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Article  |   November 2016
Brightness in human rod vision depends on slow neural adaptation to quantum statistics of light
Author Affiliations
  • Michael E. Rudd
    Howard Hughes Medical Institute and Department of Physiology and Biophysics, University of Washington, Seattle, WA, USA
    mrudd@u.washington.edu
Journal of Vision November 2016, Vol.16, 23. doi:10.1167/16.14.23
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      Michael E. Rudd, Fred Rieke; Brightness in human rod vision depends on slow neural adaptation to quantum statistics of light. Journal of Vision 2016;16(14):23. doi: 10.1167/16.14.23.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

In human rod-mediated vision, threshold for small, brief flashes rises in proportion to the square root of adapting luminance at all but the lowest and highest adapting intensities. A classical signal detection theory from Rose (1942, 1948) and de Vries (1943) attributed this rise to the perceptual masking of weak flashes by Poisson fluctuations in photon absorptions from the adapting field. However, previous work by Brown and Rudd (1998) demonstrated that the square-root law also holds for suprathreshold brightness judgments, a finding that supports an alternative explanation of the square-root sensitivity changes as a consequence of physiological adaptation (i.e., neural gain control). Here, we employ a dichoptic matching technique to investigate the properties of this brightness gain control. We show that the brightness gain control: 1) affects the brightness of high-intensity suprathreshold flashes for which assumptions of the de Vries-Rose theory are strongly violated; 2) exhibits a long time course of 100–200 s; and 3) is subject to modulation by temporal contrast noise when the mean adapting luminance is held constant. These findings are consistent with the hypothesis that the square-root law results from a slow neural adaptation to statistical noise in the rod pool. We suggest that such adaptation may function to reduce the probability of spurious ganglion cell spiking activity due to photon fluctuation noise as the ambient illumination level is increased.

Introduction
Over much of the dynamic range of human cone-mediated vision, light adaptation obeys Weber's law. Raw light intensity is transformed into a neural response that is proportional to contrast:  where ϕW is the physiological response to a flash of intensity ΔI, and I is the light level to which the system is preadapted. Put another way, the cone visual system takes the physical flash intensity ΔI as input and applies to this input the multiplicative Weber gain factor  to produce the neural response (Equation 1). This transformation begins in the cones themselves and is well suited to support color constancy when the illumination level varies.  
The situation in rod-mediated vision is considerably different. The visual input consists of a retinal rain of discrete photons. Each photon absorption produces an electrical response in the absorbing rod that lasts about 200 ms. Throughout much of the range of human rod vision, individual rods have only a small probability of absorbing more than one photon within their ∼200 ms integration time. Physiological estimates indicate that individual primate rods begin to adapt at about 5–10 R*/rod/s (isomerizations per rod per second) (Schneeweis & Schnapf, 2000). Thus, at light levels below this, any light adaptation that occurs in primate rod vision must occur in postreceptoral circuitry. The nature of this postreceptoral adaptation is still incompletely understood. 
Since the pioneering work of Barlow (1956, 1957), it has been known that psychophysical thresholds for the human rod visual system, when measured with small brief test probes, increase in proportion to the square root of adapting intensity. This differs from cone vision, in which threshold increases in direct proportion to intensity (Weber's law). A classical theory proposed independently by Rose (1942, 1948) and de Vries (1943) ascribes the square-root threshold law to the masking of dim flashes by the naturally occurring variability in photons from the adapting field. The retinal photon count within a fixed retinal area and integration time resulting from this photon rain are statistically distributed as a Poisson-random variable. Because the photon count is Poisson, the variance in a random sample of the photon rain equals the mean count. The standard deviation of the retinal photon input thus grows in proportion to the square root of the mean absorption rate. 
Under natural rod viewing conditions, this statistical variation in the retinal photon count produces an irreducible noise in the information available to an observer about the nature of environmental surfaces from which light is reflected to his or her eye. In psychophysical threshold experiments, statistical fluctuations in the retinal photon count impair the observer's ability to discriminate trials in which a dim incremental flash is added to the adapting field from trials with the same adapting field but no added flash. 
Classical photon noise account of the square-root threshold law (de Vries–Rose theory)
Let I be the mean intensity of the adapting field in a visual threshold experiment. In some proportion of trials, a brief, dim test flash of mean incremental intensity ΔI, area A, and duration T is added to the field. Photons are emitted randomly by standard incoherent light sources; hence, the number of retinal photon absorptions produced by the adapting field within time T and area A is a Poisson-distributed random variable. Thus, the mean and the variance of the photon count in trials in which no flash is presented are both IAT. In trials in which a flash is presented, the mean and variance are both (I + ΔI)AT
To statistically discriminate trials in which a flash is presented from trials in which no flash is presented, an ideal observer—one whose ability to perform the discrimination is limited only by the physical fluctuations in the photon absorption rate—would count photons absorbed over the interval AT and set a criterion number of absorptions such that any trials in which the photon count exceeds the criterion are judged to be “flash present” trials and others are judged to be “flash absent” trials. Independent of the chosen criterion, the ideal observer's signal-to-noise ratio d′ for statistically discriminating flash from no flash trials is    
The numerator of Equation 3 is the average difference between the photon counts in trials in which a flash is presented versus not presented, and the denominator is the standard deviation of this difference. In the final step of Equation 3, we make the approximation that the contribution of the photon noise from the test flash is negligible for threshold-level flash intensities. To achieve a fixed signal-to-noise ratio for statistically discriminating trials in which a flash was or was not presented—which is the definition of visual threshold—the flash intensity ΔI therefore must be increased in proportion to the square root of the mean intensity I of the adapting field. 
It is important to note that the de Vries–Rose account of the square-root threshold law, as presented here, does not assume the existence of any actual physiological light adaptation. The de Vries–Rose theory describes the behavior of an “ideal” observer: one whose ability to discriminate flash trials from blanks is limited only by the quantum statistical noise in the light signal. It is thus best thought of as a standard against which actual physiological mechanisms might be compared. It nevertheless serves to highlight the important fact that the visual sensitivity of any biological (or nonbiological) imaging device is ultimately limited by the inherent noise in the statistical photon input. In presenting the theory in this way, we have adhered more closely to the original presentation of Rose—an engineer who was interested in both biological and machine vision—than to that of de Vries, who was a physiologist and who introduced supplementary assumptions about the spatiotemporal summation parameters in human rod vision. We adopt Rose's approach because the relevant neural parameters are still not well understood, and we wish to clearly distinguish between the absolute limits on threshold set by physics and the still incompletely understood neural mechanisms. 
Physiological adaptation account of the square-root law
More recently, an alternative account of the square-root threshold law—one which does assume the existence of physiological adaptation—has been proposed that explains the law as the result of a neural adaptation that scales the response to the flash in inverse proportion to the square root of the light intensity to which the retinal area receiving the flash has recently been adapted (Brown & Rudd, 1998; Donner, Copenhagen, & Reuter, 1990; Rieke & Rudd, 2009). According to this alternative theory, there exists in rod vision a neural gain control—analogous to the Weber gain control that operates in cone vision—such that the physiological response to a small, brief flash is  where AG and TG are the area and time over which the adaptation mechanism pools the statistical photon absorptions that drive adaptation, and k is a constant of proportionality. In what follows, we refer to any gain control whose gain varies inversely with the square root of adapting intensity—as in Equation 4—as a “square-root” gain control.  
Working model of sensitivity regulation in rod vision within the square-root regime
To help motivate the psychophysical experiments reported in this paper, we also consider a more realistic psychophysical model that allows for influences of both noise and gain control in the determination of threshold and brightness. As our working hypothesis, we assume that the observer's signal-to-noise ratio for discriminating trials in which an adapting field is presented alone from trials with a superimposed small, brief flash is:    
The noise variance termsDisplay FormulaImage not available andDisplay FormulaImage not available in Equation 5 model the noise added to the mean signal strength before and after the neural gain control stage. The noise variance Display FormulaImage not available that arises prior to the gain control stage is the sum of a component that scales in proportion to the variance in the rod isomerization rate I + Display FormulaImage not available , pooled over the summation area AN and integration time TN of the gain mechanism, and an additive noise term Display FormulaImage not available that also contributes to the pregain noise in the pool. Display FormulaImage not available models the Poisson noise due to spontaneously occurring thermal isomerizations in the rods, which are physiologically indistinguishable from isomerizations produced by actual photon absorptions (Autrum, 1943; Barlow, 1956; Baylor, Matthews, & Yau, 1980; Baylor, Nunn, & Schnapf, 1984; Donner, 1992; Luo, Yue, Ala-Laurila, & Yau, 2011).  
The noise associated with the isomerization variance (photon noise plus thermal noise) may not be the spatially integrated isomerization variance per se, but rather the noise in the output of a neural cascade process in which each rod isomerization gives rise to a Poisson-distributed number of chemical events. The pooled statistical noise from the cascade, as seen at the input to the gain stage, can be modeled as a multiplied isomerization noise by the multiplicative factor η ≥ 1 in Equation 5. Such cascade processes produce statistical output distributions that are referred to variously as multiplied Poisson, doubly stochastic, or Neyman Type A distributions (Neyman, 1939; Teich, Prucnal, Vannucci, Breton, & McGill, 1982; Tripathi, 1982; Troy & Robson, 1992). The assumption that visual sensitivity is limited by such multiplied Poisson noise has been previously proposed (Reeves, Wu, & Schirillo, 1998) as an explanation of why visual sensitivity is less than would be expected if threshold was limited by the photon fluctuations from the adapting field (Denton & Pirenne, 1954; Graham & Hood, 1992). 
We purposely omitted a subscript on the gain term G in Equation 5 to allow the neural gain to depend either on the isomerization rate or on the noise in the isomerization rate. In principle, the gain factor in Equation 5 could be a square-root gain, as in Equation 4, or a Weber gain, as in Equation 1, or it could have some other functional relationship to adapting intensity or be a constant, independent of adapting intensity. But there is reason to expect, on the basis of past research (see below), that the gain factor in the square-root law regime is a square-root gain, described by Equation 4
Unlike the threshold-level signal-to-noise ratio, which can be affected by both neural gain and noise, the average suprathreshold brightness of a flash is assumed to depend only on the flash intensity ΔI and the neural gain factor G, which may or may not depend on the noise that limits threshold (or on noise at all) (Brown & Rudd, 1998). In what follows, we assume that brightness is determined by the neural response to the flash,  and we study the conditions under which two flashes, presented to the separate eyes under potentially different states of monocular adaptation, match in brightness. This provides an indirect way of studying the dependence of the monocular gain on I andDisplay FormulaImage not available . On the basis of previous experiments (described below), we expect that for small, brief test flashes, the neural gain—and thus brightness—in the square-root law regime of the rod vision will have the functional form of the square-root gain described by Equation 4.  
Previous experimental studies of square-root gain control in toads and humans
By measuring neural gain directly at several retinal processing stages in the toad, Donner et al. (1990) identified a square-root gain control that acts in the circuit reading out the rod photoreceptor signals. Those authors speculated, but did not demonstrate, that this square-root gain control depends on adaptation to photon noise, a mechanism that we refer to here as “noise adaptation.” Alternatively, the square-root gain control that they observed might be deterministic; that is, it might be a result of neural adaptation to the square root of a neural estimate of the mean adapting intensity rather than of an estimate of the photon noise level. A third possibility is that it might result from an adaptation to a neural noise that scales in proportion to the photon noise, such as the first component of the noise termDisplay FormulaImage not available in Equation 5.  
Additional evidence for a monocular square-root gain control was provided by human psychophysical experiments by Brown and Rudd (1998). They reasoned that a square-root gain control should affect the apparent brightness of suprathreshold intensity flashes, consistent with Equation 6 above. However, brightness would not be expected to change simply due to increased photon fluctuations from the adapting field in the absence of a neural gain change. Thus, unlike rod threshold experiments, brightness-matching experiments performed at the same adapting levels provided a means to directly test the square-root gain hypothesis. Brown and Rudd demonstrated that the brightness gain, like threshold, follows an inverse square-root law in a brightness-matching experiment that we describe in more detail in the Methods section of Experiment 1 below. 
These two experiments represent the entire experimental literature on square-root gain control in rod vision. Here, we present the results of a series of psychophysical brightness-matching and threshold experiments that further elucidate the properties of this gain control. In particular, we show that square-root gain control arises in human rod vision from a surprisingly slow (∼100 s) monocular neural adaptation. Furthermore, we show that increasing variability in the photon inputs decreases gain, and this gain control exhibits similarly slow kinetics. Together, these results suggest that square-root gain control is produced by statistical variability in the retinal photon absorption rate rather than the mean photon rate. 
Experiments
General methods
The experiments used a custom-built, four-channel Maxwellian view haploscope. Figure 1 illustrates the display (not to scale) as seen by the observer's left eye, right eye, and binocularly fused view. Two of the four channels were used to present the adapting fields and the other two to present test flashes. The adapting fields were 491.5 nm, 7.4° diameter disks, centered 11.4° nasally from the fovea on the observers' retinas. Test flashes were 10 ms, 491 nm, 0.55° diameter disks, presented in the center of each adapting field. 
Figure 1
 
