**Saccades quite systematically undershoot a peripheral visual target by about 10% of its eccentricity while becoming more variable, mainly in amplitude, as the target becomes more peripheral. This undershoot phenomenon has been interpreted as the strategic adjustment of saccadic gain downstream of the superior colliculus (SC), where saccades are programmed. Here, we investigated whether the eccentricity-related increase in saccades' hypometria and imprecision might not instead result from overrepresentation of space closer to the fovea in the SC and visual-cortical areas. To test this magnification-factor (MF) hypothesis, we analyzed four parametric eye-movement data sets, collected while humans made saccades to single eccentric stimuli. We first established that the undershoot phenomenon generalizes to ordinary saccade amplitudes (0.5°–15°) and directions (0°–90°) and that landing-position distributions become not only increasingly elongated but also more skewed toward the fovea as target eccentricity increases. Moreover, we confirmed the MF hypothesis by showing (a) that the linear eccentricity-related increase in undershoot error and negative skewness canceled out when landing positions were log-scaled according to the MF in monkeys' SC and (b) that the spread, proportional to eccentricity outside an extended, 5°, foveal region, became circular and invariant in size in SC space. Yet the eccentricity-related increase in variability, slower near the fovea, yielded progressively larger and more elongated clusters toward foveal and vertical-meridian SC representations. What causes this latter, unexpected, pattern remains undetermined. Nevertheless, our findings clearly suggest that the undershoot phenomenon, and related variability, originate in, or upstream of, the SC, rather than reflecting downstream, adaptive, strategies.**

^{2}) on a black background, using a 17-in. CRT-monitor with a 60-Hz refresh rate, positioned at a distance of 850 mm from the participant's eyes. All data were collected using a Dual-Purkinje-Image Eye-Tracker (Ward Technical Consulting), which samples the right eye position every millisecond with an accuracy of 10 minutes of arc. The eye-movement signal was analyzed online and reanalyzed offline, using the software developed at the Catholic University of Leuven by van Rensbergen and de Troy (1993). A more detailed description of the experimental setup is presented in Casteau and Vitu (2012).

*φ*, the corresponding collicular coordinates,

*u*and

*v*, as expressed from the rostral pole (or the representation of the fovea) along the two orthogonal axes representing, respectively, the horizontal and the deviation from the horizontal, were calculated using the following formulas: where

*B*,

_{u}*B*, and

_{v}*A*were constants, set to 1.4, 1.8, and 3, respectively (i.e., the anisotropic model).

*x*,

*y*) and collicular (

*u*,

*v*) space, but without making any assumption of their shape (e.g., whether or not they were normally distributed and unimodal), the procedure advocated by Castet and Crossland (2012) for estimating fixation stability was applied. First, the probability density function of landing positions was estimated, using the kernel density estimation (ks package in the R software; R Core Team, 2012), with a fixed kernel bandwidth (0.3° and 0.02 mm in visual and SC space, respectively); note that similar findings were obtained when the optimal bandwidth was determined using adaptive procedures, and also, as illustrated in Figure 1C, that the shape of the probability density functions matched the scatter of the observed landing positions. Then, isolines (or contour lines) were computed and were set to delineate the region(s) in space that contained 90% of the data points with the highest density estimate (see Figure 1C). This allowed us, at the same time, to filter out the small, isolated clusters, which were mostly located near the initial fixation stimulus and corresponded to initial saccades of small amplitude (≤2°; about 1.09%, 0.83%, 0.45%, and 1.16% of all initial saccades in conditions where the target appeared at an eccentricity greater than 3° in HM1, HM2, HM3, and HM-VM). When two (or more) separate clusters remained, the largest (which corresponded to the cluster with the highest density estimate at its mode) was kept for further analysis, thus providing the most representative region in space where the eyes initially landed. However, in the particular case where two clusters were detected in two opposite colliculi, as was the case when the target was displayed along the vertical meridian (see the Results section), the area considered for further analysis was the area corresponding to saccades in the correct horizontal direction (i.e., the saccades directed to the target side given the initial eye deviation from the fixation stimulus; see above); this actually corresponded to the largest area in most individuals (see Results). Note that this also allowed a more direct comparison with the landing-position distributions associated with targets displayed on other axes, which corresponded almost exclusively to saccades in the correct direction. Yet, as verified in complementary analyses, this procedure was not responsible for the specific data pattern observed with targets displayed on and near the vertical meridian (see Results).

