In contrast with the anisotropies in spatial and motion vision, anisotropies in the perception of motion duration have not been investigated to our knowledge. Here, we addressed this issue by asking observers to judge the duration of motion of a target accelerating over a fixed length path in one of different directions. Observers watched either a pictorial or a quasi-blank scene, while being upright or tilted by 45° relative to the monitor and Earth's gravity. Finally, observers were upright and we tilted the scene by 45°. We found systematic anisotropies in the precision of the responses, the performance being better for downward motion than for upward motion relative to the scene both when the observer and the scene were upright and when either the observer or the scene were tilted by 45°, although tilting decreased the size of the effect. We argue that implicit knowledge about gravity force is incorporated in the neural mechanisms computing elapsed time. Furthermore, the results suggest that the effects of a virtual gravity can be represented with respect to a vertical direction concordant with the visual scene orientation and discordant with the direction of Earth's gravity.

^{−2}(typical gravitational acceleration), while the direction was downward, upward, rightward, or leftward in different blocks of trials. Thus, target kinematics was congruent with the effects of gravity only for downward motion.

^{−2}), their speed ranged between 18° s

^{−1}and 40° s

^{−1}. These values are well within the range of speeds that are known to be best discriminated: Weber fractions (WFs) have been shown to be lowest (about 7%) for speeds between about 4° s

^{−1}and 64° s

^{−1}(De Bruyn & Orban, 1988). As for acceleration detection, it is often assessed in terms of a ratio analogous to a WF: (

*V*

_{final}−

*V*

_{initial})/

*V*

_{average}, where

*V*

_{final},

*V*

_{initial}, and

*V*

_{average}are the final, initial, and average speeds, respectively (Calderone & Kaiser, 1989; Regan, Kaufman, & Lincoln, 1986). In our experiments, this ratio ranged between 0.4 and 1.96 (corresponding to the shortest and longest durations of motion, respectively). These values are above the detection thresholds of acceleration (0.17–0.25), which are typically reported in the literature (Brouwer, Brenner, & Smeets, 2002; Calderone & Kaiser, 1989; Regan et al., 1986; Werkhoven, Snippe, & Toet, 1992).

*SD*). They were right-handed (as assessed by a short questionnaire based on the Edinburgh scale). All participants in this and the following experiments had normal or corrected-to-normal vision and gave informed consent to procedures approved by the Institutional Review Board of Santa Lucia Foundation, in conformity with the Declaration of Helsinki on the use of human subjects in research.

*X*-axis and upward

*Y*-axis in the frontal plane, plus in-depth

*Z*-axis. Scene projection was computed using on-axis linear perspective, assuming a viewpoint at [0, 3 m, −10 m] and looking at point [0, 3 m, 0]. The fixation point was located at [0, 3 m, 6 m] of this frame, while the center of mass of the target moved in a frontal plane through the origin [0, 0, 0]. We displayed a colored scene (35° by 26°, horizontal and vertical visual angles, respectively) with the image of a large room (Figure 1). Eight human figures were placed at different positions in the room to provide an approximate metric reference. Photographs of the adults were downloaded with permission from www.vyonyx.com (copyrights owned by VYONYX). The photograph of the child was downloaded with permission from www.imagecels.com (copyrights owned by Realworld Imagery). The fixation point was a red dot (0.38°) placed in the center of the rear wall. There were four circular holes in the room: on the ceiling, on the floor, and on each of the sidewalls. The distance between each pair of opposite holes was 6 m in the scale of the scene (corresponding to 20°, ±10° around fixation). A white stripe painted on the ceiling, floor, and sidewalls connected the holes.

^{−2}(33° s

^{−2}), while the direction was downward, upward, rightward, or leftward, depending on the specific block of trials (see Procedures section below). Thus, target kinematics was congruent with the effects of gravity only in the downward block. The initial speed of the target could vary in different trials, resulting in a variable total duration of the visible motion (see below). Luminance and RGB color coordinates of the background were determined over the region corresponding to the rear wall of the room: average luminance was 23 cd m

^{−2}(as measured by means of Tektronix J17 LumaColor photometer) and average RGB coordinates were 171, 113, and 78 (as determined by NBS program).

