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Article  |   February 2011
Spatiotopic apparent motion reveals local variations in space constancy
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Journal of Vision February 2011, Vol.11, 4. doi:10.1167/11.2.4
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      Martin Szinte, Patrick Cavanagh; Spatiotopic apparent motion reveals local variations in space constancy. Journal of Vision 2011;11(2):4. doi: 10.1167/11.2.4.

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Abstract

While participants made 10° horizontal saccades, two dots were presented, one before and one after the saccade. Each dot was presented for 400 ms, the first turned off about 100 ms before, while the second turned on about 100 ms after the saccade. The two dots were separated vertically by 3°, but because of the intervening eye movement, they were also separated horizontally on the retina by an additional 10°. Participants nevertheless reported that the perceived motion was much more vertical than horizontal, suggesting that the trans-saccadic displacement was corrected, at least to some extent, for the retinal displacement caused by the eye movement. The corrections were not exact, however, showing significant biases that corresponded to about 5% of the saccade amplitude. The perceived motion between the probes was tested at 9 different locations and the biases, the deviations from accurate correction, varied significantly across locations. Two control experiments for judgments of position and of verticality of motion without eye movement confirmed that these biases are specific to the correction for the saccade. The local variations in the correction for saccades are consistent with physiological “remapping” proposals for space constancy that individually correct only a few attended targets but are not consistent with global mechanisms that predict the same correction at all locations.

General introduction
In this paper, we examine what happens when an attended target moves at the time of the saccade. Does saccadic suppression (Matin, 1974; Volkmann, 1986) make the displacement invisible (Bridgeman, Hendry, & Stark, 1975)? Otherwise, if we do see the motion, is it determined by the displacement on the retina (retinotopic displacement) or by the displacement in space (spatiotopic displacement)? If the perceived motion is determined by displacement in space, how are the perceived locations corrected for the effects of the eye movements on retinal input? We address these questions with an apparent motion display, where, in its standard version, a salient stimulus is briefly presented at one location and then reappears at a different location, leading to a strong impression of motion. To use apparent motion in the context of saccades, we present the stimulus at its initial location just before the saccade and then present it at a second location just after the saccade. Because of the intervening saccade, the displacement between the two locations on the retina is very different from the displacement in the world (Figure 1). Motion during a saccade, whether of the saccade goal itself (e.g., Bridgeman et al., 1975) or the whole visual scene (Currie, McConkie, Carlson-Radvansky, & Irwin, 2000; McConkie & Currie, 1996), is strongly suppressed. Despite this suppression, a displacement can be seen if it is quite large (1/3 of the saccade amplitude or more, Bridgeman et al., 1975) or if it is orthogonal to the saccade (Niemeier, Crawford, & Tweed, 2003, although see also Bridgeman et al., 1975) or if the displaced probe is not present when the saccade lands but is turned on at least 50 ms after the saccade lands (Deubel, Schneider, & Bridgeman, 1996). In our experiment, we wanted to maximize the chance of seeing motion, so our stimuli had all three of these properties. 
Figure 1
 
Compensation for saccades. (a) Participants were instructed to fixate the green dot that exchanged position with a red dot each 600 ms. While they were fixating to the left, the first black probe (P1) appeared for 400 ms; they then saccaded to the right and a second probe (P2) of the same duration appeared below the first one 200 ms after the offset of P1. (b) On the retina, the first probe falls to the right side of the fovea while the second falls to the left side. (c) To compensate for these effects of the saccade, the visual system corrects the expected location of P1 in the opposite direction to the saccade (red dashed arrow) to obtain C1. If this correction is accurate, motion is perceived in its spatiotopic (vertical motion: from C1 to P2) rather than retinotopic direction (oblique motion: from P1 to P2) and space constancy is maintained.
Figure 1
 
Compensation for saccades. (a) Participants were instructed to fixate the green dot that exchanged position with a red dot each 600 ms. While they were fixating to the left, the first black probe (P1) appeared for 400 ms; they then saccaded to the right and a second probe (P2) of the same duration appeared below the first one 200 ms after the offset of P1. (b) On the retina, the first probe falls to the right side of the fovea while the second falls to the left side. (c) To compensate for these effects of the saccade, the visual system corrects the expected location of P1 in the opposite direction to the saccade (red dashed arrow) to obtain C1. If this correction is accurate, motion is perceived in its spatiotopic (vertical motion: from C1 to P2) rather than retinotopic direction (oblique motion: from P1 to P2) and space constancy is maintained.
Rock and Ebenholtz (1962) were the first to test the perception of apparent motion across saccades using displacements that were the same magnitude as the saccade. In this case, both pre- and post-saccadic targets fell on the same retinal location, the fovea, but despite this absence of retinal displacement, participants reported that motion was seen. This earlier finding was supported by the recent report that more complex transformational apparent motion could be seen across saccades (Fracasso, Caramazza, & Melcher, 2010). The authors of both studies concluded that apparent motion was seen between locations in space not locations on the retina although they did not test how well the perceived motion matched the actual trajectory in space. Nevertheless, the first stimulus location must have been corrected to some extent to compensate for the eye movement, rendering an impression of motion despite the absence of motion on the retina. For our stimulus, we expect that motion will be seen, as Fracasso et al. (2010) and Rock and Ebenholtz (1962), principally in spatial, not retinal, coordinates. We are interested in whether there are any systematic deviations from spatial coordinates, from accurate compensation for eye movements. In particular, we tested vertical apparent motion at 9 different locations across the visual field to determine whether the deviations from accurate correction, if any, varied as a function of location. Any local variation in the accuracy of the corrections should help differentiate among possible mechanisms of space constancy. 
What are the possible mechanisms of space constancy? We make a rough division between global correction approaches and local corrections for eye movements. For global corrections, some decision about visual stability or some overall correction or calibration is determined and applied uniformly to the visual input (see Bridgeman, Van der Heijden, & Velichkovsky, 1994 for a review). As Bridgeman (2007) and Bridgeman et al. (1994) point out, there are a number of possible corrective mechanisms. For example: 
  1.  
    Shift and compare. A copy of the motor commands to the eyes, efference copy or corollary discharge, is used to predict the retinal input following the saccade, and if there is a match to new input, it assumed that the world has been stable (e.g., Sperry, 1950; von Holst & Mittelstaedt, 1950). This has been amended to allow for some degree of mismatch before stability is rejected (Bridgeman et al., 1975).
  2.  
    Spatiotopic maps: The efference copy is used to transfer retinal input onto a map in spatial coordinates (e.g., Bischof & Kramer, 1968; Breitmeyer, Kropfl, & Julesz, 1982).
  3.  
    Reference object calibration. A memory of the saccade target “landscape” (the target and possibly a few attended items, especially those near the target; Irwin, McConkie, Carlson-Radvansky, & Currie, 1994) is used to locate the original saccade goal (e.g., Bridgeman et al., 1994; Deubel, Bridgeman, & Schneider, 1998; Deubel, Koch, & Bridgeman, 2010; Koch & Deubel, 2007). The rediscovered target then serves as a reference for spatial localization providing space constancy without using the efference copy.
In all these cases (with various caveats, see Bridgeman, 2007; Bridgeman et al., 1994), once the correction is determined, it would be assumed to hold across all locations. The eye movement is, after all, a shift of the whole visual image by the same vector and a uniform correction (e.g., Honda, 1989; Matin & Pearce, 1965) would be reasonable (ignoring deviations due to optical aberrations and the spherical retina). 
As an alternative to these global approaches, physiological studies have suggested a piecemeal compensation where corrections, based on efference copy (Duhamel, Colby, & Goldberg, 1992; Goldberg & Bruce, 1990; Sommer & Wurtz, 2004, 2006; Wurtz, 2008), are applied to the representations of only a few attended items on the retinotopic maps of saccade control centers (Gottlieb, Kusunoki, & Goldberg, 1998; Kusunoki, Gottlieb, & Goldberg, 2000). This shift of activity for each target to its expected post-saccadic location, called “remapping,” starts before the target lands and is seen even if the target is removed during or even before the saccade and so never lands there (Duhamel et al., 1992; Umeno & Goldberg, 2001). The remapping produces a crude spatiotopy for attended items (Cavanagh, Hunt, Afraz, & Rolfs, 2010; Rolfs, Jonikaitis, Deubel, & Cavanagh, 2011), and if it actually underlies the space constancy we experience, then displacements of any unattended items should not be seen (widely reported as the change blindness phenomenon; Cavanaugh & Wurtz, 2004; O'Regan, Rensink, & Clark, 1999; Rensink, O'Regan, & Clark, 1997). Critically, this blindness occurs only for non-attended items (Rensink et al., 1997). The correction in this “remapping” proposal is applied to individual attended targets, and so, unlike global correction schemes, there is no assumption that the correction be the same everywhere. It is the same efference copy vector that drives the correction for each target, but the conversion of the vector to a shift on the roughly log polar coordinates of the saccade control maps is idiosyncratic to each location and direction. Indeed, in two models of this process (Keith & Crawford, 2008; Quaia, Optican, & Goldberg, 1998), the link between the target location and the saccade vector that predicts the post-saccadic location must be learned independently for each location and saccade. Location-specific mislocalizations are a general property of peri-saccadic position judgments (Honda, 1989; Matin & Pearce, 1965) and specifically of the compression effects seen around the time of the saccade (Morrone, Ross, & Burr, 1997; Ross, Morrone, & Burr, 1997). If these effects are related to the updating of locations required for spatial constancy, they are most likely the intermediate results of shifting of coordinates rather than the final values of correction that interest us here. Nevertheless, we will compare any local variations in position judgments that we find here with those found in peri-saccadic mislocalization experiments. 
In our experiments, we test vertical displacements of 3° of visual angle, orthogonal to the horizontal saccade of 10°, allowing us to examine the accuracy of the compensation for the eye movements as a function of location. The current study contains four experiments. In the first experiment, we asked participants to adjust the relative horizontal offset between the pre- and post-saccadic dot locations until the two dots appeared vertically aligned. A demonstration of the stimulus is shown in 1. Deviations from vertical alignment on the screen were taken to characterize deviations from accurate correction. From these measurements, we construct a map of correction errors for different attended locations within 15° of the fovea. The second experiment studied the same trans-saccadic apparent motion but this time using the method of constant stimuli in conjunction with eye movement tracking. The third and fourth experiments are controls that measured perceptual biases in judging position and orientation in the absence of saccades to determine if these factors had influenced our results in the first two experiments. 
 
