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Article  |   May 2019
Stimulus dependencies of an illusory motion: Investigations of the Motion Bridging Effect
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Journal of Vision May 2019, Vol.19, 13. doi:10.1167/19.5.13
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      Maximilian Stein, Robert Fendrich, Uwe Mattler; Stimulus dependencies of an illusory motion: Investigations of the Motion Bridging Effect. Journal of Vision 2019;19(5):13. doi: 10.1167/19.5.13.

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Abstract

The Motion Bridging Effect (MBE) is an illusion in which a motion that is not consciously visible generates a visible motion aftereffect that is predominantly in the same direction as the adapter motion. In the initial study of the MBE (Mattler & Fendrich, 2010), a ring of 16 points was rotated at angular velocities as high as 2250°/s so that observers saw only an unbroken outline circle and performed at chance when asked to report the ring's rotation direction. However, when the rotating ring was replaced by a veridically stationary ring of 16 points, the stationary ring appeared to visibly spin to a halt, principally in the same direction as the initial ring's rotation. Here we continue to investigate the stimulus dependencies of the MBE. We find the MBE, measured by the correspondence between the direction of the invisible rotation of the spinning ring and perceived rotation of the stationary ring, increases as the number of points used to construct the rings decreases and grows stronger as the diameter of the rings get larger. We consider the potential contributions of temporal frequency, retinal eccentricity, luminance levels, and the separation between the points forming the rings as mediators of these effects. Data is discussed with regard to the detection of real movement and apparent motion. We conclude that the detection of the rapid rotation of the spinning ring is likely to be modulated by temporal frequency of luminance changes along the ring perimeter while the point-distance may modulate an apparent motion produced by the transition from the perceptually unbroken spinning ring to the point-defined stationary ring.

Introduction
A wealth of vision research has addressed the abilities and limits of the human visual system, and a direct approach to exploring these limits has been to focus on the reported perceptual experiences of participants. However, there is evidence for the processing of visual information, which is not accessible to conscious reports. Much of this evidence is based on priming and adaptation effects produced by unconscious information (e.g., Lin & He, 2009; Vorberg, Mattler, Heinecke, Schimdt, & Schwarzbach, 2003). Two of the present authors previously reported evidence for the processing of unconscious motion information. Mattler and Fendrich (2007) employed a ring of points that rotated so rapidly observers saw only a fused static ring, but viewing this ring primed direction judgments when observers subsequently viewed a visibly rotating ring. In 2010 Mattler and Fendrich extended their observations by reporting that viewing such a rapidly rotating ring can produce an illusory rotation in the same direction as the invisible rapid rotation in a stationary ring of points that precedes or follows the rotating ring. 
In the investigations reported in the 2010 paper, the rotating ring was 5° of visual angle in diameter, constructed of 16 points, and presented on the CRT screen of a fast phosphor oscilloscope. This “inducing ring” was rotated at angular velocities as high as 2250°/s. When the inducing ring simply appeared and vanished, observers perceived it as a flashed continuous outline circle and performed at chance when asked to report if the rotation had been clockwise or counterclockwise. However, when the inducing ring was replaced by a stationary “test ring” of 16 points, this stationary ring appeared to visibly spin to a halt, primarily in the same direction the inducing ring had been spinning. This illusionary spin was seen although the initial and final positions of the inducing ring points and the display positions of the test ring points were identical. Because the illusory rotation of the test ring usually corresponded to the direction of the inducing ring, observers were able to report this direction at substantially greater than chance levels. Subsequently, Mattler and Fendrich (2010) found there was a similar effect when the stationary test ring preceded the inducing ring. In this case, the test ring appeared to launch into motion in the direction of the inducing ring. Because these illusory motions linked the actual (although invisible) rotation of the inducing ring to the veridically stationary test ring, Mattler and Fendrich labeled this illusion the Motion Bridging Effect (MBE). 
Whereas other studies have reported ways of making a moving stimulus invisible, these studies used methods like crowding (e.g., Moutoussis & Zeki, 2006), binocular rivalry (e.g., Lehmkuhle & Fox, 1974), or continuous flash suppression (e.g., Maruya, Watanabe, & Watanabe, 2008) to mask or suppress the stimulus motion. The MBE differs from these studies because it demonstrates a motion inherently inaccessible to consciousness is not only encoded by the visual system but can subsequently manifest itself as a visible attribute. A similar phenomenon has been previously demonstrated in a different domain. Stationary gratings can be made invisible by crowding (He, Cavanagh, & Intriligator, 1996) or interocular suppression (Blake & Fox, 1974) and still generate an aftereffect. It has been shown that this is also possible if the grating is inherently inaccessible to consciousness because of its high spatial frequency (He & MacLeod, 2001). 
Although the processes underlying the MBE are still uncertain, a number of its characteristics are known. Mattler and Fendrich (2010) found the MBE, assessed by the congruence of the actual inducing ring and reported test ring spin directions, declined as the inducing ring's angular velocity increased but was still present at 2250°/s, the highest velocity they tested. They also found the MBE was maximal when there was a 90 ms ISI between the inducing and test ring presentation, and was observable with inducing ring durations as short as 15 ms, reaching an asymptotic level with durations of 60 ms. In addition, they noted that the MBE was degraded by a small (1° of visual angle) spatial mismatch between the inducing and test ring positions (produced by an expansion or upward shift of the test ring) and this degradation was complete when the spatial mismatch was increased to 3°. This outcome suggests the MBE depends on interactions that occur at an early stage of the visual pathway where neural representations map closely onto retinal locations. 
More than one factor might be responsible for the decline in the MBE as the inducing ring velocity increases. An increase in the rotation rate increases not only the linear velocity of the inducing ring but also the temporal frequency at which points cross a given position along the circumference of that ring. Temporal frequency is one basic feature that affects the detection of motion. Due to limitations in the rate at which the human visual system can track luminance changes, objects that stimulate retinal locations at high temporal frequencies may be perceived as static forms or outlines (e.g., a rotating fan blade may look like a blurred disk). The upper limit of the system's temporal response capabilities can be investigated with a flicker detection paradigm in which an observer reports the perceived flicker of a luminous patch that has its intensity modulated sinusoidally in time. Experiments using this paradigm show that sensitivity to flicker initially rises with increasing modulation frequencies and then falls off steeply (Kelly, 1961). This pattern indicates that the visual system acts like a band-pass filter that is most sensitive to frequencies ranging from approximately 10 to 30 Hz (Cornsweet, 1970), with lower and higher frequencies attenuated (see Kaufman, 1974, for a summary of early research on flicker perception). 
The temporal frequency at which a flickering light is perceived as steady is termed critical flicker frequency (CFF). While the CFF is influenced by a number of stimulus conditions, including the luminance, size, chromaticity, and sharpness of the test patch, as well as variables such as an observer's light adaptation level (Kelly, 1959; Landis, 1954), the maximum value of the CFF in humans is about 60 Hz. The apparently steady appearance of stimuli that flicker faster than the CFF can be taken to be indicative of the stability in the activity of neurons responding to the flickering stimulus. The visual system is, in effect, integrating luminance variations over short periods of time (see Barlow, 1958), with neural persistence acting to fill in the activity troughs. This filling in may occur as early as the photoreceptor level but could also be occurring at higher levels in the visual system (see Coltheart, 1980). 
However, flicker detection paradigms measure only the conscious perception of flicker and therefore only demarcate limits of conscious perception. A variety of studies indicate that some neurons can encode luminance modulation frequencies higher than the CFF. Cells in the monkey lateral geniculate nucleus respond to flicker rates well beyond the human CFF (Spekreijse, van Norren, & van den Berg, 1971), and neurons in the primary visual cortex (V1) of monkeys respond to flicker rates as high as 100 Hz when high contrast patterns are used (Williams, Mechler, Gordon, Shapley, & Hawken, 2004). In addition, macaque monkey V1 neurons respond to heterochromatic flicker at 30 Hertz (Gur & Snodderly, 1997) although macaques do not discriminate isoluminant red/green flicker at 15 Hz (Schiller, Logothetis, & Charles, 1990). In human observers, Regan (1968) has reported that stimuli with temporal frequency modulations higher than the CFF evoke potentials in the EEG (for similar findings, see Herrmann, 2001; Krolak-Salmon et al., 2003; Lyskov, Ponomarev, Sandström, Mild, & Medvedev, 1998). Akin to the results for chromatic flicker in monkeys, in a human fMRI study, Jiang, Zhou, and He (2007) found activity in visual cortical areas related to chromatic flicker at 30 Hz even when participants were unable to distinguish the presented stimulus from a static control stimulus. Moreover, Shady, MacLeod, and Fisher (2004) showed that temporal frequency modulations above the CFF can have an impact on perception. These investigators found in an adaptation experiment that patches flickering above an observer's CFF influenced sensitivity to subsequent flickering patches. 
The limited ability of observers to consciously perceive rapid rates of temporal change also sets limits on their ability to perceive rapid rates of motion. Holcombe, in a 2009 review of the visual system's temporal processing capabilities, has in fact argued that flicker can be regarded as “a degenerate case of motion” (p. 217). In the case of rapidly moving stimuli, neural persistence can generate percepts of spatially extended lines and areas rather than stimuli that are spatially displacing. This neural persistence also sets limits to the perception of apparent motion: If presented at a sufficiently rapid rate, advancing or alternating points will be seen as simultaneously present rather than advancing or jumping back and forth (Wertheimer, 1912). 
The visual system's spatial resolving capabilities with static stimuli can be evaluated by determining the minimum luminance contrast required for the detection of sinusoidal gratings of increasing spatial frequency. This determination yields the human contrast sensitivity function (e.g., Campbell & Robson, 1968). Kelly (1979) extended this method to obtain spatio-temporal contrast-sensitivity functions for moving sinusoidal gratings. He found the sensitivity function remains similar in shape but shifts to a lower spatial frequency as the grating is presented at higher velocities, and notes that at velocities greater than 100°/s there is probably no spatial frequency within the range of normal human vision that would allow a grating to have a detectable contrast. Using a direction detection task, Burr and Ross (1982) likewise found the contrast sensitivity function of moving gratings shifted towards lower spatial frequencies as the grating's velocity increases. However, Burr and Ross (1982) challenged the 100°/s limitation: They found that gratings with velocities as high as 800°/s could be detected when the spatial frequency was lowered to 0.01 c/°. They proposed that the limiting factor of motion detection, as is the case with flicker perception, is the temporal frequency of intensity modulations. In their experiment this frequency was approximately 30 Hz. Above this critical rate, motion could no longer be perceived. Thus, the sensitivity of local motion processing seems to depend on temporal rather than spatial frequency. 
Like measurements of the CFF, however, these investigations depend upon the conscious percepts of observers. They therefore do not rule out the possibility that stimulus motions with higher temporal frequencies are being encoded at the retinal level and processed at postretinal levels of the visual system although they cannot be consciously detected or discriminated. Thus, just as Shady et al. (2004) reported the processing of flicker rates above the flicker-fusion threshold, motion inaccessible to consciousness could still be processed and have an impact on perception. 
Overview
In the present study, we report four experiments that further investigate functional dependencies of the MBE. In Experiment 1 we varied the number of points used to construct the inducing and test ring. Increasing the number of points had the effect of increasing the temporal frequency of point presentations at each position along the inducing ring circumference while leaving its linear and angular velocity unchanged. In Experiment 2 we varied the diameter of the inducing and test rings. In addition to altering their retinal eccentricity, this manipulation modified the linear velocity of the inducing ring points but left the temporal frequency of point presentations along the ring circumference unchanged. In Experiment 3 we investigated the possibility that the observed effects of varying the number of points and the diameter of the rings are both mediated by a common factor: the distance between the points that formed the rings. By changing the diameter of the rings together with the number of points, we kept the point distance constant but varied temporal frequency. In Experiment 4 we addressed the effect of the inducing and test ring luminance on the MBE and found no effect of these variables. 
General methods
Participants
The participants were students at University of Goettingen with a mean age of 24.5 years. They were compensated €7 per hour for their participation. All participants undertook a visual acuity test using the Landolt ring chart and had normal or corrected-to-normal vision. In Experiments 13 participants completed three, 1 hr sessions; in Experiment 4 they completed two, 1 hr sessions. Sessions were always run on separate days. We examined 12 students in each experiment, with no student taking part in more than one. However, in Experiment 1, the data of one of the 12 students was subsequently excluded from data analyses because it was found her vision was impaired, so data from 11 subjects is reported for this experiment. 
Apparatus
Stimuli were presented on a cathode-ray oscilloscope (HAMEG HM 400) controlled by a PC with a 12-bits Digital-Analog converter. The 8 × 10 cm oscilloscope display was customized with a fast P15 phosphor (50 μs luminance decay time to 0.1%). The experiment was run in a dark room, and participants had their head positions stabilized 57 cm from the oscilloscope by a chin and forehead rest. 
Stimuli
Stimuli are illustrated in Figure 1. Participants were presented with a rapidly rotating inducing ring of luminous points and a stationary test ring of luminous points. The points were slightly blurred because pilot observations had shown this enhanced the illusionary motions being investigated. 
Figure 1
 
