal-transmitter substance, possibly calcium ion (see Hagins (1972) for a review). This substance apparently closes sodium channels in the receptor membrane, thereby causing the receptor to hyperpolarize. A hypothesis explored in detail below is that both the initial image and the short-term afterimage arise from the internal-transmitter substance. If the short-term afterimages arise from any of the sources listed above, one can understand how afterimages might mediate detection at high background intensities. When an intense background is first presented to the eye, it drives the response of the receptors or later neural stages to near their maximum level, so that even an intense increment field does not produce a detectable increase in the initial response. However, in the increment-field region, a greater amount of photopigment is bleached and consequently greater quantities of free opsin, intermediate photoproducts, and internal transmitter are released. While the background remains on, neural response remains at a high level and the presence of the increment field cannot be signaled. But, when the background is extinguished, neural response declines, and the greater concentration of photoproducts in the increment region is detected. A similar explanation is given in Sakitt (1975, 1976) and Geisler (1975, 1978). This hypothesis forms the basis of the model presented in the next section. A theory of initial-image and afterimage discrimination This section presents a model of initial-image and afterimage discrimination that is based on the models of vertebrate receptor responses proposed by Penn & Hagins (1972), Baylor et al. (1974), and others. The important point of this section is that all of the unusual properties of intensity discrimination in the dark-adapted eye that were described above are just what one predicts from the global response characteristics of vertebrate receptors. Furthermore, it is reasonable to expect the response of the entire visual system to yield these unusual properties for intensity discrimination as long as the linear and non-linear stages beyond the receptors have relatively small time constants. General assumptions. The basic components of the model are shown in Fig. 5A, which illustrates the processing of the inputs I(t) and I+(t) falling on the background and the increment regions. Fig. 5B shows typical outputs for each stage when the background and increment fields are intense and very briefly flashed. In the first stage, which is assumed to be linear, absorbed quanta cause the release of an internal-transmitter substance. If we let g(t) be the impulse-response function of the linear transformation, then the concentration of internal transmitter is given by the following equation: Y(t) = I(t) * g(t), (2) where * represents the operation of convolution. In the second stage of the model, it is assumed that the internal-transmitter substance induces a neural response through a relatively instantaneous non-linear transformation. In the present model, the non-linear transform represents nonlinearities that arise at the receptor level as well as at higher levels. It will be shown

10 174 A Light l(t) W. S. GEISLER Internal Neural transmitter response B I Y R t t t D t C t 0 t 0 t Log/ a*8 n=i= = U E n = 056-II R 0J Log Y I i I I I I Fig. 5. Model of initial-image and afterimage discrimination. A, schematic diagram showing the processing of inputs 1(t) and I+(t) to the background and increment regions, respectively. In the first stage, light is converted into internal-transmitter substance via a linear transformation. The second stage is a relatively 'instantaneous' non-linear transformation that represents non-linearities that may arise at the receptor level and at higher levels. The final stage is a comparator and decision process, which detects the increment field if the outputs from the two regions are sufficiently different. B, outputs of each stage of the model to impulse flashes that are intense enough to drive neural response to near saturation. The linear and non-linear transformations are exactly those used by Penn & Hagins (1972). The output of the comparator is the difference of the neural responses over time, from the two regions. C, neural response as a function of the log concentration of internal transmitter (lower scale), for two values of the exponent, n, in eqn. (3). Since the first stage of the model is linear, the curves in this Figure also describe peak neural response to brief flashes as a function of log intensity (upper scale). See text for explanation of the dashed lines. Ii I I I I

