November 2019
Volume 19, Issue 13
Open Access
Article  |   November 2019
Dissociation of perceived size and perceived strength in the scintillating grid illusion
Author Affiliations
  • Toyomi Matsuno
    Faculty of Economics, Hosei University, Machida, Tokyo, Japan
    toyomi.matsuno@hosei.ac.jp
  • Yuka Sato
    Faculty of Economics, Hosei University, Machida, Tokyo, Japan
Journal of Vision November 2019, Vol.19, 15. doi:https://doi.org/10.1167/19.13.15
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      Toyomi Matsuno, Yuka Sato; Dissociation of perceived size and perceived strength in the scintillating grid illusion. Journal of Vision 2019;19(13):15. doi: https://doi.org/10.1167/19.13.15.

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Abstract

The scintillating grid illusion is a geometric visual illusion where illusory dark spots are perceived on white circular patches that are located at the intersections of a grid pattern. Previous studies have measured the perceived strength of the illusory spots by varying stimulus and viewing conditions, to elucidate the mechanisms underlying the illusion; however, findings remain inconclusive. In the present study, we measured the perceived size of the illusory spots, in addition to their perceived strength, by varying the size of the geometric components and the orientation of the stimulus, to further investigate the mechanism behind this illusion. We found that the dependency of perceived spot size on the change of the stimulus parameters is dissociated from that of the perceived spot strength. The perceived size of the illusory spots linearly correlated with the size of the white patches and was less dependent on the width of the grid bars and the orientation of the stimulus, while the perceived strength was characterized by a quadratic relationship with patch size and bar width and was monotonically modulated depending on the orientation of the stimuli. These results suggest that different factors separately constrain size and strength within the processes that elicit the illusory spots. We propose that the mechanism underlying the scintillating grid illusion is based on the interruption of the surface formation process of white patches by the interference of the orientation signals of gray bars.

Introduction
The scintillating grid illusion is a visual phenomenon where illusory black spots are perceived on white round patches located at the intersections of a black grid pattern (Schrauf, Lingelbach, & Wist, 1997). The scintillating grid illusion is one of the variants of the Hermann grid illusion (Hermann, 1870) and is characterized by both shared and distinctive perceptual features, compared to the original Hermann grid illusion (Schrauf et al., 1997; Figure 1). As for the Hermann grid illusion, the illusory black spots appear at the intersections of the grids in our peripheral vision and diminish at fixated locations. Scale invariance is another shared feature. The illusion can be observed with a wide range of stimulus sizes. In addition, a distinctive feature of the scintillating grid illusion is the scintillating appearance of the illusory spots, as the name suggests. The spots appear and disappear in an irregular pattern and are perceived with higher subjective contrast and a more salient contour than the dimly spread illusory spots in the typical Hermann grid illusion. 
Figure 1
 