Movie 1
 
Stimulus demonstration.
We find, as Rock and Ebenholtz (1962) and Fracasso et al. (2010) did, that apparent motion is not seen in retinal coordinates. However, the tilt of the direction of apparent motion away from vertical that is visible to most observers in our demonstration movie (1) and reported by all our participants indicates that the correction for eye movements does not correspond to exact space constancy. The errors are on the order of 5% of the saccade amplitude and are quite noticeable as tilts away from vertical of up to about 9° of rotation. More important, these errors vary systematically as a function of the location of the stimuli in the visual field suggesting that the compensation for eye movements is not a uniform, global process. The control experiments show that these are errors intrinsic to the correction process and are not biases in position or motion verticality judgments in the absence of eye movements. Because of the timing we used in presenting the first and second stimuli (duration of 400 ms, separated by 200 ms), we believe that these errors reflect the final, stable product of correction independently of the brief mislocalizations that occur around the time of the eye movement as the correction is in progress (Honda, 1989; Matin & Pearce, 1965). Indeed, the patterns of errors we see in our results are inconsistent with and often opposite to the displacements toward the saccade target seen for peri-saccadic compression effects. 
Experiment 1
Introduction
In order to test whether trans-saccadic motion is seen in retinotopic or spatiotopic coordinates, we asked participants to adjust the horizontal position of two dots displayed sequentially one before and one after the saccade until they appeared vertically aligned (Figure 2). If motion were experienced retinotopically, the dots would have to be shifted horizontally by 10° (the size of the saccade) to appear aligned. If the motion were experienced spatiotopically, the two dots would appear to move vertically when they were aligned vertically in the display screen. If there were any inaccuracies in the correction for the saccade, then the required alignment may be shifted left or right of vertical. We tested this alignment required for perceived verticality at 9 different positions in the visual field. 
Figure 2
 
Stimuli sequences, probe locations, and data analysis. (a) Spatial layout of the stimuli for a rightward saccade trial. The magenta line represents the line of gaze during each cycle. Participants were instructed to saccade to the green dot as it exchanged locations with the red dot each 600 ms while no probes were presented. In the “saccade probe cycle,” participants continued to saccade to the green dot as it changed positions, but now black probe dots were presented before and after the saccade either top-first or bottom-first (only top-first is shown here). (b) The pairs of probes could appear at 9 different spatial locations equally spaced by 10° horizontally and 5° vertically. (c) InExperiment 1, participants used a mouse to adjust the horizontal location of the probes, which were originally displaced vertically by 3° but could be displaced horizontally by ±0.5° or ±1° producing an initial random left or right physical tilt. (d) To compensate for the effect of the saccade, the visual system corrects the expected location of the first black probe in the opposite direction to the saccade (red dashed arrow). The correction vector (CRWD) could be either too long (hypermetric error, displayed here) or too short (hypometric error, not displayed) reflecting, respectively, an over- or undercompensation of the saccade vector (SRWD). After the saccade (middle panel), this error leads to the perception of a tilted motion from the corrected location of the first probe (orange dot) to the second probe location. Participants adjust the two dot locations until the perceived motion is vertical. The error of correction (right panel), that is, the difference between the top and bottom probe horizontal positions (“Δm”), is measured and displayed (red dot) at each tested location (top scale), as an offset relative to the accurate correction for the saccade (gray dashed dot). The bottom scale represents the direction (positive as hypermetric, negative as hypometric error) and the amount of the error.
Figure 2
 
Stimuli sequences, probe locations, and data analysis. (a) Spatial layout of the stimuli for a rightward saccade trial. The magenta line represents the line of gaze during each cycle. Participants were instructed to saccade to the green dot as it exchanged locations with the red dot each 600 ms while no probes were presented. In the “saccade probe cycle,” participants continued to saccade to the green dot as it changed positions, but now black probe dots were presented before and after the saccade either top-first or bottom-first (only top-first is shown here). (b) The pairs of probes could appear at 9 different spatial locations equally spaced by 10° horizontally and 5° vertically. (c) InExperiment 1, participants used a mouse to adjust the horizontal location of the probes, which were originally displaced vertically by 3° but could be displaced horizontally by ±0.5° or ±1° producing an initial random left or right physical tilt. (d) To compensate for the effect of the saccade, the visual system corrects the expected location of the first black probe in the opposite direction to the saccade (red dashed arrow). The correction vector (CRWD) could be either too long (hypermetric error, displayed here) or too short (hypometric error, not displayed) reflecting, respectively, an over- or undercompensation of the saccade vector (SRWD). After the saccade (middle panel), this error leads to the perception of a tilted motion from the corrected location of the first probe (orange dot) to the second probe location. Participants adjust the two dot locations until the perceived motion is vertical. The error of correction (right panel), that is, the difference between the top and bottom probe horizontal positions (“Δm”), is measured and displayed (red dot) at each tested location (top scale), as an offset relative to the accurate correction for the saccade (gray dashed dot). The bottom scale represents the direction (positive as hypermetric, negative as hypometric error) and the amount of the error.
Methods
Participants
Four volunteers from the Université Paris Descartes took part in the first experiment (2 authors and 2 participants naive to the purpose of the experiment, age 23–61 years; 1 female). All had normal or corrected-to-normal vision and gave their informed consent. Experiments were carried out according to the ethical standards specified in the Declaration of Helsinki. 
Stimuli
Participants were seated in a silent and dimly lit room with the head positioned on a chin rest, 63 cm in front of a computer screen. Stimuli were red, green, and black 0.7° diameter dots on a gray background, presented on a 22″ Formac ProNitron 22800 screen with a spatial resolution of 1440 by 1050 pixels (or 36.7° by 27.6°) and a vertical refresh rate of 100 Hz. The experiment was controlled by an Apple MacPro Dual Intel-Core Xeon computer. Manual responses were acquired via a standard keyboard and mouse. The experimental software controlling stimulus display and response collection was implemented in Matlab (MathWorks, Natick, MA), using the Psychophysics Toolbox (Brainard, 1997; Pelli, 1997). 
Procedure
Two dots, one green and one red, were presented at 5° to the right and 5° to the left of screen center. The red and green dots exchanged position each 600 ms. Participants were instructed to always fixate the green dot and follow it as accurately as possible as it moved back and forth. After a few back and forth cycles with no probes presented, participants were able to synchronize their saccades with the exchange of the two fixation dots (“no probe cycle” in Figure 2a). Once they judged that they were moving their eyes in synchrony with the green dot, they were instructed to press a button on the keyboard to start a trial. As they pressed the button, two circular black probes were presented sequentially, one before and one after the saccade (“saccade probe cycle” in Figure 2a). Each probe was presented for 400 ms, with a gap of 200 ms between them. If the participant's eye movements were synchronized to the exchange of red and green fixation dots, the first probe would be turned off about 100 ms before the saccade and the second turned on about 100 ms after the saccade. These assumed timings were not verified in this experiment but were in Experiment 2. Each “saccade probe cycle” was alternated with a “no probe cycle,” allowing the equiprobable presentation of probes during leftward or rightward saccade trials (see 2 and 3). These probes could appear at nine equiprobable locations on the screen equally spaced by 10° horizontally and 5° vertically from center of the screen (Figure 2b). The first black probe is originally displaced vertically by 3° from the second probe, but this displacement could be one of four other combinations of horizontal shifts of the two probes by ±0.5° or ±1.0°, producing a left or right physical tilt (Figure 2c), randomizing the initial tilt of the probes across trials. Finally, the order of appearance of these two probes was equiprobably top-first or bottom-first, producing equal numbers of upward or downward displacement trials. 
 
Movie 2
 
Stimulus used during a rightward saccade-downward motion trial of the Experiment 1 and 2.
 
Movie 3
 
Stimulus used during a rightward saccade-upward motion trial of the Experiment 1 and 2.
As they moved their eyes back and forth, participants were instructed to use the computer mouse to adjust the horizontal position of the two probes (simultaneously in an opposite direction, the bottom probe in the direction of the mouse and the top probe in the opposite direction) until they perceived motion between them to be vertical. Once participants were satisfied with the adjustment, they pressed a button to indicate their final setting and another trial began. Three participants ran 5 blocks of 36 trials each (180 trials) and one participant ran 6 blocks (216 trials). Moreover, to determine the stability across time of the observed effect, two participants ran the same 5 blocks (180 trials) and 6 blocks (216 trials) a second time, after a 3-month delay. 
Results
Participants adjusted the horizontal positions of the probes to make their displacement appear vertical. If the compensation for saccades is accurate, the final setting will be accurate on the screen. If compensation is not perfect, however, the adjustment, “Δm,” away from actual vertical measures the over- or undercompensation of the saccade vector (hypermetric or hypometric correction). We analyzed four sets of nine mean adjustments arising from the combination of the two directions of saccades (rightward and leftward) with the two directions of the probes (bottom-first and top-first). Figure 3 shows the different combination of the directions of saccade and directions of motion with the adjustment observed across participants for probes presented centrally. For this specific position, we always observed overcompensation of the saccade vector leading for specific combination of variable to clockwise (Figures 3a and 3d) or counterclockwise perceived motion (Figures 3b and 3c), when probes are vertically aligned on the screen. We then correct the sign of the “Δm” each time for the saccade direction and motion direction to recover whether the error was a hypermetric correction or a hypometric correction. 
Figure 3
 
Combinations of saccade and motion directions. In each panel, the colored arrow shows the direction of the saccade and the gray arrows show the direction of the motion. The red and blue dots represent the recovered or corrected location of the first probe relative to the second for all participants and for the central probe presentation (error bar indicates SEM). In all these cases, the error observed was an overcompensation of the saccade vector: the recovered location of the first probe is displaced beyond its actual spatial location in the direction opposite to the saccade. To characterize these correction errors, we first use the horizontal shift the subjects add to the display (“Δm”) in order to make it appear vertical. We then correct the sign of the “Δm” each time for the saccade direction and motion direction to recover whether the error was a hypermetric correction (as shown here) or a hypometric correction. For downward motion and rightward saccades (a), this overcompensation of the saccade vector produces a clockwise rotation of the perceived direction, as it does for the leftward saccade and upward motion (d). When we combine alternating left and right saccades with alternating up and down motions as we do in demonstration 1 (either (a) and (d), or (b) and (c)), the tilt will therefore appear the same on each saccade. In the experiment, we test only one combination of saccade and motion direction at a time (see2 and3, respectively, for the conditions of (a) and (b)).
Figure 3
 