11 ROD AND CONE THRESHOLDS 175 below, that the present model makes several strong predictions which are essentially independent of the form of the non-linear transformation. However, in order to derive complete quantitative predictions it was assumed that the output of the nonlinear transformation is given by the equation: R(t) =Rmax Y (t)n 3 Y(t)n + y n'( where n is an exponent whose value is between 0 and 1.0, Y1 is the half-saturation constant, and RnIax is the maximum neural response. The prediction of eqn. (3) for peak neural response as a function of flash intensity is the familiar function, R Rmax In In+Iln' where Ii is the flash intensity that produces half-saturation. The final stage of the model is a comparator and decision stage. The simplest assumption is that the comparator calculates the difference, D(t), of the neural responses over time from the two regions, and if the difference exceeds some criterion, &, the increment field is detected. The curves plotted in Fig. 5B are the predictions of the model for the parameter values used by Penn & Hagins (1972) to fit rod responses in the rat. The condition depicted is one in which impulse flashes are intense enough to drive neural response to near saturation (Rmax). While the responses are near saturation, their difference is nearly zero. However, neural response decays more slowly in the increment region since twice as much internal transmitter was released in that region. Thus, the difference in response reaches a prolonged peak during the decay or 'afterimage' phase. As can be seen in Fig. 5 B, there is also a brief initial peak that occurs while the responses in the two regions are rising. For impulse flashes below the half-saturation intensity, the difference in neural response has a somewhat broader initial peak and no afterimage peak. The initial peak that occurs with intense flashes poses a problem for all models of the present type. Since it is always the same height as the afterimage peak (when the impulse intensities exceed the half-saturation value), the increment field is never more easily detected in the afterimage than in the initial image. However, any mechanism that will tip the balance in favour of detection during the decay of neural response will solve the problem. One such mechanism, which is highly likely to exist, is simply that the stages following the non-linear transformation integrate, to some extent, the neural signals they receive. This mechanism will work, since integration will suppress the brief initial peak to a greater extent than the prolonged afterimage peak. Therefore, we assume that D(t) = h(t) * R+(t) - h(t) * R(t), (4) where h(t) is the impulse-response function of the final integrator. In accordance with the hypothesis that the decay of neural response is primarily controlled by the decay rate of internal transmitter, we require that the time window of integration represented by h(t) be narrower than that represented by g(t). General predictions. Certain general predictions hold for the whole class of models

12 176 W. S. GEISLER defined by eqns. (2), (3) and (4), and they can be tested without estimating parameters. Fig. 5C is very useful for understanding the predictions for conditions in which the background and increment fields are flashed for the same duration. Owing to the linearity assumption, the ratio of the concentration of internal transmitter in the increment region to that in background region will remain constant during and after the flash (i.e. Y+(t)/Y(t) = constant). Therefore, the difference in the output from the background and increment regions can be determined simply by fixing the distance between a pair of vertical lines in Fig. 5C, then sliding them to the right and then the left in accord with the rise and decay rate given by the impulse-response function. (The relatively small amount of integration that occurs after the non-linear transformation has little effect except to prevent the increment field from being detected while responses in the background and increment regions are rising.) Using this graphical construction, several strong predictions of the model become apparent. (1) The increment field should be detected after the peak response (in the afterimage) when the background intensity exceeds I,. This means that afterimages should begin mediating threshold (with equal duration background and increment fields) at the point at which the Weber fraction of the initial-image branch reaches its minimum. (2) For background intensities above I,, the Weber fraction should be constant. In other words, afterimage discrimination should follow Weber's law. This follows since the smallest Weber fraction occurs when the background intensity is I,, and this same Weber fraction will be maintained at all higher intensities, as long as the observer waits until the response to the background decays to 0O5. MacLeod (1974) used a similar argument to explain why Weber's law holds for the discrimination of long-term afterimages. (3) At threshold (and at any fixed multiple of threshold) the latency until the increment field appears in the afterimage should increase with background intensity. The first two of these predictions appear to hold approximately for both rods and cones (see the solid circles in Figs. 2 and 6A). The third prediction, as noted earlier, has been confirmed for rods and cones by simple observations. (Reaction-time measures of the latency until detection will be reported in a subsequent paper.) It is interesting that the above predictions are essentially independent of the form of the non-linear transformation. The crucial assumption is that there is a linear transformation followed by a relatively instantaneous, saturating, non-linear transformation. Quantitative predictions. For ease of computation, it was decided to first let g(t) be an exponential-decay function (a single RC stage). The impulse-response function of the final integrator was also assumed to be an exponential-decay function, but with a small time constant, tr. The actual value of this time constant (if it is small) is not critical for the predictions of the model. The increment-threshold results shown in Fig. 2 and the comparable data reported earlier for the cones (Geisler, 1978) are replotted in Fig. 6. The predictions of the single-rc-stage model are given by the continuous curves. The relative separations of the predicted afterimage branches are solely determined by the time constant t8 of the initial linear stage. The smaller is t., the larger is the predicted spread of the afterimage branches. The exponents in the nonlinear transformation (eqn. (3)) that were used to obtain the predictions are less than 1-0, but