Display sequence in Experiments 14. On a given trial the inducing ring rotated either clockwise (as shown in the figure) or counterclockwise. Note the points on the oscilloscope were bright on a dark background.
Figure 1
 
Display sequence in Experiments 14. On a given trial the inducing ring rotated either clockwise (as shown in the figure) or counterclockwise. Note the points on the oscilloscope were bright on a dark background.
In all the reported experiments, 12, 16, or 20 equally spaced points were used to construct the inducing and test rings. The position of these points could be adjusted in steps of 0.25 angular degrees, allowing them to be placed in 1440 potential positions along the ring circumference. The inducing ring was rotated by advancing all of its points by a specified number of positions every millisecond, so that it was updated with an effective frame rate of 1000 Hz. When the inducing ring was formed from 16 or 20 points, it could be rotated clockwise or counterclockwise with angular velocities of 250°, 750°, 1500°, and 2250°/s. These velocities entailed sequential point position advances of 1, 3, 6, and 9 steps. Due to programming constraints, when the inducing ring was constructed from 12 points, the highest angular velocity was 2500°/s rather than 2250°/s which entailed a step size of 10. The rotation of the inducing ring always started and ended with points placed at the same set of ring circumference positions. These were also the positions of the test ring points when the test ring was presented. At the higher rotation velocities (above 250°/s) the inducing ring appeared to be an unbroken outline circle. To produce a display free of visible switching artifacts, the CRT electron beam was switched off during its transit between the successive point positions. The diameter of the rings, number of points used to form the rings, and actual rotation rates employed in each experiment are described in each experiment's specific methods sections. 
As indicated in Figure 1, the inducing ring duration was usually 121 ms. However, in Experiments 1 and 3 when the inducing ring velocity was 250°/s, this duration was 91 ms when 16 points were displayed and 145 ms when 20 points were displayed. These deviations from the standard display time were necessitated by the combined constraints imposed by the relatively slow inducing ring rotation and the need to end its display interval when its point positions matched the position of the test ring points. Note that these variations in the inducing ring display time had no effect on our measures of the MBE since at the 250°/s velocity the MBE could not be measured because the inducing ring rotation was visible even when no test ring was presented. 
Statistical analysis
In all the experiments, we used signal detection methods to analyze performance (Macmillan & Creelman, 2004). We defined hits as clockwise responses to a clockwise rotation and false alarms as clockwise responses to a counterclockwise rotation. Hit and false alarm rates were estimated separately for each subject in each condition and corrected with the log-linear rule (Hautus, 1995). We evaluated d′ measures of the discrimination ability of participants with repeated measures analyses of variances (ANOVAs). All reported ANOVA p values were corrected using Greenhouse-Geisser estimates of sphericity but for the sake of readability, the uncorrected degrees of freedom are reported. Differences between specific conditions were evaluated with posthoc Bonferroni-corrected two-tailed t tests. Performance levels in specific conditions were compared to a chance level of 50% with one-tailed Bonferroni-corrected t tests that evaluated whether d′ exceeded zero. 
Luminance of the stimuli
Because of its small size, special methods are required to measure the brightness of a point on an oscilloscope screen. We employed a Minolta LS-100 luminance meter with a close-up lens that allowed us to focus on a small region of the oscilloscope display. To obtain a measurable luminance signal, we presented a rectangular matrix of 12, 16, or 20 contiguous points with their intensity, on-time, and repetition rates set to match the intensity, on-time, and repetition rates of the points in our inducing ring or test ring displays. Luminance estimates of the time averaged brightness of our display points were obtained by measuring the luminance of these matrices. 
The brightness of the test ring points in the 12 and 16 point display conditions was about 1.50 cd/m2 on a dark background. Point luminance in the 20 point condition was slightly lower (1.20 cd/m2). The time averaged brightness of the inducing ring points varied with point number and velocity (see Table 1). We address the possible effect of these luminance variations in Experiment 4
Table 1
 
Time averaged brightness of inducing ring points in Experiments 1 and 3 in cd/m2. Note: The replication of the point refresh rates in the 250°/s angular velocity conditions necessitated time averaging across flicker rates that were below the flicker fusion frequency.
Table 1
 
Time averaged brightness of inducing ring points in Experiments 1 and 3 in cd/m2. Note: The replication of the point refresh rates in the 250°/s angular velocity conditions necessitated time averaging across flicker rates that were below the flicker fusion frequency.
Experiment 1
In this experiment we confirmed the previously reported Motion Bridging Effect (Mattler & Fendrich, 2010) and investigated the effect of varying the number of points used to form the inducing and test rings. Changing the number of points effectively changes the temporal frequency of point presentations at each position along the inducing ring circumference (see Table 2) while keeping the linear and angular velocity constant. 
Table 2
 
Temporal frequency of point presentations at each position along the circumference of the inducing ring in Experiments 1 and 3 in Hz.
Table 2
 
Temporal frequency of point presentations at each position along the circumference of the inducing ring in Experiments 1 and 3 in Hz.
The limiting factor in the perception of the direction of sinusoidal gratings is the temporal modulation rate of the stimulus (e.g., Burr & Ross, 1982). This might hold true for the MBE as well, although the MBE is found when the temporal frequency of point presentations along the inducing ring circumference substantially exceeds conventional flicker fusion thresholds (de Lange Dzn, 1958; Kelly, 1961). A decrease in the MBE as the number of points increases, with the consequent increase in their temporal frequency, would support this premise. 
Method
Stimuli
Twelve, 16, or 20 equally spaced points formed the circumference of the inducing and test rings (see Figure 2). The point separations for these three point numbers were respectively 1.44, 1.08, and 0.86° of visual angle measured along the ring circumference. The number of points in the inducing and test rings was identical in any given trial. The diameter of the rings was 5.5° of visual angle. Rotation rates of 250°, 750°, 1500°, and 2250°/s (2500°/s in the 12 point condition) were employed. 
Figure 2
 