Combinations of saccade and motion directions. In each panel, the colored arrow shows the direction of the saccade and the gray arrows show the direction of the motion. The red and blue dots represent the recovered or corrected location of the first probe relative to the second for all participants and for the central probe presentation (error bar indicates SEM). In all these cases, the error observed was an overcompensation of the saccade vector: the recovered location of the first probe is displaced beyond its actual spatial location in the direction opposite to the saccade. To characterize these correction errors, we first use the horizontal shift the subjects add to the display (“Δm”) in order to make it appear vertical. We then correct the sign of the “Δm” each time for the saccade direction and motion direction to recover whether the error was a hypermetric correction (as shown here) or a hypometric correction. For downward motion and rightward saccades (a), this overcompensation of the saccade vector produces a clockwise rotation of the perceived direction, as it does for the leftward saccade and upward motion (d). When we combine alternating left and right saccades with alternating up and down motions as we do in demonstration 1 (either (a) and (d), or (b) and (c)), the tilt will therefore appear the same on each saccade. In the experiment, we test only one combination of saccade and motion direction at a time (see2 and3, respectively, for the conditions of (a) and (b)).
We tested rightward and leftward saccades independently and rather than combining their data, we treated them as 18 different test locations as we were interested in the correction for eye movements at all 18 retinal locations of the first probe, the one appearing before the saccade. We then ran an ANOVA for these eighteen retinal positions with the two directions of the probes (downward vs. upward) as independent variables and with the data reversion explained above. The ANOVA showed a strong main effect of the different retinal positions on the perceived vertical alignment (F(17, 51) = 4.26, p < 0.001). However, there was no significant difference of perceived alignment for downward vs. upward displacement (F(1, 51) = 0.03, p > 0.85) nor an interaction of this variable with the retinal positions of the probes (F(17, 51) = 1,26, p > 0.25). We thus collapsed the data across upward vs. downward displacement in all further analyses and in the data figures. 
Group and individual data are shown in Figures 4 and 5, respectively. Results are presented in retinal coordinates (i.e., in coordinates relative to the fovea) and are plotted on maps of horizontal error of alignment as a function of saccade direction. Each graph shows, for the nine retinal locations tested during leftward and the nine tested during rightward saccades, the accurately corrected location of the first probe that would support space constancy in gray (gray dashed dots in Figures 4 and 5) and the horizontal deviation from that location (red and blue dots in Figures 4 and 5) corresponding to the horizontal distance between the two probes (“Δm”) after the participant's adjustment. Group results indicate that the corrected location of the first probe was significantly over- or underestimated by different amounts depending on its location. 
Figure 4
 
Horizontal error of alignment derived from the adjustments required to make the motion between the two probes appear vertical. The data are averaged across the 4 participants and shown separately for rightward and leftward saccades at each of the 9 tested locations. The gray dots show horizontal locations where the first stimulus should be perceived to support accurate vertical motion in world coordinates. Data are presented in retinal coordinates with the central cross showing the location of the fovea, following the saccade. The small red and green dots indicate the fixation and saccade target. Scales at the top and at the left of each map reflect the horizontal and vertical distances from the fovea tested with each of 9 probe pairs. The scale below each map gives the mean value of alignment errors for each spatial location tested. Note that these scales are reversed for rightward and leftward saccades, with positive values always representing overcompensation or hypermetric correction and negative values representing undercompensation or hypometric correction. The alignment scales are also magnified by a factor four relative to the retinal scale to make these deviations clearly visible. Horizontal error bars indicate SEM.
Figure 4
 
Horizontal error of alignment derived from the adjustments required to make the motion between the two probes appear vertical. The data are averaged across the 4 participants and shown separately for rightward and leftward saccades at each of the 9 tested locations. The gray dots show horizontal locations where the first stimulus should be perceived to support accurate vertical motion in world coordinates. Data are presented in retinal coordinates with the central cross showing the location of the fovea, following the saccade. The small red and green dots indicate the fixation and saccade target. Scales at the top and at the left of each map reflect the horizontal and vertical distances from the fovea tested with each of 9 probe pairs. The scale below each map gives the mean value of alignment errors for each spatial location tested. Note that these scales are reversed for rightward and leftward saccades, with positive values always representing overcompensation or hypermetric correction and negative values representing undercompensation or hypometric correction. The alignment scales are also magnified by a factor four relative to the retinal scale to make these deviations clearly visible. Horizontal error bars indicate SEM.
Figure 5
 
Individual results. Each row corresponds to rightward and leftward saccade trials for each participant. Conventions used here are described in Figure 4.
Figure 5
 
Individual results. Each row corresponds to rightward and leftward saccade trials for each participant. Conventions used here are described in Figure 4.
As expected from the Rock and Ebenholtz (1962) and Fracasso et al. (2010) results, the group and individual data for all retinal locations tested suggest that the direction of apparent motion between the two probes is perceived more in world coordinates than in retinal coordinates. For example, for the presentation of an apparent motion at the vertical and horizontal center of the screen in the case of a rightward saccade (see coordinates [0°, −5°] in rightward map in Figure 4), we observed a hypermetric correction of 0.55° ± 0.13° (an overcompensation). This perceived misalignment of about one-half a degree is about 5% of the 10° shift of the probes' projections on the retina. The magnitudes of the correction errors are shown in Figure 6 and indicate that across participants, a 10-deg saccade leads to less than 4% (0.38° ± 0.09° deg for 10-deg saccade) of errors from exact space constancy for targets presented within 15° of the fovea. These variations from accurate correction of the saccade vary significantly (p < 0.001) across the 9 locations and two directions of saccades. Note that, if the motion had been seen in retinal coordinates, the correction would have been an offset of 10° of visual angle and the rotation from vertical, the apparent direction of motion, would have been 73°, much closer to horizontal (at 90°) than vertical (at 0°). Thus, this first experiment demonstrated that participants perceived the probe displacement in roughly spatiotopic rather retinotopic coordinates and that the correction for the eye movements has a small but significant error that depends on location. 
Figure 6
 
Magnitude of correction errors for individuals and group. Each bar represents the mean values of alignment bias independently of the sign of the effect (absolute value) averaged over the nine spatial locations and two saccade directions. Error bars indicate SEM.
Figure 6
 
Magnitude of correction errors for individuals and group. Each bar represents the mean values of alignment bias independently of the sign of the effect (absolute value) averaged over the nine spatial locations and two saccade directions. Error bars indicate SEM.
The variation in vertical alignment averaged across participants explained about 40% of the total within-participant variance and we next analyzed the direction and the magnitude of the local correction biases for each participant. Figure 5 shows the intersubject variability in the amount and the direction of biases between locations tested and saccade directions. For probes where the correction crosses the vertical midline, the correction errors were hypermetric in all participants (data in horizontal coordinates −5° on the rightward maps and +5° on the leftward maps in Figure 5). However, when the correction for the eye movement left the probe in the same hemifield, the pattern of local biases varied substantially from participant to participant (data in horizontal coordinates −15° and +5° in rightward maps; −5° and +15° in leftward maps in Figure 5). 
This variability was not just measurement noise as the individual patterns showed significant intertrial reliability. In particular, two participants were tested two times with 3 months between the two sessions. The horizontal errors of alignments observed for each of the nine spatial locations tested twice were between each other similar and highly correlated (AC: R 2 = 0.72, one-tailed t-test, p < 0.001; MS: R 2 = 0.86, one-tailed t-test, p < 0.001) across this large interval, demonstrating the robustness of the individual patterns. 
Finally, from the group and individual data, there is a tendency for first probe location to appear displaced toward the fixation target (red dots in Figures 4 and 5). Unlike the very large peri-saccadic compression effects (e.g., Ross et al., 1997, often equal to the saccade amplitude), this compression effect is centered on the initial fixation not the saccade target and is much smaller (about 5% of the saccade amplitude). It is observed on group results for both directions of saccade (see Figure 4) and is more or less pronounced for each participant (see Figure 5). 
Discussion
The results of Experiment 1 show that trans-saccadic, vertical, apparent motion is seen in a roughly spatiotopic rather that a retinotopic reference frame, consistent with previous results (Fracasso et al., 2010; Rock & Ebenholtz, 1962) and extending it beyond Rock and Ebenholtz's conditions to motion orthogonal to the saccade direction at location other than saccade target. These results suggest that the location of a pre-saccadic, attended target has been corrected for the saccade so that when the target then reappears in a new location, its displacement from the first location is seen roughly in world coordinates. The salience of the motion direction allowed us to measure deviations from world coordinates quite accurately, deviations that we attribute to significant (F(17, 51) = 4.26, p < 0.001) local biases in correction. When programming a 10-deg saccade, attended targets within 15° of the fovea are corrected to their post-saccadic retinal locations with errors representing less than 4% of the saccade amplitude. Moreover, beyond the local variation in the correction errors common to all participants, we also observed local biases specific to each that were quite stable across as much as a 3-month interval. 
Experiment 2
Introduction
The results of the first experiment clearly suggest that trans-saccadic motion is seen in roughly spatiotopic rather than retinotopic coordinates. However, because we did not measure eye movements, one might argue that participants made saccades too early or too late and had an opportunity to see the two probes during a single fixation. In this second experiment, we used a method of constant stimuli with the trans-saccadic apparent motion of the first experiment and then analyzed only those trials with appropriate eye movements. We restricted the tested locations to only the three along the horizontal midline rather than all 9 tested in Experiment 1, but again for both leftward and rightward saccades. This procedure allows us to determine first whether the observed motion remains roughly spatiotopic on trials when the eye movement intervenes between the first and second stimuli and, second, whether the observed direction of motion is affected by the position of each saccade landing. 
Methods
Participants
Two participants from first experiment took part in the second experiment (1 author and 1 participant, age 24–25 years). 
Instruments and stimuli
Participants were seated in a quiet and dimly lit room with the head positioned on a chin rest, 63 cm in front of a computer screen. Stimuli, display, and computer configurations were identical to those in the first experiment. Movements of the right eye were measured using an EyeLink 1000 Desktop Mount (SR Research, Osgoode, Ontario, Canada) with a sampling at 1 kHz. The experimental software controlling stimulus display and response collection was implemented in Matlab (MathWorks, Natick, MA), using the Psychophysics (Brainard, 1997; Pelli, 1997) and EyeLink (Cornelissen, Peter, & Palmer, 2002) toolboxes. Manual responses were acquired via a standard keyboard. Saccades were analyzed offline using a two-dimensional velocity space algorithm developed by Engbert and Mergenthaler (2006). Psychometric functions were fitted using the Psignifit toolbox (Wichmann & Hill, 2001a, 2001b). 
Procedure
Each trial began with two “no probe cycles” (Figure 2a) in which two dots, one green and one red, swapped positions with the same temporal configuration as in Experiment 1. These two initial sequences help participants to synchronize their saccades with the exchange of the two dots and to prepare themselves for the main sequence. Following these two sequences, a “saccade probe cycle” (Figure 2a) was displayed once. In this third sequence, two black probes were displayed sequentially with one of 11 equiprobable horizontal gaps between them, from −1.75° (bottom probe to the left of the top probe) to +1.75° (bottom probe to the right of the top probe) following a constant stimuli procedure. We tested three horizontal locations of the nine spatial locations used in the first experiment. Thus, probes could appear at three equiprobable locations on the horizontal midline of the screen, equally spaced by 10° from the center of the screen. The overall displacement between the two black probes ranged from 3° (when probes were displayed vertically) to ∼3.5° (when probes were horizontally displaced by ±1.75°). Finally, as in Experiment 1, the probes' order of presentation was equiprobably bottom-first (downward motion) or top-first (upward motion). 
After the “saccade probe cycle,” a tone indicated that participants were to report the tilt of the motion perceived: clockwise or counterclockwise. One participant ran 10 blocks of 132 trials each (1320 trials) and the other ran 12 blocks of 132 trials each (1584 trials). We selected only trials in which correct saccades were detected, that is trials without blinks, starting and ending in an area within a 2° radius circle centered on the fixation target and on the saccade target. This selection left 90.15% of all trials for the first participant (AC: 1190/1320 trials kept) and 96.02% of all trials for the second participant (MS: 1521/1584 trials kept). Each session was composed of 2 or 3 blocks and always lasted less than 20 min. During each session, the eye-tracking system was recalibrated every 8 min. 
Results
In addition to the selection based on spatial accuracy and lack of blinks, we also selected trials based on accurate timing. We instructed participants to initiate their saccade close to the moment at which the green fixation dot changed its position on the screen, i.e., during the 200 ms that separated the probe presentations (Figure 7). To be sure that apparent motion was tested trans-saccadically, we sorted trials in 3 categories: “early saccade,” “late saccade,” and “correct saccade.” On “early saccade” trials, saccades landed before the disappearance of the first probe (see light green lines in Figure 7), whereas on “late saccade” trials, saccades start after the appearance of the second probe. In both cases, the participant might be able to see both probes on one fixation, so the motion would not be strictly trans-saccadic. On the other hand, “correct saccade” trials (see dark green lines in Figure 7) were appropriate for tests of trans-saccadic motion. The horizontal eye positions as a function of the type of saccade trials are displayed in Figure 7. For the first participant, AC, correct saccade trials represent 74.0% of all trials (880/1190), early saccade trials 25.7% (306/1190), and late saccade trials 0.3% (4/1190). For the second participant, MS, the proportions were, respectively, 60.3% (917/1521), 39.4% (600/1521), and 0.3% (4/1521). 
Figure 7
 