The dichoptic stimuli used in the experiments as seen by the observer's left eye, right eye, and in binocular view. The regions outlined by the colored ovals illustrate the display elements as projected to the observer's left and right retinas. In binocular view, the display appeared as a composite of these elements, surrounded by darkness and aligned by the observer's binocular fusion of the red fixation dot.
Figure 1
 
The dichoptic stimuli used in the experiments as seen by the observer's left eye, right eye, and in binocular view. The regions outlined by the colored ovals illustrate the display elements as projected to the observer's left and right retinas. In binocular view, the display appeared as a composite of these elements, surrounded by darkness and aligned by the observer's binocular fusion of the red fixation dot.
Test and adapting fields were produced by imaging an aperture on the observer's retina, through which the observer viewed the defocused image of a 468-nm 1500-mcd LED (Digi-Key Electronics P465-ND, Panasonic Electronic Components LNG992CFBW) through a 491.5-nm narrowband interference filter. Retinal defocusing was achieved by focusing the image in the plane of the observer's pupil. Retinal images were stopped down by projecting them through a 2-mm optical aperture, imaged in the pupillary plane. 
Two 640-nm fixation dots, each a few arcmin in width, were projected on each of the observer's retinas. Fixation was maintained by binocularly fusing the fixation dots shown to the left and right eyes. This fusion resulted in a proper spatial alignment of the adapting fields and test flashes. Binocular fusion and proper fixation were reliable and stable after a period of training. Alignment of the optical paths with the observer's pupils was achieved through a bite bar arrangement. 
Stimulus presentation was controlled using custom programs written in the IGOR Pro programming language (WaveMetrics, Inc., Lake Oswego, OR) and a 16-bit interface (Instrutech Corporation, Great Neck, NY). The overall range of the adapting field and test flash intensities was set for each experiment through the use of glass neutral density filters (ThorLabs, Newton, NJ). Additional test flash luminance adjustments were made on the basis of the observer's responses through computer-controlled voltage changes to the LEDs. Responses were achieved by button presses and recorded on the computer. 
Luminance calibration was accomplished by placing a light meter (Graseby Optronics 350, Graseby Electro-Optics, Orlando, FL) in the position of the observer's pupil. Power measurements were converted to quanta per unit area-time, and photon absorption rates in the rods were estimated on the basis of the following assumptions: 1° visual angle = 0.3 mm on the retina (Wandell, 1995), retinal rod density = 1.35 × 105 rods/mm2 at 3.43 mm temporal eccentricity (Rodieck, 1998), and 28% corneal quanta exciting rods (Donner, 1992). Calibration was checked periodically throughout the study. 
We report light intensities in estimated units of R*/rod/s. For our narrowband 491.5-nm stimuli, V′ ≅ 0.915 (Le Grand, 1968) and 1 R*/rod/s ≅ 0.085 sc Td (Walraven, Enroth-Cugell, Hood, MacLeod, & Schnapf, 1990, p. 55, their equation 3), based on the assumptions listed above. This conversion factor has uncertainty due to uncertainties in the assumptions. 
All experimental protocols were approved by the Human Subjects Review Board of the University of Washington. The experiments were conducted with each observer's informed consent in conformance with the Declaration of Helsinki. 
At the beginning of each session, the observer dark-adapted for 40 min while sitting in a small experimental room, the walls of which were painted black. At the end of the 40-min dark-adaptation period, the observer placed his or her teeth on the bite bar and initiated an experimental session by simultaneously pressing button controls held in the left and right hands. This turned on the adapting fields and fixation dots. The observer fixated the dots to fuse the two eyes' images. Fixation was maintained for 10 min (Experiments 1 and 2) or 6 min (Experiments 3 and 4) to light-adapt the observer's retinas to the field intensities specified by the experimental protocol for each experiment. At the end of the light-adaptation period, the observer initiated the brightness measurements by again pressing both buttons. 
An experimental session typically required a few hours to complete. The observer was allowed to take breaks as often as needed. If a break was taken, the observer was required to repeat the light-adaptation procedure while on the bite bar before resuming measurements. 
Experiment 1: Effects on flash brightness of adding full-field random flicker to the adapting field: First half second following flicker onset
Experiment 1 was based in part on the study by Brown and Rudd (1998), who demonstrated a square-root law for brightness matches over the same range of adapting intensities that yielded the square-root threshold law. Those authors identified the intensity of a small, brief test flash presented on an adapting field that matched in brightness a fixed-intensity comparison flash with no adapting field presented simultaneously to the observer's other eye. Brightness matches were proportional to the square root of the adapting field on which the test was presented. This result is consistent with the assumption that the neural response to the test flash depends on a monocular gain factor that varies with the square root of the adapting field intensity (Equation 4). The flash brightness in the dark-adapted eye was presumably determined by the ratio of the comparison flash intensity to a fixed “dark light” or “dark noise” level (Barlow, 1977; Baylor et al., 1980; Baylor et al., 1984; Donner, 1992; Fechner, 1860; Shapley & Enroth-Cugell, 1984) in that eye. 
In the current study, we instead manipulated the adapting field in the eye opposite to the one in which the flash intensity was adjusted. The intensity of the flash presented on the variable field remained fixed, as did the adapting field in the eye that saw the adjustable flash (Figure 2). This method had the advantage that any changes in the neural gain that influenced the brightness of the test flash were applied to a fixed-intensity flash, making it easier to establish that brightness changes were due to changes in neural gain. Furthermore, presenting the adjustable flash on a fixed-intensity adapting field, rather on no adapting field, provided better control of the adaptive state of the match eye. Whereas Brown and Rudd (1998) presented the left- and right-eye adapting fields in the same location of visual space in the observer's binocular view (thus ensuring that square-root adaptation was monocular), we separated the adapting fields to ensure that spatiotopic cortical adaptation did not contribute to any square-root law brightness adaptation that we observed. Both procedures produced results consistent with Equation 4
Figure 2
 
The brightness matching method used in Experiment 1. The neural gain of a retinal patch in one of the observer's two eyes—the “test” eye—was manipulated by varying the photon statistics of the light from the large disk-shaped adapting field in that eye, including both the mean and variance of the photon count from the field. The brightness gain of the test eye was assessed by superimposing on the adapting field a small (0.55°), brief (10 ms) test flash of fixed intensity and finding the value of a simultaneous incremental flash of variable intensity presented on a steady field of fixed intensity 0.1 R*/rod/s to the observer's other eye—the “match” eye—that matched the test flash in brightness.
Figure 2
 
The brightness matching method used in Experiment 1. The neural gain of a retinal patch in one of the observer's two eyes—the “test” eye—was manipulated by varying the photon statistics of the light from the large disk-shaped adapting field in that eye, including both the mean and variance of the photon count from the field. The brightness gain of the test eye was assessed by superimposing on the adapting field a small (0.55°), brief (10 ms) test flash of fixed intensity and finding the value of a simultaneous incremental flash of variable intensity presented on a steady field of fixed intensity 0.1 R*/rod/s to the observer's other eye—the “match” eye—that matched the test flash in brightness.
As illustrated in Figure 2, the brightness of the test flash depended not only on the test flash intensity, but also on the intensity of the background field. The situation is analogous to that of classical simultaneous contrast, in which the brightness of a static target depends on the Weber ratio of the target and background (Rudd, 2010, 2013, 2014; Wallach, 1948). However, for small and brief targets at rod light levels, brightness depends on the ratio of the incremental target intensity to the square root of the intensity of the background (adapting field) on which the flash is presented. Previous research conducted with static targets shows that target appearance can be influenced by whether the observer is explicitly instructed to make luminance matches or matches based on the target/background luminance ratio (Arend & Spehar, 1993a, 1993b; Rudd, 2010). Importantly, we did not see any evidence of such multiple modes of appearance in our experiments with our briefly flashed stimuli. 
In addition to testing the square-root brightness matching law with steady adapting fields, we ran interleaved conditions in which random full-field flicker was added to the adapting field in the test eye (only) with the mean field intensity held constant. Comparing brightness results achieved with and without added flicker enabled us to determine whether adaptation in the square-root regime depends on noise or only on the mean field intensity. We further examined whether the temporal delay between addition of noise to the field and the onset of the test flash mattered for three different temporal delays within the first half second after turning on the random full-field flicker. 
Methods
Two subjects participated. MER was one of the authors of the study. HCC was a paid female undergraduate research assistant. 
Within each experimental session, the adapting field in one eye—hereafter, the “match” eye—was fixed at the value 0.1 R*/rod/s. The average intensity of the adapting field in the other eye—the “test” eye—was set to a value of 0.01, 0.1, or 1.0 R*/rod/s. Equal numbers of sessions were carried out using either the left or right eye as the test eye. Left- and right-eye sessions were run in a counterbalanced order, usually on different days. 
Within each session, several consecutive experimental runs were performed. Each run had a total duration of 16 s and consisted of two 8-s segments. During the first segment, no flicker was added to the field in the test eye. During the second segment, random, temporal low-pass, full-field, truncated Gaussian flicker was added to the field in the test eye. The Gaussian flicker was truncated at the value 0 (no negative luminances) and was spectrally flat with a low-pass temporal cutoff at 50 Hz. The segment without flicker always preceded the segment with flicker, and the two types of segments alternated regularly throughout a session. 
Simultaneous flashes (10 ms, 491.5 nm, 0.55°) were presented in the centers of the match and test eye adapting fields at temporal delays of 4000, 4225, or 4450 ms following each segment onset. During segments in which random flicker was added to the field in the test eye, the standard deviation of the flicker was set to 3% contrast during the first half of the 8-s segment, then increased to 36% contrast throughout the second half (Figure 3). The test flash appeared either 0, 225, or 450 ms after the increase in flicker contrast in the flicker-added segments. The decision to set the flicker contrast at 3% during the first 8-s segment on the flicker-added trials, rather than at 0%, was made to match our procedure to that of a parallel physiological study, the results of which are not reported here. The flash parameters were chosen both to optimize detection by the rod system and to optimize the chances of obtaining square-root behavior, which is known to hold only for flashes of sufficiently brief duration (Barlow, 1957; Sharpe, Stockman, Fach, & Markstahler, 1993). Brown and Rudd (1998) obtained square-root laws for both threshold and brightness matches using 10-ms flashes. 
Figure 3
 
Time course of the stimulus presentation in Experiments 1 and 2. Each experimental run consisted of two 8-s segments. The test and match flashes were presented simultaneously 0, 225, or 450 ms into the second (4 s) half of each segment. In the second segment (only), 3% contrast full-field flicker was added to the adapting field in the test eye but not to the adapting field in the match eye during the first half of the segment, and the flicker contrast was increased to 36% during the second half. The figure illustrates the flash delay (τ = 0, 225, or 450 ms) relative to the flicker contrast increase.
Figure 3
 