*R*

_{sacc}) and its direction (

*φ*

_{sacc}) were computed based on the (

*x*,

*y*) coordinates of the mode in visual space, with

*R*

_{sacc}corresponding to the square root of the sum of squares of

*x*and

*y*and

*φ*

_{sacc}corresponding to the angle formed by

*R*

_{sacc}(see Figure 1D, left panel). To estimate saccade accuracy as a function of the effective eccentricity (

*R*

_{tg}) and direction (

*φ*

_{tg}) of the target, both

*R*

_{tg}and

*φ*

_{tg}were recalculated on each trial, taking into account the initial eye deviation relative to the fixation stimulus (exactly as was done for initial landing positions); these were then averaged over all trials in a given condition for comparison with the corresponding saccade parameters. Two estimates of saccade accuracy in visual space were derived: (a) the

*amplitude error*(or difference in degrees of visual angle between the radius of the mode and mean target eccentricity) and (b) the

*gain*(or ratio of radius of the mode and mean target eccentricity). Because Ottes et al.'s (1986) afferent-mapping function projects

*R*- and

*φ*-retinal coordinates onto corresponding

*u*- and

*v*-collicular coordinates, the

*u*-coordinate of the estimated mode was used to approximate saccade amplitude in SC space. To compute the corresponding amplitude error, the u-coordinate of the mode was subtracted from the mean across trials of the u-coordinate of the effective target eccentricity (relative to the initial fixation position) but after shifting the target on the collicular saccade axis (see Figure 1D, right panel). The latter correction was done to free amplitude-error estimates from possible direction errors; indeed, the u-value that corresponds to a given target eccentricity (or saccade amplitude) varies depending on target (saccade) direction.

*area*subtended by the selected isoline cluster was also computed in both visual and SC space. This was used as a first index of the spread of initial landing position distributions. To further characterize visual and collicular landing position distributions, an ellipse was fitted to the selected isoline area. The estimated

*length of major and minor axes of the ellipse*provided additional estimates of the spread of the distribution, along and perpendicular to the target axis, as we will see; following van Opstal and van Gisbergen (1989b), we will refer to these as

*eccentricity*and

*direction axes*. Estimating major- and minor-axis lengths also allowed us to compute the area subtended by the ellipse and to estimate in turn the quality of the fit, by comparison with the estimated area of the isoline cluster before the ellipse was fit. This was relatively good and comparable across individuals and data sets, as well as between visual and SC spaces (mean residuals: 0.00242 deg

^{2}and 0.00012 mm

^{2}, 0.00884 deg

^{2}and 0.00013 mm

^{2}, 0.00442 deg

^{2}and 0.00010 mm

^{2}, 0.01589 deg

^{2}and 0.00045 mm

^{2}, respectively in HM1, HM2, HM3, and HM-VM data sets). The slope of the relationship between the area subtended by the fitted ellipse and the 90% isoline area was close to 1 (visual space: 0.995, 0.985, 0.996, and 0.987; SC space: 1, 0.999, 1 and 0.987; for HM1, HM2, HM3, and HM-VM, respectively), and the proportion of variance explained (

*R*

^{2}) was greater than or equal to 0.9995 in both visual and SC space.

*ratio of the ellipse's major and minor axes*was then computed to estimate the elliptic versus round shape of visual and collicular distributions. Finally, the level of asymmetry of visual and collicular distributions and their skewness (or signed asymmetry) along eccentricity and direction axes, and hence in terms of saccade amplitude and direction, were estimated. The absolute

*asymmetry*corresponded to the Euclidian distance between the estimated center of the selected isoline cluster (see below) and its mode. The

*amplitude skewness*in visual space corresponded to the difference between the eccentricity of the center and the eccentricity of the mode of the selected isoline cluster; in SC space, it corresponded to the difference between corresponding u-coordinates. It was negative when the distribution was skewed toward the (representation of the) fovea and positive when skewed in a direction away from the (representation of the) fovea.

*Direction skewness*in visual space corresponded to the distance, in degrees of visual angle, between the axis of the center and the axis of the mode of the selected isoline cluster; in SC space, it corresponded to the difference between corresponding v-coordinates. It was negative when the distribution was skewed toward the (representation of the) horizontal meridian or a position below it. Three different estimates of the center of the selected isoline cluster were used: (a) the estimated center of the fitted ellipse, (b) the mean of the sampling points that were used for computation of the isoline area, and (c) the raw mean of all initial landing positions, but after exclusion of the population of small-amplitude saccades that kept the eye in the region of the initial fixation stimulus. All three center estimates yielded similar asymmetry/skewness patterns as a function of the tested variables, but the first two were a bit less sensitive, leading to slightly smaller estimates of asymmetry and skewness compared with the third, raw mean–based estimate of the distribution's center. Only the results for the latter are presented below.

^{2}× Direction). For HM data sets, the starting fixed structure comprised both the linear and the quadratic components of the effect of target eccentricity. In all analyses, the predictor(s), that is, eccentricity in HM data sets and eccentricity and direction in the HM-VM data set, were entered as continuous variables. Because the smallest eccentricity in the range of tested eccentricities was different from zero and differed between data sets, this variable was centered on its mean. The fixed effects of optimal LME models are fully reported in Tables A1–A10 of Appendix 1. Only significant (

*p*≤ 0.05) intercept and slope estimates are reported in the Results section; for each model, the optimal random structure is presented in the corresponding table's legend. Note that nonsignificant effects and interactions are often dropped when gradually converging toward an optimal fixed structure; they were thus not systematically reported in the tables, all depending on the outcome of the optimal fixed-structure procedure.