(a) Hermann grid illusion; (b) scintillating grid illusion.
Figure 1
 
(a) Hermann grid illusion; (b) scintillating grid illusion.
Previous psychophysical studies have revealed four other characteristics of the scintillating grid illusion. First, the strength of the illusion largely depends on the straightness and continuity of the grid bars. Geier, Séra, and Bernáth (2004) and Levine and McAnany (2008) tested stimuli of sinusoidally curved grid lines and demonstrated that the illusion was diminished or disappeared with a small amplitude of the curvature. Qian, Kawabe, Yamada, and Miura (2012) also reported that the orientation information of the grid bars plays an important role in the illusion, demonstrating that the illusion is weakened when spatial gaps or offsets between the bars were introduced. Second, the scintillating grid illusion is tuned to a specific spatial configuration of the patches, grid bars, and their background, suggesting that the illusion depends on the spatial interaction of such stimulus components. Schrauf et al. (1997) showed that the strength of the illusory percept depends on the size ratio and luminance contrast among bars, patches, and their background. Qian, Yamada, Kawabe, and Miura (2009) studied the illusion by varying the contour shape of white patches and the orientation of the grid bars, and they reported that the relative orientation between the grid and patch contour is critical. The spatial alignment of the white patches and grid bars is also critical to elicit the illusion. Schrauf and Spillmann (2000) found that displacement of the white disk from the grid intersections weakens the illusion, while stereoscopic integration of the misaligned stimuli into an aligned stimulus partially counteracts the reduction. Third, the scintillating grid illusion is optimally induced with a transient stimulation of the retina. Schrauf, Wist, and Ehrenstein (2000) investigated the temporal properties of the illusion and revealed that the rated strength was maximum at exposure durations of about 200 ms and decreased at shorter durations. Fourth, attentional allocation diminishes the visibility of the illusion. VanRullen and Dong (2003) manipulated the attentional focus using spatial precueing and dual task paradigms and found that not only eye fixation but even covert attention diminishes the scintillation at the surrounding intersections. 
Some mechanisms have been proposed to explain the above-mentioned characteristics of the scintillating grid illusion, following the debates on the Hermann grid illusion, but our understanding remains incomplete. Early studies on the Hermann grid illusion hypothesized that the lateral inhibition of retinal ganglion cells that have center-surround antagonistic receptive field organization are involved in the appearance of the illusory smudges at the grid intersections (Baumgartner, 1960; Spillmann & Levine, 1971). According to this theory, the illusion is a result of the differences in responses of the ganglion cells whose receptive field centers fall onto the intersections, compared to those whose receptive field centers fall onto the nonintersection parts of the grid. Schrauf et al. (1997) suggested that lateral inhibition mechanisms are also involved in the scintillating grid illusion, based on their findings that shared stimulus properties elicit the two illusions. In a similar vein, Yu and Choe (2006) proposed a neural model of the scintillating grid illusion based on retinal inhibitory interactions, that involve recurrent processes, and simulated the scintillating effect with this model. Retinal mechanisms, however, do not explain some characteristics of the two grid illusions. For example, Schiller and Carvey (2005) remarked that this explanation is neither consistent with the orientation dependency nor the scale invariance of the Hermann grid illusion. As an alternative theory, they proposed that the illusion is mediated by the orientation-selective neural mechanisms of early visual cortical processing. Schrauf et al. (1997, 2000) also suggested that cortical mechanisms contribute to the scintillating grid illusion, in addition to retinal processing. Qian et al. (2012) suggested that the activity of orientation-selective cells in the early visual cortex in response to the straight bars of the grid would be a plausible underlying mechanism explaining their psychophysical data on the scintillating grid illusion. They claimed that the illusory spots were percepts elicited by the weak neural representations of gray bars. 
In this study, by measuring the perceived size of the illusory spot, we aimed to further investigate the properties of the scintillating grid illusion and add new insights to the ongoing discussion. All previous experimental studies on the scintillating grid illusion mentioned above have focused on the strength of the illusory percepts in order to characterize the phenomenon. Therefore, participants of such studies evaluated the visibility of the illusion in the dimension of subjective luminance contrast, using methods such as cancellation techniques or magnitude rating. On the other hand, no studies have directly focused on the perceived shape or size of the illusory spots. The illusory spots of the scintillating grid illusion are perceived with a salient circular contour that does not coincide with the edges of the stimulus components. This disassociation indicates that the contour size of the illusory spots is constrained by some undefined mechanism. In studies of visual illusions, not only the measurement of perceived brightness but also the investigation of illusory contour and surface formation processes have contributed to our understanding of the visual system (Murray & Herrmann, 2013; Spillmann & Dresp, 1995). The measurement of the perceived size of the illusory spots in the scintillating grid illusion and the determinant factors would thus help to further elucidate the mechanisms underlying grid illusions and visual perception in general. We manipulated three spatial properties of the scintillating grid stimulus in our experiments: patch size, grid bar width, and stimulus orientation. If the contour of the illusory spots was formed by relying on some spatial properties of stimulus components, changes in the size of either white patches or grid bars or both would affect the perceived size of the illusory spots. In particular, suggestions that the grid illusion depends on the orientation signals of the grid bars (e.g., Qian et al., 2012; Schiller & Carvey, 2005) would predict that the location shift of bar edges expands and shrinks the size of the illusory spots. Alternatively, the size of the illusory spots could depend on the strength of the illusion, and a reduction of the illusory signals may simply constrain the border of the illusory spots. In that case, the stimulus parameter dependency of the perceived size of the illusory spots would be consistent with that of the perceived strength. Thereby, stimulus rotation that does not alter the physical size of the stimulus components but decreases the perceived strength of the illusion would shrink the perceived size of the illusory spots. 
We conducted six main experiments and one replication experiment. In the first three experiments, participants evaluated the perceived size of the illusory spots of the scintillating grid illusion while spatial properties of the stimuli, that is, patch size (Experiment 1), bar width (Experiment 2), and grid orientation (Experiment 3) were varied. In the other three main experiments, participants rated the perceived strength of the illusion; the stimuli were identical to the first three experiments, so that the spatial parameter dependencies of perceived strength could be compared to that of perceived size. Prior to these experiments, a pretest was conducted in which the perceived size of physically presented black circles, instead of illusory ones, was measured, to collect baseline data for perceived size and to familiarize observers with the size adjustment procedure. In addition to these experiments, a replication experiment of Experiments 1 to 3 was conducted with newly recruited participants (Experiment 7). In the replication experiment, the perceived size of the illusory spots was evaluated with variations of all three stimulus conditions (patch size, bar width, and grid orientation) and slightly modified probe stimuli. This experiment was conducted to assess the possible response biases caused by the experimental design and the properties of the probe stimuli in Experiments 1 to 3. 
Method
Participants
Twelve observers (five females and seven males) ranging in age from 19 to 37 years (mean age = 21.75, SD = 4.69) participated in Experiments 1 to 6. Two of the participants were authors. The others were undergraduate students at Hosei University who were naïve to the purpose of the experiments. Elimination of the authors' data did not alter the main results, and all the data were therefore merged for the analyses. The other 12 naïve observers (eight females and four males, 18 to 23 years old, mean age = 18.92, SD = 1.32) participated in the replication experiment (Experiment 7). The students received course credits for participation. All participants had normal or corrected-to-normal visual acuity. The experiments were conducted according to the guidelines outlined in the Declaration of Helsinki, and the experimental design was approved by the Research Ethics Committee of the Department of Economics, Hosei University (201702-01 for Experiments 1 to 6, 2019S05 for Experiment 7). Written informed consent was obtained from each participant before the experiments. 
Apparatus and stimuli
The stimuli were generated on an Intel Core i7-based computer and displayed on a 27-in. TFT monitor (EIZO, ColorEdge CG277) using custom-built software written in Visual C++. The stimulus images were displayed without antialiasing. The monitor resolution and the refresh rate were set to 2,560 × 1,440 pixels and 60 Hz. Participants observed the monitor at a viewing distance of 68 cm using a chin- and headrest in a dark experimental booth. Stimulus luminance was measured using a colorimeter (Konica-Minolta, CS-100A). A keyboard (Experiments 1 through 3) and a mouse (Experiments 4 through 6) were used as response devices. 
The scintillating grid stimulus consisted of eight horizontal and eight vertical gray bars on a black background, with a white disk at each intersection. The stimulus image subtended 17.6° × 17.6° of visual angle and was presented on a white screen. The center-to-center separation between adjacent bars was 1.97° of visual angle. The standard stimulus had a bar width of 0.394° and a patch diameter of 0.551°. The ratio between the bar width and the patch diameter of the standard stimulus was about 1:1.4, where the perceived strength of the illusion was thought to be maximized (Schrauf et al., 1997). The width (Experiments 1 and 4) and diameter (Experiments 2 and 5) of the stimulus increased and decreased around these values. The lower and upper limits of the patch diameter and grid bar width were determined by the results of a pilot study with two authors and one naïve participant, where the visible ranges of the illusion were measured using a method of adjustment (see Supplementary Files S1, S2 (raw data), and S3 (R and Stan scripts)). The orientation of the stimulus varied from 0° to 45° of rotation angle in Experiments 3 and 6 and was orthogonal in the other experiments. The luminance of the bars, disks, and the black background was approximately 10 cd/m2, 103 cd/m2, and 0 cd/m2, respectively. 
Procedure
In the main experiment, each participant completed four experimental sessions to measure the perceived size of the physical (pretest) or illusory spots (Experiments 1 through 3), and three sessions to measure the perceived strength of the illusion (Experiments 4 through 6). Experiments 1 to 3 preceded Experiments 4 to 6 for all the participants, while the order within each set of three experiments was varied among participants. Participants took a self-paced break of a few minutes after each session. After the first four sessions, participants took a break of at least 5 minutes, until they decided to restart the experiment. In the replication experiment (Experiment 7), each participant completed one experimental session. 
Participants were not required to fixate during the task, because the grid illusion is substantially reduced or abolished with steady fixation (Schrauf et al., 1997). No time limit was imposed on the participants for observing stimuli and giving responses. 
Measurement of the perceived size of the illusory spots:
The perceived size of the illusory spots was measured by using a method of adjustment. At the start of each trial, a test stimulus and a comparison stimulus were presented to the left and right of the center of the white screen (Figure 2). The comparison stimulus consisted of a square contour line of the same size as the outer boundary of the test stimulus, and 64 probe squares that had the same color as the black background of the scintillating grid stimulus and were located at points corresponding to the intersections of the grid stimulus. The initial size of the probe squares was randomly varied from 0.098° to 0.590° (5–30 pixels) across trials. Participants were required to adjust the size of the probe squares until the apparent size of the square was equal to the size of the illusory black spots. To accurately adjust the probe size with 1-pixel accuracy on the monitor, the probe stimuli in Experiments 1 to 3 were square-shaped, despite the circular shape of the illusory spots, thereby keeping the shape of the probe stimuli identical. All probe squares in a display had the same size. The left and right arrow keys of the keyboard were used to increase and decrease both the height and width of probe squares in units of 1 pixel. A no-response period of approximately 200 ms was inserted after each key press. When participants pushed the space bar twice in succession to terminate the adjustment period in a trial, the stimuli disappeared, and a 2000-ms intertrial-interval followed. 
Figure 2
 