Time course of the stimulus presentation in Experiments 1 and 2. Each experimental run consisted of two 8-s segments. The test and match flashes were presented simultaneously 0, 225, or 450 ms into the second (4 s) half of each segment. In the second segment (only), 3% contrast full-field flicker was added to the adapting field in the test eye but not to the adapting field in the match eye during the first half of the segment, and the flicker contrast was increased to 36% during the second half. The figure illustrates the flash delay (τ = 0, 225, or 450 ms) relative to the flicker contrast increase.
A block of trials comprised three consecutive runs with flash delays of 0, 225, and 450 ms relative to the increase in temporal contrast in the test eye. The order of the flash delays was randomized independently within each block. The intensity of the flash presented on the variable intensity adapting field in the test eye was fixed throughout the entire experiment at the value 20 R*/rod/s. The initial intensity of the “match” flash presented to the other eye on an adapting field of fixed intensity was set to a value consistent with a square-root brightness match to the test flash. When the average intensity of the adapting field in the test eye was 0.1 R*/rod/s—and thus equal to the constant intensity of the field in the match eye—the match flash intensity was initialized at the value 20 R*/rod/s. When the average intensity of the adapting field in the test eye was 0.01 R*/rod/s—a log unit lower than the adapting field in the match eye—the initial match flash intensity was 63.3 R*/rod/s: 0.5 log unit higher than the test flash. When the average intensity of the adapting field in the test eye was 1.0 R*/rod/s—a log unit higher than the intensity of the adapting field in the match eye—the initial match flash intensity was 6.33 R*/rod/s: 0.5 log unit lower than the test flash. We chose our initial settings to be as unbiased as possible in light of previous results indicating a square-root law for brightness. Brown and Rudd (1998) chose initial settings that were either higher or lower with an identical outcome. Further checks on our procedure are described below. 
Within each experimental session, six interleaved staircase procedures were carried out to find the match flash intensities producing a brightness match to the test flash in the flicker and no-flicker segments for each of the three flash delays. Following each flash pair presentation, the observer indicated which of the two flashes appeared brighter by pressing the response button held in the hand corresponding to the side in which the brighter flash appeared (right or left). The computer then either increased or decreased by 2% the match flash intensity in the next trial with that combination of flicker/no-flicker and flash delay, depending on whether the match flash was judged to be dimmer or brighter than the test flash. This procedure was repeated until the observer's decisions regarding which flash was brighter reversed 50 times. 
The computer averaged the values of the match flash intensities corresponding to each reversal, and the average was taken as an estimate of the match flash intensity that best matched the test flash in brightness for that specific condition (flash delay and temporal contrast). The values of 2% and 50 reversals were chosen by trial and error during pilot studies. Importantly, the pilot studies also verified that estimates of the best matching intensity drifted in the direction predicted by the square-root brightness matching law when the initial match flash intensity was set to a value far from this prediction. 
If the observer failed to respond after a particular flash pair, no response was recorded and no adjustment was made to the match flash intensity in the subsequent presentation. This procedure allowed the observer to exit the bite bar if necessary and resume measurements after first readapting to the adapting fields in both eyes. 
Between successive experimental trials, steady adapting fields with the same average intensities as those used in the experimental trials were presented continuously to both eyes in order to maintain a constant state of adaptation to the mean light level throughout the experiment. 
Nine experimental sessions were carried out for each of the three average adapting field intensities in the test eye—0.01, 0.1, and 1.0 R*/rod/s—with either the left or right eye designated as the test eye. In total, nine brightness matches were performed within each cell of the Flicker vs. No Flicker × Test Eye × Mean Adapting Field Intensity × Flash Delay factorial design. 
Results
Separate data analyses were carried out for the brightness matches made to flashes presented on steady and flickering adapting fields. For each analysis, the data from left and right test eye sessions were averaged. 
We first report the data from the no-flicker conditions. These data establish a baseline from which the effects of flicker can be assessed. The matches made at the three different flash delays did not differ statistically at any of the three adapting field intensities for either of the two observers (as expected because there was no flicker). The data were therefore collapsed across the three delays for the purpose of analyzing the effect of adapting field intensity on brightness. 
The average brightness matches made on steady (nonflickering) fields at each of the mean adapting field intensities are presented in Table 1. In Figure 4A, we plot these matches as a function of the intensity of the adapting field in which the match was performed. The square-root gain hypothesis (Equation 4) predicts that the matches should fall on the diagonal black line, which has a slope of −0.5. The predicted slope is negative because the flash with adjustable luminance was presented to the eye opposite to that in which the adapting field intensity was varied. 
Table 1
 
Statistical significance and effect size for the brightness enhancement of a 20 R*/rod/s test produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed t-tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Table 1
 
Statistical significance and effect size for the brightness enhancement of a 20 R*/rod/s test produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed t-tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Figure 4
 
Brightness matches for small, brief test flashes follow the square-root law. (A) Matches made on a 0.1 R*/rod/s adapting field of fixed intensity to a 20 R*/rod/s test flash presented on a nonflickering adapting field in the other—match—eye as a function of the adapting field intensity in the test eye. The matches for both observers fall on a line of slope −0.5 as predicted by the hypothesis that the monocular gain applied to the test flash varies inversely with the square root of adapting intensity. (B) Matches made under the same conditions, but with 36% contrast, random, full-field Gaussian flicker added to the adapting field in the test eye (only). The square-root matching law is still followed approximately, but small increases in the intensity of the matches were observed at the higher adapting field intensities relative to the matches made to test flashes presented on nonflickering fields. Thus, adding full-field flicker to the adapting field increased the test flash brightness by a small amount (see text for further details).
Figure 4
 
Brightness matches for small, brief test flashes follow the square-root law. (A) Matches made on a 0.1 R*/rod/s adapting field of fixed intensity to a 20 R*/rod/s test flash presented on a nonflickering adapting field in the other—match—eye as a function of the adapting field intensity in the test eye. The matches for both observers fall on a line of slope −0.5 as predicted by the hypothesis that the monocular gain applied to the test flash varies inversely with the square root of adapting intensity. (B) Matches made under the same conditions, but with 36% contrast, random, full-field Gaussian flicker added to the adapting field in the test eye (only). The square-root matching law is still followed approximately, but small increases in the intensity of the matches were observed at the higher adapting field intensities relative to the matches made to test flashes presented on nonflickering fields. Thus, adding full-field flicker to the adapting field increased the test flash brightness by a small amount (see text for further details).
The square-root brightness matching prediction held to a high degree of precision. For observer HCC, the slope of a least-squares regression model fit to the data had a slope of −0.499 and an associated R2 value of 1.000; for MER, the slope was −0.513 and R2 = 1.000. The average of the two slopes is −0.506. 
The average brightness matches made in the conditions in which full-field flicker was added to the field are also presented in Table 1. These matches are plotted against the average adapting field intensity in Figure 4B. Only one significant effect of flash delay relative to the onset of flicker was found for either observer at any of the three adapting field intensities: MER, 0.1 R*/rod/s field, F(2, 87) = 3.545, p = 0.033. Because no general pattern of delay effects was observed, and the one significant effect had a considerable likelihood of having arisen by chance (Bonferroni-corrected alpha for nine tests = 0.37), the results were collapsed across delays. 
To first approximation, the square-root brightness law also held for test flashes presented on a flickering field. However, statistical tests indicated the presence of a small tendency for the flash brightness to increase on the flickering field, especially at the higher adaptive field intensities. To show this, we first divided the brightness matches made at each of the adapting field intensities by the match predicted by the square-root law. This normalization allowed direct comparison of matches made at different adapting field intensities. The normalized matches from the flicker and no-flicker conditions were compared in one-tailed t tests using flicker and no-flicker brightness matches within the same experimental run. 
The results demonstrate a clear, albeit small, tendency for the flicker to increase the brightness of test flashes presented at short delays following an increase in the contrast of the full-field flicker (effect size in Table 1). This tendency was most pronounced for test flashes presented on the higher intensity adapting fields (Table 1). In general, effect sizes were larger for HCC than for MER; HCC also reported that the random flicker was more perceptually salient. 
Additional conditions using 6.325 and 2.0 R*/rod/s test flashes
We repeated the experiment using test flash standards of intensity 6.325 and 2 R*/rod/s: 0.5 and 1.0 log units less intense than the original 20 R*/rod/s standard. Both observers had difficulty making brightness matches on a 0.1 R*/rod/s field to either a 6.325 or 2.0 R*/rod/s test flash presented on a 1.0 R*/rod/s field because the test flash intensities were too near threshold. Hence, these conditions were not run. We also eliminated the conditions involving the 0.1 R*/rod/s adapting intensity because a brightness match made to a flash presented in a field of that intensity should be a luminance match if the experimental procedure is valid as we had already verified with the 20 R*/rod/s standard. 
Neither observer showed a significant effect of temporal delay for matches made to either standard (all ps > 0.500), so we collapsed the data across delays to analyze the effects of added flicker on test flash brightness. 
Table 2 presents the matches made on a 0.1 R*/rod/s field to 2.0 and 6.325 R*/rod/s test flashes (0.301 and 0.801 in log units) presented on a 0.01 R*/rod/s adapting field along with the implied brightness-matching slopes. The results confirm that a square-root brightness-matching law held for both observers and both standards. 
Table 2
 
Matches made on a −1 log unit adapting field to 0.301 and 0.801 log unit test flashes on a −2 log unit field with the implied brightness matching slopes.
Table 2
 
Matches made on a −1 log unit adapting field to 0.301 and 0.801 log unit test flashes on a −2 log unit field with the implied brightness matching slopes.
We also analyzed the data from the no-flicker conditions in an alternative way by regressing the matches (in log units) made to each of the three standards presented on the 0.01 R*/rod/s adapting field against the actual standard intensities. The condition for a brightness match is that the neural responses ΦM and ΦT to flashes in the match and test eyes should be equal. If the monocular gain varies inversely with the square root of adapting intensity (Equation 4), a brightness match should be obtained when  where ΔIM and IM are the flash and adapting field intensities in the match eye, and ΔIT and IT are the flash and adapting intensities in the test eye.  
Solving Equation 7 for ΔIM and taking the logarithm of the resulting expression yields    
The square-root brightness-matching hypothesis thus predicts that a least-squares regression model of match flash intensity versus the test flash intensity (both in log units) should have slope 1 and intercept −0.5. For MER, the actual least-squares regression slope was 1.011 and the intercept −0.517. For HCC, the slope was 0.996 and the intercept −0.494. The average slope for the two observers was 1.004 and the average intercept −0.506. The R2 associated with the least-squares linear model was 1.000 for the combined data. 
We next examined the matches made in the trials with random full-field flicker added to the adapting field in the test eye. No significant effect of temporal delay was found for either observer for matches made to either standard (all ps > 0.10), so we again collapsed the data across delays. 
We then tested for an increase in test flash brightness following an increase in flicker contrast for the 2.0 and 6.325 R*/rod/s standards. For both observers and both standards, the flicker significantly increased the test brightness (Table 3). The effect was especially pronounced for low-intensity flashes in the case of HCC, who exhibited greater sensitivity to the added flicker than did MER in general. 
Table 3
 
Statistical significance and effect size for the brightness enhancement of 2.0 and 6.325 R*/rod/s tests produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Table 3
 
Statistical significance and effect size for the brightness enhancement of 2.0 and 6.325 R*/rod/s tests produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Discussion
Experiment 1 replicates the earlier findings of a square-root law for brightness matching (Brown & Rudd, 1998) and further demonstrates that adding full-field flicker to the adapting field increases the test brightness within the first half second after adding flicker. 
The matches obtained with steady adapting fields are consistent with the experimental hypothesis that brightness gain is controlled separately in each eye by a neural mechanism whose gain varies inversely with the square root of adapting intensity (Equation 4). The square-root gain property was observed even when a 63.25 R*/rod/s test flash was presented on a 0.01 R*/rod/s adapting field. In that case, the test flash produced 2,000 times more photon absorptions than the adapting field during the 10-ms window in which the flash was presented (or 200 times more photon absorptions during the ∼100-ms integration time of a rod photoreceptor). Thus, even if the square-root brightness law were somehow thought to depend on square-root threshold behavior, the conditions of our experiment would violate a key assumption of the classic de Vries–Rose explanation of the latter, namely, the assumption that the photon fluctuations from the adapting field should dominate the photon fluctuations from the test flash as in Equation 3. The fact that matches made to 6.325 and 2.0 R*/rod/s test flashes on a 0.01 R*/rod/s field also obey the square-root law for brightness demonstrates that the law holds over at least a 1.5 log unit range of test flash intensities at this adapting intensity. 
In principle, the brightness gain could be controlled by the mean field intensity, or the standard deviation of the noise fluctuations from the field, or both. However, if the gain was inversely proportional to the standard deviation of either the photon noise or multiplied photon noise then the test brightness should have decreased when flicker was added to the field. Instead, brightness (and thus the monocular gain) actually increased slightly during the first half second following the addition of flicker to the adapting field. In Experiment 3, we show that this flicker-induced increase in the brightness gain is a transient phenomenon: At longer delays, adding flicker to the field decreases the flash brightness. 
Experiment 2: Visual threshold measured on steady and flickering fields
In Experiment 1, the brightness gain increased slightly during the first half second following the addition of full-field flicker to the adapting field. The de Vries–Rose theory (Equation 3)—which assumes no neural gain control—predicts that adding photon noise to the adapting field should decrease visual sensitivity if the test flash intensity remains fixed. Whether the more realistic psychophysical model expressed by Equation 5—which includes both gain control and noise—also predicts that adding flicker noise will reduce sensitivity depends on whether the photon noise–dependent component ηIANTN of the total noise that limits threshold dominates the other pre- of post- gain control terms in the denominator of Equation 5. Traditionally, this has been assumed to be the case in the square-root law threshold regime of rod vision. 
If indeed photon noise dominates, then the gain terms in the numerator and denominator of Equation 5 will cancel. As a result, adding flicker to the field should decrease the observer's d′ independent of any changes in the neural gain. If the post-gain additive noise term instead dominates the threshold-limiting noise, then adding full-field flicker to the adapting field should not affect the observer's sensitivity. Importantly, this is true even if adding flicker changes the neural gain and thus the flash brightness. In fact, because the results of Experiment 1 indicate that adding flicker to the field increases the gain over the first half second following the addition of flicker, we would expect the addition of flicker to increase sensitivity during this period if the noise-limiting threshold is internal additive noise rather than the noise associated with photon fluctuations. Thus, by examining whether adding flicker to the field decreases or increases visual sensitivity near threshold, we can assess whether external or internal noise limits threshold in the square-root law regime. 
To test these two alternative hypotheses about the source of the noise that limits threshold, in Experiment 2 we measured visual threshold at each of the three adapting field intensities employed in Experiment 1, both with and without random full-field flicker added to the field. 
Method
The same two observers who participated in Experiment 1 participated in Experiment 2. The visual stimulus was identical to that used in Experiment 1 except where noted. 
Within each experiment session, the mean intensities of the adapting fields presented to the observer's two eyes were equal and set to 0.01, 0.1, or 1.0 R*/rod/s. As in Experiment 1, an experimental session consisted of a series of runs, each of which contained two presentation segments, lasting 8 s each. During the first segment, the intensities of the left and right adapting fields remained steady and equal. During the second segment, random full-field temporal flicker was added to the adapting fields in both eyes. The flicker intensity was distributed according to a truncated Gaussian with a 0 R*/rod/s cutoff (no negative intensities). During the first 4 s of each 8-s noise segment, the temporal noise contrast was 3%; during the last 4 s, it was 36%. The temporal contrast noise was low-pass with a 50-Hz temporal cutoff frequency. 
Within each segment, a 10-ms, 491-nm, 0.55° diameter test flash was presented in the center of the adapting field in either the left or right eye with equal probability. The test eye was chosen randomly in each trial. In flicker trials, the flash onset occurred 225 ms after the increase in noise contrast. On no-flicker trials, the flash onset occurred 4225 ms into the trial. After each flash presentation, the observer depressed the response button held in the hand corresponding to the side on which he or she judged the flash to have been presented. 
The session continued until the flash had been presented at least 50 times to each eye in both the flicker and no-flicker conditions. At the end of each session, d′ was calculated separately for the flicker and no-flicker conditions by converting the proportions of correct and incorrect trials to z scores for Hits and False Alarms and computing d′ = z(Hit Rate) – z(False Alarm Rate). 
Between sessions, the flash intensity was adjusted iteratively to achieve a target d′ value of about 1.0 in the no-flicker condition (MER: d′ = 0.94; HCC: d′ = 1.37). We repeated this procedure four times at each adapting field intensity. In the no-flicker trials, d′ did not vary across successive sessions carried out at the same flash intensity for either observer at any of the three adapting field intensities. For the flicker trials, d′s were averaged across the four repetitions to estimate the overall d′ for the same flash intensities that produced a d′ of 0.94 or 1.37 (depending on the observer) in the no-flicker trials. 
Results
The results are shown in Table 4. The visual thresholds in the table are the flash intensities that produced the target d′ on the steady adapting field (i.e., in the no-flicker condition). The corresponding d′ values for the flicker trials indicate the reduction in discriminability produced by adding full-field flicker to the adapting field. p-values correspond to one-tailed t tests (df = 3) of the hypothesis that discriminability is lowered when flicker is added to the field. This discriminability reduction was highly significant for both observers at all three adapting field intensities. 
Table 4
 