Trial selection. This graph shows fifty representative horizontal eye traces of individual probe cycles for each participant. The bars at the bottom of the figure represent the onset, the offset, and the colors of the saccade target, the fixation target, and the first and second probes. To be sure that apparent motion was tested trans-saccadically, we sorted trials into 3 categories, “early saccade” (light green lines), “late saccade” (black lines, not displayed because they represent less than 1% of trials for both participants), and “correct saccade” trials (dark green lines) where saccades end before the onset of the second probe (see text). Bars at the right of the figure represent the proportion of trials in each category for each participant.
Figure 7
 
Trial selection. This graph shows fifty representative horizontal eye traces of individual probe cycles for each participant. The bars at the bottom of the figure represent the onset, the offset, and the colors of the saccade target, the fixation target, and the first and second probes. To be sure that apparent motion was tested trans-saccadically, we sorted trials into 3 categories, “early saccade” (light green lines), “late saccade” (black lines, not displayed because they represent less than 1% of trials for both participants), and “correct saccade” trials (dark green lines) where saccades end before the onset of the second probe (see text). Bars at the right of the figure represent the proportion of trials in each category for each participant.
In this experiment, the two probes were horizontally displaced from each other to a variable degree and participants were instructed to report the tilt direction of apparent motion perceived while they followed a green dot moving back and forth between the two fixation locations. We quantified the horizontal deviation from accurate saccade correction by determining the horizontal offset between the two probes required to obtain a level of 50% clockwise or counterclockwise reports of motion. This level is determined as the point of subjective verticality (PSV) where the horizontal displacement between probes led to the perception of a trans-saccadic vertical apparent motion. We measure PSV values for all trials as well as for only “correct saccade” trials. 
Two representative psychometric curves for “correct saccade” trials are displayed in Figure 8a for each participant and drawn again on maps of horizontal error of alignment in Figure 8b. As in Experiment 1, these results give us the opportunity to observe the direction and the amount of correction for the saccade. Thus, for probes presented 10° to the left or to the right of screen center (Figure 8b), the corrections were always hypometric (undercompensated) for AC and almost always hypometric for MS (3 out 4 cases). As in Experiment 1, the corrections for probes presented between fixation and saccade target (5° left or right of screen center) were hypermetric (overcompensated) for both participants. The average absolute biases (where 0° bias represents accurate, spatiotopic correction) across all locations and saccade directions for both participants were 0.66° ± 0.04° (AC: 0.61° ± 0.27°; MS: 0.70° ± 0.32°). These biases in saccade correction showed a significant variation across locations (F(5, 5) = 9.71, p < 0.05). 
Figure 8
 
Results of Experiment 2. (a) Percentage of “clockwise” report of two representative locations tested with rightward (red curves) and leftward (blue curves) “correct saccade” trials for both participants. Psychometric functions were fitted to estimate the point of subjective verticality (PSV), i.e., the amount of horizontal offset of probes leading to equal response of clockwise and counterclockwise deviations from vertical (see red and blue arrows). Shaded areas represent 95% confidence interval computed by bootstrapping. Black arrows represent correction biases observed in Experiment 1 for the same participant at the same location and saccade direction. (b) Maps of horizontal errors of alignment. Each row corresponds to rightward and leftward saccade trials for both participants. Conventions used here are described in Figure 4. The locations of the 4 representative cases displayed in (a) are shown here with light red dots for rightward saccade and with light blue dots for leftward saccade trials.
Figure 8
 
Results of Experiment 2. (a) Percentage of “clockwise” report of two representative locations tested with rightward (red curves) and leftward (blue curves) “correct saccade” trials for both participants. Psychometric functions were fitted to estimate the point of subjective verticality (PSV), i.e., the amount of horizontal offset of probes leading to equal response of clockwise and counterclockwise deviations from vertical (see red and blue arrows). Shaded areas represent 95% confidence interval computed by bootstrapping. Black arrows represent correction biases observed in Experiment 1 for the same participant at the same location and saccade direction. (b) Maps of horizontal errors of alignment. Each row corresponds to rightward and leftward saccade trials for both participants. Conventions used here are described in Figure 4. The locations of the 4 representative cases displayed in (a) are shown here with light red dots for rightward saccade and with light blue dots for leftward saccade trials.
How similar are the results in this experiment compared to those of the first experiment? How valid were the tests of spatiotopic apparent motion based on the subjective saccade synchronization in Experiment 1? First, there is a significant correlation between the adjustment settings of the first experiment (for these same three locations) and the PSV values of the second experiment for both participants. This is true when using PSV values based on all trials (AC: R 2 = 0.85, one-tailed t-test, p < 0.05; MS: R 2 = 0.80, one-tailed t-test, p < 0.05) as well as only “correct saccade” trials (Figure 9a; AC: R 2 = 0.76, one-tailed t-test, p < 0.05; MS: R 2 = 0.74, one-tailed t-test, p < 0.05). The magnitude of the deviation from accurate correction increased in the second experiment to 0.66° ± 0.04° (average for both participants) versus 0.38° ± 0.09° in the first experiment (for these same three locations along the horizontal meridian), but this change was not significant for either participant. We also observed here a small tendency of compression toward fixation target as described in Experiment 1, for both directions of saccade for one participant (AC) but only one direction of saccade for the other (leftward saccade, MS). These results suggest that the distribution of correction errors obtained in the first experiment (Figures 4 and 5) reflected the underlying processes of correction for saccades and were not due to the method of measurement nor to inaccuracies of saccade execution. 
Figure 9
 
Correlations. (a) Correlation of the observed biases obtained with the method of adjustment in Experiment 1 and those from method of constant stimuli of Experiment 2 for the “correct saccade” trials only. The 6 points represent the combination of the 2 directions of saccade and the 3 spatial locations tested. Filled dots correspond to the two representative conditions displayed in Figure 8a. (b) Correlation between saccade landing position relative saccade target and PSV values observed for rightward (red square) and leftward (blue square) representative saccade trials for each participant. PSV values are plotted against the average landing position relative saccade target for four bins of equal sample size. Error bars indicate 95% confidence interval.
Figure 9
 
