In an influential theoretical model, human sensorimotor control is achieved by a Bayesian decision process, which combines noisy sensory information and learned prior knowledge. A ubiquitous signature of prior knowledge and Bayesian integration in human perception and motor behavior is the frequently observed bias toward an average stimulus magnitude (i.e., a central-tendency bias, range effect, regression-to-the-mean effect). However, in the domain of eye movements, there is a recent controversy about the fundamental existence of a range effect in the saccadic system. Here we argue that the problem of the existence of a range effect is linked to the availability of prior knowledge for saccade control. We present results from two prosaccade experiments that both employ an informative prior structure (i.e., a nonuniform Gaussian distribution of saccade target distances). Our results demonstrate the validity of Bayesian integration in saccade control, which generates a range effect in saccades. According to Bayesian integration principles, the saccadic range effect depends on the availability of prior knowledge and varies in size as a function of the reliability of the prior and the sensory likelihood.

*Bayesian brain*hypothesis (Knill & Pouget, 2004; Petzschner et al., 2015), the estimation of environmental parameters can be mathematically formulated as the product of the prior belief distribution

*Q*(Θ) on the parameter Θ and the likelihood distribution

*L*(S|Θ):

_{Θ|S}of the posterior distribution is the weighted average of the prior mean μ

_{Θ}and the likelihood mean μ

_{S|Θ}, where the weights are given by the relative uncertainties (i.e., variances) of the distributions:

_{Θ}, that is, from a person’s central a priori expectation (Figure 1A). On the other hand, a weak central-tendency bias results from weak influence of the prior on the posterior estimate, that is, when the sensory likelihood is clearly more reliable than the prior information. In recent years, the Bayesian framework has emerged as a universal account for the influence of prior knowledge in human sensorimotor behavior. Prior-evoked central-tendency effects, i.e., the characteristic overestimation of small stimulus magnitudes and underestimation of large stimulus magnitudes within the range of possible true values of Θ, have been reported across sensory modalities, for different environmental parameters, and across different motor tasks (Körding & Wolpert, 2004; Vilares et al., 2012; Jazayeri & Shadlen, 2010; Sciutti et al., 2014; Sato & Kording, 2014; Olkkonen et al., 2014; Kok et al., 2013; Grau-Moya et al., 2012; Wiener et al., 2016; Trommershäuser et al., 2003; Petzschner & Glasauer, 2011).

*range effect*within a simple saccade-targeting experiment published by Kapoula (1985) (see also Kapoula & Robinson, 1986). In this study, subjects made goal-directed saccades from a fixation stimulus toward a small square-shaped target stimulus that appeared randomly across trials at one out of five different distances ranging from 2.7

^{○}to 9.5

^{○}in one block of the experiment and from 7

^{○}to 21.9

^{○}in a different block. The main finding was that subjects systematically overshot the targets at the near end of the range of target distances and undershot the targets at the far end. Since the two test ranges overlapped, it was found that subjects either under- or overshot the overlapping target distances depending on whether the distances where framed within the proximal or the distal range. Kapoula’s work is widely cited and has become a standard reference for a similar effect in eye movements during reading (McConkie et al., 1988; Rayner, 1998).

^{○}and 13

^{○}or 10.5

^{○}and 20.5

^{○}of eccentricity. In this study, target-presentation times of 50 ms were used in order to decrease the frequency of secondary saccades. Gillen et al. (2013) found no evidence of a saccadic range effect, but they did replicate the finding of a general saccadic undershoot across target distances. The same results were found in two further studies (Gillen & Heath, 2014; Heath et al., 2015), where no range effect was found for prosaccades, and in a study by Vitu (1991), who tested the range effect in saccadic eye movements using isolated words at different distances as saccade targets and found, again, no evidence for a range effect.

*flat prior with unlimited support*), the posterior estimate would not be biased toward a central location regardless of the actual precision of the sensory likelihood (Figure 1B). Thus, an uninformative flat prior represents a special case for which the Bayesian model predicts the absence of a central-tendency bias.

*range effect*should not be used as a synonym for over- and undershoots as the landing positions of generally hypometric saccades toward a near target need not necessarily overshoot the target even if its position was perceptually overestimated in the first place. The critical test of a central-tendency bias in saccades is the prediction of a relative difference of landing positions for targets at identical absolute positions that are framed as either relatively near or relatively far depending on the range of target eccentricities in the experiment. For example, within a range of proximal target eccentricities from 2° to 12

^{○}, a target at 10

^{○}should elicit a larger undershoot than the identical target within a set of distal targets ranging from 8

^{○}to 18

^{○}, if the position of these targets is underestimated in the proximal condition and overestimated in the distal condition depending on the availability of priors that favors distances shorter than 10

^{○}(proximal range) or longer (distal range) than 10

^{○}.