Increment thresholds measured on steady fields of −2, −1, and 0 log units (0.01, 0.1. and 1.0 R*/rod/s) together with the corresponding ds for flash detection on adapting fields with or without added full-field flicker. Notes: p-values denote significance of d′ reduction when full-field flicker was added to the adapting field (see text for further details).
Table 4
 
Increment thresholds measured on steady fields of −2, −1, and 0 log units (0.01, 0.1. and 1.0 R*/rod/s) together with the corresponding ds for flash detection on adapting fields with or without added full-field flicker. Notes: p-values denote significance of d′ reduction when full-field flicker was added to the adapting field (see text for further details).
We also checked the validity of the square-root threshold law for thresholds measured on the steady adapting fields by calculating the slopes of the log threshold function over the upper and lower one log unit adapting field ranges from −2 to −1 log R*/rod/s and from −1 to 0 log R*/rod/s. Over the lower range, the slopes were 0.520 (MER) and 0.525 (HCC). Over the upper range, they were 0.606 (MER) and 0.553 (HCC). 
Discussion
The results in Table 4 demonstrate that adding flicker noise to the adapting field reduces the discriminability of flash versus no-flash trials. We do not believe that this discriminability reduction was due to a decrease in the brightness gain applied to the test flash in the flicker trials because in Experiment 1 we found that adding flicker to the field did not decrease the test flash brightness; rather it increased brightness, even for low intensity flashes. 
The results of Experiments 1 and 2 together are consistent with a parameterization of the model presented in Equation 5 in which the dominant component of the noise that limits threshold discrimination is proportional to the photon noise summed across a spatiotemporal pool of rod photoreceptor responses. This noise arises prior to a gain control whose gain increases slightly during the first half second following the addition of random full-field flicker to the adapting field. 
Despite the fact that early noise apparently gets through to both the neural stage that controls the brightness gain and the neural stage that controls visual sensitivity (which may be different), none of our results to this point are consistent with a noise gain control having the inverse square-root form of Equation 4. Thus, the noise gain control that accounts for the small brightness enhancement seen within the first half second following an increase in added flicker contrast cannot account for the square-root brightness gain control observed in Experiment 1 and in the experiments of Brown and Rudd following long-term adaptation. However, in Experiment 3 we show that the initial increase in brightness gain observed in Experiment 1 shortly after adding flicker to the adapting field evolves into a brightness gain decrease after more prolonged flicker adaptation. In Experiment 4, we further show that the time course of this brightness gain decrease is roughly the same as the time course of square-root law adaptation to changes in mean intensity under conditions in which no artificial flicker is added to the adapting field. This suggests that both effects might be explained by adaptation to changes in the photon noise level. 
Experiment 3: Effects on flash brightness of adding full-field random flicker to the adapting field: Longer delays following flicker onset
In Experiment 1, the test brightness increased in the first half second following the addition of random full-field flicker to the adapting field. However, most studies of the square-root threshold law, as well as Brown and Rudd's (1998) study of the square-root brightness law, have involved much longer adaptation times, presumably sufficient for complete adaptation. In Experiment 3, we probed brightness gain over a 6-min time interval following the cessation of adaptation to full-field flicker to examine whether noise gain control could contribute to a square-root gain change at longer adaptation times. 
Method
Two observers participated in Experiment 3. MS was a male undergraduate in his 20s, and JSR was a female lab technician in her 30s. Both were paid for their participation. 
The experiment began with a 6-min adaptation period, during which the mean intensities of the adapting fields in both eyes remained fixed at 0.1 R*/rod/s. The 6-min period was chosen on the basis of pilot studies showing that adaptation was complete after this period for these stimuli. Throughout the entire adaptation period, the field intensity in the test eye (only) was modulated by random full-field flicker, comprising random samples of binary noise presented at 98% contrast, updated every 10 ms. 
At the end of the adaptation period, the random flicker was turned off, but the adapting fields in the test and match eyes remained on at the same mean intensity. Beginning either at the moment that the flicker in the test eye ceased (in half of the trials) or 1 s after the flicker cessation (in the other half of the trials), pairs of simultaneous 10-ms, 20 R*/rod/s flashes were presented in the centers of the adapting fields in the two eyes, once every 2 s for a period of 6 min. After each flash pair presentation, the observer made a forced-choice judgment regarding which of the two flashes appeared brightest. 
Results
In Figure 5, the results are plotted as the proportion of trials, as a function of time, in which each observer judged the flash presented to the flicker-exposed eye to be brighter than the flash presented to the non-flicker-exposed eye. The results have been averaged over sessions in which the flicker was presented to either the left or right eye, and responses have been binned into 10-s intervals. 
Figure 5
 
Time course of recovery from adaptation to 36% binary-random full-field flicker. Proportion of trials in which a 20 R*/rod/s test flash presented on a steady 0.1 R*/rod/s adapting field in the test eye was judged to be brighter than a 20 R*/rod/s match flash, presented on a steady 0.1 R*/rod/s adapting field in the match eye, as a function of time following 6 min of adaptation to binary-random full-field flicker of mean intensity 0.1 R*/rod/s in the test eye. Immediately following the cessation of the random flicker in the test eye, flashes presented to that eye are judged to be brighter than flashes of equal intensity presented simultaneously to the other eye about 0% of the time, indicating that adaption to the full-field flicker reduced the monocular gain in the test eye. This gain reduction dissipated over a period of about 100 s for both observers.
Figure 5
 
Time course of recovery from adaptation to 36% binary-random full-field flicker. Proportion of trials in which a 20 R*/rod/s test flash presented on a steady 0.1 R*/rod/s adapting field in the test eye was judged to be brighter than a 20 R*/rod/s match flash, presented on a steady 0.1 R*/rod/s adapting field in the match eye, as a function of time following 6 min of adaptation to binary-random full-field flicker of mean intensity 0.1 R*/rod/s in the test eye. Immediately following the cessation of the random flicker in the test eye, flashes presented to that eye are judged to be brighter than flashes of equal intensity presented simultaneously to the other eye about 0% of the time, indicating that adaption to the full-field flicker reduced the monocular gain in the test eye. This gain reduction dissipated over a period of about 100 s for both observers.
Immediately following the cessation of flicker in the test eye, the flash presented to that eye was essentially never judged to be brighter than a flash of identical intensity presented to the eye that had not been exposed to the random flicker. Thus, the 6-min exposure to random flicker decreased the brightness gain in the test eye. The perceptual dimming of the test flash gradually dissipated over a course of about 100 s. Following the recovery of the brightness gain from exposure to full-field flicker, both observers exhibited a small brightness rebound effect, such that the flash presented to the flicker-adapted eye appeared brighter in slightly more than half of the trials. 
Because the adapting fields in the two eyes had the same mean intensity throughout the experiment, the perceptual dimming of the test flash must have been due to a neural gain mechanism that adapts to changes in the level of the noise fluctuations from the field rather than to the mean adapting intensity alone. This noise gain control mechanism operates over the same range of mean adapting intensities that produce the square-root threshold and brightness laws for nonflickering stimuli in rod vision. 
Discussion
Experiment 3 demonstrates the existence of a neural gain mechanism that adapts to statistical fluctuations in the photon absorption rate, not just to the mean rate. Unlike the slight noise gain increases observed in Experiment 1, the long-term effect of increasing the photon noise from the field in Experiment 3 was to decrease the neural gain. Thus, the noise gain control observed in Experiment 3 has properties similar to those of the gain control mechanism proposed by Donner et al. (1990) to account for square-root gain control in the toad retina. Furthermore, the time course of the noise adaptation in humans is close to the 90-s time course observed in physiological studies of square-root law adaptation in the toad. This similarity is somewhat surprising given that toads are cold-blooded animals and humans are warm-blooded; hence, the kinetics of the underlying processes might be expected to differ. The similarity in kinetics suggests that the fundamental constraint on the time course of adaptation may not be set by biophysics, but by inherent properties of natural image statistics that place evolutionary constraints on the time course of adaptation in rod vision and that favor this time scale. We return to this point in the General discussion
Because adding full-field flicker to the adapting field results increases the photon noise level from the field, the same gain control mechanism that produced the adaptation to the full-field flicker could, in principle, be responsible for the long-run square-root brightness changes observed in Experiment 1. That is, the brightness changes may have been produced by a monocular gain mechanism that adapts on the basis of a statistical estimate of the standard deviation of the photon noise from the adapting field. If so, the estimate must have been based on a noise sample taken over a time window of roughly 100 s. 
Experiment 4: Time course of square-root adaptation to changes in mean adapting intensity
Experiment 3 demonstrated that brightness in the square-root law regime is influenced by a noise gain control with a total adaptation time of about 100 s. If square-root adaptation to changes in the mean light intensity is mediated by this noise gain control, then it too should exhibit slow kinetics. Experiment 4 tests this prediction. 
To accomplish this, we first adapted the observer's retina in the test eye to a 0.1 R*/rod/s field, roughly the midpoint in log units of the square-root law range. We then measured the time course of gain changes when the field intensity was either increased or decreased by a factor of four. The increased and decreased adapting intensities were also both well within the square-root law regime of human rod vision, as demonstrated in Experiment 1
Method
Two observers, MH and AR, participated in Experiment 4. Both were males in their 20s. They were paid for their participation. 
Within the square-root law regime, increasing the intensity of the adapting field in the test eye by a factor of four will, in the long run, decrease the brightness of a fixed-intensity test flash (as measure by the intensity of a matching flash in the other eye) by a factor of two. Similarly, decreasing the intensity of the adapting field in the test eye by a factor of four will, in the long run, increase the test flash brightness by a factor of two. 
We first studied the time course of the brightness change elicited by increasing the adapting field intensity (i.e., the time course of light adaptation). While maintaining fixation, the observer first adapted for 6 min to steady 0.1 R*/rod/s adapting fields in both eyes. At the end of the 6-min adaptation period, the mean intensity of the adapting field in the test eye was increased to 0.4 R*/rod/s. Beginning either at the moment that the adapting field intensity was increased (in half the trials) or 1 s after the field intensity increased (in the other half of the trials), simultaneous 10-ms flashes were presented every 2 s in the centers of the adapting fields in both eyes. The intensity of every test flash was 20 R*/rod/s, and the intensity of every match flash was equal to a fixed standard value that depended on the particular experimental condition (see below). Following the presentation of each flash pair, the observer indicated which of the two flashes appeared brightest by a button press. 
To measure the time required for light adaptation to be halfway complete, we set the match standard to 15 R*/rod/s: a value midway between the intensities corresponding to a square-root brightness match before (20 R*/rod/s) and long after (10 R*/rod/s) the adapting field in the test eye was increased from 0.1 to 0.4 R*/rod/s. We defined the time at which light adaptation was 50% complete as the time at which the test flash matched this 15 R*/rod/s standard in brightness: that is, the time at which the test flash was judged to be dimmer than that standard in 50% of the trials. 
More generally, the percentage of light adaptation at time t is given by the formula  where MLA(t) is the intensity of the standard that matches the test at time t, MLA(0) is the fixed test intensity (20 R*/rod/s), and MLA(∞) is the predicted value of the match after complete adaptation (10 R*/rod/s).  
The percentage of dark adaptation, DA(t)%, following a decrease in field strength from 0.1 to 0.025 R*/rod/s, is calculated similarly using the formula  where MDA(0) is 20 R*/rod/s, and MDA(∞) is 40 R*/rod/s.  
From Equations 9 and 10, we computed the match standards MLA(t%) corresponding to %LA(t) = 5, 20, 35, 50, 65, 80, and 95 to be 19.5, 18.0, 16.5, 15.0, 13.5, 12.0, and 10.5 R*/rod/s, and the match standards MDA(t%) corresponding to %DA(t) = 5, 20, 35, 50, 65, 80, and 95 to be 21, 24, 27, 30, 33, 36, and 39 R*/rod/s. We then used these calculated match standards as the fixed match standards in our experiment to find the times t% corresponding to these particular percentages of light and dark adaptation. 
Results
First, we present the results of the light-adaptation experiments. Figure 6A plots the proportion of trials in which the fixed 20 R*/rod/s test flash was judged to be brighter than a 15 R*/rod/s match standard as a function of time after the intensity of the adapting field in the test eye was increased. For both observers, the proportion of “brighter than” judgments decreased monotonically with time and reached an asymptote of 0% by the end of the 6-min response period. Light adaptation in this experiment required several minutes to complete. The time at which the test flash was judged to be brighter than the 15 R*/rod/s standard 50% of the time was taken to be the time at which the two flashes matched in brightness. This time was similar for the two subjects: 108 s for MH and 100 s for AR. 
Figure 6
 