*u*-collicular coordinate, the data points corresponding to different target directions overlapped more (for comparison, see Figure 5B–C) and showed an overall increase as

*u*-values went from about 1 mm to near 0 mm. In addition, as shown in Figure 8B, the relationship for targets on the horizontal meridian in HM-VM was quite similar to that obtained in the other HM data sets; all nicely overlapping curves showed a gradually increasing spread of landing positions as

*u*-values decreased, starting again from about 1 mm. This suggests that how far, from the rostral pole, the target projects may be a critical variable in determining the variability of saccadic endpoints. Yet this clearly cannot be the only explanatory variable for the gradual increase of the spread of landing positions from the 0° to the 90° axis, because this was still visible even after controlling for the target's

*u*-coordinate (see Figure 8A). The fact that small-eccentricity targets are more likely to yield bilateral population activity in SC space near the vertical meridian (see Figures 2B–C and 4B) may contribute as well. However, this can also not be the only explanation because bilateral landing position distributions were observed mainly in the 90° condition.

*u*-coordinate on the rostrocaudal axis. In Figure 9C, the ratio of major-to-minor axis length in SC space was thus replotted as a function of the target's

*u*-coordinate in HM-VM and HM data sets (left and right panels, respectively). This indicates that landing-position distributions tended to be relatively circular (ratio only slightly greater than 1) when targets were projected further than about 1 mm from the rostral pole. For targets projected within less than 1 mm, the distributions were more like ellipses that were gradually more elongated as targets projected closer to the rostral pole but still more so closer to the representation of the vertical meridian. Thus, the target's coordinate on the rostrocaudal axis seemed to play a role, but again could not be the only explanation for the effect of target direction. Similarly, the fact that landing-position distributions tended to be bilateral for small-amplitude saccades along the 90°, and to some extent the 80°, axis, although likely contributing, could not account by itself for the remaining differences as a function of target direction, because these ranged from the oblique to the vertical conditions.

*u*-axis (see Appendix 1, Tables A6B–A7B). The negative intercept estimate for amplitude skewness was significantly different from zero in all data sets except HM1 (−0.003, −0.006, and −0.006 in HM2, HM3, and HM-VM, respectively), whereas that for direction skewness was significant only in HM-VM (estimate: 0.002). Amplitude skewness showed no significant variation with eccentricity except in HM3, where both the linear and the quadratic components were significant (estimates: −0.001 and 0.0001, respectively). In HM-VM, there was an interaction between eccentricity and direction (estimate: −0.00004).

*u*-axis. Yet, this asymmetry was relatively invariant across eccentricities, except again near the foveal and vertical-meridian representations in some data sets. Thus, the peripheral increase in visual negative skewness could be explained by nonhomogeneous efferent mapping, but negative skewness itself cannot. Negative skewness is also not consistent with our initial assumption that jitter in the location of point images is rotation symmetrical.

*p*value]: 0.953 [0.0007], 0.983 [0.0431], 0.989 [0.0004], and 0.950 [0.0050] in HM1, HM2, HM3, and HM-VM, respectively). It overall remained unaffected by target eccentricity; only the quadratic effect of eccentricity in HM2 (slope estimate: −0.00113,

*p*= 0.0000) and the interactions between eccentricity and direction in HM-VM were significant (slope estimates: −0.00011,

*p*= 0.0001, and 0.00002,

*p*= 0.0433, for Eccentricity × Direction and Eccentricity

^{2}× Direction, respectively).

*p*< 0.0000 and 0.1680,

*p*< 0.0000, respectively). Yet the interaction between bin and the quadratic component of the effect of eccentricity was not significant in these two data sets, and none of the interactions were significant in HM1, where eccentricities ranged between 0.5° and 6°. This suggests that the slower eccentricity-related increase in spread within, compared with outside, the extended foveal region was not due to small-amplitude saccades being launched with longer latencies.

*p*s ≥ 0.0683); the interaction between bin and the linear component of the effect of eccentricity was significant in all data sets (estimates: 0.078, 0.053, 0.045, and 0.040 in HM1, HM2, HM3, and HM-VM, respectively), and the interaction between bin and the quadratic component of the effect of eccentricity was significant in HM1 (estimate: −0.013). This finding, which corroborates previous reports showing a reduction over time of the linear relationship between saccades' hypometria and target eccentricity (de Bie et al., 1987), further argues against an overall gain-adjustment account of the undershoot phenomenon, as further discussed below.

*u*), axis. Actually, when applying the correction they proposed for computation of the direction,

*v*, parameter, we obtained data patterns that were nearly identical to those reported above (not shown here).

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