An illustration of the scintillating grid stimulus and the probe squares used in the experiments.
Figure 2
 
An illustration of the scintillating grid stimulus and the probe squares used in the experiments.
Prior to the experiment, participants were instructed to repeat the following task until they confirmed the size match: looking around the center of the left test image, identifying the illusory spots, and then switching their gaze to the center of the right comparison stimulus to adjust the size of the probe square at a corresponding position in the visual field. When the illusory spot was invisible, they were instructed to continue to push the left arrow key until the size of the probe squares reached the minimum and it became invisible. Trials where a participant set the size of the probe squares to between 0 and 2 pixels (0.0°–0.039° of visual angle) in Experiments 1 to 3 were excluded from the subsequent data analyses, as these values deviated more than 3 SD from the mean of all trials (mean = 18.3 pixels, SD = 5.36). The extraordinarily small responses could be due to the stimuli being invisible or participants failing to minimize the probe stimulus size for invisible responses. Seven trials (2.9% of all trials) from three participants (P02, P07, and P09) in Experiment 1, eight trials (3.3% of all trials) from two participants (P07, P09) in Experiment 2, and one trial (0.3% of all trials, participant P09) in Experiment 3 were excluded from the analyses. 
Pretest
In the pretest, the perceived size of physical black spots was evaluated by matching them to the size of probe squares, for the purpose of collecting baseline data and to familiarize the participants with the adjustment procedure. The test stimulus was prepared by removing the grid bars from the scintillating grid stimulus described above and adding black circles (1.0 cd/m2) on white patches (Figure 3). The diameter of the white patches varied between 0.354°, 0.551°, and 0.748° of visual angle. The diameter of the black circles was set to be smaller than that of the white patches. Three diameters of black circles for each patch size were tested (0.197°, 0.256°, 0.315° for small white patches, 0.197°, 0.354°, 0.512° for medium white patches, and 0.197°, 0.453°, 0.709° for large white patches). 
Figure 3
 