Psychometric functions for light and dark adaptation in the square-root law regime. (A) Light adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance had recently been increased from 0.1 to 0.4 R*/rod/s, was judged to be brighter than a 15 R*/rod/s standard flash, presented on an unchanged 0.1 R*/rod/s adapting field in the other (match) eye, as a function of time following the adapting field increase in the test eye. The 15 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the match flash 50% of the time. The time at which this match occurs is the time at which light adaptation to the new test adapting field is 50% complete because the match standard midway intensity between the 20 R*/rod/s standard that matched the test flash before the adapting field in the test eye was increased and the 10 R*/rod/s standard that matches the test flash when square-root law adaptation is complete. A match to the 15 R*/rod/s standard occurred at flash delays of 100 and 108 s for the two experimental observers. (B) Psychometric functions like the one shown in panel A produced by subject AR with each of seven match standards that were chosen so that they would match the test in brightness when light adaptation was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete (see Experiment 4, Methods for details). (C) Same as panel B but for subject MH. (D) Dark adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance was decreased from 0.1 to 0.25 R*/rod/s, was judged to be brighter than a 30 R*/rod/s standard match flash, presented on an unchanged 0.1 R*/rod/s adapting field in the observer's other eye, as a function of time following the decrease in adapting field intensity. The 30 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the standard 50% of the time. The time at which this occurs marks the 50% adaptation time to the changed adapting field intensity because the 30 R*/rod/s match standard is midway in intensity between the 20 R*/rod/s standard that matches the test flash before the decrease in adapting field intensity and the 40 R*/rod/s match standard that would match the test flash after square-root adaptation to the decreased adapting field intensity was complete. A match to the 30 R*/rod/s standard occurred at flash delays of 117 and 121 s following the adapting field intensity decrease for the two experimental observers. (E) Functions like the one shown in panel D produced by subject AR with seven match standards, which where chosen so that they would match the test in brightness when adaptation to the decreased field intensity was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete. (F) Same as panel E but for subject MH. Proportion of “brighter than” responses have been binned within 10-s intervals in all figure panels.
Figure 6
 
Psychometric functions for light and dark adaptation in the square-root law regime. (A) Light adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance had recently been increased from 0.1 to 0.4 R*/rod/s, was judged to be brighter than a 15 R*/rod/s standard flash, presented on an unchanged 0.1 R*/rod/s adapting field in the other (match) eye, as a function of time following the adapting field increase in the test eye. The 15 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the match flash 50% of the time. The time at which this match occurs is the time at which light adaptation to the new test adapting field is 50% complete because the match standard midway intensity between the 20 R*/rod/s standard that matched the test flash before the adapting field in the test eye was increased and the 10 R*/rod/s standard that matches the test flash when square-root law adaptation is complete. A match to the 15 R*/rod/s standard occurred at flash delays of 100 and 108 s for the two experimental observers. (B) Psychometric functions like the one shown in panel A produced by subject AR with each of seven match standards that were chosen so that they would match the test in brightness when light adaptation was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete (see Experiment 4, Methods for details). (C) Same as panel B but for subject MH. (D) Dark adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance was decreased from 0.1 to 0.25 R*/rod/s, was judged to be brighter than a 30 R*/rod/s standard match flash, presented on an unchanged 0.1 R*/rod/s adapting field in the observer's other eye, as a function of time following the decrease in adapting field intensity. The 30 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the standard 50% of the time. The time at which this occurs marks the 50% adaptation time to the changed adapting field intensity because the 30 R*/rod/s match standard is midway in intensity between the 20 R*/rod/s standard that matches the test flash before the decrease in adapting field intensity and the 40 R*/rod/s match standard that would match the test flash after square-root adaptation to the decreased adapting field intensity was complete. A match to the 30 R*/rod/s standard occurred at flash delays of 117 and 121 s following the adapting field intensity decrease for the two experimental observers. (E) Functions like the one shown in panel D produced by subject AR with seven match standards, which where chosen so that they would match the test in brightness when adaptation to the decreased field intensity was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete. (F) Same as panel E but for subject MH. Proportion of “brighter than” responses have been binned within 10-s intervals in all figure panels.
Figure 6B and C plots the proportion of trials in which each observer judged the test flash in the dynamically light-adapting eye to be brighter than each of the seven standards MLA(t%). Note that the functions have the required property that the time over which the test flash appears brighter than a given match standard 50% of the time increases as a function of adaptation percentage. 
Next, we present the results of the dark adaptation experiments. Figure 6D plots the proportion of trials in which each of the two observers judged the test flash to be brighter than the 30 R*/rod/s standard, corresponding to %DA(t) = 50, as a function of time after the adapting field intensity decreased. The time at which these psychometric functions crossed the 50% mark are the times required to adapt halfway to the new adapting field intensity. As in the case of light adaptation, this time was very similar for the two subjects: 121 s and 117 s for MH and AR, respectively. 
Figure 6E and F plots the proportions of trials in which each observer judged each of the seven dark adaptation standards to appear brighter than the fixed test flash as a function of time following the fourfold decrease in the intensity of the adapting field in the test eye. The times at which each of the psychometric plots crossed the 50% mark are the times t% required to reach the value of %DA(t) corresponding to the match standard MDA(t%) that generated that plot. 
Figure 7 summarizes the results of Experiment 4 by plotting the times corresponding to the crossings of the 50% mark for each of the psychometric curves in Figure 6. An interesting and unanticipated outcome of the experiment was that each of the plots in Figure 7 is a straight line. In other words, percentages of light adaptation and dark adaptation both grow linearly in time. A linear regression model of the data accounts for 97.3% of the variance in the match times in the case of AR and 99.5% in the case of MH. The excellent fits of the linear model perhaps justify extrapolating to times at which the square-root light-adaptation process would be run to completion, which is only slightly beyond the measured times for 95% adaptation. For AR, the time at which light adaptation is estimated to complete is 139 s. For MH, it is 178 s. 
Figure 7
 
Time course of light and dark adaptation in the square-root law regime. Value of the standard flash in the match eye that matched in brightness a 20 R*/rod/s flash presented to the test eye as a function of time following an increase or decrease in adapting field intensity in the test eye for each of the two observers. (A) Data from subject AR. (B) Data from subject MH. Colored lines and R2 values correspond to least-squares linear regression models of the data (maroon = light adaptation, blue = dark adaptation). Extrapolation of the linear regression models to longer times predicts that square-root light adaptation would complete at the time at which the light adaptation function intersects with the lower edge of the graph (match standard = 10 R*/rod/s), and square-root dark adaptation would complete at the time at which the dark adaptation function intersects with the upper edge of the graph (match standard = 40 R*/rod/s) because the match standards corresponding to these boundaries were established in Experiment 1 to match the test flash after long-run square-root adaptation.
Figure 7
 