^{○}–12

^{○}and 8.5

^{○}–18.5

^{○}). Furthermore, target presentation times were either 50 ms as in Gillen et al. (2013) or 500 ms, that is, longer than a typical saccadic reaction time, as in Kapoula (1985) and Nuthmann et al. (2016). Most important, the probability of target locations within the test ranges was not uniformly distributed but randomly drawn from a centered Gaussian probability distribution leading to a higher probability of target locations near the center of the test range and a decreasing probability of target locations near the boundaries of the test ranges. We predict that the underlying informative prior structure leads to the development of a central-tendency bias in saccadic eye movements. In more detail, we expect that subjects systematically overestimate the position of targets at the near end of both test ranges and systematically underestimate the distance of targets at their far ends, leading to corresponding modulations of saccadic responses and, most important, to the prediction of different saccadic landing positions at overlapping target distances. Overlapping targets in Experiment 1 (between 8.5

^{○}and 12

^{○}) should elicit stronger undershoot in the proximal condition of the experiment (as we predict target-position estimates to be biased toward the center of the proximal range at 7

^{○}) than in the distal condition (as the target-position estimates should be biased toward the center of the distal range at 13.5

^{○}).

^{○}to 20

^{○}. In order to manipulate the reliability of the sensory likelihood, the targets of this experiment consisted of a cloud of points, which were randomly scattered according to a two-dimensional Gaussian (centered at the true target position) with either a low standard deviation (i.e.,

*narrow likelihood condition*) or a high standard deviation (i.e.,

*wide likelihood condition*). Furthermore, the probability distribution of the varying target positions corresponded to either a broad Gaussian distribution (i.e.,

*weak prior*) or a narrow Gaussian (i.e.,

*strong prior*) for either half of the participants. Following Equation 2, we expected that saccadic errors toward the same saccade target distances within the same range of target eccentricities are modulated by both the reliability of the prior and the likelihood. More specifically, we expected that the combination of a weak prior and precise sensory information (narrow likelihood) generates the smallest centering bias, and vice versa, we expected that an informative prior together with increased sensory uncertainty (wide likelihood) leads to the strongest bias.

^{○}× 0.5

^{○}) either on the left or the right side of the monitor. A fixation check within an invisible squared box centered on the fixation cross with a side length of 1

^{○}ensured an appropriate gaze position before the presentation of the target. A successful fixation check triggered the onset of a variable foreperiod interval between 250 and 550 ms, which was randomly drawn for each trial from a uniform distribution. After the foreperiod, the fixation cross disappeared and the target stimulus was simultaneously flipped to the screen without any time delay.

^{○}. The target appeared at variable distances that were randomly drawn from two different (truncated) Gaussian prior distributions: The proximal prior, \(d = {\cal N}(\mu ,\sigma )={\cal N}(7^{\circ },2.6^{\circ }) \hbox{ with } 2^{\circ }\lt d \lt 12^{\circ }\), was sampled from a normal distribution with a mean target distance of 7

^{○}and a standard deviation of 2.6

^{○}, truncated at 2

^{○}and 12

^{○}. Thus, when the sampled value fell outside the truncated range, another sample was drawn. Targets in the distal prior condition, \(d = {\cal N}(13.5^{\circ },2.6^{\circ }) \hbox{ with } 8.5^{\circ }\lt d \lt 18.5^{\circ }\), were drawn from a normal distribution with a mean distance of 13.5

^{○}and a standard deviation of 2.6

^{○}, truncated at 8.5

^{○}and 18.5

^{○}.

^{1}

^{○}on the left side of the monitor. The target appeared after a constant foreperiod of 750 ms.

^{○}, 6

^{○}, 8

^{○}, 10

^{○}, 12

^{○}, 14

^{○}, 16

^{○}, 18

^{○}, or 20

^{○}from the starting fixation position. The standard deviation of the underlying Gaussian was either 0.1

^{○}(narrow likelihood) or 2.5

^{○}(wide likelihood). The frequencies of the targets to appear at one of the nine positions across the test range were nonuniformly distributed and differed between the two prior conditions (weak vs. informative) according to Table 1.

^{○}) was employed. In the next two sessions, wide and narrow likelihood trials were equally often presented in a randomized order.

*ps*< 0.5). Note that we pooled the data from the 50-ms and 500-ms presentation-time conditions as it turned out that presentation time had no significant effect on the spatial accuracy of the saccades in our experiment (see the results of the mixed-effect model below).

^{○}to 12

^{○}, all but one subject showed the expected negative linear trend in saccadic errors with increasing target distance. Moreover, the predicted overestimation of near target positions and underestimation of far target positions in the proximal range is reflected by a systematic overshoot of the near targets and undershoot of far targets in all subjects that exhibited the negative linear trend. For the distal prior condition (targets between 8.5

^{○}and 18.5

^{○}), the result is not as ideal-typical as the results of the proximal prior condition. Six out of 10 participants show the expected negative trends, with three participants further showing an overshoot of the near targets within the distal range. Four subjects (1,3,5, and 7) show no systematic modulation of saccadic errors by target distance within the distal range.