Stimulus presented at the left side of the display in the pretest.
Figure 3
 
Stimulus presented at the left side of the display in the pretest.
Each observer participated in a practice session of nine trials and a test session of 27 trials. Practice trials, where each condition was tested once, were conducted before a test session. During a test session, each combination of the patch and spot diameters was tested three times. The order of trials for each stimulus condition was pseudorandomized. 
Experiment 1
Experiment 1 examined the effect of the patch size on the perceived size of the illusory spots. The diameter of the circular white patch of the scintillating grid stimulus varied between 0.354°, 0.453°, 0.551°, 0.650°, and 0.748° of visual angle. The bar width of the stimuli was fixed at 0.394° of visual angle, and the stimulus was not rotated (0°) within a session. Each observer participated in a practice session of five trials, in which each patch size was tested once, and a test session of 20 trials, in which each patch size was tested four times. During both practice and test sessions, the trials for each stimulus condition were pseudorandomly intermixed. 
Experiment 2
Experiment 2 examined the effect of bar width on the perceived size of the illusory spots. The width of the gray bars varied between 0.197°, 0.295°, 0.394°, 0.492°, and 0.590° of visual angle. The patch size was fixed at 0.551° of visual angle, and the stimulus was not rotated within a session. Each observer participated in a practice session of five trials, in which each bar width was tested once, and a test session of 20 trials, in which each condition was tested four times. During both practice and test sessions, the trials for each stimulus condition were pseudorandomly intermixed. 
Experiment 3
In Experiment 3, the rotation angle of the stimuli varied between 0° and 45°, rotated clockwise in 9° steps. The patch size and bar width were fixed at 0.551° and 0.394° of visual angle. In addition to the scintillating grid stimulus, the frame surrounding the probe squares and the positions of the probe squares were rotated corresponding to the stimulus. Each probe square remained upright, in order to assure accurate adjustment in units of 1 pixel, as in the other experiments (Figure 2). Each observer participated in a practice session consisting of six trials, in which each of the rotation angles was tested once, and a test session of 24 trials, in which each condition was tested four times. During both practice and test sessions, the trials for each stimulus condition were pseudorandomly intermixed. 
Rating of the illusion strength
Participants rated the strength of the illusory spots as in previous studies (e.g., Qian et al., 2012; Schrauf et al., 1997; Schrauf & Spillmann, 2000). At the start of each trial, a test stimulus appeared to the left of the white screen. After a 2000-ms interval, response scales with captions were additionally presented to the right of the screen (Figure 4). Participants were instructed to rate the strength of the illusion on a 7-point scale from 0 to 6, where 0 was the weakest and 6 was the strongest. When participants indicated their selection with a mouse click, the display turned blank and a 2000-ms intertrial interval followed. The stimuli and the number of trials in Experiments 4 to 6 were identical to those in the corresponding experiments of the perceived size measurement. 
Figure 4
 
The display used in Experiments 3, 4, and 5. The rating circles were presented with descriptions that instructed participants to rate the strength of the illusion on a 7-point scale ranging from 0 (weak) to 6 (strong) in Japanese.
Figure 4
 
The display used in Experiments 3, 4, and 5. The rating circles were presented with descriptions that instructed participants to rate the strength of the illusion on a 7-point scale ranging from 0 (weak) to 6 (strong) in Japanese.
Replication experiment
Experiment 7 retested the effects of the stimulus conditions on the perceived size of the illusory spots with newly recruited participants. The procedure of the test was changed from Experiments 1 to 3 in the following three aspects, in order to eliminate a possible response bias. First, all three stimulus conditions, patch size, bar width, and stimulus orientation, were varied among trials within an experimental session. In Experiments 1 to 3, only one aspect of the stimulus changed between trials, and the participant's attentional focus on the specific stimulus components might have biased the results. In Experiment 7, 18 combinations of three patch sizes (0.492°, 0.571°, 0.650°), three bar widths (0.335°, 0.413°, 0.492°), and two rotation angles (0° and 18°) were thus intermixed within a session. The range of the stimulus parameters was about 40% of that used in Experiments 1 to 3, considering the visibility of the illusion with the combination of these three factors. Second, circular probe stimuli were used instead of square-shaped probes, to test possible idiosyncratic effects of the matching procedure between perceived circular illusory spots and rectangular probe squares in Experiments 1 to 3 (Figure 5). The size of the circular probe was varied by changing the horizontal and vertical diameter with 1-pixel accuracy, keeping the circular shapes as similar to each other as possible. Third, a rectangle frame surrounding the black probe stimuli was filled with gray color with the same luminance as the grid bar color (10 cd/m2). In Experiments 1 to 3, the probe stimuli were placed on a white background of the same color as the patches, which could have biased the attentional focus of the participants toward the white patches during the matching procedure. 
Figure 5
 