The various display conditions in Experiment 13. When the diameter of the rings was varied, the visual angle of its diameter is indicated. Note that the sizes of the rings depicted here are not precisely scaled to their sizes in the actual displays. Also note the points on the oscilloscope were bright on a dark background.
Figure 2
 
The various display conditions in Experiment 13. When the diameter of the rings was varied, the visual angle of its diameter is indicated. Note that the sizes of the rings depicted here are not precisely scaled to their sizes in the actual displays. Also note the points on the oscilloscope were bright on a dark background.
Task
Participants reported the direction of any perceived rotation (clockwise or counterclockwise). Responses were registered with the arrow keys of a conventional computer keyboard, with the right arrow indicating a clockwise rotation and the left arrow a counterclockwise rotation. Trial blocks were started with a press of the space key. No feedback was given about the correctness of responses. 
Procedure
Participants were instructed to maintain their gaze on a central fixation point during the trials. This fixation point brightened for 750 ms at the start of each trial to indicate that the inducing ring was about to appear. In the conditions in which the test ring was presented, it followed the inducing ring after a 60 ms interstimulus interval (ISI) and remained visible for 500 ms. Subjects' reports of the inducing ring direction were recorded starting 300 ms after the offset of the test ring. A response was required for the experiment to proceed. A new trial started 1 second after the response. 
Design
Fourteen trial blocks were run in each session with the first two treated as practice and excluded from data analysis. The number of points in the inducing ring was changed in the seventh and 11th blocks following a Latin Square design. There were 48 trials in each block, with an additional eight practice trials (which were not analyzed) added at the start of the seventh and 11th blocks. Blocks in which only the inducing ring was presented were alternated with blocks in which the inducing ring was followed by the test ring. The angular velocity and the rotation direction of the inducing ring were varied quasirandomly within each block. 
The combination of two test ring states (present vs. absent), four inducing ring angular velocities (250°, 750°, 1500°, 2250°/2500°/s) and three point numbers (12, 16, 20) produced 24 experimental conditions. There were 72 trials in each of these conditions: 36 with a clockwise inducing ring rotation and 36 in counter-clockwise rotation. 
Results
Results and confidence intervals for sensitivity to the inducing ring direction in all the conditions of Experiment 1 are presented in Figure 3A. This data is also presented as mean percent correct accuracy rates in Appendix Table A1. To take into account the very different response profiles found when the test ring is present and absent, we calculated two separate, two-way ANOVAs, one for each of the test ring conditions (present vs. absent). These analyses are presented in Table A2 in the Appendix. They evaluate the effect of angular velocity and the number of points that formed the inducing and test ring on observers' sensitivity (d′) to the inducing ring rotation. We will refer to these factors as Velocity and Point-number
Figure 3
 
(A) Mean sensitivity (d′) for 11 participants of Experiment 1 as a function of inducing ring velocity and point-number in the inducing ring only conditions (the dashed lines) and the inducing ring + test ring conditions (the solid lines). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Figure 3
 
(A) Mean sensitivity (d′) for 11 participants of Experiment 1 as a function of inducing ring velocity and point-number in the inducing ring only conditions (the dashed lines) and the inducing ring + test ring conditions (the solid lines). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Inducing ring only
When the inducing ring was presented alone, the main effects of both Velocity and Point-number were significant and there was a significant Velocity × Point-number interaction (Appendix Table A2). Mean sensitivity declined as Velocity increased (d′ = 3.85, d′ = 0.61, d′ = 0.10, and d′ = 0.06 at 250°, 750°, 1500°, and 2250°/2500°/s, respectively) and as Point-number increased (d′ = 1.36, d′ = 1.14, and d′ = 0.96 with 12, 16, and 20 points, respectively). As can be seen in Figure 3A, sensitivity rates were high in the 250°/s condition, dropped precipitously in the 750°/s condition, and were at chance in the two high velocity conditions. This pattern of outcomes accords with informal reports that the inducing ring perimeter fused into a visually unbroken outline circle at the higher velocities but was perceived as a set of spinning points at the lowest velocity (250°/s). 
The effect of Point-number was restricted to the two lower ring velocities. At the higher velocities, sensitivity remained near zero irrespective of the point number, accounting for the Point-number × Velocity interaction. Two-tailed t tests (with a Bonferroni-corrected alpha level of 0.0042) were performed to compare the across subject mean values of d′. These tests indicate that with the 250°/s velocity the sensitivity was lower in the 20 point condition than in the 16 point condition, whereas with the 750°/s velocity the sensitivity was lower in both the 20 and 16 point conditions than the 12 point condition. With the 1500°/s and 2250°/2500°/s velocities, the point conditions did not produce any significant differences in d′
A fundamental aspect of the MBE is that the improvement in sensitivity produced by the test ring presentation occurs even when subjects perform at chance in the inducing ring only condition. To investigate the velocity and point-number levels at which subjects were unable to discriminate the rotation direction of the inducing ring, we directly compared the discrimination ability of participants in the inducing ring only condition to a chance level of 50% (d′ = 0). To do this we calculated 12 one-tailed t tests (with a Bonferroni adjusted alpha level of 0.0042), one for each combination of velocity and point-number. We found that mean sensitivity was significantly greater than zero with every point-number when the angular velocity was 250°/s (p < 0.0001 in all cases), and with the 750°/s velocity when the inducing ring was formed by 12 or 16 points (p < 0.0001 and p = 0.0026, respectively). The mean d′ did not exceed zero in the 20 point 750°/s condition or with any point number in the two high velocity rotation conditions. Except for the 12 point 1500°/s condition, this was the case with the two high velocities even when we employed an uncorrected 0.05 significance level. 
Inducing ring + test ring
To address the characteristics of the MBE, similar analyses were carried out with the inducing ring + test ring conditions. However, it is only meaningful to speak of the MBE when the appearance of the test ring enables the discrimination of the inducing ring direction. The 250°/s velocity therefore was excluded from these analyses since at this velocity the inducing ring rotation is readily visible even when no test ring is presented. 
In a 3 × 3 ANOVA with Velocity (750°, 1500°, 2250°/2500°/s) and Point-number (12, 16, 20 points) as factors the main effects of both Velocity and Point-number were significant as well as the Velocity × Point-number interaction (Appendix Table A2). As in the inducing ring only conditions, mean sensitivity declined with increasing angular velocities (d′ = 2.48, d′ = 1.42, and d′ = 0.75 for the 750°, 1500°, and 2250°/2500°/s velocities, respectively), and with increasing point numbers (d′ = 2.03, d′ = 1.59, and d′ = 1.03 with 12, 16, and 20 points, respectively). However, it can be seen in Figure 3A that the pattern of the data is very different from that found when no test ring was present. Rather than plummeting towards zero, observer's sensitivity rates remained quite high at the 750°/s velocity and declined in a relatively linear fashion as the velocity increased, remaining well above zero at 1500°/s and 2250°/2500°/s. In addition, point number continued to have an effect at all three velocities, although there is a reduction in this effect as the velocity increased, accounting for the Velocity × Point-number interaction. 
The reliability of the observed pattern of effects was evaluated with two-tailed posthoc t tests that compared specific values of d′ in the various point-number conditions (using a Bonferroni-corrected alpha level of 0.0056). In all three velocity conditions sensitivity was significantly poorer in the 20 point condition than in the 12 point condition (see Figure 3A). In addition, d′ values were significantly lower in the 20 point condition than in the 16 point condition with the 750°/s and 1500°/s velocities. No difference between the 12 point and the 16 point condition was found at any velocity when we used the Bonferroni-corrected α values. However, d′ values were lower in the 16 point condition than in the 12 point condition at 750°/s, and at 2250°/2500°/s when we employed an uncorrected 0.05 significance level. 
Velocity and Point-number both modulate the temporal frequency of the point presentations at specific localities on the inducing ring circumference: As the rotation rates go up, points cross a given locality more frequently, and as the point number goes up, fewer steps are needed for successive points to cross a given locality. To address the hypothesis that the decline in the strength of the MBE as the inducing ring velocity increases and point-number decreases, is attributable to the increased temporal frequency of point presentations, we calculated the temporal frequency for each combination of point number and angular velocity (see Table 2) and correlated these rates with observer's mean sensitivity (d′) in that condition. A significant relationship between the MBE and the temporal frequency of the inducing ring was found, r = −0.92, p < 0.001, with the MBE decreasing as temporal frequency increased (see Figure 3B). 
Summary and discussion: Experiment 1
In Experiment 1 we replicated the MBE and confirmed its dependence on angular velocity (Mattler & Fendrich, 2010). When the inducing ring was presented by itself, observers' ability to detect its rotation fell to chance at velocities of 1500°/s and higher, but remained well above chance at velocities of up to 2500°/s when the rotating ring was followed by the stationary test ring. Interestingly, at these high velocities the temporal frequency of point presentations along the circumference of the inducing ring could be as high as 125 Hz, which not only exceeds conventional estimates of the CFF (Kaufman, 1974) but also exceeds the reported limits for the conscious detection of motion (Burr & Ross, 1982; Kelly 1979). 
Experiment 1 also demonstrates that the conscious perception of motion in the inducing ring only conditions and the MBE in the inducing ring + test ring conditions were dependent on the number of points that formed the rings. Increasing the number of points reduced an observer's ability to discriminate the inducing ring's rotation direction both when it was presented by itself and when it was followed by the test ring, though in the latter case performance remained better than chance at every tested velocity and point-number. Because increasing the point number acted to increase the temporal frequency without altering velocity, the effect of point number is consistent with the premise that the conscious perception of motion was limited by the temporal frequency of the point presentations along the inducing ring circumference. When only the inducing ring was presented, this effect was largely overridden by ceiling effects at the 250°/s velocity (since participant's responses were mostly correct) and basement effects at the 1500°/s and higher velocities (since participants always performed at chance), but at the 750°/s velocity, performance in the 12 point (25 Hz) condition was substantially better than in the 16 (33 Hz) and 20 (42 Hz) point conditions (see Figure 3A). These results are consistent with Burr and Ross's study (1982) in which the detection of the motion of sinusoidal gratings begins to deteriorate at around 10 Hz and completely breaks down at around 30 Hz. 
When both the inducing and test rings were presented, the effect of point-number was far more pervasive, clearly demonstrating that the illusory motion of the MBE, like the actual inducing ring motion, is modulated by the number of points in the inducing ring. Increasing the number of points reliably reduced the MBE with the 750°, 1500°, and 2250°/2500°/s velocities. As is the case for the percepts of the actual motion, this effect may be attributable to an increase in the temporal frequency. However, in the case of the MBE the upper limit of the effective range of temporal frequency would be above 125 Hz, considerably higher than it is in the case for flicker and motion perception. 
Varying the point number has the additional effect of generating complex changes in the inducing ring's spatial frequency spectrum. The effect of altering the point-number could therefore also be attributed to a dependency of the MBE on the inducing ring's spatial frequency composition. While we cannot rule out the spatial frequency spectrum of the inducing ring as a factor that contributes to the observed effects, we regard this issue as unlikely. Burr and Ross (1982) regarded spatial frequencies as meaningful for motion detection only because they determined the temporal frequency of the intensity modulations that occurred when their grating stimuli moved. Moreover, when the inducing ring rotates at a high velocity, there is little opportunity for the spatial frequency components associated with the number of inducing ring points to have any effect because the inducing ring fuses due to visual persistence into an apparently continuous outlined circle irrespective of point separation. This implies the moving points are rendered visually simultaneous at every position where they are displayed so that the amplitude of the spatial frequency components associated with the inducing ring point separations will be reduced to virtually zero. Based on these considerations, the findings of Experiment 1 are more likely to be the effect of temporal frequency changes rather than any change in the spatial frequency spectrum of the inducing ring. 
Experiment 2
In this experiment we investigated the effect of changing the diameter of the inducing and test ring on the illusory motion percept. Changing the diameter of the inducing ring has two major effects: It changes both the retinal eccentricity of the points that form the ring and changes their linear velocity as the ring rotates (see Table 3). However, the change in ring diameter will not change the angular velocity of the rotating points or the temporal frequency of the point presentations at specific positions along the ring circumference. 
Table 3
 