Correlations. (a) Correlation of the observed biases obtained with the method of adjustment in Experiment 1 and those from method of constant stimuli of Experiment 2 for the “correct saccade” trials only. The 6 points represent the combination of the 2 directions of saccade and the 3 spatial locations tested. Filled dots correspond to the two representative conditions displayed in Figure 8a. (b) Correlation between saccade landing position relative saccade target and PSV values observed for rightward (red square) and leftward (blue square) representative saccade trials for each participant. PSV values are plotted against the average landing position relative saccade target for four bins of equal sample size. Error bars indicate 95% confidence interval.
Next, we used the data from the early and late saccade trials from this second experiment to examine how imprecise saccade timing would affect the results in the first experiment where eye movements were not monitored. There was a fair proportion of early saccades in the second experiment (25 to 40%), and if early saccades had similar frequency in the first experiment, they would not have been discarded and may have affected the results. Because early and late saccade trials for both participants were not equally distributed across all combinations of variables, we did not fit psychometric curves to determine the PSV values of “incorrect” saccade trials. Instead, we subtracted PSV values obtained with all saccade trials from those obtained with only correct saccade trials to determine the magnitude of the effect of all the “incorrect” saccade trials (i.e., “all saccade trials” = “incorrect saccade trials” + “correct saccade trials”). Using this subtraction method, we observe that “incorrect” saccade trials lead to significantly smaller biases (0.12° ± 0.02°, t(2) = 20.91, p < 0.001) than “correct” saccade trials (0.66° ± 0.04°) and almost no bias when motion probes were presented at the central location between fixation and saccade target (0.02° ± 0.001°). Thus, on trials where a direction judgment could be made without a saccade intervening between the first and second stimuli, there was, unsurprisingly, little or no misjudgment of the vertical displacement. In this case, the only effect of a mixture of inappropriately early or late saccades in Experiment 1 would be to reduce the correction biases we measured. The smaller effect with inappropriate timing is one possible source for the larger deviations measured here in the “correct saccade” trials than were measured in Experiment 1
Finally, we analyzed whether the saccade landing location affected the direction judgments. In particular, the deviations from vertical motion that we measured may not be due to errors of saccade correction but errors of saccade landing. In other words, the compensation for saccades might be accurate, but errors in the saccade landing may introduce an error in the expected post-saccadic location of the initial probe stimulus relative to its pre-saccadic location on the monitor. A similar question has been studied recently (Collins, Rolfs, Deubel, & Cavanagh, 2009; Ostendorf, Liebermann, & Ploner, 2010) concerning the perceived motion or displacement of a subsequent test relative to the saccade target itself. These authors reported that the judgments of probe position were independent of saccade landing location. To test this question in our display, we analyzed the PSV values as a function of the deviation of the saccade landing site from the saccade target, dividing our data into four quartiles of deviation. If the deviation of the saccade landing site from the fixation target was source of the misjudgment of the motion direction, the PSV values in each landing site quartile should shift by the same amount as the landing site. Instead, we find relative independence. Figure 9b presents the 4 PSV values obtained by fitting data separated into quartiles of saccade landings for 2 representative spatial locations tested. The figure shows relatively flat functions, indicating that the perceived deviations from vertical did not depend on the landing sites of the saccades. We did not observe any significant correlation for any of the 6 combinations of saccade directions and spatial locations (AC: 0.72 > R 2 > 0.14, two-tailed t-test, 0.63 > p > 0.15; MS: 0.80 > R 2 > 0.07, two-tailed t-test, 0.73 > p > 0.06). Moreover, if trial-to-trial oculomotor errors caused the deviations of apparent motion from vertical, saccades should, in general, show more undershoot in the case of hypometric results and overshoot in the case of hypermetric results. However, the saccade amplitudes observed for locations with hypometric and hypermetric results have Gaussian distributions with quite similar median values (AC: saccade amplitude medians of 10.51° and 10.62° for locations with hyper- vs. hypometric corrections for vertical motion, respectively; MS: 10.75° and 10.78°, respectively). Kruskal–Wallis non-parametric comparisons of these distributions show no significant differences (AC: p > 0.72; MS: p > 0.91). Overall, we see no evidence that the deviations from vertical motion are a result of saccade landing errors. 
Discussion
In this second experiment, we observed deviations from accurate compensation of the saccade of about 6% of the 10° saccade amplitude for the trials where the motion was strictly trans-saccadic. These errors are similar in magnitude to those observed in Experiment 1 despite the differences in methods. The eye movement recordings in this second experiment showed that participants made early saccades in about a third of the trials, and on those trials, they could have registered both probe locations following the saccade. However, there appears to be little or no misjudgment of vertical motion on those early saccade trials so that the inevitable inclusion of some inaccurately timed saccades in Experiment 1 would have, if anything, reduced the strength of the deviations from spatiotopy but could not create those deviations. Altogether these results confirm that trans-saccadic apparent motion is seen in a roughly spatiotopic rather than in a retinotopic reference frame and that the compensation for eye movements is fairly accurate with errors in the range of 5% to 6% of the saccade length with, in addition, a significant variation in error as a function of location. 
Finally, the analysis of saccade landing sites revealed that trial-to-trial oculomotor inaccuracy did not contribute to the deviations in perception of vertical apparent motion. As other studies suggest (Collins et al., 2009; Ostendorf et al., 2010), oculomotor errors are taken into account when the corrections for a saccade are generated so that the visual system can accurately predict where the target should be relative to the landing site. 
No-saccade control experiments
Introduction
The analysis used in the two previous experiments assumed that all errors in judging vertical orientation of the motion were due to errors in accurately correcting for the saccades. However, judgments of vertical motion and of the position of the individual probes may be biased at different locations even in the absence of saccades. In these two control experiments, we measure the magnitude of these judgment errors when no saccades are made. Our first control was matched in procedure (9 locations, method of adjustment) and participants to Experiment 1 while our second control was matched in procedure (3 locations, method of constant stimuli) and participants to Experiment 2. The results reveal non-saccade-related biases that, in total, account for about 25% of the errors measured in the first experiments, indicating that compensation for the saccades is somewhat more accurate than those measurements suggested. 
We first measure any biases in judging vertical orientation of motion at different locations to determine if any of the deviations measured in the first experiment can be attributed to errors in verticality judgments of the motion in the absence of saccades. Studies investigating verticality perception for lines (without motion) at different eccentricities suggest that participants are quite accurate (for a review, see Mittelstaedt, 1983), at least when their heads are positioned vertically. We now extend these tests to the verticality of apparent motion using exactly the same display as in the first experiment but this time holding fixation throughout (Figure 10a). 
Figure 10
 
Verticality motion control experiment. (a) Spatial layout of the stimuli for a sample rightward fixation trial. Participants were instructed to keep fixation on the green dot throughout the trial, composed of alternating “no probe cycles” and “probe cycles.” After each “probe cycle,” they moved a mouse to adjust the horizontal locations of both probes in subsequent cycles, continuing until they perceived motion as vertical. The magenta line represents the supposed line of gaze during the trial. Probe position and duration were identical to those of Experiment 1. (b) Absolute values of errors for the 4 participants in the saccade task of Experiment 1, in the verticality of motion judgments during fixation, and for the alignment errors with saccades after correction for verticality or motion bias. Error bars indicate SEM.
Figure 10
 
Verticality motion control experiment. (a) Spatial layout of the stimuli for a sample rightward fixation trial. Participants were instructed to keep fixation on the green dot throughout the trial, composed of alternating “no probe cycles” and “probe cycles.” After each “probe cycle,” they moved a mouse to adjust the horizontal locations of both probes in subsequent cycles, continuing until they perceived motion as vertical. The magenta line represents the supposed line of gaze during the trial. Probe position and duration were identical to those of Experiment 1. (b) Absolute values of errors for the 4 participants in the saccade task of Experiment 1, in the verticality of motion judgments during fixation, and for the alignment errors with saccades after correction for verticality or motion bias. Error bars indicate SEM.
We then measure position biases, specifically, the tendency to estimate the location of stimuli as being closer to the fovea than they really are. In our trans-saccadic motion experiments, the first and second stimuli are at different locations relative to the fovea (for example, the first to the right of fixation and the second to the left) so that the foveal biases (Mateeff & Gourevich, 1983) of those two locations may combine to contribute to an apparent deviation from vertical. 
Methods
Participants
The four participants of Experiment 1 took part in the verticality motion control experiment (2 authors and 2 participants, age 24–61 years). The two participants of Experiment 2 took part in position control experiment (1 author and 1 participant, age 24–25 years). 
Instruments and stimuli
The stimuli, experimental room, display, toolboxes, and computer configurations were identical to those of the first two experiments. Fixation of the right eye was monitored with the same eye-tracking system as in Experiment 2
Procedure
All participants took part in the verticality control experiment after the first experiment. The two participants who took part in position control experiment completed this last experiment following all the others. For both verticality motion and position controls, each trial began with two circles—one green and one red—displayed on a gray background. Contrary to the saccade tasks of the first two experiments, participants were now instructed to always fixate the green circle that could equiprobably be located 5° to the left or to the right of screen center and remained there throughout the trial. 
In the verticality motion control experiment, once fixation was held for at least 200 ms, two circular black probes were presented equiprobably at one of the nine different locations tested during Experiment 1 (Figure 2b). All stimuli sizes and durations of display were identical to those used in both previous experiments. The only major difference was that the green and red circles never exchanged positions. Each “probe cycle” alternated with a “no probe cycle” (Figure 10a) of the same duration (1.2 s), during which no probe was presented. As in Experiment 1, probes were randomly initially presented vertically aligned or with a physical tilt (Figure 2c). Because no significant difference was found between bottom-first (upward motion) and top-first (downward motion) order of presentations of the probes in Experiment 1, they were here always displayed top-first (downward motion). As participants fixate the green circle, they were instructed to use the computer mouse to adjust the horizontal positions of the probes after each downward motion until they perceived the motion to be vertical. Fixation was monitored online and trials were interrupted and randomly replayed later if the eyes crossed a circular boundary with a radius of 2° centered on the green circle or if a blink was detected. All participants ran 8 blocks of 18 trials each (144 trials). Each session was composed of 2 or 3 blocks and always lasted less than 20 min. During each session, the eye-tracking system was recalibrated every 8 min. When we compare this fixation task with the saccade task (Experiment 1) in the Result section below, we correct alignment errors observed during the saccade task for verticality bias measured here. To make the correction, we assume that the direction of the motion can only be perceived in the saccade task following the saccade when information from both locations is available. Following a leftward saccade, then, the apparent motion is seen as occurring to the right of fixation, and vice versa. We therefore subtracted the verticality bias obtained when participants judged motion on the right of fixation from the trans-saccadic measures for leftward saccade trials and vice versa. 
In the position control experiment, once fixation was detected for at least 200 ms, a circular black probe was presented one time during 400 ms (Figure 11a). We tested 3 different horizontal locations equally spaced by 10° from center of the screen (same locations tested in Experiment 2). From these three locations on the horizontal midline of the screen, the probe was horizontally displaced with one of 11 equiprobable horizontal offsets from −1.75° (to the left) to +1.75° (to the right) following a constant stimuli procedure. Finally, this probe appeared equiprobably at a top or at a bottom vertical position, respectively, 1.5° above or below the horizontal center of the screen to test the two motion locations used in Experiments 1 and 2. After each trial, participants report whether the probe was displayed to the left or to the right of a position corresponding to half of the distance between the red and the green dot from the fixation point. One participant ran 8 blocks of 132 trials each (1056 trials) and the other ran 10 blocks (1320 trials). For the analysis of the data, we selected only trials in which fixation was maintained in a 2° radius circle boundary centered on a green dot and trials without blink during all the duration of the trial. This selection left 95.93% of all trials for the first participant (AC: 1013/1056 trials kept) and 99.47% of all trials for the second participant (MS: 1313/1320 trials kept). Each session was composed of 2 to 4 blocks and always lasted less than 15 min. During each session, the eye-tracking system was recalibrated every 8 min. In the Results section below, we correct the absolute magnitude of alignment errors observed in the saccade tasks of Experiments 1 and 2, respectively, with the absolute magnitude of the bias observed in the no-saccade control task for verticality of motion bias and for position bias. We never correct the saccade task results with all the no-saccade task results taken together, because the procedures followed in each of these experiments were different. For Experiment 1 and the control for verticality of motion bias, we used a method of adjustment, while for Experiment 2 and the control for position bias, we used a method of constant stimuli with an eye-tracking correction. 
Figure 11
 