Example of the test displays in Experiment 7.
Figure 5
 
Example of the test displays in Experiment 7.
Each observer participated in a practice session of four trials, randomly selected from the 18 stimulus conditions, and a test session of 36 trials, in which each condition was tested twice. The direction of the stimulus rotation (clockwise or counterclockwise) was counterbalanced within a session. Thirteen trials (3.0% of all trials) of “invisible” responses, where the probe size was set to 0, were excluded from the analyses. 
Data analyses
Collected data were analyzed using a Bayesian generalized linear mixed model with stimulus conditions (bar width, patch size, or stimulus orientation) as fixed factors and participant as a random factor in each experiment. 
For the measurement of perceived size (pretest, Experiments 1 to 3, and 7), we estimated the linear effects of the stimulus parameters by fitting linear mixed regression models to the data as follows:  
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}\tag{1}y = \sum {x_i}{\beta _i} + \gamma + \varepsilon ,\gamma \sim N\left( {\mu ,{\sigma _\gamma }^2} \right),\varepsilon \sim N\left( {0,{\sigma _\varepsilon }^2} \right)\end{equation}
where y is the perceived size of the illusion, x is the properties of the ith stimulus condition, β is the slope of the regression line, γ is the random effect of participants that is normally distributed around the intercept of the regression line (μ) with a variance of σγ2, and ε is a normally distributed error term with a variance of σε2.  
In Experiments 4 to 6, we collected ordinal data (rating values from 0 to 6) and ordered logit regression models were therefore fitted to the data. Based on the patterns of stimulus parameter dependencies for the perceived strength of the illusion shown in previous experiments (Schrauf et al., 1997), visual inspection of our data, convergence assessments, and WAIC values (Watanabe, 2010; Table S2), we included the quadratic terms for the stimulus conditions in the models for the analyses of Experiments 4 and 5 (Equation 2), but not for Experiment 6 (Equation 3):  
\begin{equation}\tag{2}\ln \left( {{{{\theta _k}} \over {1 - {\theta _k}}}} \right) = {\beta _1} \cdot {\left( {x - {\beta _2}} \right)^2} - {c_k} + \gamma ,\gamma \sim N\left( {0,{\sigma _\gamma }^2} \right)\end{equation}
 
\begin{equation}\tag{3}\ln \left( {{{{\theta _k}} \over {1 - {\theta _k}}}} \right) = \beta \cdot x - {c_k} + \gamma ,\gamma \sim N\left( {0,{\sigma _\gamma }^2} \right)\end{equation}
where θk is the probability that the rating value is not greater than k-1, x defines the properties of the stimulus condition, βs are the regression coefficients, γ is the random effect of participants, and ck is the cut point value determining the partition of the kth category.  
In all analyses, uniform prior distributions were assumed for each parameter. Each Markov chain Monte Carlo analysis was composed of four chains of 5000 steps each. The draws of the first 500 steps in each chain were discarded from the analyses as warm up draws, and a posterior mean of each model parameter was calculated from 18000 simulated posterior observations. Convergence of the chains was confirmed with the criterion of R̂ values (Gelman et al., 2013). These analyses were conducted using R version 3.6 (R Core Team, 2019) and the Stan software via the rstan package (Carpenter et al., 2017; Stan Development Team, 2019). 
Results
Pretest
In the pretest, the dark gray spots on white patches were matched with probe squares with a width that was 5% to 15% smaller (Figure 6), probably because of the differences in shape and area. The discrepancy slightly increased with the size of the gray spots. 
Figure 6
 
Perceived size of the black spot (left) and its ratio (right) to the actual spot size in the pretest. Each plot represents averaged responses of a participant, and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 6
 
Perceived size of the black spot (left) and its ratio (right) to the actual spot size in the pretest. Each plot represents averaged responses of a participant, and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
The perceived size data were analyzed by fitting a linear mixed model with sizes of black spots and white patches as fixed factors and participants as a random factor. The perceived size linearly increased as a function of physical spot size with a slope of about 0.8. The slope for patch size was near zero in analyses of both perceived size and ratio to spot size, suggesting that the participants' judgement of the size of the spots was not influenced by the contextual white patches. 
Experiments 1 through 3: Measurement of the perceived size of the illusory spots
The results for Experiments 1 to 3 revealed the dependency of the perceived size of the scintillating grid illusion on different stimulus components (Figure 7, left panels, Table 2). In comparison with the other two stimulus conditions, changes in patch size modulated the perceived size of the illusory spots of the scintillating grid illusion to a larger degree. The perceived size of the illusory spots monotonically increased as a function of the patch size of the stimulus in Experiment 1 (Figure 7, top left). The estimated size of the perceived illusory spots changed from 0.225° to 0.517° (0.292° increment), corresponding to the 0.394° change in the size of the white patch from the smallest to the largest. The estimated ratio of the perceived size of the illusory spots to the patch size was relatively constant (0.626–0.704) throughout the range of the patch sizes we used (Figure 8, Table 2). These results suggest that the size of the illusory spots was constrained by the size of the patch contours. 
Figure 7
 
Results of Experiments 1 to 6. Each point represents averaged responses of a participant (TM and SK were authors) and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 7
 
Results of Experiments 1 to 6. Each point represents averaged responses of a participant (TM and SK were authors) and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Table 1
 