Linear velocity of the rotating points in the 3 diameter conditions of Experiment 2 in degrees of visual angle per second.
Table 3
 
Linear velocity of the rotating points in the 3 diameter conditions of Experiment 2 in degrees of visual angle per second.
While the MBE occurs at rotational velocities so high that visual persistence prevents motion from being consciously perceived, this rotation may nevertheless be capable of driving speed-selective cortical cells (Simoncelli & Heeger, 1998) tuned to high velocity motions. Velocity-tuned cells in the medial temporal cortex of monkeys that are specifically responsive to rotational movements have been reported by Tanaka and Saito (1989). If such cells play a role in mediating the MBE, increasing the linear velocity of the inducing ring points could decrease the MBE by raising velocities above the response limits of these cells. Alternatively, the MBE could increase with increasing eccentricity because the modulation sensitivity of the visual system increases with eccentricity (Hartmann, Lachenmayr, & Brettel, 1979; Tyler, 1985, 1987). 
Method
Stimuli and procedures were similar to those described for Experiment 1 with the following changes: In Experiment 2 the inducing and test rings were always constructed from 16 points, but the diameter of the rings was varied block-wise in three steps: 3.5°, 5.5°, and 7.5° of visual angle (see Figure 2). The point-distance was respectively 0.69°, 1.08°, and 1.47° of visual angle (measured along the ring circumference) for the three diameters. The diameter of the test ring always matched that of the inducing ring. As in the first experiment, 14 trial blocks were run in each session with the first two treated as practice and excluded from data analysis. The diameter of the inducing ring was changed after the sixth and 10th blocks following a Latin Square design. There were 48 trials in each block, with an additional eight practice trials (which were not analyzed) added at the start of the seventh and 11th blocks. 
The inducing ring duration was always 91 ms and the test ring duration 500 ms. Inducing ring rotation velocities of 250°, 750°, 1500°, and 2250°/s were presented within each block in a quasirandom order. For the corresponding linear velocities see Table 3. The combination of two test ring states (present vs. absent), four inducing ring angular velocities, and three diameters produced 24 experimental conditions. There were 72 trials in each of these conditions: 36 with a clockwise inducing ring rotation and 36 in counterclockwise rotation. 
Results
Results and confidence intervals for all the conditions of Experiment 2 are presented as d′ values in Figure 4A. Figure 4B shows the relationship between the MBE and temporal frequency. Mean percentage correct performance values are presented in Appendix Table A3. As in Experiment 1, separate ANOVAs were calculated for the test ring present and test ring absent conditions. These ANOVAs are presented in Appendix Table A4
Figure 4
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 2 as a function of inducing ring velocity and ring diameter in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% confidence intervals. Points and confidence intervals are slightly offset horizontally to improve their visibility.
Figure 4
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 2 as a function of inducing ring velocity and ring diameter in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% confidence intervals. Points and confidence intervals are slightly offset horizontally to improve their visibility.
Inducing ring only
When only the inducing ring was presented, the across subject sensitivity (d′) to the rotation direction was examined with a 4 × 3 ANOVA with Velocity and Diameter as the factors. The main effect of Velocity was significant, but we found no significant main effect of Diameter and no significant interaction between Velocity and Diameter (Appendix Table A4). The effect of Velocity followed the pattern observed in Experiment 1, as shown in Figure 4A. Sensitivity rates were high in the 250°/s condition, abruptly dropped in the 750°/s condition, and fell to chance levels in the two high velocity conditions. 
Twelve one-tailed t tests (with a Bonferroni-corrected alpha level of 0.0042) confirmed the consistent above chance performance in the 250°/s velocity conditions (p < 0.0001 in all cases) and chance performance in the high velocity 1500° and 2250°/s conditions. In the 750°/s condition, d′ was slightly but significantly greater than zero with the two larger ring diameters (d′ = 0.48 and d′ = 0.33, p = 0.0013 and p = 0.0037, for the 5.5° and 7.5° rings, respectively). Performance did not differ from chance for any ring diameter in the two high velocity conditions. Except for the 7.5° ring diameter in the 2250°/s condition (p = 0.0073), which we regard as a false positive, this was the case even when we employed an uncorrected 0.05 significance level. 
Inducing ring + test ring
Similar analyses were carried out for the inducing ring + test ring conditions to address the characteristics of the MBE. As in Experiment 1, trials with 250°/s velocity were excluded from these analyses because at this velocity discrimination of the inducing ring rotation did not depend upon the test ring presentation. In a 3 × 3 ANOVA with Velocity and Diameter as factors, the main effect of both these factors was significant but they did not interact (Appendix Table A4). 
As in Experiment 1, the pattern of the data is very different from that found when no test ring was present. As expected, mean sensitivity declined with increasing angular velocities in a relatively linear fashion (d′ = 2.39, d′ = 1.28, and d′ = 0.74 at 750°, 1500°, and 2250°/s, respectively). In addition, as the ring diameter increased the MBE grew stronger (d′ = 1.00, d′ = 1.46, and d′ = 1.95 at 3.5°, 5.5°, and 7.5° of visual angle). However, despite the fact that the 3.5° diameter ring produced the lowest accuracy rates at all velocities, only with the 2250°/s velocity did that rate come close to chance. The reliability of the effect of ring diameter was confirmed by Bonferroni-corrected two-tailed posthoc t tests (α = 0.0056) which indicated that at all three velocities the MBE was significantly weaker with the 3.5° ring diameter than the 7.5° ring diameter. Additionally, the MBE was weaker with the 5.5° ring diameter than with 7.5° ring diameter in the 750°/s condition and weaker with 3.5° ring diameter than the 5.5° diameter in the 2250°/s condition. 
Discussion: Experiment 2
In Experiment 2 we found that the MBE depends on the diameter of the inducing and test ring, with a larger ring diameter producing a stronger MBE. Increasing the diameter of the rings both increased the retinal eccentricity of the points that formed the rings and increased the linear velocity of the points as the inducing ring rotated. The increase in linear velocity produced by increases in the inducing ring's angular velocity is reliably associated with a decrease in the MBE. Therefore, it would be reasonable to expect any effect of the increase produced by the larger ring diameter would also be a reduction of the MBE. Since we observed the reverse—larger ring diameters increased the MBE—we think it is likely that the increase in retinal eccentricity was responsible for this effect. While it is possible that the increase in velocity did in fact act to reduce the MBE but this reduction was overwhelmed by an opposing effect produced by the eccentricity change, our data suggest that the increase in linear velocity had no effect at all. When we calculated the linear velocity for each combination of diameter and angular velocity (see Table 3) and correlated these rates with observers' mean sensitivity (d′) no significant relationship between the MBE and linear velocity was found (r = −0.41, p > 0.05). 
The ring diameters we employed placed the ring circumferences in two anatomically distinct regions of the retina. Because the foveal region has an anatomical diameter of approximately 5° (Millodot, 2018; Schubert, 2014; Wandell, 1995), the inducing and tests rings were foveal in the 3.5° diameter condition but parafoveal in the 7.5° condition. Therefore, the increase in the MBE with increasing ring size might be attributable to the para- and perifoveal areas being more capable of processing rapid motions due to the increased dominance of transient cells in these regions relative to the fovea (for Y cell dominance in cats, see Hoffmann, Stone, & Sherman, 1972; for the M cell increase in monkeys, see Schein & de Monasterio, 1987; on the implications for human perception see Breitmeyer & Ganz, 1976). While there is data indicating that sensitivity to motion is similar in the fovea and more peripheral regions of the retina (McKee & Nakayama, 1984) or that the fovea is actually more sensitive to motion than the periphery (Finlay, 1982), Sekuler (1975) concluded that the periphery has a distinct preference for high rates of temporal modulation. One can construe this as a shift in the temporal frequency response function for moving stimuli towards higher values as retinal eccentricity increases. Along with other evidence, Sekuler notes that when Armstrong (reported in Sekuler, 1975) presented targets with linear velocities up to 100°/s on an oscilloscope, the motion of these targets was only perceived when the target was 10° eccentric. Baker and Braddick (1985) argue, based on an increase of the spatial limit of apparent motion (dmax) with eccentricity, that high velocity motions that are not visible in central vision can become visible in the periphery of the retina. The argument that the periphery of the retina is sensitive for high temporal modulation is further supported by findings from flicker research. Tyler (1985), for example, reports the peripheral CFF can be substantially higher than the foveal CFF (see also Hartmann et al., 1979; and Tyler, 1987). If the temporal frequency is a primary determinant of the strength of the MBE, as our Experiment 1 results suggest (see also Figure 4B), these findings make a higher temporal sensitivity in the peripheral retina a likely candidate for the reported effect of ring diameter. 
There is, however, an additional factor that needs to be considered. In both Experiments 1 and 2 the stimulus conditions that led to an increase in the MBE were associated with an increase in the spatial separation between the points that formed the inducing and test rings. This finding raises the possibility that the observed effects were attributable to this increased point separation. In Experiment 3 we examined this possibility. 
Experiment 3
In Experiment 1 we found reducing the number of points in the inducing and test rings increased the strength of the MBE. In Experiment 2 we found that enlarging the rings increased the MBE. The fact that both these manipulations increased the spatial separation between the points forming these rings raised the possibility that this change in separation was mediating the change in performance in both cases. In the discussion of Experiment 1 we noted that the spatial frequency components associated with the separation of the points that form the inducing ring do not seem likely to directly impact the MBE because when its rotation is rapid, visual persistence transforms the ring into a visually continuous outline. Nevertheless, because point spacing was a confounding variable in both Experiments 1 and 2 we felt its potential contribution to the experimental outcomes needed to be empirically evaluated. This was done in Experiment 3 by covarying the diameter and the number of points in the inducing and test rings in a manner that caused the spacing between the points forming the rings to remain constant. Experiment 3 is therefore akin to Experiment 1 in that at each tested angular velocity, temporal frequency is varied, but this is accomplished while holding point separations constant. 