Time course of light and dark adaptation in the square-root law regime. Value of the standard flash in the match eye that matched in brightness a 20 R*/rod/s flash presented to the test eye as a function of time following an increase or decrease in adapting field intensity in the test eye for each of the two observers. (A) Data from subject AR. (B) Data from subject MH. Colored lines and R2 values correspond to least-squares linear regression models of the data (maroon = light adaptation, blue = dark adaptation). Extrapolation of the linear regression models to longer times predicts that square-root light adaptation would complete at the time at which the light adaptation function intersects with the lower edge of the graph (match standard = 10 R*/rod/s), and square-root dark adaptation would complete at the time at which the dark adaptation function intersects with the upper edge of the graph (match standard = 40 R*/rod/s) because the match standards corresponding to these boundaries were established in Experiment 1 to match the test flash after long-run square-root adaptation.
Similarly, a linear model of the dark-adaptation data accounts for 99.4% of each observer's data. By extrapolating the linear models to the time at which 100% of the adaptation would have occurred, we estimated the projected time for complete dark adaptation in our experiment to be 188 s in the case of AR and 213 s in the case of MH. For each subject, the projected time for complete dark adaptation was longer than the projected time for complete light adaptation (about 35% longer in the case of MH and 20% longer in the case of AR). The projected light- and dark-adaptation times were correlated for the two observers, with MH taking about 28% longer to light-adapt than AR did and about 13% longer to dark-adapt. The finding that dark adaptation was slower than light adaptation is not unexpected because neural responses at many levels of visual processing speed up when the input level is higher. The asymmetry in light- and dark-adaptation times is also consistent with psychophysical studies of the rates of light and dark adaptation not specifically targeting square-root adaptation in rod vision (see Bartlett, 1964; Rushton, 1965, for reviews). 
A second unanticipated finding of this experiment was that square-root adaptation had a gradual or delayed onset, as indicated by the long time required to achieve 5% adaptation and the failure of the linear model to account for the initial period of adaptation. This effect is captured by the positive x-intercepts of the linear regression fits in Figure 7. The sluggish onset held for both light and dark adaptation. 
Discussion
The classical, de Vries-Rose, theory of the square-root threshold law ascribes that law to the masking of dim targets by photon fluctuations from the adapting field, which may be confused with the presentation of an actual test flash. According to their theory, changes in visual sensitivity should occur instantaneously when the adapting field is changed because this produces an instantaneous change in the level of the photon noise from the field. If the integration time of the rods is taken into account, we would expect any changes in sensitivity to be complete in about 200 ms. Our experiments instead demonstrate that square-root adaptation continues for about 2–3 min following a change in adapting intensity with light adaptation taking, on average, about 159 s to complete, compared to 200 s for dark adaptation. 
The results of Experiment 4 further demonstrate that light and dark adaptation both unfold linearly in time over a substantial portion of the time course of adaptation. However, the linear time course of adaptation begins only after a substantial delay, which differs markedly for different observers. The finding of an apparent delay-to-adapt suggests that square-root adaptation is subject to a hysteresis effect of some sort, which may reflect the existence of a threshold or internal criterion to begin adapting. The linear time course of adaptation and the hysteresis effect are likely to provide important clues about the underlying neural processes, but we currently have no explanation of either finding. 
General discussion
Direct evidence for noise gain control in the square-root regime
Adaptation in the square-root law regime of human rod vision can be driven by a sparse photic input that produces no more than one photon absorption per rod, on average, over the approximately 200-ms rod integration time. At the low end of the square-root law intensity range, fewer than one in 100 rods absorbs a photon within this window, and an even smaller fraction of the rods generates spontaneous photon-like noise events (Baylor et al., 1984). The neural activity in the rod pool thus consists of a random sample of quantal rod responses generated by individual photon absorptions and occasional spontaneous noise events. 
Our results reveal for the first time the existence of a form of light adaptation that is driven by statistical fluctuations in the isomerization rate within this rod pool. Donner et al. (1990) previously demonstrated a form of adaptation in the toad retina for which the neural gain varies inversely with the square root of the mean adapting level, and they proposed that this square-root adaptation results from noise adaptation. But their physiological experiments did not specifically manipulate stimulus noise. The only previous human psychophysical study of square-root adaptation (Brown & Rudd, 1998) also did not manipulate stimulus noise. Therefore, neither of those experiments could distinguish between a neural adaptation that is controlled by photon noise, per se, and an adaptation that is controlled by changes in the mean intensity. The noise gain control that we reveal here shares slow (∼100 s) kinetics with gain changes evoked by changes in luminance, suggesting that it may also be responsible for square-root adaptation when the mean adapting level, and hence the level of the photon fluctuation noise from the adapting field, is changed. 
The classical theory of photon noise–limited threshold cannot explain square-root brightness matching
Our results also reinforce the earlier conclusion of Brown and Rudd (1998) that the square-root law is not just a threshold detection phenomenon, but is rather a general light adaptation phenomenon that applies to both threshold and brightness. This adaptation must be monocular because it acts separately on stimuli delivered to each eye rather than on the combination of those stimuli (Brown & Rudd, 1998; see their figure 2A). 
In Experiment 1 of the current study, the square-root law governed brightness matches even when a 20 R*/rod/s test flash was presented on a 0.01 R*/rod/s adapting field. Thus, the square-root law applies to test flashes that are at least 24 times threshold, judging from the threshold measurements performed in Experiment 2. The de Vries–Rose theory cannot plausibly apply to this situation, not only because that theory applies only to threshold judgments (i.e., signal/noise discriminations) and not to brightness (gain), but also because the model assumptions break down when the photon noise contributed by the flash is too large to be neglected. For a 20 R*/rod/s test flash superimposed on a 0.01 R*/rod/s adapting field, the contribution of the flash to the total photon count variance—evaluated over a 0.2-s rod integration time and the spatial area of the test flash—would exceed the contribution of the adapting field by a factor of 100, thus leading to a strong violation of the de Vries–Rose assumptions. This deficiency in the ideal observer model's ability to account for our results might be alleviated if a noise pooling area larger than our 0.55° test flashes were assumed. We think a larger pooling area is likely, but the actual extent of spatial pooling has not been measured. We return to this issue below. 
The slow time course of square-root law adaptation measured in Experiment 4 also does not fit into the de Vries–Rose picture, according to which flash detection should be masked by photon fluctuations from the adapting field only during the spatiotemporal window in which the test flash might or might not be presented in a given experimental trial. Adaptation to a changed adapting level should be instantaneous if the de Vries–Rose account was the correct explanation of square-root behavior. 
Unexpected temporal properties of light and dark adaptation in the square-root law regime
In Experiment 4, light and dark adaptation both evolved linearly in time over the range of adaptation times examined: from the 5% adaptation time to 95% adaptation time. Extrapolating this linear time course to the 0% adaptation time implies a 20- to 60-s delay before adaptation begins (see intercepts of regression lines in Figure 7). Further exploration of the period of the first 5% of adaptation would be required to determine whether such an extrapolation is valid. But a delay to begin adapting might make sense in light of the extreme sparseness of the quantal events driving adaptation under the conditions of the experiment. Perhaps a criterion degree of evidence regarding the change in adapting level must be accumulated before gain changes begin because, if adaptation were to begin too soon, such changes might be predicated on an unreliable estimate of the ambient light level (or the ambient noise level; see below). 
In Experiment 3, the time course of adaptation to a cessation of full-field flicker was measured with a series of forced-choice “which is brighter?” judgments carried out with stimuli of identical luminance in the adapted and nonadapted eyes. This differs from the match-to-standard method that revealed the linear time course of adaptation in Experiment 4. So it is not surprising that Experiment 3 did not reveal the same linear adaptation time course or apparent delay that was seen in Experiment 4
Temporal evolution of the brightness gain over the course of square-root law adaptation
The brightness enhancement that we observed during the early phase of flicker adaptation in Experiment 1 is consistent with previous results demonstrating that flickering stimuli appear brighter than steady stimuli. This phenomenon—the Brücke–Bartley effect (Bartley, 1938; Brücke, 1864)—is strongest for modulation rates of 8–10 Hz in photopic vision. The effect becomes progressively weaker at low luminance, and the frequency at which it peaks shifts to lower frequencies (van de Grind, Grüsser, & Lunkenheimer, 1973). For a 4° square-wave modulated stimulus with an intensity below 1 cd/m2, the effect peaks at modulation rates <1 Hz (Rabelo & Grüsser, 1961). 
We are not aware of any previous studies of the Brücke–Bartley effect in the pure scotopic range. In any case, the brightness enhancement observed here is not exactly the same as Brücke–Bartley because our effect was obtained with an incremental flash superimposed on a randomly modulated adapting field rather than with a flickering test. But the two phenomena are likely related. The weakness of the brightness enhancement in our experiments is perhaps to be expected given that our observers reported that the flicker was not very perceptually salient, consistent with the grainy or cloudy appearance of dim scotopic stimuli. The flicker was even harder to notice in the brief flash than in the adapting field. Nevertheless, frequencies above those that were perceptually visible may have contributed to changes in the neural gain. 
Much literature exists on the topic of visual adaptation to flickering adapting fields (for reviews, see Kohn, 2007; Webster, 2011). Both retinal and cortical and monocular and binocular mechanisms are involved (Baccus & Meister, 2002, 2004; Chander & Chichilnisky, 2001; Dhruv, Tailby, Sokol, & Lennie, 2011; Pokorny, 2011; Ross & Speed, 1991; Shady, MacLeod, & Fisher, 2004; Smith, 1971; Solomon, Peirce, Dhruv, & Lennie, 2004; Zhuang & Shevell, 2015). Our work appears to be unique, however, in exploring flicker adaptation at mean luminance levels that are known to produce square-root adaptation, which is essential to our goal of understanding the possible influence of noise adaptation on square-root gain control. 
An interesting outstanding question is whether the same mechanism that is responsible for the brightness gain increase at short flash delays relative to flicker onset is also response for the gain decrease seen at longer delays. A plausible unitary explanation of both effects is that increasing the photon noise from the field—either by experimentally adding random flicker or by increasing the mean intensity of a “steady” photon-noisy field—increases the response of underlying neural mechanisms that encode brightness, which then causes neural adaptation to this added noise. Such behavior has been observed in theoretical models of brightness encoding based on integrate-and-fire neuron dynamics (Rudd & Brown, 1996, 1997a, 1997b), but other neural mechanisms might produce similar behavior. Neural noise adaptation of this type might also contribute to the classic Brücke–Bartley effect at these light levels as well as to the related Broca–Sulzer effect, in which flash brightness first increases, then decreases, as the duration of a steady flash is increased (Broca & Sulzer, 1902; van de Grind et al., 1973; White, Collins, & Rinalducci, 1976). 
Whatever its cause, the Broca–Sulzer effect probably contributed to the apparent delay to adapt observed in Experiment 4 because any initial brightness increase that is produced by a luminance increase would have to be overcome before square-root gain control, as defined by Equation 4, could manifest. A similar transient darkness enhancement follows a luminance decrease (Glad & Magnussen, 1972; White, Irvin, & Williams, 1980) and thus would be expected to delay the onset of the long-run brightness gain increase associated with dark adaptation. 
Adding full-field flicker to the adapting field as a means to manipulate the photon noise level
In Experiments 1, 2, and 3, we added random full-field flicker to the adapting field to manipulate the standard deviation of the photon noise from that field independently of the mean field intensity. The noise produced by this added random full-field flicker differs from the photon noise produced by a field of fixed intensity in that it generates spatial correlations in mean intensity across the adapting field, as well as temporal correlations within the 10-ms intervals between updates in the flicker noise. It also increases the level of the spatiotemporally uncorrelated Poisson photon fluctuations. We used this method to manipulate the photon noise level because our apparatus was not set up to produce random dot or scintillation noise. 
However, because the field with added flicker produced a very sparse retinal photon rain on the retina, it did produce scintillation noise on a fine scale. Furthermore, the temporal contrast of the full-field flicker was only 36%, and it was spectrally flat below the 50-Hz cutoff. This is not a strong visual stimulus, and the observers reported that it was difficult to see the flicker at any but the highest mean adapting intensity of 1 R*/rod/s. Thus, we are confident that were not studying adaptation to a flicker that drove the visual system strongly, which is the usual case in studies of flicker adaptation. 
What our results directly demonstrate is that there exists a mechanism whose gain adapts to the level of this combined stimulus noise, independent of the mean field intensity. The results of Experiments 3 and 4 together indicate that the time required for this mechanism to recover from flicker adaptation is about the same as the time required for complete adaptation to changes in mean intensity within the square-root law regime of human rod vision. Thus, a parsimonious account of the entire pattern of results is that the same noise gain mechanism that adapts to the field with added flicker adapts to changes in the standard deviation of the photon noise from the adapting field when no flicker is added to the field. This hypothesis accounts for both the square-root gain changes produced when the intensity of a nonflickering adapting field is changed and the similarities in the time courses of adaptation to changes in the mean level and recovery from flicker adaptation. Although we did not measure the time required to adapt to the onset of full-field flicker, our working hypothesis is that this time is on the same order as the time required to recover from adaptation to full-field flicker (i.e., about 100 s). Further experiments would be required to test this assumption. 
Comparison of the present results to data on retinal light adaptation in the macaque
The neural mechanisms that underlie the brightness phenomena studied here are not presently understood, but recent studies of neural gain control and noise in the macaque retina are relevant. Ala-Laurila and Rieke (2014) found that the spike rate variance of macaque ON parasol ganglion cells increases as a linear function of the mean light level at the low end of the range of adapting levels investigated here (0.010–0.015 R*/rod-s). Their findings are consistent with an absence of retinal light adaptation at these low light levels, but they also suggest that photon noise may get through to higher stages of the monocular pathway, at least at the low end of the square-root law adaptation range. Thus, in principle, photon noise could drive neural adaptation at a postreceptor processing stage. 
Schwartz and Rieke (2013) investigated the quantitative properties of light adaptation in macaque ON parasol ganglion cells over the same range of light levels investigated in the present study. At the lowest adapting levels within this range, they found further evidence for the increase in the ON parasol spike rate noise reported by Ala-Laurila and Rieke (2014). At higher light levels, the variance in the spike rate noise was approximately independent of adapting level, as would be expected if the dominant noise source was a fixed internal neural noise or, alternatively, if a retinal square-root gain control exactly counteracted the effects of intensity-dependent changes in the photon noise from the adapting field prior to the generation of ganglion cell spikes. Within the luminance range over which the spike rate variability was constant, the gain of both the spike output and input current to the ON parasols exhibited Weber adaptation rather than square-root adaptation. 
What are we to make of this apparent discrepancy in the quantitative properties of ON parasol and psychophysical adaptation (i.e., Weber vs. square-root) at these light levels? One possibility is that the ON parasol ganglion cells do not form the basis of the neural signal on which brightness perception is based in rod-mediated vision. A second is that the functional architecture of the macaque and human retinas differ in some fundamental way. A third—and perhaps most likely—possibility is that human observers exhibit square-root adaptation, Weber adaptation, or an intermediate form of adaptation, depending on the stimulus properties other than adapting intensity (see below). Thus, the conditions of the physiological recordings may have somehow favored Weber adaptation over square-root adaptation. 
In the toad, Weber gain control in ganglion cells is operative only at the high end of the rod range. Square-root gain control determines the ganglion cell gain within a midrange of rod-mediated adapting intensities. Weberian ganglion cell gain is produced only at high adapting intensities for which the rods themselves exhibit Weber gain control (Donner et al., 1990). In human psychophysics, the situation seems to be fundamentally different. Human rods do not exhibit Weber gain control at scotopic intensities, and Weber or near-Weber threshold behavior is exhibited only for test stimuli that are considerably larger and of longer duration than the ones used here (and in the macaque retinal gain experiments). 
Table 5 summarizes the TVI slopes (slopes of log–log plots of incremental threshold vs. adapting intensity) obtained in a number of previous psychophysical studies of visual threshold. The square-root threshold law corresponds to a slope of 0.5 and Weber's law to a slope of 1. The data in the table shows that the TVI slope varies between these two values, primarily as a function of test diameter and, to a lesser extent, test duration. TVI slope data from a few additional studies are tabulated in Barlow (1957, table 1) and follow the same trend as those shown in Table 5, supporting the conclusion that Weber's law holds approximately only for test stimuli much larger than the ones that we used in our experiments. 
Table 5
 
TVI slopes measured with tests of various sizes and durations in previous studies. Note: Sharpe et al. (1993) denotes Sharpe, Stockman, Fach, & Markstahler (1993).
Table 5
 