Position control experiment. (a) Spatial layout of the stimuli for a sample rightward simulated trial. In two successive randomly presented trials, participants keep fixation on the green dot located at the right or left of the screen center. The magenta line represents the supposed line of gaze during trials. Participants evaluate the probe position displayed above and then below the horizontal midline with a physical horizontal offset following a method of constant stimuli. (b) Absolute values of errors for the 2 participants in the saccade task of Experiment 2, in the position judgment during fixation, and for the alignment errors with saccade after correction of position bias. Error bars indicate SEM.
Figure 11
 
Position control experiment. (a) Spatial layout of the stimuli for a sample rightward simulated trial. In two successive randomly presented trials, participants keep fixation on the green dot located at the right or left of the screen center. The magenta line represents the supposed line of gaze during trials. Participants evaluate the probe position displayed above and then below the horizontal midline with a physical horizontal offset following a method of constant stimuli. (b) Absolute values of errors for the 2 participants in the saccade task of Experiment 2, in the position judgment during fixation, and for the alignment errors with saccade after correction of position bias. Error bars indicate SEM.
Results and discussion
For the verticality of motion control experiment, we found significant variation in the adjustments for verticality across the retinal locations tested (F(17, 51) = 2.14, p < 0.05). The absolute size of the deviations was about half that measured with saccades in the first experiment (fixation task: 0.20° ± 0.02°; saccade task: 0.38° ± 0.09°, ns, t(6) = 1.94, p = 0.65). The verticality motion biases were, however, mostly independent of or slightly in the direction opposite to the alignment deviations measured with saccades (0.08 > R 2 > 0.002, two-tailed t-test, 0.86 > p > 0.26), so that when we corrected the measures obtained for each participant in the first experiment with those obtained in this fixation task (Figure 10b), the absolute magnitude of correction errors was unaffected (before correction: 0.38° ± 0.09; after correction: 0.43° ± 0.02°) and we still found a significant variation across the retinal locations tested (F(17, 51) = 6.36, p < 0.001). These biases in judging verticality of apparent motion, therefore, do not explain the errors of correction for saccades observed in the first experiment. 
Our second control task, for position, measured the effect of position biases (Mateeff & Gourevich, 1983) at the different locations corresponding to those presented in the apparent motion tests. For example, if the first stimulus in the apparent motion sequence is perceived closer to the fovea than it was, and following the saccade, the second stimulus is also perceived closer to the new position of the fovea, then the perceived motion direction will deviate from vertical. 
To simulate the effect of position biases on the trans-saccadic motion stimuli, we used separate trials with eyes fixed for the pre-saccadic (e.g., to the left of fixation) and post-saccadic locations (e.g., to the right) to measure the bias at all appropriate locations (Figure 11a). From these data, we computed the combined bias for the conditions tested in Experiment 2. First, we observed small but significant correlations between the results obtained in the saccade trials and the simulated intersaccadic position biases for both our participants (AC: R 2 = 0.51, one-tailed t-test, p < 0.01; MS: R 2 = 0.36, one-tailed t-test, p < 0.05). Second, we evaluate the magnitude of errors corrected for position bias (Figure 11b) and found a decrease from 0.66° ± 0.04° of alignment errors in the saccade task to 0.50° ± 0.03° of alignment errors corrected for position bias. These results suggest that the position bias explained about 23% of the amplitude of the alignment errors observed in Experiment 2 and about 40 to 50% of the variance observed. The horizontal errors of alignment (Experiment 2) now corrected for position bias still show significant variation (F(5, 5) = 5.02, p < 0.05) across the 6 different locations (3 locations, two directions of saccade), although this ANOVA with only 2 participants can only be considered as suggestive. 
To summarize the results in these two control tasks without eye movements, we found systematic biases in judgments of verticality of motion between two successive stimuli and in judgments of the perceived location of single dots. We estimated the proportion of the saccade correction errors explained by these biases and found that the vertical motion bias had little effect whereas position biases were correlated with the deviations reported in Experiment 2 (R 2 of 0.51 and 0.36 for the two participants). When corrected for position bias, correction errors still represented about 5% of saccade length and showed variation across locations. 
General discussion
Normally, we see objects moving only when they actually move in the world and not whenever our eyes move. Our visual system corrects for eye movements. However, what happens if something moves at the same time as our eyes move? Change blindness (Grimes, 1996; Henderson & Hollingworth, 1999) demonstrates that we are unable to see displacements that occur at the time of a saccade, unless it is the displacement of an attended item (O'Regan et al., 1999). Even then, if an attended target is displaced along the direction of the saccade and the target is present when the saccade lands, significant shifts may be undetected (Bridgeman et al., 1975). Nevertheless, under some conditions, shifts at the time of the saccade are seen: for large displacements (Bridgeman et al., 1975), or displacements orthogonal to the saccade (Niemeier et al., 2003), or displacements where the target reappears after the saccade has landed (Deubel et al., 1996). Our stimuli had all three of these properties, maximizing the visibility of the trans-saccadic motion. The question we addressed was whether the perceived motion would be veridical, that is, appropriate to the displacement in spatial coordinates as opposed to retinal coordinates. Our two first experiments demonstrated that a vertical motion straddling a horizontal saccade was seen roughly in spatial coordinates. This result extends the Rock and Ebenholtz (1962) report of simple spatiotopic apparent motion to a condition where the stimulus motion was orthogonal to the saccade and at non-foveal locations. This allowed us to measure the accuracy of the correction for the saccade at different positions in the visual field. 
We determined the accuracy of the correction at several locations (9 in the first experiment, 3 in the second) within about 15 deg around the fovea for both leftward and rightward saccades. The deviations between the perceived motion and the actual motion as it occurs in space suggest that biases in the correction represent about 5% of the saccade length and that they vary significantly across locations. We observed variations in these local biases across participants and these idiosyncratic patterns were stable over test–retest intervals of up to 3 months. The second experiment showed that these errors held up when measured using a different method (constant stimuli instead of adjustment) and after sorting trials to limit the analysis to trials where the saccade occurred between the presentation of the two positions of the apparent motion stimulus. Two control experiments evaluated the contribution of biases of verticality of motion direction and position seen in the absence of eye movement. The local biases in judging verticality had little or no effect whereas the position biases could account for about a quarter of the amplitude of the local biases in saccade correction. 
What does this mean for the mechanisms of saccade correction?
Following Rock and Ebenholtz (1962) and Fracasso et al. (2010), our results show that we can see a motion in the world that is coincident with an eye movement and that we do not see it in retinal coordinates but in roughly spatial coordinates. Of most interest was the local variation in the correction for the saccades, a result that argues against global approaches to space constancy and for local corrections. It is very unlikely that the probes themselves, presented at different locations, were the source of the position-dependent variation in corrections for the saccades. In our stimulus, the saccade target itself never moved and was present when the saccade landed. The second motion probe appeared about 100 ms after the saccade so it is difficult to argue that the probe location affected the correction for the saccades. Even if it did, the effect should have been larger for probes nearer the fovea (Irwin et al., 1994), but we did not see this pattern as some of the largest errors of correction were for the more distant probes (Figures 4 and 5). 
We suggest therefore that our results support local correction mechanisms like the remapping process seen in single cell activity on saccade control centers (Gottlieb et al., 1998; Kusunoki et al., 2000; Wurtz, 2008). The correction in this remapping case is applied to individual attended targets (Rolfs et al., 2011), and even though the same efference copy vector drives the correction for each target, the conversion of the vector to a shift on the roughly log polar coordinates of the saccade control maps is idiosyncratic to each location and direction (Cavanagh et al., 2010). The two models of this process (Keith & Crawford, 2008; Quaia et al., 1998) require that the link between the target location and the saccade vector that predicts the post-saccadic location must be learned independently for each location and saccade. 
As explained above, our non-uniform results for the different spatial locations tested challenge any alternative explanations sustaining a global or uniform process of saccade correction for the entire visual field. This includes also explanations in terms of incomplete saccadic suppression or a lingering attentional trace at the retinotopic location (Golomb, Chun, & Mazer, 2008; Golomb, Pulido, Albrecht, Chun, & Mazer, 2010). These contributions from retinotopic factors would be seen as hypometric errors (undercompensations) that should be similar for all locations. In contrast, we observed many hypermetric errors (for example, for the central test location as in 1, 2, and 3) and significant local variation. 
The results argue against a simplistic global correction mechanism where the compensation for the eye movement is the same at all locations. Nevertheless, any detailed implementation of a global correction, even models that do not call on extra-retinal information (Bridgeman, 2007; O'Regan et al., 1999), will have to deal with the application of the global correction to local targets that are not the saccade target. This may introduce local variations of correction as well, so our results act more to constrain global models of compensation than to rule them out completely. 
An additional property of correction for saccades was reported by Fracasso et al. (2010) who showed that the apparent motion across a saccade not only keeps track of the pre-saccadic location but also the pre-saccadic shape. Specifically, for a target that changed shape between its pre- and post-saccadic presentations, the shape change was seen as a transformational motion (Tse, Cavanagh, & Nakayama, 1998) rather than a shape replacement. This result suggests that correction for saccades not only updates location but also keeps track of target shape. There is recent evidence of this in an fMRI study of remapping (Knapen, Swisher, Wolfe, Tong, & Cavanagh, 2010) that showed a trans-saccadic transfer of a shaped region of BOLD activation that was isomorphic to the target shape. 
What about mislocalization and compression?
Our results are far from the first to show an effect of location on the perceived position errors of brief probes. Several key papers have established that peri-saccadic mislocalization and compression effects (Honda, 1989; Lappe, Awater, & Krekelberg, 2000; Matin & Pearce, 1965; Morrone et al., 1997) can be as large as the saccade itself and that their amplitude depends on the test position. We will outline first how our tests differ from those of the mislocalization and compression experiments and we then consider what would happen if, indeed, some of these compression effects contributed to the correction errors we report. First the differences: our 400-ms stimuli are not very brief compared to the typical 25 ms or less in the peri-saccadic mislocalization and compression tasks. In addition, the timing is very different. Our first probe is removed about 100 ms before and the second probe presented about 100 ms after the saccade whereas peri-saccadic mislocalization is minimal at these intervals. For these reasons, we suggest that our results reflect the end product of the correction for eye movements and not the rapidly changing intermediate states that may be revealed by brief presentations within 50–75 ms of the saccade. 
Nevertheless, given that our effect is so much smaller (5% rather than 100% of saccade amplitude) perhaps our smaller effect is just the size the compression effects would have at these long durations. Here the evidence is completely the opposite. For example, for probes presented midway between the fixation and the saccade target (see 2 and 3), our results suggest that the first probe is seen, after the saccade has landed, as having been further from the saccade target, not closer as the compression effect would predict. In addition, the compression studies would predict compression toward the saccade target (Morrone et al., 1997; Ross et al., 1997); however, if anything, we found a small tendency to compression toward the fixation target. So we conclude that our local variations in saccade correction are stable, final products of the correction process and unrelated to the peri-saccadic mislocalization effects. 
Why do we not see these deviations from space constancy every time we move our eyes?
Finally, given that we find an inaccuracy of about 5% in the correction for saccades, we must ask why do we not notice these errors in everyday life. First, the most noticeable errors may be those for the saccade target itself, which lands on or near the fovea at the end of the saccade. In our experiments, we never measured locations closer than 5° to the fovea, but a recent article by Collins et al. (2009) did. They also report a bias of about 5% for displacements of the saccade target itself. Specifically, if the saccade target was removed during the eye movement and replaced at its original location with a delay of 200 ms, it appeared to have moved forward—in the direction of the saccade, a hypermetric correction. So why are even these fovea displacements not seen? Deubel et al. (2010) have shown that we are very insensitive to trans-saccadic target displacements if the target is present when the saccade lands, as it is of course for our everyday saccades around real scenes. These authors also showed that we are much more sensitive to the same displacements if there is a blank temporal gap of 50 ms or more between the saccade landing (with no target present) and the representation of the target. In our stimuli, and in those of Collins et al. (2009), there was a post-saccadic delay before the presentation of the new location of the target so this should accentuate its visibility. However, we do suggest that these inaccuracies of spatial constancy may be noticed for attended targets even without blanking, although of course they go unnoticed for unattended parts of the visual field (Cavanaugh & Wurtz, 2004; O'Regan et al., 1999). For example, with a saccade from the left edge of this page to the right edge, a small but consistent displacement of the text in the direction opposite to the saccade might be visible. The displacement is, at best, subtle and easily ignored. In our experiments (see 1), we make this subtle displacement clearly visible by removing the stimulus during the saccade and replacing it only later, about 100 ms after the saccade. 
Conclusion
Using an apparent motion test, we demonstrate that trans-saccadic displacement is perceived in roughly spatial rather than retinal coordinates. This spatiotopic apparent motion provides a simple method for measuring the accuracy of the processes that correct for eye movements and shows that there is a significant local variation in the corrections. These local variations suggest that space constancy depends on local corrections consistent with the physiological remapping of individual attended targets. 
Acknowledgments
We are grateful to J. K. O'Regan for his comments on our work, to members of the Centre Attention and Vision (AV, CB, FVV, MR, RA, TBD, and TK), and to E. Parison. This research was supported by a Chaire d'Excellence Grant to PC and a French Ministère de l'Enseignement Supérieur et de la Recherche Grant to MS. 
Commercial relationships: none. 
Corresponding author: Martin Szinte. 
Email: martin.szinte@gmail.com. 
Address: Laboratoire Psychologie de la Perception, Center Attention & Vision, Université Paris Descartes – CNRS, 45, rue des Saints-Pères, Paris 75006, France. 
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Figure 1
 