Method
Fourteen trial blocks of 48 trials, half with the inducing ring followed by the test ring, were run in each of three sessions, following the sequence and procedures described for Experiment 1. Stimuli were similar to those employed in Experiment 1 with the following modifications. As in the first Experiment, 12, 16, or 20 equally spaced points formed the circumference of the inducing and test rings, with changes in the point number at the start of the seventh and 11th block following a Latin-square design. However, these changes were coupled with changes to the inducing ring diameter so that an interpoint separation of 1.08° of visual angle on the circumference of the ring was maintained. The required ring diameters were 4.125°, 5.5°, and 6.875° of visual angle in the 12, 16, and 20 point conditions, respectively (see Figure 2). As in Experiments 1 and 2, clockwise and counterclockwise inducing ring rotation directions with angular velocities of 250°, 750°, 1500°, and 2250°/s (2500°/s in the 12 point condition) were presented in a quasirandom fashion. 
The combination of two test ring states (present vs. absent), four inducing ring angular velocities, and three point-number/diameters produced 24 experimental conditions. There were 72 trials in each of these conditions: 36 with a clockwise inducing ring rotation and 36 in counterclockwise rotation. 
Results
Results and confidence intervals for all conditions of Experiment 3 are presented in Figure 5A. Figure 5B shows the relationship between the MBE and temporal frequency. Mean percent correct rates are presented Appendix Table A5. As in the previous experiments, separate ANOVAs were calculated for the test ring present and test ring absent conditions. These analyses are presented in Appendix Table A6
Figure 5
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 3 as a function of inducing ring velocity and point-number/diameter combination in the inducing ring only (the dashed line) and inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Figure 5
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 3 as a function of inducing ring velocity and point-number/diameter combination in the inducing ring only (the dashed line) and inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Inducing ring only
When the inducing ring was presented alone, the effect of Velocity replicated the pattern found in the previous experiments with mean sensitivity declining sharply to chance levels as the inducing ring Velocity increased. A 4 × 3 ANOVA with Velocity and Point-number/diameter as factors revealed a significant main effect of Velocity and a significant main effect of Point-number/diameter (Appendix Table A6). The effect of Point-number/diameter reflects a decline in performance as Point-number/diameter increased. However, while sufficient to produce a significant main effect in the ANOVA, the effect of Point-number/diameter was quite small and could not be confirmed at any velocity with Bonferroni-corrected two-tailed posthoc t tests (α = 0.0042). When we employed an uncorrected 0.05 significance level, point-number/diameter did produce significant posthoc d′ differences but only when the inducing ring velocity was 250°/s, with performance in the 20 point condition poorer than in the 12 and 16 point conditions. Posthoc comparisons at the higher velocities all failed to yield significant differences even at 0.05 level. 
Bonferroni-corrected one-tailed t tests (α = 0.0042) confirmed that sensitivity was significantly greater than zero with every point-number when the angular velocity was 250°/s (p < 0.0001 in all cases), but did not differ from chance for any point number at any of the higher velocities. When we employed an uncorrected 0.05 significance level, d′ exceeded 0 in every 750°/s velocity condition, but at higher velocities the only case of a data point exceeding chance was in the 16 point condition when the ring diameter was 5.5° and velocity was 2250°/s, t(11) = 2.28, p = 0.0217. We regard this outcome as a probable false positive. 
Inducing ring + test ring
To evaluate the effect of point separations on the MBE, a similar ANOVA was performed on data of conditions where the test ring followed the inducing ring (Appendix Table A6). As in the first two experiments, the 250°/s condition was excluded from these analyses because it was not appropriate for measuring the MBE, so there were three Velocity and three Point-number/diameter conditions. As expected, the main effect of Velocity was significant with a profile similar to that observed in the inducing + test ring conditions of the previous experiments. Sensitivity to the inducing ring direction was high in the 750°/s condition, and declined relatively smoothly with increasing angular velocities (d′ = 2.51, d′ = 1.36, and d′ = 0.82 at 750°, 1500°, and 2250°/2500°/s velocities, respectively), but remained well above chance at the two highest velocity. However, there was no significant main effect of Point-number/diameter and no significant interaction between Velocity and Point-number/diameter. 
Thus, despite the fact that point-number (Experiment 1) and diameter (Experiment 2) both modulated the strength of the MBE when presented separately, when covaried so that the interpoint distance remained constant, they seemed to have no effect. This result supports the hypothesis that the interpoint distance was the critical variable in the previous experiments, although, as discussed below, this hypothesis raises problems and other interpretations are feasible. 
Discussion: Experiment 3
Experiment 1 suggested that the increases in the temporal frequency of activations on the inducing ring circumference might be a critical determinant of the MBE's decline with increasing angular velocity. In Experiment 2 it was found that increasing the retinal eccentricity of the inducing ring circumference enhanced the MBE. In Experiment 3, when these variables were covaried to keep the separation between the points that formed the inducing and test rings constant, the magnitude of the MBE did not change. As this manipulation required the pairing of an increase in the ring diameter with an increase in the number of points forming the rings, the simplest way to account for this result is to posit that the enhancement produced by the increase in diameter and decrement produced by the increase in point number simply canceled. This account seems unlikely because it requires that the cancelation be coincidentally perfect when the interplay of point-number and diameter also holds the interpoint distance constant. We return to this consideration in our final discussion. Here, we note that the alternative hypothesis that the distance between the inducing ring points was directly responsible for the effects observed in Experiments 1 and 2 also needs to be considered. 
Accepting this premise, however, implies that the changes in temporal frequency produced by our point number manipulations in Experiment 3 played little or no role in determining the MBE strength (see Figure 5B). This implication does not fit well with the existing evidence (e.g., Burr & Ross, 1982) that motion perception is in general dependent on the processing of temporal frequencies. There must, in fact, be some temporal frequency at which neural persistence makes the neural responses to intensity changes too shallow to differentiate a rotating from a stationary ring. The absence of an effect of point-number in Experiment 3 suggests that with temporal frequencies as high as 125 Hz (see Table 2) the flattening of neural modulations is not yet sufficient to degrade the MBE. This is a surprising outcome. Moreover, in combination with the implication of Experiment 2 that linear velocity does not affect the MBE, it leaves the relatively steady decline in the MBE with increasing angular velocity unaccounted for. The results of Experiment 3 therefore raise puzzling issues that need to be addressed. We will return to these issues in our final discussion. 
Experiment 4
In Experiments 1 and 3, a side effect of varying the number of points in the inducing ring was a change in the time-averaged luminance of the inducing ring and test ring points. Because luminance changes flicker sensitivity (Kelly, 1961), these luminance variations could in principle have contributed to the observed effects of point-number. Experiment 4 evaluated this possibility by varying the luminance of inducing and test rings to determine what effect, if any, these luminance changes might have had on the experimental outcomes. Two experimental sessions were run on separate days. In Session 1, the luminance of the inducing ring points was varied in four steps and the test ring luminance was held constant. In session 2, there were two inducing ring luminances combined with four test ring luminances. Across the two sessions we employed luminance levels that encompassed all the measured test ring point luminances (1.20 to 1.50 cd/m2) and time averaged inducing ring luminances in the 1500° and 2250°/2500°/s angular velocity conditions of Experiments 1 and 3 (0.072 to 0.149 cd/m2; cf. Table 1). 
Method: Session 1
Eighteen blocks of 32 trials were run during the first session. Participants were presented with inducing rings and test rings like those employed in the previous experiments (see Figure 1). The inducing rings were always constructed from sixteen equally spaced points, 5.5° in diameter, and presented for 91 ms. As in the previous experiments, the test ring was presented for 500 ms. There were four time-averaged inducing ring luminances: 0.073, 0.094, 0.118, and 0.188 cd/m2. The luminance was constant within each block but varied across the trial blocks using a repeated Latin Square design. The luminance of the test ring points was held constant at 1.088 cd/m2. Two inducing ring velocities were presented within each block: On 25% of the trials (16 total in each condition) there was a slow 250°/s rotation, and on 75% of the trials (48 in each condition) there was a rapid 1500°/s ring rotation. No test ring was presented in the initial block; then blocks in which the test ring was presented and blocks with no test ring were alternated. The first two blocks and the slow rotation rate trials were treated as practice and not analyzed. 
Method: Session 2
Seventeen trial blocks with 32 trials per block were run. The test ring luminance was varied in four steps (0.733, 1.088, 1.441, and 1.779 cd/m2,) and each test ring luminance level was presented with two inducing ring luminance levels (0.073 and 0.188 cd/m2). The inducing ring-test ring luminance combinations were varied quasirandomly between blocks. Two inducing ring velocities were presented within each block: On 25% of the trials (16 total in each condition) there was a slow 250°/s ring rotation, and on 75% of the trials (48 in each condition) there was a rapid 1500°/s ring rotation. The first block and the slow rotation rate trials were treated as practice and were not analyzed. 
Results
Results and confidence intervals for all conditions of Session 1 are presented in Figure 6. The across subject sensitivity (d′) to the rotation direction was calculated for each condition and examined with a 2 × 4 ANOVA with Test ring (inducing ring only vs. inducing ring + test ring condition) and Inducing ring luminance as the factors. As expected, the main effect of Test ring was significant, F(1, 11) = 100.37, p < 0.001. The mean sensitivity was higher when the inducing ring was followed by the test ring (d′ = 1.67 compared to d′ = 0.25) but we found no significant main effect of Inducing ring luminance, F(3, 33) = 0.34, p = 0.759, and no interaction between Test ring and Inducing ring luminance, F(3, 33) = 0.06, p = 0.944. As can be seen in Figure 6, accuracy rates were stable across the full range of inducing ring luminances. This is true both when the inducing ring is presented on its own (54.1% to 55.4%) and when it is followed by the test ring (77.1% to 78.3%). 