^{−1}), resulting in a total duration of motion

*T*= 800 ms and in an average speed of 25° s

^{−1}over the displayed trajectory. During this phase, participants were instructed to watch each video so as to memorize the standard flight duration. After this phase, 360 test trials (ISI = 2500 ms) were presented with the same direction of motion as the standard trials but with a variable initial speed and total duration of motion. These stimuli could have one of nine possible durations (within a range of 500–1100 ms, centered on 800 ms), randomized across trials. Each exact duration was an integer multiple of the monitor frame duration. The average target speed ranged between 18° s

^{−1}and 40° s

^{−1}(corresponding to the longest and shortest durations, respectively). In each test trial, after 500 ms from target disappearance, a question mark appeared over the fixation point, prompting the participants to provide a response in 2 s. (If they responded before or after the allocated time window, the trial was rejected and repeated at the end of the experiment.) They indicated whether the test stimulus was longer or shorter in duration than the standard stimuli by pressing the right or left mouse button, respectively. Each stimulus duration was presented 40 times (9 durations × 40 repetitions = 360 test trials) in each block. Participants were asked to fixate the central red dot during the presentation of both standard and test stimuli. No performance feedback was provided.

*Y*= 1, and a “Shorter” response as

*Y*= 0. Because the responses were asymmetrically distributed about

*P*(

*Y*= 1) = 0.5, they were fitted with the log–log link function:

*P*() is the probability of response for test duration equal to

*x,*and

*α*

_{p}and

*β*

_{p}are the intercept and slope of the model, respectively. The log–log model fitted the data better than a logistic model according to the Akaike information criterion (AIC; Akaike, 1973): at the population level, the difference in AIC was 131. Although the choice of the log–log model was primarily suggested by its good fit, there was also an a priori reason to prefer it over the logistic model. As pointed out by Miller and Ulrich (2001), Weber's law provides a very general argument that ideal psychometric functions should be positively skewed (as in the log–log model), rather than symmetric functions (as in the logistic or probit models), because a given change in stimulus value should have a greater effect at the bottom of the stimulus range than at the top.

*β*

_{p}provides a measure of the precision of discrimination (the higher the slope, the greater the precision). The point of subjective equivalence (PSE) estimates the accuracy of the judgment. It was computed from the psychometric function of Equation 2 according to

*u*

_{ i }is the random effect for subject

*i,*

**X**is the design matrix, and

**is the vector of coefficients of the fixed effects. In our case, the GLMM included a single random effect parameter (the random intercept) and eight parameters of fixed effects estimating the intercept, the flight duration (that is, the slope with the downward condition as the baseline), three dummy variables corresponding to the three remaining conditions (leftward, rightward, and upward), and the interaction between flight duration and the three dummy variables. For each parameter, we computed the following Wald statistics:**

*β**SE*is its estimated standard error, and we derived the corresponding two-sided

*p*-values (Agresti, 2002). The same statistics was used to test the statistical significance of the parameters of the population psychometric functions. Because the standard deviation of the random location factor turned out to be significantly different from zero (

*z*= 20.5,

*p*< 0.001), the use of a mixed model was statistically justified (Agresti, 2002). To compare the precision of discrimination across conditions and experiments, the values of the slope of the population responses were normalized by dividing each value by the slope of the downward condition.

*T*. This can be derived from the psychometric function as Δ

*T*= 0.5(

*T*

_{0.75}−

*T*

_{0.25}), where

*T*

_{0.75}and

*T*

_{0.25}are the values of flight duration yielding 0.75 and 0.25 probabilities of “Longer” responses. The Weber fraction then is WF = Δ

*T*/

*T*

_{standard}, where

*T*

_{standard}is the standard duration (800 ms).

*α*= 0.05, after adjusting for multiple comparisons according to the false discovery rate procedure (Benjamini & Hochberg, 1995).

*p*< 0.05) steeper (higher slope) for downward motion than for all other tested directions of motion, indicating a higher precision of judgment or, equivalently, a better discrimination of the stimulus duration for the downward motion. This trend was observed also at the level of single subjects: the response slope was higher for downward motion than that for the other directions in most (6/7) participants. In one participant, the slope for downward was the second highest after that for leftward.