Compensation for saccades. (a) Participants were instructed to fixate the green dot that exchanged position with a red dot each 600 ms. While they were fixating to the left, the first black probe (P1) appeared for 400 ms; they then saccaded to the right and a second probe (P2) of the same duration appeared below the first one 200 ms after the offset of P1. (b) On the retina, the first probe falls to the right side of the fovea while the second falls to the left side. (c) To compensate for these effects of the saccade, the visual system corrects the expected location of P1 in the opposite direction to the saccade (red dashed arrow) to obtain C1. If this correction is accurate, motion is perceived in its spatiotopic (vertical motion: from C1 to P2) rather than retinotopic direction (oblique motion: from P1 to P2) and space constancy is maintained.
Figure 1
 
Compensation for saccades. (a) Participants were instructed to fixate the green dot that exchanged position with a red dot each 600 ms. While they were fixating to the left, the first black probe (P1) appeared for 400 ms; they then saccaded to the right and a second probe (P2) of the same duration appeared below the first one 200 ms after the offset of P1. (b) On the retina, the first probe falls to the right side of the fovea while the second falls to the left side. (c) To compensate for these effects of the saccade, the visual system corrects the expected location of P1 in the opposite direction to the saccade (red dashed arrow) to obtain C1. If this correction is accurate, motion is perceived in its spatiotopic (vertical motion: from C1 to P2) rather than retinotopic direction (oblique motion: from P1 to P2) and space constancy is maintained.
Figure 2
 
Stimuli sequences, probe locations, and data analysis. (a) Spatial layout of the stimuli for a rightward saccade trial. The magenta line represents the line of gaze during each cycle. Participants were instructed to saccade to the green dot as it exchanged locations with the red dot each 600 ms while no probes were presented. In the “saccade probe cycle,” participants continued to saccade to the green dot as it changed positions, but now black probe dots were presented before and after the saccade either top-first or bottom-first (only top-first is shown here). (b) The pairs of probes could appear at 9 different spatial locations equally spaced by 10° horizontally and 5° vertically. (c) InExperiment 1, participants used a mouse to adjust the horizontal location of the probes, which were originally displaced vertically by 3° but could be displaced horizontally by ±0.5° or ±1° producing an initial random left or right physical tilt. (d) To compensate for the effect of the saccade, the visual system corrects the expected location of the first black probe in the opposite direction to the saccade (red dashed arrow). The correction vector (CRWD) could be either too long (hypermetric error, displayed here) or too short (hypometric error, not displayed) reflecting, respectively, an over- or undercompensation of the saccade vector (SRWD). After the saccade (middle panel), this error leads to the perception of a tilted motion from the corrected location of the first probe (orange dot) to the second probe location. Participants adjust the two dot locations until the perceived motion is vertical. The error of correction (right panel), that is, the difference between the top and bottom probe horizontal positions (“Δm”), is measured and displayed (red dot) at each tested location (top scale), as an offset relative to the accurate correction for the saccade (gray dashed dot). The bottom scale represents the direction (positive as hypermetric, negative as hypometric error) and the amount of the error.
Figure 2
 
Stimuli sequences, probe locations, and data analysis. (a) Spatial layout of the stimuli for a rightward saccade trial. The magenta line represents the line of gaze during each cycle. Participants were instructed to saccade to the green dot as it exchanged locations with the red dot each 600 ms while no probes were presented. In the “saccade probe cycle,” participants continued to saccade to the green dot as it changed positions, but now black probe dots were presented before and after the saccade either top-first or bottom-first (only top-first is shown here). (b) The pairs of probes could appear at 9 different spatial locations equally spaced by 10° horizontally and 5° vertically. (c) InExperiment 1, participants used a mouse to adjust the horizontal location of the probes, which were originally displaced vertically by 3° but could be displaced horizontally by ±0.5° or ±1° producing an initial random left or right physical tilt. (d) To compensate for the effect of the saccade, the visual system corrects the expected location of the first black probe in the opposite direction to the saccade (red dashed arrow). The correction vector (CRWD) could be either too long (hypermetric error, displayed here) or too short (hypometric error, not displayed) reflecting, respectively, an over- or undercompensation of the saccade vector (SRWD). After the saccade (middle panel), this error leads to the perception of a tilted motion from the corrected location of the first probe (orange dot) to the second probe location. Participants adjust the two dot locations until the perceived motion is vertical. The error of correction (right panel), that is, the difference between the top and bottom probe horizontal positions (“Δm”), is measured and displayed (red dot) at each tested location (top scale), as an offset relative to the accurate correction for the saccade (gray dashed dot). The bottom scale represents the direction (positive as hypermetric, negative as hypometric error) and the amount of the error.
Figure 3
 
Combinations of saccade and motion directions. In each panel, the colored arrow shows the direction of the saccade and the gray arrows show the direction of the motion. The red and blue dots represent the recovered or corrected location of the first probe relative to the second for all participants and for the central probe presentation (error bar indicates SEM). In all these cases, the error observed was an overcompensation of the saccade vector: the recovered location of the first probe is displaced beyond its actual spatial location in the direction opposite to the saccade. To characterize these correction errors, we first use the horizontal shift the subjects add to the display (“Δm”) in order to make it appear vertical. We then correct the sign of the “Δm” each time for the saccade direction and motion direction to recover whether the error was a hypermetric correction (as shown here) or a hypometric correction. For downward motion and rightward saccades (a), this overcompensation of the saccade vector produces a clockwise rotation of the perceived direction, as it does for the leftward saccade and upward motion (d). When we combine alternating left and right saccades with alternating up and down motions as we do in demonstration 1 (either (a) and (d), or (b) and (c)), the tilt will therefore appear the same on each saccade. In the experiment, we test only one combination of saccade and motion direction at a time (see2 and3, respectively, for the conditions of (a) and (b)).
Figure 3
 