Figure 6
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 1) as a function of time averaged inducing ring luminance levels in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). The error bars show 95% CI. The solid gray line indicates the chance level of accuracy (d′ = 0).
Figure 6
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 1) as a function of time averaged inducing ring luminance levels in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). The error bars show 95% CI. The solid gray line indicates the chance level of accuracy (d′ = 0).
Results and confidence intervals for session 2 are shown in Figure 7. We once again performed a 2 × 4 ANOVA with Inducing ring luminance and Test ring luminance as factors. Neither of these factors had a significant main effect and there was no significant interaction between them: F(1, 11) = 0.11, p = 0.751 for the main effect of Inducing ring luminance, F(3, 33) = 0.95, p = 0.398 for the main effect of Test ring luminance, and F(3, 33) = 2.82, p = 0.064 for the interaction. The nonsignificant interaction might point to a slight trend towards larger MBEs in conditions in which inducing ring and test ring have comparable luminances. Overall, however, Figure 7 shows that neither the inducing ring nor test ring luminance had any effect on accuracy rates. 
Figure 7
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 2) as a function of test ring luminance levels with each of two time averaged inducing ring luminance levels. The error bars show 95% CI. Only the inducing ring + test ring conditions are shown. The solid gray line indicates the chance level of accuracy (d′ = 0).
Figure 7
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 2) as a function of test ring luminance levels with each of two time averaged inducing ring luminance levels. The error bars show 95% CI. Only the inducing ring + test ring conditions are shown. The solid gray line indicates the chance level of accuracy (d′ = 0).
Discussion: Experiment 4
Although the luminance variations associated with the number of points displayed could in principle have confounded the results of Experiments 1 and 3, in Experiment 4 we found that luminance changes that encompassed the full range of these variations had no effect on participants' performance. We conclude that luminance was not a confounding factor. It needs to be noted, though, that the range of luminance levels we investigated was quite small. It is therefore entirely possible that luminance variations over a wider range could modulate the MBE. 
General discussion
When participants judged the rotation direction of a rapidly rotating ring of points (the inducing ring), they performed at chance when the investigated angular velocities were 1500°/s and higher. However, when an additional stationary ring (the test ring) was presented after a short delay, an illusory rotation of this test ring allowed them to report the inducing ring's rotation direction at rates substantially better than chance even at the highest velocity employed (2500°/s). This effect was previously reported by Mattler and Fendrich (2010) and termed the Motion Bridging Effect. The experiments reported here examine the effect of two variables on the MBE: the number of points that made up the inducing and test rings and the diameter of those rings. We found an increase in the MBE (measured by the accuracy of reports of the inducing ring direction) when the number of points used to construct the rings was reduced. In addition, we found an increase in the MBE when the diameter of the rings was increased. However, both of these manipulations influenced the distance between the points that formed the inducing and test rings. When this distance was held constant by covarying the point number and ring diameter, no effect on the MBE was observed. We will speculate that processes related to the perception of both real and apparent motion may be contributing to this pattern of results. 
Burr and Ross (1982) have proposed that all motion perception is limited by the temporal frequency of intensity modulations on the retina. If the temporal frequency is too high (e.g., over 30 Hz for sinusoidal gratings), the spatial-temporal properties of the stimulus cannot be translated into visible motion. If this argument is extended to the MBE, then both the decline in the MBE with increasing angular velocity and the decline with an increase in the number of points used to form the inducing ring can be explained. However in the case of the MBE, the temporal frequency limit must be at least 125 Hz, which is well above the rates that normally mediate motion detection and substantially above the human flicker fusion frequency. In addition, to account for the performance gain in the MBE when the diameter of the rings is increased, one must also posit that its dependence on temporal frequency is scaled by retinal eccentricity, with high temporal frequencies more readily processed at more peripheral positions. This scaling can also explain the stability of the MBE in Experiment 3 if one accepts that the decline in the MBE produced by the increase in point-number is canceled by the improved processing of high temporal frequencies as the ring diameter is increased. However, this explanation requires that the decline and improvement in the MBE balance one another almost perfectly when a combined effect of these variables also acts to hold the separation between the points that form the inducing ring constant. The presumption that these effect null-points co-occur by chance strains the credibility of this account. 
Apparent motion
An alternative approach to explaining the observed pattern of outcomes is to propose that the MBE is modulated by the spatial separation between the test ring rather than inducing ring points. In our experiments, the separation of the points that formed the test ring always matched the separation of the points in the inducing ring, and because the test ring was always veridically stationary, this separation was visually salient irrespective of the inducing ring velocity. We speculate that an essential component of the MBE is an apparent motion percept generated by the inducing ring to test ring transition. 
In their 2010 paper, Mattler and Fendrich (2010) linked the MBE to apparent motion because the MBE consists of an intermediate visual percept that is synthesized from the sequence of two rings, one with a continuous outline and the other constructed of points. They note that a synthesis of the two rings by an updating process could account for both the deceleration when the continuous ring precedes the pointed ring and the acceleration when the pointed ring precedes the continuous outlined ring. Here we consider the link to apparent motion in more detail. 
It has long been known that the spatial separation between the successive stimuli used to produce apparent motion can modulate the motion percepts (e.g., Korte, 1915; Wertheimer, 1912). Korte proposed that as the spatial distance between two stimuli increases their temporal distance also needs to increase to sustain an apparent motion percept (Korte's third law). More recent proposals have argued that depending on the presentation conditions, the relationship between spatial and temporal distance needed to maintain optimal apparent motion can be either a positive coupling or a tradeoff, so that an increase in spatial distance must be coupled with a decrease in temporal distance (Gepshtein & Kubovy, 2007). 
A phenomenon related to apparent motion in which there is also an interplay of spatial and temporal factors has been termed Singularbewegung (Wertheimer, 1912), polarized gamma movement (Kanizsa, 1979), or the line-motion illusion (LMI; Downing & Treisman, 1997). In LMI experiments, a square flashed on a screen precedes the presentation of an adjacent line that appears as smoothly expanding outward from the square. Contemporary research on the LMI was initiated by Hikosaka, Miyauchi, and Shimojo (1993) who proposed that the square draws attention to its spatial location and facilitates neural processing at the end of the line adjacent to that location leading to the precept of expansion. Downing and Treisman (1997) discounted the attentional explanation of the LMI by showing a complementary phenomenon in which the line appears to shrink into a square. This result cannot be accounted for by attention because no cue to bias attention to a specific location was presented. The authors therefore interpret the LMI as an apparent motion linking two successive states that are treated as representations of a single object, making the LMI an instance of what Tse, Cavanagh, and Nakayama (1998) have termed “transformational” apparent motions. The LMI, like more traditional instances of apparent motion, exhibits spatial distance dependencies: Ratings of perceived velocity are increased when the line length is increased (Hubbard & Ruppel, 2011). 
There are noteworthy similarities between the displays used to demonstrate the LMI and our MBE displays. In the case of the LMI, there is an immediate transition between a stationary line and one or more stationary markers, and a smooth illusory motion links the two stationary states, ensuring perceptual continuity. In case of the MBE, there is a transition from a perceptually stationary outline ring to multiple veridically stationary points. We propose that when this transition happens, the segments of the inducing ring flanking the points of the test ring contract into those points. 
Given this hypothesis, a question that arises is why observers see all the test ring points rotate together, rather than seeing the inducing ring segments adjacent to each of the test ring points, contract locally into those points. One possible explanation for the common direction of the test ring points in the MBE is that it represents an instance of the apparent motion yoking effect initially reported by Ramachandran and Anstis (1983). These investigators presented bistable dots pairs that could be seen to be undergoing either a horizontal or vertical oscillation when repetitively switched between their alternative states. When groups of these dot pairs are presented and switched simultaneously, the perceived direction of the oscillation is identical in all the pairs. In addition, other investigations have shown that when multiple line-motion stimuli are presented together, their motion signals can pool and be integrated to form a joint global motion percept (Tang, Dickinson, Visser, Edwards, & Badcock, 2013; Tse & Logothetis, 2002). These yoking and integration effects could result in all the inducing ring segments contracting in the same direction to generate the rotation percept. 
The hypothesis that the test ring motion is a case of transformational apparent motion leaves, however, a core aspect of the MBE unexplained: It does not explain why the test ring generally rotates in the same direction as the inducing ring's invisible rotation. The transition from the continuous inducing ring to the point-defined test ring is ambiguous in respect to which direction the line will transform, so one might expect the test ring's rotation direction to be random. The fact that its direction usually matches the direction of the inducing ring indicates the inducing ring direction must have been encoded somewhere in the visual system and acted to bias the perceived rotation of the test ring. As we have previously noted, this implies the registration of temporal frequencies well beyond the range that the human visual system is conventionally thought to be able to process. The hypothesis that the test ring motion is a form of transformational apparent motion does not alter these implications. The MBE would therefore represent a case where a motion that is inherently inaccessible to consciousness is nevertheless modifying subsequent percepts. We note that if the biasing effect of the inducing ring acts on every test ring point this would support the percept that these points all move in a common direction. Moreover, if the ability of the inducing ring to bias the direction of the test ring motion varies with the spacing of the test ring points, this could account for the potential dependence of the MBE on that spacing. While this idea is speculative, the spatial dependencies found with apparent motion and the LMI lend credence to the supposition that the perceived rotation of the test ring could be a function of spacing of the points in that ring. 