*z*= 0.536,

*p*= 0.59). Next, we computed the values of the slope of the population responses obtained for each motion direction (Figure 2b). The slope was significantly higher in downward motion than upward (

*z*= 4.468,

*p*< 0.001), rightward (

*z*= 3.919,

*p*< 0.001), and leftward motions (

*z*= 2.454,

*p*= 0.016). Absolute values of the slope were 0.0083 ± 0.00032 (mean ±

*SD*, over all subjects), 0.0064 ± 0.00026, 0.0072 ± 0.00029, and 0.0066 ± 0.00027 for down, up, left, and right, respectively. On average, the slope for downward motion was 23, 13, and 20% higher than the slope for upward, leftward, and rightward motions, respectively. For downward motion, average discrimination threshold (Δ

*T*) was 87 ± 3 ms (mean ±

*SD*, over all subjects) and average Weber Fraction (WF) was 0.109 ± 0.004. For upward motion, Δ

*T*= 110 ± 4 ms, and WF = 0.137 ± 0.005.

*SD*) for downward, 824 ± 50 ms for upward, 826 ± 49 ms for leftward, and 805 ± 65 ms for rightward, indicating fairly accurate estimates of stimulus duration. This shows that the method we used to present reference and test stimuli did not create any bias.

*SD*). None of them had previously participated in Experiment 1. All were right-handed, except an ambidextrous subject.

^{−2}) along a straight path between two opposite landmarks, as in Experiment 1. The direction of target motion was downward, upward, rightward, or leftward, depending on the specific block of trials (order counterbalanced across subjects).

*SD*) for downward, 802 ± 52 ms for upward, 800 ± 36 ms for leftward, and 802 ± 50 ms for rightward.

*z*= 0.253,

*p*= 0.80). The slope for downward motion was significantly higher than that for upward motion (

*z*= 3.834,

*p*< 0.001, see Figure 4b), but it was not significantly different from that for rightward (

*z*= 1.387,

*p*= 0.22) or leftward (

*z*= 0.090,

*p*= 0.93) motion. Thus, when pictorial cues were removed from the scene, average discrimination performance for downward motion remained superior (by 19%) to that for upward motion, but it lost the superiority over horizontal motions. This trend was observed in the psychometric functions of most (6/7) participants. The average WFs were 0.095 ± 0.003 (mean ±

*SD*over all subjects) and 0.108 ± 0.004 for downward and upward motions, respectively.

*SD*). None of them, except subject A.M., had participated in previous experiments. All were right-handed, except an ambidextrous subject.

^{−2}) along a straight path between two opposite landmarks. The direction of target motion was down-and-rightward, down-and-leftward, up-and-rightward, or up-and-leftward, depending on the specific block of trials (order counterbalanced across subjects).

*SD*over all subjects) for down-and-leftward, 776 ± 37 ms for up-and-rightward, 774 ± 45 ms for down-and-rightward, and 770 ± 42 ms for up-and-leftward.

*z*= 1.6,

*p*= 0.19) nor did it differ significantly between the two conditions involving an upward motion (up-and-rightward versus up-and-leftward,

*z*= 0.5,

*p*= 0.66). Therefore, we pooled the data for each pair of conditions and found that there was no significant difference of the slope between downward and upward directions (

*z*= 1.7,

*p*= 0.08, Figure 6a). Although the difference was not significant, the slope for downward was slightly higher (by 7%) than that for upward. This superiority was observed in 5/7 participants. The average WF was 0.096 ± 0.008 (mean ±

*SD*over all subjects) for downward motions.

*SD*). All were right-handed, except a left-handed subject.

*Latin Square*design (session 0 = session 4). The order of the sessions was randomized across subjects. Overall, for each motion direction, a given stimulus duration was presented 50 times in each experiment, yielding a total of 450 test trials (9 durations × 50 repetitions). All other design parameters, testing procedures, and data analyses were identical to those described for Experiments 1–3.