*t*values larger than 2 were considered significant at the 5% level (Kliegl et al., 2013). Table 2 shows the fixed-effect results of the linear mixed model.

^{○}were systematically overshot and targets above 7

^{○}were undershot. Moreover, as the intercept did not deviate from the center of the proximal range, we conclude that the transition of saccadic overshoot responses to saccadic undershoots appears at the center of the range of target distances in this experimental condition. Thus, the results for the proximal conditions demonstrate a clear range effect in saccades and agree almost ideal-typical with the predictions of the Bayesian saccade-planning model. Target presentation time did not modulate saccadic accuracy, suggesting that the precision of the sensory localization of the position of the target did not differ between the 50-ms and the 500-ms presentation condition. The mixed model analysis for the distal condition revealed a significant negative trend in saccadic errors as a function of target distance plus a significantly negative intercept. No other effects were significant. These results confirm that saccadic errors within the distal range depend on the distance of target within the range such that increasing target distances elicit increasing undershoots of target positions. However, as reflected in the negative intercept of the model and visualized in Figure 4, saccadic responses toward targets below 13

^{○}(i.e., the center of the distal range) lack a systematic overshoot component. However, as already pointed out, the absence of an overshoot component could be due to a general oculomotor undershoot strategy, especially to distant target stimuli. Thus, the key test for the presence of a range effect in the saccadic responses in this experiment is the comparison of saccadic errors in overlapping saccade-target distances of the proximal and the distal ranges. Figure 4 suggests that the landing-position errors of saccades toward the same absolute target distances are substantially different between the distal and proximal conditions and were systematically biased toward the centers of the respective range. In order to test this difference, we ran a separate linear regression analysis on saccadic errors within the overlapping range of target distances (i.e., from 8.5

^{○}to 12

^{○}) employing target distance and range (proximal vs. distal) as predictors. Results are reported in Table 3. We found again a highly significant negative linear trend in saccadic error by target distance (i.e., increasing undershoots with increasing target distance). Most important, we also found a highly significant effect of the range of target distances such that targets from 8.5

^{○}to 12

^{○}in the distal condition elicited larger undershoot errors than the corresponding targets in the proximal condition. This range-contingent difference of saccadic landing positions on identical targets can not be explained by a general undershoot tendency of the saccadic system but matches our key prediction for a saccadic range effect as a result of biased target-position estimates based on the availability of informative prior knowledge.

^{○}–20

^{○}), reliability of the prior (weak vs. informative), reliability of the likelihood (narrow vs. wide), and all possible interactions. Subjects were included as random intercepts. Since we expected the smallest range effect for the combination of weak prior information and precise sensory measurements (i.e., narrow likelihood), the levels of the factor prior were represented numerically as weak = 0 and informative = 1, and the levels of the factor likelihood as narrow = 0 and wide = 1, so that the fixed-effects estimates of the linear mixed model could be interpreted relative to the baseline condition “weak prior and narrow likelihood.” Estimates with

*t*values larger than 2 were considered significant at the 5% level. Table 4 shows the coefficients of the fixed effects of our analysis.

^{○}and 12

^{○}(average target distance of 7

^{○}) and a distal range between 8.5

^{○}to 18.5

^{○}(average target distance 13

^{○}). We found that targets appearing at identical positions within both ranges elicited different saccadic landing-position errors. Moreover, the differences in the saccadic errors between the proximal and the distal range agreed well with the prediction of the Bayesian model as we found that saccadic landing positions at distal targets in the proximal range largely undershoot their counterparts in the proximal range, suggesting that targets at the far end of the proximal range are biased toward the center of the proximal range while identical targets in the distal range are biased toward the center of the distal range. This finding demonstrates that saccadic responses are not fully determined by the absolute spatial position of the target but are systematically biased according to the range of other targets in the experiment, hence showing the existence of a saccadic range effect.

*uniform prior*; see also Nuthmann et al. 2016; Vitu, 1991). Against the backdrop of our findings and in the light of Bayesian decision theory, an opposite conclusion would be more appropriate: The fact that saccadic errors in the previous studies did not express a significant difference between target locations actually reflects that the relational properties associated with the target eccentricities were learned (i.e., equal probabilities) and appropriately employed by the subjects. As we have pointed out before (see Figure 1), the consequence of a uniform prior plus clearly visible targets (low sensory uncertainty) is the absence or near absence of a range effect. This direct consequence of a Bayesian target-localization process during saccade control is particularly instructive, since the debate about the existence of a saccadic range effect is based on mixed findings from experiments with uniform priors. Our results provide important insights into the perceptual processes that contribute to the generation of goal-directed saccades, and the idea of Bayesian saccade planning allows a reevaluation of previous studies on the saccadic range effect.

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