Conscious and unconscious motion processing
The processing of movement is generally thought to rest upon direction selective cells that have been found in the visual cortex of macaque and owl monkeys (Albright, 1984; Zeki, 1980). A well-known phenomenon that can be related to the reduced responsiveness of direction selective cells in V1 is the motion aftereffect (MAE; see Anstis, Verstraten, & Mather, 1998, for a review). The MAE is characterized by an illusory percept of a motion in the reverse direction of a previously inspected actual linear or rotary motion. It is generally attributed to a damping by neural adaptation of the activity of the direction sensitive cells that mediate the perception of the inspected motion, creating an imbalance that favors cells tuned to motion in opposite direction. The rotating dot displays used in the present experiments can produce a MAE if the rate of rotation is sufficiently slow (Mattler & Fendrich, 2010) but we think the origins of MAE and MBE are likely to be quite different. Besides the obvious difference in the direction of the perceived illusory motion, the build-up and decay of the MBE is far more rapid than the MAE. The MAE typically builds during extended exposures to the adapting stimulus and slowly fades away (Hershenson, 1989, 1993). In contrast, Mattler and Fendrich found the MBE occurred with exposures to the inducing ring as brief as 30 ms and reached its maximum strength after only 60 ms. While the duration of the MBE was not formally measured, they note that it generally “braked to a standstill” in the course of their 500 ms test ring presentation. Additionally the temporal frequencies of the stimuli presented in MBE experiments clearly exceed the temporal frequency limits of adapting stimuli in MAE experiments. The duration of the MAE is highest when the adapter frequency is 1–5 Hz, begins to rapidly decline at 20 Hz, and when the adapter is presented with a temporal frequency of 50 Hz, no MAE is seen (Pantle, 1974). These considerations lead us to conclude that the MBE and MAE are not related effects. 
Studies of the MAE and other phenomena, however, show (as does the MBE) that movement can be processed without awareness of the motion direction. In a functional MRI study, Moutoussis and Zeki (2006) found a drifting grating at 15° eccentricity generated motion specific activations in brain areas as high as MT. Surprisingly, this was also the case when the grating was crowded by two additional flickering gratings and participants could no longer judge the direction of movement of the target grating. Additional evidence is provided by binocular rivalry studies demonstrating that the duration of the MAE does not depend on whether the drifting adaptation grating is binocularly suppressed (Lehmkuhle & Fox, 1974, O'Shea & Crassini, 1981). Other studies have shown that the MAE still occurs in reduced form when the adapter is rendered unconscious by continuous flash suppression (Kaunitz, Fracasso, & Melcher, 2011; Maruya et al., 2008). 
Apparent motion has also been investigated in the context of binocular rivalry: Wiesenfelder and Blake (1991) showed that subjects still perceived apparent motion when the first frame of a two-frame sequence was suppressed. Blanco and Soto (2009) have reported an instance of the LMI in which the flashed stimulus that generates the illusion is not consciously visible. In their experiment, a gray circle was displayed in one of four positions, followed by a mask in all four of these positions. When only the circle and masks were shown, participants were unable to report the position where the circle had been presented. However, when a gray line was presented subsequent to the masks, it was perceived as moving outward from the circle's location. All of these cases differ from the MBE, however, in that they involve an inherently detectable stimulus that only fails to reach consciousness because it is masked or suppressed. 
An instance where an inherently invisible moving stimulus has been shown to have perceptual consequences has, however, been reported by Glasser, Tsui, Pack, and Tadin (2011), but in this case the motion was undetectable because of its brevity rather than velocity. In this study a high contrast 1 cycle/° sinusoidal grating drifted across a 16° diameter Gabor patch at 15°/s for intervals as short as 25 ms. Even when the presentations were too brief to allow observers to identify the drift direction of this grating, they produced a brief but detectable motion aftereffect in a subsequently low contrast stationary grating. However, this aftereffect corresponds to a classic MAE—a perceived motion in the direction opposite the preceding real motion—whereas the aftereffect in the case of the MBE is in the same direction as the preceding real motion. In addition, the aftereffect reported by Glasser et al. decreased steadily as the ISI between the moving and subsequent stationary stimulus grew larger than zero. The MBE, in contrast, increases with ISI from zero up to 90 ms (Mattler & Fendrich, 2010). These differences suggest that the neural mechanisms responsible for the aftereffects reported by Glasser et al. and those responsible for the MBE are fundamentally different. 
In addition, global motion percepts that would otherwise be apparent become imperceptible when the number of motion directions shown becomes too high (e.g., Greenwood & Edwards, 2009). Lee and Lu (2014) demonstrated that such imperceptible global motions can elicit a motion aftereffect. Observers viewed a doughnut shaped field of drifting Gabor patches. Subsets of these Gabors conveyed global motion directions (e.g., expansion) but because a multiplicity of motion directions was shown, these global motions were not consciously detectable. Nevertheless, when a set of stationary Gabors replaced a subset of the drifting Gabors that had signified a specific global motion direction, a motion aftereffect that reversed the global motion direction was seen. Like the MBE, the MAE reported by Lee and Lu demonstrates the processing of motion information that is not consciously seen due to visual system processing limitations. However, in the case of the global motions addressed by Lee and Lu, these limitations occur at a stage where visible motions are integrated. Although observers are not able to link the displayed Gabors to derive a conscious global motion percept, the grating drift within each individual Gabor can be readily seen. In contrast, the motion responsible for the MBE fails to reach consciousness because of limitations in ability of the visual system to process a primary sensory attribute—luminance modulations. The MBE demonstrates that motion information can be derived from stimuli with temporal frequencies that have generally been regarded as beyond the range of human sensory encoding. 
We have argued that two processes contribute to the MBE: (a) a transformational apparent motion generated by the transition from the perceptually continuous outline of the rapidly rotating inducing ring to the discrete stationary points of the test ring, and (b) a direction biasing process mediated by the inducing ring motion. This view of the MBE makes an interesting prediction. If the motion seen when the inducing ring is replaced by the test ring is a form of transformational apparent motion, then one would expect a motion percept to be present even if the inducing ring is a genuinely stationary outline. Under these conditions there can, of course, be no MBE because the MBE is defined by congruence between the inducing and test ring motion directions. Nevertheless, if observers see a coherent rotational motion of the test ring points similar to that observed with the MBE displays, this effect would argue strongly that the motion seen in the case of the MBE displays is mediated by a transformational component. On the other hand, if no motion or a directionally incoherent motion is seen, this would argue against the role of a transformational component in the MBE. In addition, studies in which the character of the inducing stimulus is changed could provide further information about the generality of the MBE and the suggested connection to apparent motion. Displays which produce linear or radial apparent motions might also be capable of giving rise to MBE like percepts. We are currently exploring these possibilities. 
Summary
The MBE demonstrates the neutral encoding of direction information from a ring of points that spins so rapidly its rotation is not consciously detectable (Mattler & Fendrich, 2010). When this spinning ring (the inducing ring) is followed by a stationary ring of points (the test ring), observers report seeing the test ring to rotate. This rotation is primarily in the same direction as the inducing ring, although this direction is not detectable when the inducing ring is presented by itself. The present study reveals that the MBE, indexed by the congruence of the inducing and test ring's rotation directions, weakens as number of points used to form the rings is increased. This finding supports the premise that the MBE is limited by the temporal frequency of point presentations produced as the points of the inducing ring advance around the ring perimeter. In addition, we found the MBE strengthens as the diameter of the rings is increased, which can be attributed to the more efficient registration of high temporal frequencies at more peripheral retina locations. However, when we covaried the number of points and the ring diameter in a manner that maintained a constant separation between the points that formed the inducing and test rings, the strength of the MBE was also constant. To account for this pattern of results, we propose that the MBE reflects the joint action of two processes, one that registers the direction of the inducing ring's rotation (and declines as a function of temporal frequency) and one that generates the illusory test ring spin (and is sensitive to the spacing between the points that form the test ring). We further propose that the second process represents an instance of a transformational apparent motion, generated by the transition from the inducing ring's perceptually continuous outline to the discrete stationary points of the test ring. We think the directional congruence that defines the MBE is produced by a biasing of the test ring direction by registered inducing ring direction information. Finally, we suggest the spacing of the test ring points (which was always the same as the spacing of the inducing ring points) may modulate the ability of the encoded inducing ring direction information to influence the perceived test ring direction, accounting for the dependence of the MBE on the point separations. Research to evaluate this speculative analysis is now in progress. 
Acknowledgments
The authors thank Johannes Bommes and Marilena Reinhardt for their valuable support in recruiting participants and collecting data. 
We acknowledge support by the Open Access Publication Funds of Goettingen University. 
Data from Experiment 2 were presented at the 39th European Conference on Visual Perception, Barcelona, Spain. 
Commercial relationships: none. 
Corresponding author: Uwe Mattler. 
Address: Department of Experimental Psychology, University of Goettingen, Goettingen, Germany. 
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Appendix
Table A1
 