TVI slopes measured with tests of various sizes and durations in previous studies. Note: Sharpe et al. (1993) denotes Sharpe, Stockman, Fach, & Markstahler (1993).
Unlike the situation with the toad, there is no noticeable tendency in human psychophysics for the TVI slope to increase at higher adapting levels within the domain of rod-mediated vision. For example, the slopes of 0.75–0.8 measured by Sharpe, Stockman et al. (1993) for 6° stimuli are constant over several log units and do not tend to approach slopes of 1.0 at high luminances. Furthermore, TVI slope does not appear to depend critically on the intensity of the test flash itself (Chen, MacLeod, & Stockman, 1987; Sharpe, Stockman, et al., 1993). In our Experiments 1 and 2, the square-root law held at threshold for flash intensities as low as 0.85 R*/rod/s and for brightness matches for intensities as high as 63.2 R*/rod/s. 
Of course, the quantitative law that governs threshold for a stimulus of a given size and duration does not necessarily have to be the same law that governs the systems-level gain for a stimulus of the same size and duration. Nevertheless, in the few cases for which threshold and brightness have both been collected in the same observer, the slope that governs brightness (i.e., gain) has always been found to be the same as the one that governs threshold. This is true of the present study; it is true of the square-root brightness and threshold study of Brown and Rudd (1998); and—most importantly for purposes of generalization—it is true of a third study by Sharpe, Whittle, and Norby (1993), in which threshold and haploscopic brightness were both measured as a function of adapting luminance with 1.85° targets (having durations of 50 and 200 ms). In the study by Sharpe, Whittle, and Norby (1993), the slopes of log–log plots of threshold and brightness versus adapting intensity were both found to equal 0.69. 
Comparison of our results with the psychophysical results of Krauskopf and Reeves (1980) and Reeves and Grayhem (2016)
Krauskopf and Reeves (1980) measured increment thresholds with a large, 200-ms test, first on a 491-nm field, then again 400 ms after the field was turned off. The thresholds measured on the field roughly followed Weber's law (TVI slopes of about 1 or a bit less), whereas the thresholds measured after the field were removed roughly followed the square-root law. Reeves and Grayhem (2016) replicated this result and further showed that thresholds measured with a small, 5-arcmin test roughly follow the square-root law when measured on a field, but fall to a level that is independent of the adapting intensity after the field is extinguished. 
Reeves and his colleagues interpreted the 0.5 log unit drop in threshold that occurred whenever the adapting field was extinguished to be the result of removing photon noise from the field. This idea is consistent both with the deVries–Rose theory and our physiologically motivated threshold model (Equation 5). We agree that the 0.5 log unit threshold drop is probably the result of the removal of the photon noise. But, in light of our current findings, it is natural to ask why threshold falls rapidly when the adapting intensity changes while the brightness gain—as measured in our Experiment 4—changes more slowly. One idea is that there is something fundamentally different about changing the magnitude of the adapting intensity within the square-root law region of the threshold curve versus extinguishing the adapting field altogether. Another is suggested by our threshold model. 
According to Equation 5, gain and threshold can exhibit a different functional dependence on the adapting level as long as the noise that limits threshold is dominated by noise sources arising prior to the gain control stage. In that case, the same gain factor that controls the neural response to the flash also controls the noise, and the two effects cancel in the expression for the observer's threshold signal-to-noise ratio. Thus, the results of Reeves and Grayhem (2016) can be reconciled with ours as long as the pre-gain noise dominates the post-gain noise even when the noise from the field is removed (i.e., when I = 0 in Equation 5). 
Krauskopf and Reeves (1980) further suggested that the near-Weber thresholds that they measured on the field were due to a combination of photon noise that accounts for one half of the Weber slope of 1 on a log–log plot and an additional square-root gain control that accounts for the other half and whose gain changes more slowly than 400 ms. They proposed that this slow square-root gain control explains the square-root threshold law for large targets that they measured after the field was turned off. 
One might be tempted to identify this slow square-root gain control with the one that we studied here. But we think that it is unlikely that Weber's law results from a sum (in log units) of one square-root factor that depends on photon noise and a second square-root gain factor. First, it is hard to see how such a theory would account for the slopes intermediate between one half and 1 that are obtained with a test flash of intermediate size (Table 5). Second, as explained above, any gain factor will cancel out of the expression for visual threshold as long as pre-gain noise dominates post-gain noise, as it must for the Krauskopf–Reeves explanation of the threshold drop that is observed when the adapting field is extinguished to be viable. 
We think it is more likely that greater-than-square-root slopes result from a decrease in the efficiency with which large targets are spatially integrated by the neural mechanisms underlying detection as adapting level is increased. This could be due either to a shrinkage of the target integration area as the adapting level is increased (Barlow, 1958; Sharpe, Stockman et al., 1993; Sharpe, Whittle et al., 1993) or to the increased presence of subtractive inhibition at higher adapting intensities (Adelson, 1982; Geisler, 1983). These two potential explanations are not mutually exclusive. 
It is clear from this brief literature survey that we have an incomplete understanding of the factors that produce square-root versus Weber adaptation in both human and macaque rod vision. This precludes trying to directly connect mechanistic studies of adaptation in the macaque retina with human psychophysical studies. In what follows, we consider two theoretical questions that arise on the assumption that square-root adaptation occurs within the retina in primates. Our discussion of these points is necessarily speculative given the present knowledge of the underlying physiology. The first concerns the function of square-root law adaptation, and the second concerns its possible retinal locus
Functional significance of square-root gain control
As a general rule, the visual systems of both vertebrates and invertebrates seem to have evolved a strategy for light adaptation in which higher-level neural mechanisms adapt at lower light levels than do mechanisms located earlier in the visual processing stream. In primate rod vision, this strategy makes sense because high-level mechanisms integrate over larger spatial and temporal rod pools than do low-level mechanisms; thus they receive more reliable input in sparse photic environments. 
Weber adaptation occurs in individual primate rods at adapting intensities above 5–10 R*/rod/s (Schneeweis & Schnapf, 2000). At these light levels, Weber adaptation helps to guard against saturation in both individual rods and postreceptor mechanisms. However, there remains a danger that mechanisms that integrate the outputs of many rods over space and time could saturate at light levels below 5–10 R*/rod/s unless some kind of additional postreceptoral gain control operates on the output of the rod pool. One such mechanism is the Weber gain control that is revealed with large test flashes in human psychophysics and that appears to dominate ganglion cell recordings; another is square-root adaptation. 
Fully dark-adapted human observers are sufficiently light-sensitive to detect test flashes producing only a few photon absorptions at the observer's retina (Hecht, Shlaer, & Pirenne, 1942). Thus, in the dark—where neither Weber nor square-root gain control operates—the ganglion cell must generate approximately one spike or more per absorbed photon within the pool of rods from which it receives (indirect) input (Ala-Laurila & Rieke, 2014; Barlow, Levick, & Yoon, 1971). In the absence of any gain reduction, a mean light level of 0.01 R*/rod/s would produce a firing rate of at least 10 Hz, and an adapting field of 1 R*/rod/s would produce a 1-kHz firing rate. This is clearly not feasible. How then should gain be controlled? 
A square-root gain control operating on the rod pool prior to the ganglion cell spike generation mechanism would have the right quantitative properties to hold constant the standard deviation of the fluctuations in the ganglion cell input current when adapting luminance is increased. Such a mechanism could bridge light levels at which noise in the rod pool is dominated by spontaneous thermal isomerizations in the rods and those at which individual rods begin to light-adapt. A square-root gain control located late in the rod circuitry would both keep constant the probability of generating a spurious, photon-noise induced, ganglion cell spike and normalize the retinal output in such a way that any given ganglion cell spike rate would always correspond to a fixed signal-to-noise ratio for detection (Rudd & Brown, 1996). Thus, it makes sense that square-root adaptation would have evolved in a retina that is sensitive enough to pass signals from individual photons in the dark and in which detection reliability was a prime consideration motivating evolutionary design. 
The Weber adaptation that occurs in individual rod photoreceptors at high levels—and at least approximately in human and macaque under some stimulus conditions—likely serves an altogether different visual function. Weber adaptation is a stronger form of adaptation that compresses the very large dynamic range of potential environmental luminances into the much smaller range of luminance contrasts. The dynamic range of luminance contrast (i.e., the Weber fraction) is only half as large, in log units, as that of a neural signal whose gain is regulated by square-root law adaptation. The recoding of raw luminance into Weber contrast results in a neural representation that is well suited for maintaining lightness constancy—stable encoding of the reflectances of surfaces in the visual environment under changes in illumination level. Whereas changes in the ambient illumination of a visual scene dramatically alter the point-wise luminances in the retinal image, they leave local Weber ratios unaffected. Thus, a visual system that exhibits Weber gain control will transform the raw luminance input into a neural code that remains stable under the challenge of frequently encountered ambient changes in environmental illumination: a situation that that also pertains in cone vision. 
Square-root adaptation lacks this critical property of preserving the visual response under changes in illumination level. The quantitative differences in the two types of adaptation are consistent with the interpretation that square-root and Weber adaptation have fundamentally different purposes. Under the conditions in which square-root law adaptation occurs, the need for reliable detection apparently trumps the need for lightness constancy. When environmental objects are small, and hence reflect only a small number of photons to the observer's retina, their reliable detection is the visual system's prime consideration. As they become larger, the system can afford to reduce neural gain in order to achieve the fixed Weber ratio with changing illumination level required for lightness constancy. The situation is somewhat analogous to the trade-off that allows only achromatic vision at low luminance but trichromatic vision at higher luminances. In both cases, more advanced perceptual capabilities, such as constancy and color vision, become feasible only when the photon flux is sufficient to produce a sufficiently reliable input signal. 
Limitations on the neural locus of adaptation to photon noise imposed by the need for reliable gain settings
The extreme sparseness of the photon signals encountered at the low adapting luminances studied here sets fundamental constraints on the neural level at which retinal gain changes might be reliably controlled by changes in the standard deviation of the photon absorptions occurring within a photoreceptor pool. A rod bipolar cell—the first retinal processing stage at which rod signals are spatially pooled—integrates over a pool of about 20 rods. At a mean adapting intensity of 0.01 R*/rod/s—roughly the light level at which square-root gain control kicks in—an individual bipolar cell receives ∼20 photon-induced rod responses over an adaptation time of 100 s. The standard deviation of these photon-induced events is 4.5 R*/rod/s. This number is probably too small to support reliable adaptation to quantal fluctuation noise in the rod pool, although adaptation to the mean light level could be reliably achieved at the level of the bipolars. In other words, if adaptation to noise occurred as early as the bipolar cells, the gain would bounce around too wildly, and the adaptation would fail to serve the presumed function of sufficiently stabilizing the post-bipolar network noise. 
At the next stage of spatial pooling, individual AII amacrine cells pool over roughly 500 rods. Over an adaptation time of 100 s, an AII amacrine thus “sees” about 500 photon-induced rod responses. The standard deviation of these quantal responses would be about 20, a number that would probably be sufficient to support reliable adaptation to photon noise. 
At the next stage of pooling, a parasol ganglion cell integrates over several thousand rods. At an adapting light level of 0.01 R*/rod/s, an individual ganglion cell receives input from 2000–4000 photon-induced rod responses over a 100-s window. The standard deviation of the photon fluctuation noise, as seen by a parasol cell over this time window, is about 50. This number is clearly sufficient to produce reliable adaptation to photon noise. Thus, for reliable adaptation, a square-root gain control mechanism that adapts to the level of photon-induced noise in its input would have to be located at least as late in the retinal rod network as the AII amacrines. 
In light of these theoretical considerations, an additional finding of the Schwartz and Rieke (2013) study in macaques, not yet discussed, is intriguing. Although the time courses that those authors observed for the Weber gain changes in AII amacrines, ON parasol input currents, and parasol spike outputs were all quite rapid (completing in <1 s), the flash responses in the ganglion cell input current and spike output were both superimposed on a baseline level that exhibited a much slower time course (Schwartz & Rieke, 2013, their figure 5, supplement 1). We fit this slow change in baseline with an exponential model and estimated the characteristic time of the exponential baseline shift be ∼100 s. This time is quite similar to the time scales associated with square-root adaptation in Experiments 3 and 4 of the current study. Thus, although Weber adaptation dominated under the conditions of these physiological experiments, the data from these same experiments also suggest the presence of other forms of adaptation that could contribute to the perceptual effects studied here. How this would happen is unclear at present, and a plausible neural model of our findings would have to explain both the linear time course and initial, sluggish phase of psychophysical square-root light adaptation. 
Acknowledgments
The authors wish to thank Bill Geisler, Adam Reeves, and an anonymous reviewer for helpful comments on earlier drafts of the paper; Huynh Chheang Chhor, Mahdi Hedayat, Abdullah Rahmani, Jessica Rowlan, and Max Schreiber for serving as psychophysical observers; and Eric Martinson, Paul Newman, and Bryan Venema for technical assistance. 
This work was funded by a Howard Hughes Medical Institute Investigator Award to Dr. Fred Rieke. 
Commercial relationships: none. 
Corresponding author: Michael E. Rudd. 
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Figure 1
 
The dichoptic stimuli used in the experiments as seen by the observer's left eye, right eye, and in binocular view. The regions outlined by the colored ovals illustrate the display elements as projected to the observer's left and right retinas. In binocular view, the display appeared as a composite of these elements, surrounded by darkness and aligned by the observer's binocular fusion of the red fixation dot.
Figure 1
 
The dichoptic stimuli used in the experiments as seen by the observer's left eye, right eye, and in binocular view. The regions outlined by the colored ovals illustrate the display elements as projected to the observer's left and right retinas. In binocular view, the display appeared as a composite of these elements, surrounded by darkness and aligned by the observer's binocular fusion of the red fixation dot.
Figure 2
 
The brightness matching method used in Experiment 1. The neural gain of a retinal patch in one of the observer's two eyes—the “test” eye—was manipulated by varying the photon statistics of the light from the large disk-shaped adapting field in that eye, including both the mean and variance of the photon count from the field. The brightness gain of the test eye was assessed by superimposing on the adapting field a small (0.55°), brief (10 ms) test flash of fixed intensity and finding the value of a simultaneous incremental flash of variable intensity presented on a steady field of fixed intensity 0.1 R*/rod/s to the observer's other eye—the “match” eye—that matched the test flash in brightness.
Figure 2
 