Estimated model parameters for the pretest.
Table 1
 
Estimated model parameters for the pretest.
Table 2
 
Estimated model parameters for Experiments 1, 2, and 3.
Table 2
 
Estimated model parameters for Experiments 1, 2, and 3.
Figure 8
 
The ratio of the perceived size of the illusory spot to patch size in Experiment 1. Each point represents the averaged ratio of one participant, and the dashed line represents the mean across participants. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 8
 
The ratio of the perceived size of the illusory spot to patch size in Experiment 1. Each point represents the averaged ratio of one participant, and the dashed line represents the mean across participants. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
On the other hand, changes in bar width and the rotation angle of the stimuli had much less or no influence on the perceived size of the illusory spots (Figure 7, middle and bottom left, Table 2). In Experiment 2, the estimated size of the illusory spot slightly increased as a function of bar width, showing only a 0.028° increment, from 0.352° to 0.380°, in response to the 0.394° change in bar width from the thinnest to the widest. In Experiment 3, the estimated perceived size slightly decreased with the increase in rotation angle, showing a 0.073° decrement from 0.325° to 0.399° throughout the 0° to 45° rotation conditions. The patch size used in Experiments 2 and 3 was fixed at 0.551°. The perceived size of the illusory spots estimated from the fitted model in Experiment 1 was 0.371°, 95% CI [0.334, 0.408], with a patch size of 0.551°, and the results of Experiments 2 and 3 were consistent with this estimation. 
The illusory contour of the scintillating grid illusion appeared to be more salient, in comparison with the contours in the Hermann grid illusion. Supporting this subjective impression, judgements of size were consistent among trials. The variance of the error term of the fitted models, σε2 = 0.053, 95% CI [0.048, 0.058] in Experiment 1, 0.055 [0.050, 0.060] in Experiment 2, and 0.042 [0.038, 0.046] in Experiment 3, was comparable to that of the pretest, σε2 = 0.049 [0.045, 0.053], where the size of the physically presented dark spots with salient contours was measured, suggesting that the contours of the illusory spots were as definite as the contours defined by physical luminance contrast. 
Experiments 4 through 6: Rating of the illusion strength
The results of the strength ratings of the illusory spots were not consistent with the perceived size data (Figure 7, Table 3). The rated strength data from Experiment 4 and Experiment 5 were well fitted by the quadratic functions of the patch size and bar width, suggesting that there is an optimum combination of stimulus parameters for the illusion. These results are consistent with previous studies (Qian et al., 2009; Schrauf et al., 1997). Schrauf et al. (1997) reported that the rated strength of the scintillating grid illusion was maximal when the patch size was approximately 1.4 times larger than the bar width. That corresponds to 0.551° of patch size in Experiment 4 and 0.394° of bar width in Experiment 5, where the bar width and patch size were fixed at 0.394° (Experiment 4) and 0.551° (Experiment 5), respectively. These optimal parameter values coincided well with the estimated peaks of the quadratic function in our analyses, 0.548°, 95% CI [0.540, 0.557], of patch size in Experiment 4 and 0.380°, [0.375, 0.385], of bar width in Experiment 5. 
Table 3
 
Estimated model parameters for Experiments 4, 5, and 6.
Table 3
 
Estimated model parameters for Experiments 4, 5, and 6.
The strength of the illusion monotonically declined as a function of rotation angle of the scintillating grid stimulus, which is consistent with earlier claims that orientation signal processing plays an important role in the percept of the illusion (Qian et al., 2012; Qian et al., 2009; see Supplementary File S1 for a detailed comparison with previous studies). 
Experiment 7: Replication of Experiments 1 to 3
The additional experiment with newly recruited participants replicated the findings of Experiments 1 to 3, despite the differences in experimental design and probe stimuli (Figure 9, Table 4). The perceived size of the illusory spots linearly increased with the enlargement of the patch size. The estimated slopes of each stimulus condition in this replication experiment (0.794, 0.054, and −0.003 for patch size, bar width, and rotation angle) were consistent with the slopes in Experiments 1 to 3 (0.741, 0.072, and −0.002). These results confirmed that the results of Experiments 1 to 3 were not influenced by response biases caused by the experimental design or the specific appearance of the probe stimuli. 
Figure 9
 
Results of Experiment 7. Each plot represents averaged responses from each stimulus condition. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 9
 
Results of Experiment 7. Each plot represents averaged responses from each stimulus condition. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Table 4
 