13 ROD AND CONE THRESHOLDS 177 are within the range reported for vertebrate receptors. However, this correspondence does not imply that the nonlinearities that control intensity discrimination arise within the receptors. Assuming that one scotopic or photopic troland second is -0 ~0 0) ~0 -J Log background (td) +O 0 0 -) -0 T0 CAr 0, 0 -J Log background (scotopic td) Fig. 6. A, symbols are the photopic increment thresholds obtained in the dark-adapted eye, for various background durations (from Geisler (1978)). The continuous curves are the predictions of the single-rc-stage model. Parameter values: n = 0-8, ts = 300 msec, Y1 = 6-5, 8 = 0-02, tr = 20 msec. B, symbols are a replot of the scotopic increment thresholds in Fig. 2. The continuous curves are the predictions of the single- RC-stage model. Parameter values: n = 07, t, = 400 msec, Y1 = 0 05, 6 = 0 04, tr = 20 msec. Rmx was simply set to 1 0 since changing its value is equivalent to changing 6. Background duration (msec): *-*, 50; O-Q, 100; *-*, 200; A-A, 500. equivalent to two to four absorbed quanta per receptor (for cones it is probably less), the half-saturation constants estimated from the fits are two to three orders of magnitude smaller than those reported for rat rods (Penn & Hagins, 1972) or turtle cones (Baylor & Fuortes, 1970). Predictions were also obtained with Penn & Hagins' impulse-response function, which is obtained by the convolution of two pairs of exponential-decay functions, where the functions in each pair share the same time constant. Not surprisingly, this

14 178 W. S. GEISLER version of the model is also able to make reasonable predictions. However, in order to obtain reasonable fits to the afterimage branches, the time constants had to be somewhat larger than those used by Penn & Hagins. Also the time constant of the final integration had to be lengthened over that needed for the single RC-stage model. Since both the single- and four-rc-stage models can account for the data reasonably well, it follows that the threshold data reported here do not allow a fine analysis of the form of the impulse-response function. CONCLUSIONS The adequate fit of the models and the fairly reasonable values of the best-fitting parameters support the hypothesis that the short-term afterimages measured in the rod and cone experiments arise from internal-transmitter substance within the receptors or from a substance with very similar properties. The models presented here are, of course, oversimplified. For example, they do not take into account the slower decay of internal transmitter that apparently occurs with high energy flashes. If the decay rate becomes significantly slower above 1000 absorbed quanta per rod (Penn & Hagins, 1972), then the effects of this slow decay rate should be evident at intensities just above those used in the present experiment. In fact, Sakitt's (1976) measurements of the persistence of short-term rod afterimages are in agreement with this prediction. This research was supported in part by a grant from the University Research Institute at the University of Texas and by N.I.H. grant EY REFERENCES ADELSON, T. (1977). Decay of rod signals following bright flashes. J. opt. Soc. Am. 67, ALPERN, M. (1971). Rhodopsin kinetics in the normal eye. J. Phymiol. 217, ALPERN, M., RUs~roN, W. A. H. & ToRn, S. (1970). Signals from cones. J. Physiol. 207, AGUILAR, N. & STILES, W. S. (1954). Saturation of the rod mechanism of the retina at high levels of stimulation. Optica Acta 1, BARLow, H. B. (1972). Dark and light adaptation: psychophysical. In Handbook of Sensory Physiology, VII/4. Visual Psychophysics, ed. JAMEsoN, D. & HURVICH, L. M., pp Berlin: Springer. BAYLOR, D. A. & FUORTES, M. G. F. (1970). Electrical responses of single cones in the retina of turtle. J. Phy^iol. 207, BAYLOR, D. A. & HODGKIN, A. L. (1973). Detection and resolution of visual stimuli by turtle photoreceptors. J. Physiol. 234, BAYLOR, D. A. & HODGKIN, A. L. (1974). Changes in time scale and sensitivity in turtle photoreceptors. J. Physiol. 242, BAYLOR, D. A., HODGKIN, A. L. & LAMB, T. D. (1974). The electrical response of turtle cones to flashes and steps of light. J. Physiol. 242, BRINDLEY, G. S. (1959). The discrimination of afterimages. J. Physiol. 147, DUCROZ, J. J. & RusrroN, W. A. H. (1966). The separation of cone mechanisms in dark adaptation. J. Physiol. 183, GEISLER, W. S. (1975). Visual adaptation and inhibition. Ph.D. dissertation, Indiana University, University Microfilms no , 137. GEISLER, W. S. (1978). Adaptation, afterimages and cone saturation. Vision Res. 18, GEISLER, W. S. (1979). Evidence for the equivalent-background hypothesis in cones. Vision Res. (In the Press.)

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