The brightness matching method used in Experiment 1. The neural gain of a retinal patch in one of the observer's two eyes—the “test” eye—was manipulated by varying the photon statistics of the light from the large disk-shaped adapting field in that eye, including both the mean and variance of the photon count from the field. The brightness gain of the test eye was assessed by superimposing on the adapting field a small (0.55°), brief (10 ms) test flash of fixed intensity and finding the value of a simultaneous incremental flash of variable intensity presented on a steady field of fixed intensity 0.1 R*/rod/s to the observer's other eye—the “match” eye—that matched the test flash in brightness.
Figure 3
 
Time course of the stimulus presentation in Experiments 1 and 2. Each experimental run consisted of two 8-s segments. The test and match flashes were presented simultaneously 0, 225, or 450 ms into the second (4 s) half of each segment. In the second segment (only), 3% contrast full-field flicker was added to the adapting field in the test eye but not to the adapting field in the match eye during the first half of the segment, and the flicker contrast was increased to 36% during the second half. The figure illustrates the flash delay (τ = 0, 225, or 450 ms) relative to the flicker contrast increase.
Figure 3
 
Time course of the stimulus presentation in Experiments 1 and 2. Each experimental run consisted of two 8-s segments. The test and match flashes were presented simultaneously 0, 225, or 450 ms into the second (4 s) half of each segment. In the second segment (only), 3% contrast full-field flicker was added to the adapting field in the test eye but not to the adapting field in the match eye during the first half of the segment, and the flicker contrast was increased to 36% during the second half. The figure illustrates the flash delay (τ = 0, 225, or 450 ms) relative to the flicker contrast increase.
Figure 4
 
Brightness matches for small, brief test flashes follow the square-root law. (A) Matches made on a 0.1 R*/rod/s adapting field of fixed intensity to a 20 R*/rod/s test flash presented on a nonflickering adapting field in the other—match—eye as a function of the adapting field intensity in the test eye. The matches for both observers fall on a line of slope −0.5 as predicted by the hypothesis that the monocular gain applied to the test flash varies inversely with the square root of adapting intensity. (B) Matches made under the same conditions, but with 36% contrast, random, full-field Gaussian flicker added to the adapting field in the test eye (only). The square-root matching law is still followed approximately, but small increases in the intensity of the matches were observed at the higher adapting field intensities relative to the matches made to test flashes presented on nonflickering fields. Thus, adding full-field flicker to the adapting field increased the test flash brightness by a small amount (see text for further details).
Figure 4
 
Brightness matches for small, brief test flashes follow the square-root law. (A) Matches made on a 0.1 R*/rod/s adapting field of fixed intensity to a 20 R*/rod/s test flash presented on a nonflickering adapting field in the other—match—eye as a function of the adapting field intensity in the test eye. The matches for both observers fall on a line of slope −0.5 as predicted by the hypothesis that the monocular gain applied to the test flash varies inversely with the square root of adapting intensity. (B) Matches made under the same conditions, but with 36% contrast, random, full-field Gaussian flicker added to the adapting field in the test eye (only). The square-root matching law is still followed approximately, but small increases in the intensity of the matches were observed at the higher adapting field intensities relative to the matches made to test flashes presented on nonflickering fields. Thus, adding full-field flicker to the adapting field increased the test flash brightness by a small amount (see text for further details).
Figure 5
 
Time course of recovery from adaptation to 36% binary-random full-field flicker. Proportion of trials in which a 20 R*/rod/s test flash presented on a steady 0.1 R*/rod/s adapting field in the test eye was judged to be brighter than a 20 R*/rod/s match flash, presented on a steady 0.1 R*/rod/s adapting field in the match eye, as a function of time following 6 min of adaptation to binary-random full-field flicker of mean intensity 0.1 R*/rod/s in the test eye. Immediately following the cessation of the random flicker in the test eye, flashes presented to that eye are judged to be brighter than flashes of equal intensity presented simultaneously to the other eye about 0% of the time, indicating that adaption to the full-field flicker reduced the monocular gain in the test eye. This gain reduction dissipated over a period of about 100 s for both observers.
Figure 5
 
Time course of recovery from adaptation to 36% binary-random full-field flicker. Proportion of trials in which a 20 R*/rod/s test flash presented on a steady 0.1 R*/rod/s adapting field in the test eye was judged to be brighter than a 20 R*/rod/s match flash, presented on a steady 0.1 R*/rod/s adapting field in the match eye, as a function of time following 6 min of adaptation to binary-random full-field flicker of mean intensity 0.1 R*/rod/s in the test eye. Immediately following the cessation of the random flicker in the test eye, flashes presented to that eye are judged to be brighter than flashes of equal intensity presented simultaneously to the other eye about 0% of the time, indicating that adaption to the full-field flicker reduced the monocular gain in the test eye. This gain reduction dissipated over a period of about 100 s for both observers.
Figure 6
 
Psychometric functions for light and dark adaptation in the square-root law regime. (A) Light adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance had recently been increased from 0.1 to 0.4 R*/rod/s, was judged to be brighter than a 15 R*/rod/s standard flash, presented on an unchanged 0.1 R*/rod/s adapting field in the other (match) eye, as a function of time following the adapting field increase in the test eye. The 15 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the match flash 50% of the time. The time at which this match occurs is the time at which light adaptation to the new test adapting field is 50% complete because the match standard midway intensity between the 20 R*/rod/s standard that matched the test flash before the adapting field in the test eye was increased and the 10 R*/rod/s standard that matches the test flash when square-root law adaptation is complete. A match to the 15 R*/rod/s standard occurred at flash delays of 100 and 108 s for the two experimental observers. (B) Psychometric functions like the one shown in panel A produced by subject AR with each of seven match standards that were chosen so that they would match the test in brightness when light adaptation was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete (see Experiment 4, Methods for details). (C) Same as panel B but for subject MH. (D) Dark adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance was decreased from 0.1 to 0.25 R*/rod/s, was judged to be brighter than a 30 R*/rod/s standard match flash, presented on an unchanged 0.1 R*/rod/s adapting field in the observer's other eye, as a function of time following the decrease in adapting field intensity. The 30 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the standard 50% of the time. The time at which this occurs marks the 50% adaptation time to the changed adapting field intensity because the 30 R*/rod/s match standard is midway in intensity between the 20 R*/rod/s standard that matches the test flash before the decrease in adapting field intensity and the 40 R*/rod/s match standard that would match the test flash after square-root adaptation to the decreased adapting field intensity was complete. A match to the 30 R*/rod/s standard occurred at flash delays of 117 and 121 s following the adapting field intensity decrease for the two experimental observers. (E) Functions like the one shown in panel D produced by subject AR with seven match standards, which where chosen so that they would match the test in brightness when adaptation to the decreased field intensity was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete. (F) Same as panel E but for subject MH. Proportion of “brighter than” responses have been binned within 10-s intervals in all figure panels.
Figure 6
 
Psychometric functions for light and dark adaptation in the square-root law regime. (A) Light adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance had recently been increased from 0.1 to 0.4 R*/rod/s, was judged to be brighter than a 15 R*/rod/s standard flash, presented on an unchanged 0.1 R*/rod/s adapting field in the other (match) eye, as a function of time following the adapting field increase in the test eye. The 15 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the match flash 50% of the time. The time at which this match occurs is the time at which light adaptation to the new test adapting field is 50% complete because the match standard midway intensity between the 20 R*/rod/s standard that matched the test flash before the adapting field in the test eye was increased and the 10 R*/rod/s standard that matches the test flash when square-root law adaptation is complete. A match to the 15 R*/rod/s standard occurred at flash delays of 100 and 108 s for the two experimental observers. (B) Psychometric functions like the one shown in panel A produced by subject AR with each of seven match standards that were chosen so that they would match the test in brightness when light adaptation was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete (see Experiment 4, Methods for details). (C) Same as panel B but for subject MH. (D) Dark adaptation: Proportion of trials in which a 20 R*/rod/s test flash, presented on an adapting field whose luminance was decreased from 0.1 to 0.25 R*/rod/s, was judged to be brighter than a 30 R*/rod/s standard match flash, presented on an unchanged 0.1 R*/rod/s adapting field in the observer's other eye, as a function of time following the decrease in adapting field intensity. The 30 R*/rod/s standard matches the test flash in brightness when the test flash is judged to be brighter than the standard 50% of the time. The time at which this occurs marks the 50% adaptation time to the changed adapting field intensity because the 30 R*/rod/s match standard is midway in intensity between the 20 R*/rod/s standard that matches the test flash before the decrease in adapting field intensity and the 40 R*/rod/s match standard that would match the test flash after square-root adaptation to the decreased adapting field intensity was complete. A match to the 30 R*/rod/s standard occurred at flash delays of 117 and 121 s following the adapting field intensity decrease for the two experimental observers. (E) Functions like the one shown in panel D produced by subject AR with seven match standards, which where chosen so that they would match the test in brightness when adaptation to the decreased field intensity was either 5%, 20%, 35%, 50%, 65%, 80%, or 95% of the way complete. (F) Same as panel E but for subject MH. Proportion of “brighter than” responses have been binned within 10-s intervals in all figure panels.
Figure 7
 
Time course of light and dark adaptation in the square-root law regime. Value of the standard flash in the match eye that matched in brightness a 20 R*/rod/s flash presented to the test eye as a function of time following an increase or decrease in adapting field intensity in the test eye for each of the two observers. (A) Data from subject AR. (B) Data from subject MH. Colored lines and R2 values correspond to least-squares linear regression models of the data (maroon = light adaptation, blue = dark adaptation). Extrapolation of the linear regression models to longer times predicts that square-root light adaptation would complete at the time at which the light adaptation function intersects with the lower edge of the graph (match standard = 10 R*/rod/s), and square-root dark adaptation would complete at the time at which the dark adaptation function intersects with the upper edge of the graph (match standard = 40 R*/rod/s) because the match standards corresponding to these boundaries were established in Experiment 1 to match the test flash after long-run square-root adaptation.
Figure 7
 
Time course of light and dark adaptation in the square-root law regime. Value of the standard flash in the match eye that matched in brightness a 20 R*/rod/s flash presented to the test eye as a function of time following an increase or decrease in adapting field intensity in the test eye for each of the two observers. (A) Data from subject AR. (B) Data from subject MH. Colored lines and R2 values correspond to least-squares linear regression models of the data (maroon = light adaptation, blue = dark adaptation). Extrapolation of the linear regression models to longer times predicts that square-root light adaptation would complete at the time at which the light adaptation function intersects with the lower edge of the graph (match standard = 10 R*/rod/s), and square-root dark adaptation would complete at the time at which the dark adaptation function intersects with the upper edge of the graph (match standard = 40 R*/rod/s) because the match standards corresponding to these boundaries were established in Experiment 1 to match the test flash after long-run square-root adaptation.
Table 1
 
Statistical significance and effect size for the brightness enhancement of a 20 R*/rod/s test produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed t-tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Table 1
 
Statistical significance and effect size for the brightness enhancement of a 20 R*/rod/s test produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed t-tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Table 2
 
Matches made on a −1 log unit adapting field to 0.301 and 0.801 log unit test flashes on a −2 log unit field with the implied brightness matching slopes.
Table 2
 
Matches made on a −1 log unit adapting field to 0.301 and 0.801 log unit test flashes on a −2 log unit field with the implied brightness matching slopes.
Table 3
 
Statistical significance and effect size for the brightness enhancement of 2.0 and 6.325 R*/rod/s tests produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Table 3
 
Statistical significance and effect size for the brightness enhancement of 2.0 and 6.325 R*/rod/s tests produced by adding full-field random flicker to the adapting field. Notes: p-values are for one-tailed tests, degrees of freedom in parentheses. ± indicates standard error of the mean. See text for definition of effect size.
Table 4
 
Increment thresholds measured on steady fields of −2, −1, and 0 log units (0.01, 0.1. and 1.0 R*/rod/s) together with the corresponding ds for flash detection on adapting fields with or without added full-field flicker. Notes: p-values denote significance of d′ reduction when full-field flicker was added to the adapting field (see text for further details).
Table 4
 
Increment thresholds measured on steady fields of −2, −1, and 0 log units (0.01, 0.1. and 1.0 R*/rod/s) together with the corresponding ds for flash detection on adapting fields with or without added full-field flicker. Notes: p-values denote significance of d′ reduction when full-field flicker was added to the adapting field (see text for further details).
Table 5
 
TVI slopes measured with tests of various sizes and durations in previous studies. Note: Sharpe et al. (1993) denotes Sharpe, Stockman, Fach, & Markstahler (1993).
Table 5
 
TVI slopes measured with tests of various sizes and durations in previous studies. Note: Sharpe et al. (1993) denotes Sharpe, Stockman, Fach, & Markstahler (1993).
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