Estimated model parameters for Experiment 7.
Table 4
 
Estimated model parameters for Experiment 7.
Discussion
The seven experiments presented here reveal that the perceived size of the scintillating grid illusion is mainly constrained by the size of the white patch area where the illusion appears, and that such stimulus parameter dependencies for perceived size are not consistent with those of perceived strength. We found that the perceived size of the illusory spots monotonically increased as the patch size increased, keeping the ratio to the patch size relatively constant. The effect of bar width and rotation angle on the perceived size was relatively limited. The results for perceived strength showed patterns that are different from those of perceived size. The illusion was strongest when the ratio between bar width and patch size was about 1:1.4 (Schrauf et al., 1997), and the deviation of either patch size or bar width from this ratio decreased the strength of the illusion. Monotonical rotation of the stimulus decreased the strength of the illusion to a large degree. 
The dissociated result patterns for the measurements of perceived size and the strength ratings suggest a mechanism where the location of the contour of the illusory spots is determined independently from the factors that determine the illusory contrast percepts. Previous studies that focused on the strength of the illusion have discussed the importance of the condition of the gray bars in the occurrence of the illusion (Geier et al., 2004; Levine & McAnany, 2008; Qian et al., 2012; Qian et al., 2009). For example, Qian et al. (2012) proposed that orientation processing of the bars is critical for the illusion, and that our visual system perceives the neural representation of the gray bars as an illusory smudge within the patch. The results of the rated strength of the illusion in our study are also consistent with this view, demonstrating a weakened illusion with the oblique stimuli, in which the orientation signals of the grid components were attenuated in comparison with the vertical and horizontal grids. These results support the view that the orientation signal of the grid bars plays an important role in eliciting the illusory spots. 
On the other hand, neither the locations of the edges of the bars nor the bar orientation largely affected the contour size of the illusory spots. If orientation signals corresponding to the grid bars directly determined the appearance of the illusory spots, the size of the perceived illusory spots depends on the width of the grid bars and the location of their borders. Our data revealed that this is not the case. The perceived size of the illusory spots was relatively insensitive to changes of the grid components of the stimuli, suggesting that retinotopic activation in response to the orientation signals does not directly determine the spatial configuration of the illusory spots. To fully explain our results, we need to assume a process that also takes the surface area of the white patches into account, in addition to the orientation signals of the bars. 
One possible explanation that reconciles the different stimulus element dependencies of perceived size and strength would be brightness spreading (Spillmann & Dresp, 1995; Weil & Rees, 2011). Brightness spreading, or area completion, is a phenomenon whereby the illusory surface properties are induced and spread to the illusory or physical contours, and the blank area is perceptually filled. Based on this phenomenon, our results may be explained as a combination of the emergence of black illusory smudges and their spreading. The orientation-selective mechanism may be involved in the emergence of the illusory smudges with vague and indefinite contours and sizes, and such illusory smudges may be spread under the constraints of the contour edges of the white patches. This process, however, cannot explain our results. In perceptual spreading, illusory color typically propagates until the surface is bordered by a luminance-defined contour (van Lier, Vergeer, & Anstis, 2009; Weil & Rees, 2011). In contrast, here, the white patches were not completely filled by black color in the scintillating grid illusion. The estimated size of the perceived illusory spot was always smaller than the patch size (about 60%–70%), irrespective of the stimulus conditions. That means the area near the edges of the white patches remains intact. In addition, our results suggest that the contour of the illusory spots is perceived as salient as the physically presented dark spots. These results are not consistent with an explanation based on brightness spreading. 
As an alternative explanation, we propose a mechanism for the scintillating grid illusion that is based on the interference between processes related to the grid bars and the surface of the white patches. The mechanism assumes that the orientation signals of the grid bars interfere with and interrupt the surface filling-in process of the overlapping white patches, resulting in unfilled surface areas. The idea comes from the visual phenomenon known as area suppression (Paradiso & Nakayama, 1991; Stoper & Mansfield, 1978). Area suppression has been reported as a specific type of visual masking phenomenon: Surface perception of a preceding target stimulus is suppressed by a contour of a mask stimulus that spatially overlaps, or is near, the target. For example, Paradiso and Nakayama (1991, Figure 1 and 8) demonstrated that a filled white disk target followed by thin grid lines or quadrangular four-line segments was perceived as a hollowed circle. Area suppression has been described as a process where the propagation of filling-in starts at the edge of the target and is stopped by the mask stimulus before the area is filled. Although area suppression is a phenomenon of temporal interference between briefly presented stimuli, similar interruptions of filling-in processes may intermittently cause hollowed white patches in the scintillating grid illusion at the moment when the representation of the stimulus components would be updated with eye movements. 
The assumed process that the grid bars interfere with the processing of the overlapping visual elements may be shared with some known similar grid illusions. Qian and Mitsudo (2016) found that the perceived shape of circular patches placed on the midpoints between grid intersections were deformed and appeared to be oval (the “eggs illusion”). Kawabe, Qian, Yamada, and Miura (2010) demonstrated that the straight edges of diamond shapes placed on the intersections of the grid are perceptually distorted to be jagged (the “jaggy diamond illusion”). Such interferences could occur depending on local interactions of visual elements at the intersections of bars and overlapping items (Qian & Mitsudo, 2016), or on global mechanisms that relate to periodic gratings (Schrauf et al., 1997; von der Heydt, Peterhans, & Dursteler, 1992) or amodal-integrated representations of fragmented bars (Ban et al., 2013; Sugita, 1999). 
An explanation based on area suppression is consistent with several characteristics of the scintillating grid illusion that may not be fully explained by other mechanisms. First, the explanation is consistent with the scintillating feature of the illusion. Surface filling-in of the white patches and its interruption are dynamic and transient processes that could intermittently occur depending on the update of visual information. Instability of the illusory spot may be due to such an intermittent nature of the process. Second, a reduction of the illusory spots at fixated and attended points (Schrauf et al., 1997; VanRullen & Dong, 2003) would be in line with the interference explanation. Fixation and attention to the white patches resolve the interference and promote the processing of surface information, and surface processing of the fixated or attended white patches may therefore be immune to interruption by the distracting orientation signals. Third, the interruption of surface processing also explains the distinctiveness of the subjective contrast change in the illusion. The relatively high-contrast salient appearance of the illusory spots, in comparison with other brightness illusions (e.g., Purves, Williams, Nundy, & Lotto, 2004), is another distinct feature of the scintillating grid illusion. While other brightness illusions are phenomena of surface color modulation, the scintillating grid illusion could indicate a lack of the surface itself. In other words, the scintillating grid illusion would not be a phenomenon where the black spot is illusory “visible” but where the surface of the white patch is “invisible.” Although the assumed mechanism does not make any prediction about the appearance of the area that was not filled in, the black color of the illusion seemingly derives from the background color. In agreement with this idea, illusory spots turned bluish when the background was filled with dark blue in our informal observations (Figure 10). Fourth, the interruption of surface formation processes could explain the scale-invariance of the illusion. The scintillating grid illusion is perceived within a large range of retinal stimulus sizes (Schrauf et al., 1997); this makes it untenable to explain the illusion by mechanisms such as lateral inhibition, which assumes the involvement of specific cells with limited receptive field sizes (Schiller & Carvey, 2005). Surface filling-in processes are known to take place at a variety of stimulus sizes (Paradiso & Hahn, 1996), and their interruption may therefore explain the illusion at different spatial scales. 
Figure 10
 