Combinations of saccade and motion directions. In each panel, the colored arrow shows the direction of the saccade and the gray arrows show the direction of the motion. The red and blue dots represent the recovered or corrected location of the first probe relative to the second for all participants and for the central probe presentation (error bar indicates SEM). In all these cases, the error observed was an overcompensation of the saccade vector: the recovered location of the first probe is displaced beyond its actual spatial location in the direction opposite to the saccade. To characterize these correction errors, we first use the horizontal shift the subjects add to the display (“Δm”) in order to make it appear vertical. We then correct the sign of the “Δm” each time for the saccade direction and motion direction to recover whether the error was a hypermetric correction (as shown here) or a hypometric correction. For downward motion and rightward saccades (a), this overcompensation of the saccade vector produces a clockwise rotation of the perceived direction, as it does for the leftward saccade and upward motion (d). When we combine alternating left and right saccades with alternating up and down motions as we do in demonstration 1 (either (a) and (d), or (b) and (c)), the tilt will therefore appear the same on each saccade. In the experiment, we test only one combination of saccade and motion direction at a time (see2 and3, respectively, for the conditions of (a) and (b)).
Figure 4
 
Horizontal error of alignment derived from the adjustments required to make the motion between the two probes appear vertical. The data are averaged across the 4 participants and shown separately for rightward and leftward saccades at each of the 9 tested locations. The gray dots show horizontal locations where the first stimulus should be perceived to support accurate vertical motion in world coordinates. Data are presented in retinal coordinates with the central cross showing the location of the fovea, following the saccade. The small red and green dots indicate the fixation and saccade target. Scales at the top and at the left of each map reflect the horizontal and vertical distances from the fovea tested with each of 9 probe pairs. The scale below each map gives the mean value of alignment errors for each spatial location tested. Note that these scales are reversed for rightward and leftward saccades, with positive values always representing overcompensation or hypermetric correction and negative values representing undercompensation or hypometric correction. The alignment scales are also magnified by a factor four relative to the retinal scale to make these deviations clearly visible. Horizontal error bars indicate SEM.
Figure 4
 
Horizontal error of alignment derived from the adjustments required to make the motion between the two probes appear vertical. The data are averaged across the 4 participants and shown separately for rightward and leftward saccades at each of the 9 tested locations. The gray dots show horizontal locations where the first stimulus should be perceived to support accurate vertical motion in world coordinates. Data are presented in retinal coordinates with the central cross showing the location of the fovea, following the saccade. The small red and green dots indicate the fixation and saccade target. Scales at the top and at the left of each map reflect the horizontal and vertical distances from the fovea tested with each of 9 probe pairs. The scale below each map gives the mean value of alignment errors for each spatial location tested. Note that these scales are reversed for rightward and leftward saccades, with positive values always representing overcompensation or hypermetric correction and negative values representing undercompensation or hypometric correction. The alignment scales are also magnified by a factor four relative to the retinal scale to make these deviations clearly visible. Horizontal error bars indicate SEM.
Figure 5
 
Individual results. Each row corresponds to rightward and leftward saccade trials for each participant. Conventions used here are described in Figure 4.
Figure 5
 
Individual results. Each row corresponds to rightward and leftward saccade trials for each participant. Conventions used here are described in Figure 4.
Figure 6
 
Magnitude of correction errors for individuals and group. Each bar represents the mean values of alignment bias independently of the sign of the effect (absolute value) averaged over the nine spatial locations and two saccade directions. Error bars indicate SEM.
Figure 6
 
Magnitude of correction errors for individuals and group. Each bar represents the mean values of alignment bias independently of the sign of the effect (absolute value) averaged over the nine spatial locations and two saccade directions. Error bars indicate SEM.
Figure 7
 
Trial selection. This graph shows fifty representative horizontal eye traces of individual probe cycles for each participant. The bars at the bottom of the figure represent the onset, the offset, and the colors of the saccade target, the fixation target, and the first and second probes. To be sure that apparent motion was tested trans-saccadically, we sorted trials into 3 categories, “early saccade” (light green lines), “late saccade” (black lines, not displayed because they represent less than 1% of trials for both participants), and “correct saccade” trials (dark green lines) where saccades end before the onset of the second probe (see text). Bars at the right of the figure represent the proportion of trials in each category for each participant.
Figure 7
 
Trial selection. This graph shows fifty representative horizontal eye traces of individual probe cycles for each participant. The bars at the bottom of the figure represent the onset, the offset, and the colors of the saccade target, the fixation target, and the first and second probes. To be sure that apparent motion was tested trans-saccadically, we sorted trials into 3 categories, “early saccade” (light green lines), “late saccade” (black lines, not displayed because they represent less than 1% of trials for both participants), and “correct saccade” trials (dark green lines) where saccades end before the onset of the second probe (see text). Bars at the right of the figure represent the proportion of trials in each category for each participant.
Figure 8
 
Results of Experiment 2. (a) Percentage of “clockwise” report of two representative locations tested with rightward (red curves) and leftward (blue curves) “correct saccade” trials for both participants. Psychometric functions were fitted to estimate the point of subjective verticality (PSV), i.e., the amount of horizontal offset of probes leading to equal response of clockwise and counterclockwise deviations from vertical (see red and blue arrows). Shaded areas represent 95% confidence interval computed by bootstrapping. Black arrows represent correction biases observed in Experiment 1 for the same participant at the same location and saccade direction. (b) Maps of horizontal errors of alignment. Each row corresponds to rightward and leftward saccade trials for both participants. Conventions used here are described in Figure 4. The locations of the 4 representative cases displayed in (a) are shown here with light red dots for rightward saccade and with light blue dots for leftward saccade trials.
Figure 8
 
Results of Experiment 2. (a) Percentage of “clockwise” report of two representative locations tested with rightward (red curves) and leftward (blue curves) “correct saccade” trials for both participants. Psychometric functions were fitted to estimate the point of subjective verticality (PSV), i.e., the amount of horizontal offset of probes leading to equal response of clockwise and counterclockwise deviations from vertical (see red and blue arrows). Shaded areas represent 95% confidence interval computed by bootstrapping. Black arrows represent correction biases observed in Experiment 1 for the same participant at the same location and saccade direction. (b) Maps of horizontal errors of alignment. Each row corresponds to rightward and leftward saccade trials for both participants. Conventions used here are described in Figure 4. The locations of the 4 representative cases displayed in (a) are shown here with light red dots for rightward saccade and with light blue dots for leftward saccade trials.
Figure 9
 
Correlations. (a) Correlation of the observed biases obtained with the method of adjustment in Experiment 1 and those from method of constant stimuli of Experiment 2 for the “correct saccade” trials only. The 6 points represent the combination of the 2 directions of saccade and the 3 spatial locations tested. Filled dots correspond to the two representative conditions displayed in Figure 8a. (b) Correlation between saccade landing position relative saccade target and PSV values observed for rightward (red square) and leftward (blue square) representative saccade trials for each participant. PSV values are plotted against the average landing position relative saccade target for four bins of equal sample size. Error bars indicate 95% confidence interval.
Figure 9
 
Correlations. (a) Correlation of the observed biases obtained with the method of adjustment in Experiment 1 and those from method of constant stimuli of Experiment 2 for the “correct saccade” trials only. The 6 points represent the combination of the 2 directions of saccade and the 3 spatial locations tested. Filled dots correspond to the two representative conditions displayed in Figure 8a. (b) Correlation between saccade landing position relative saccade target and PSV values observed for rightward (red square) and leftward (blue square) representative saccade trials for each participant. PSV values are plotted against the average landing position relative saccade target for four bins of equal sample size. Error bars indicate 95% confidence interval.
Figure 10
 
Verticality motion control experiment. (a) Spatial layout of the stimuli for a sample rightward fixation trial. Participants were instructed to keep fixation on the green dot throughout the trial, composed of alternating “no probe cycles” and “probe cycles.” After each “probe cycle,” they moved a mouse to adjust the horizontal locations of both probes in subsequent cycles, continuing until they perceived motion as vertical. The magenta line represents the supposed line of gaze during the trial. Probe position and duration were identical to those of Experiment 1. (b) Absolute values of errors for the 4 participants in the saccade task of Experiment 1, in the verticality of motion judgments during fixation, and for the alignment errors with saccades after correction for verticality or motion bias. Error bars indicate SEM.
Figure 10
 
Verticality motion control experiment. (a) Spatial layout of the stimuli for a sample rightward fixation trial. Participants were instructed to keep fixation on the green dot throughout the trial, composed of alternating “no probe cycles” and “probe cycles.” After each “probe cycle,” they moved a mouse to adjust the horizontal locations of both probes in subsequent cycles, continuing until they perceived motion as vertical. The magenta line represents the supposed line of gaze during the trial. Probe position and duration were identical to those of Experiment 1. (b) Absolute values of errors for the 4 participants in the saccade task of Experiment 1, in the verticality of motion judgments during fixation, and for the alignment errors with saccades after correction for verticality or motion bias. Error bars indicate SEM.
Figure 11
 
Position control experiment. (a) Spatial layout of the stimuli for a sample rightward simulated trial. In two successive randomly presented trials, participants keep fixation on the green dot located at the right or left of the screen center. The magenta line represents the supposed line of gaze during trials. Participants evaluate the probe position displayed above and then below the horizontal midline with a physical horizontal offset following a method of constant stimuli. (b) Absolute values of errors for the 2 participants in the saccade task of Experiment 2, in the position judgment during fixation, and for the alignment errors with saccade after correction of position bias. Error bars indicate SEM.
Figure 11
 
Position control experiment. (a) Spatial layout of the stimuli for a sample rightward simulated trial. In two successive randomly presented trials, participants keep fixation on the green dot located at the right or left of the screen center. The magenta line represents the supposed line of gaze during trials. Participants evaluate the probe position displayed above and then below the horizontal midline with a physical horizontal offset following a method of constant stimuli. (b) Absolute values of errors for the 2 participants in the saccade task of Experiment 2, in the position judgment during fixation, and for the alignment errors with saccade after correction of position bias. Error bars indicate SEM.
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