*z*= 0.9,

*p*= 0.45) nor did it differ significantly between the two upward conditions (up-and-rightward versus up-and-leftward,

*z*= 1.7,

*p*= 0.14). Critically, there was no significant difference of the slope between downward and upward conditions (

*z*= 1.47,

*p*= 0.14, Figure 6b), although the slope for downward was slightly higher (by 5%) than that for upward as in the previous experiment.

*SD*). All were right-handed, except a left-handed subject. Four of them had participated in previous experiments (the author A.M. plus subjects G.C., M.Z., and A.Z. who had been involved in Experiment 4).

*z*= 2.4,

*p*= 0.017, see Figure 6c). This trend was observed in 5/7 subjects. The average WFs were 0.067 ± 0.03 (mean ±

*SD*over all subjects) and 0.074 ± 0.02 for downward and upward motions, respectively. Thus, despite the tilt of the observers relative to the picture, the up/down anisotropy in the responses was still present, although the size of the effect was about one half that found in the canonical upright position (Experiment 1).

*SD*over all subjects) for downward and 794 ± 27 for upward.

*SD*). All were right-handed, except for a left-handed subject. Three of them had participated in previous experiments (the author A.M. plus subjects G.C. and M.Z. who had been involved in Experiments 4 and 5).

*z*= 3.2,

*p*= 0.002), leftward (

*z*= 3.08,

*p*= 0.002), and rightward (

*z*= 4.1,

*p*< 0.001). On average, the slope for pictorial downward was 14, 14, and 18% higher than the slope for upward, leftward, and rightward, respectively. The slope associated with downward motion was the highest among all other conditions in 5/7 tested subjects. Average WFs were 0.065 ± 0.02 (mean ±

*SD*, over all subjects), 0.076 ± 0.029, 0.077 ± 0.038, and 0.079 ± 0.031 for downward, upward, leftward, and rightward motions, respectively.

*SD*) for downward, 762 ± 37 ms for upward, 768 ± 42 ms for leftward, and 771 ± 35 ms for rightward, indicating fairly accurate estimates of stimulus duration.

*P*), orientation of the observer (

*O*), and orientation of target motion (

*T*) relative to the physical vertical. Accordingly,

*β*

_{p(Upward)}/

*β*

_{p(Downward)}(the

*ratio*of the slopes of upward and downward psychometric functions) was modeled as

*O, T,*and

*P*could take the value of 1 or 0 depending on whether or not they corresponded to the default in a given experiment (e.g.,

*O*=

*T*=

*P*= 0 in Experiment 1). The regression parameters (

*a*–

*d*) were obtained by fitting the data of all experiments together (except Experiment 4 that was redundant with Experiment 3). The resulting weighing coefficients were 43, 37, and 20% (of the overall response) for

*O, T,*and

*P,*respectively. The results changed very little if Experiment 3 (involving a non-significant down/up difference) was excluded from the analysis.

*O*) dominate is in line with much previous work on the perceptual discrimination of scenes, people, and actions (e.g., Chang et al., 2010; Kushiro et al., 2007; Troje, 2003). On the other hand, the substantial contribution of visual references intrinsic to the scene, such as the direction of target motion (

*T*) and the presence of additional pictorial cues (

*P*), agrees with the previous observation that viewing a photograph with strong polarization cues, which indicate relative “up” and “down” directions in the picture, can alter the perceived direction of absolute “up” and “down” directions in the real world (Jenkin et al., 2004).

^{−2}toward the subject, presented stereoscopically in an immersive virtual environment, was more accurate when the ball was launched downward from above (obeying gravity) than when it was launched upward from below (violating gravity). In addition, the observation that the down/up anisotropy was enhanced by embedding motion in a pictorial scene nicely parallels the previous observation that a similar pictorial context facilitated interception of gravitational acceleration over the interception of an unnatural acceleration, whereas a blank scene reduced such bias (Miller et al., 2008).

*D*=

*t*

_{2}−

*t*

_{1}of a vertical fall, subject to gravity, is given by

*v*

_{1}and

*v*

_{2}are the speed at time

*t*

_{1}and

*t*

_{2}, respectively, and

*g*is the gravitational acceleration. Whereas speed estimates might be derived from visual motion processing, the value of gravitational acceleration could be internalized either in exact or approximate form (see Zago, McIntyre, Senot, & Lacquaniti, 2008).