The scintillating grid illusion on a black and a bluish background.
Figure 10
 
The scintillating grid illusion on a black and a bluish background.
Conclusions
The findings of the current study demonstrate that the perceived size and perceived strength of the illusory spots in the scintillating grid illusion depend on different aspects of the stimulus configuration. In particular, we found that the size of the illusion was constrained by the contour size of the white patches at intersections, but not by the width of the grid bars. We propose a process that involves surface filling-in as the mechanism underlying the scintillating grid illusion; this explanation does however require further verification. Any putative mechanisms, ours and others, that might be tested in the future should take into account both the effect of the intensity of the orientation signals of the grid bars and constraints on the size of the white patches. 
Acknowledgments
This study was financially supported by MEXT KAKENHI Grant Number 19K12739. 
Commercial relationships: none. 
Corresponding author: Toyomi Matsuno. 
Address: Faculty of Economics, Hosei University, Machida, Tokyo, Japan. 
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Figure 1
 
(a) Hermann grid illusion; (b) scintillating grid illusion.
Figure 1
 
(a) Hermann grid illusion; (b) scintillating grid illusion.
Figure 2
 
An illustration of the scintillating grid stimulus and the probe squares used in the experiments.
Figure 2
 
An illustration of the scintillating grid stimulus and the probe squares used in the experiments.
Figure 3
 
Stimulus presented at the left side of the display in the pretest.
Figure 3
 
Stimulus presented at the left side of the display in the pretest.
Figure 4
 
The display used in Experiments 3, 4, and 5. The rating circles were presented with descriptions that instructed participants to rate the strength of the illusion on a 7-point scale ranging from 0 (weak) to 6 (strong) in Japanese.
Figure 4
 
The display used in Experiments 3, 4, and 5. The rating circles were presented with descriptions that instructed participants to rate the strength of the illusion on a 7-point scale ranging from 0 (weak) to 6 (strong) in Japanese.
Figure 5
 
Example of the test displays in Experiment 7.
Figure 5
 
Example of the test displays in Experiment 7.
Figure 6
 
Perceived size of the black spot (left) and its ratio (right) to the actual spot size in the pretest. Each plot represents averaged responses of a participant, and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 6
 
Perceived size of the black spot (left) and its ratio (right) to the actual spot size in the pretest. Each plot represents averaged responses of a participant, and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 7
 
Results of Experiments 1 to 6. Each point represents averaged responses of a participant (TM and SK were authors) and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 7
 
Results of Experiments 1 to 6. Each point represents averaged responses of a participant (TM and SK were authors) and a dashed line represents the average among participants. The solid lines show the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 8
 
The ratio of the perceived size of the illusory spot to patch size in Experiment 1. Each point represents the averaged ratio of one participant, and the dashed line represents the mean across participants. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 8
 
The ratio of the perceived size of the illusory spot to patch size in Experiment 1. Each point represents the averaged ratio of one participant, and the dashed line represents the mean across participants. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 9
 
Results of Experiment 7. Each plot represents averaged responses from each stimulus condition. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 9
 
Results of Experiment 7. Each plot represents averaged responses from each stimulus condition. The solid line shows the regression estimates, and the gray shading represents the 95% Bayesian confidence intervals of the estimates.
Figure 10
 
The scintillating grid illusion on a black and a bluish background.
Figure 10
 
The scintillating grid illusion on a black and a bluish background.
Table 1
 
Estimated model parameters for the pretest.
Table 1
 
Estimated model parameters for the pretest.
Table 2
 
Estimated model parameters for Experiments 1, 2, and 3.
Table 2
 
Estimated model parameters for Experiments 1, 2, and 3.
Table 3
 
Estimated model parameters for Experiments 4, 5, and 6.
Table 3
 
Estimated model parameters for Experiments 4, 5, and 6.
Table 4
 
Estimated model parameters for Experiment 7.
Table 4
 
Estimated model parameters for Experiment 7.
Supplement 1
Supplement 2
Supplement 3
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