Mean percentage of correct direction reports in Experiment 1.
Table A1
 
Mean percentage of correct direction reports in Experiment 1.
Table A2
 
ANOVA outcomes for Experiment 1. dfn = numerator df; dfd = denominator df.
Table A2
 
ANOVA outcomes for Experiment 1. dfn = numerator df; dfd = denominator df.
Table A3
 
Mean percentage of correct direction reports in Experiment 2.
Table A3
 
Mean percentage of correct direction reports in Experiment 2.
Table A4
 
ANOVA outcomes for Experiment 2. dfn = numerator df; dfd = denominator df.
Table A4
 
ANOVA outcomes for Experiment 2. dfn = numerator df; dfd = denominator df.
Table A5
 
Mean percentage of correct direction reports in Experiment 3.
Table A5
 
Mean percentage of correct direction reports in Experiment 3.
Table A6
 
ANOVA outcomes for Experiment 3. dfn = numerator df; dfd = denominator df.
Table A6
 
ANOVA outcomes for Experiment 3. dfn = numerator df; dfd = denominator df.
Figure 1
 
Display sequence in Experiments 14. On a given trial the inducing ring rotated either clockwise (as shown in the figure) or counterclockwise. Note the points on the oscilloscope were bright on a dark background.
Figure 1
 
Display sequence in Experiments 14. On a given trial the inducing ring rotated either clockwise (as shown in the figure) or counterclockwise. Note the points on the oscilloscope were bright on a dark background.
Figure 2
 
The various display conditions in Experiment 13. When the diameter of the rings was varied, the visual angle of its diameter is indicated. Note that the sizes of the rings depicted here are not precisely scaled to their sizes in the actual displays. Also note the points on the oscilloscope were bright on a dark background.
Figure 2
 
The various display conditions in Experiment 13. When the diameter of the rings was varied, the visual angle of its diameter is indicated. Note that the sizes of the rings depicted here are not precisely scaled to their sizes in the actual displays. Also note the points on the oscilloscope were bright on a dark background.
Figure 3
 
(A) Mean sensitivity (d′) for 11 participants of Experiment 1 as a function of inducing ring velocity and point-number in the inducing ring only conditions (the dashed lines) and the inducing ring + test ring conditions (the solid lines). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Figure 3
 
(A) Mean sensitivity (d′) for 11 participants of Experiment 1 as a function of inducing ring velocity and point-number in the inducing ring only conditions (the dashed lines) and the inducing ring + test ring conditions (the solid lines). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Figure 4
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 2 as a function of inducing ring velocity and ring diameter in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% confidence intervals. Points and confidence intervals are slightly offset horizontally to improve their visibility.
Figure 4
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 2 as a function of inducing ring velocity and ring diameter in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% confidence intervals. Points and confidence intervals are slightly offset horizontally to improve their visibility.
Figure 5
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 3 as a function of inducing ring velocity and point-number/diameter combination in the inducing ring only (the dashed line) and inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Figure 5
 
(A) Mean sensitivity (d′) for 12 participants of Experiment 3 as a function of inducing ring velocity and point-number/diameter combination in the inducing ring only (the dashed line) and inducing ring + test ring conditions (the solid line). (B) Mean d′ values from the inducing ring + test ring conditions for rotation rates of 750°/s and greater as a function of temporal frequency. The solid gray line indicates the chance level of accuracy (d′ = 0). The error bars show 95% CI. Points and confidence intervals are slightly offset horizontally to improve their visibility in (A).
Figure 6
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 1) as a function of time averaged inducing ring luminance levels in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). The error bars show 95% CI. The solid gray line indicates the chance level of accuracy (d′ = 0).
Figure 6
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 1) as a function of time averaged inducing ring luminance levels in the inducing ring only conditions (the dashed line) and the inducing ring + test ring conditions (the solid line). The error bars show 95% CI. The solid gray line indicates the chance level of accuracy (d′ = 0).
Figure 7
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 2) as a function of test ring luminance levels with each of two time averaged inducing ring luminance levels. The error bars show 95% CI. Only the inducing ring + test ring conditions are shown. The solid gray line indicates the chance level of accuracy (d′ = 0).
Figure 7
 
Mean sensitivity (d′) for 12 participants of Experiment 4 (Session 2) as a function of test ring luminance levels with each of two time averaged inducing ring luminance levels. The error bars show 95% CI. Only the inducing ring + test ring conditions are shown. The solid gray line indicates the chance level of accuracy (d′ = 0).
Table 1
 
Time averaged brightness of inducing ring points in Experiments 1 and 3 in cd/m2. Note: The replication of the point refresh rates in the 250°/s angular velocity conditions necessitated time averaging across flicker rates that were below the flicker fusion frequency.
Table 1
 
Time averaged brightness of inducing ring points in Experiments 1 and 3 in cd/m2. Note: The replication of the point refresh rates in the 250°/s angular velocity conditions necessitated time averaging across flicker rates that were below the flicker fusion frequency.
Table 2
 
Temporal frequency of point presentations at each position along the circumference of the inducing ring in Experiments 1 and 3 in Hz.
Table 2
 
Temporal frequency of point presentations at each position along the circumference of the inducing ring in Experiments 1 and 3 in Hz.
Table 3
 
Linear velocity of the rotating points in the 3 diameter conditions of Experiment 2 in degrees of visual angle per second.
Table 3
 
Linear velocity of the rotating points in the 3 diameter conditions of Experiment 2 in degrees of visual angle per second.
Table A1
 
Mean percentage of correct direction reports in Experiment 1.
Table A1
 
Mean percentage of correct direction reports in Experiment 1.
Table A2
 
ANOVA outcomes for Experiment 1. dfn = numerator df; dfd = denominator df.
Table A2
 
ANOVA outcomes for Experiment 1. dfn = numerator df; dfd = denominator df.
Table A3
 
Mean percentage of correct direction reports in Experiment 2.
Table A3
 
Mean percentage of correct direction reports in Experiment 2.
Table A4
 
ANOVA outcomes for Experiment 2. dfn = numerator df; dfd = denominator df.
Table A4
 
ANOVA outcomes for Experiment 2. dfn = numerator df; dfd = denominator df.
Table A5
 
Mean percentage of correct direction reports in Experiment 3.
Table A5
 
Mean percentage of correct direction reports in Experiment 3.
Table A6
 
ANOVA outcomes for Experiment 3. dfn = numerator df; dfd = denominator df.
Table A6
 
ANOVA outcomes for Experiment 3. dfn = numerator df; dfd = denominator df.
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