There is little direct psychophysical evidence that the visual system contains mechanisms tuned to head-centered velocity when observers make a smooth pursuit eye movement. Much of the evidence is implicit, relying on measurements of bias (e.g., matching and nulling). We therefore measured discrimination contours in a space dimensioned by pursuit target motion and relative motion between target and background. Within this space, lines of constant head-centered motion are parallel to the main negative diagonal, so judgments dominated by mechanisms that combine individual components should produce contours with a similar orientation. Conversely, contours oriented parallel to the cardinal axes of the space indicate judgments based on individual components. The results provided evidence for mechanisms tuned to head-centered velocity—discrimination ellipses were significantly oriented away from the cardinal axes, toward the main negative diagonal. However, ellipse orientation was considerably less steep than predicted by a pure combination of components. This suggests that observers used a mixture of two strategies across trials, one based on individual components and another based on their sum. We provide a model that simulates this type of behavior and is able to reproduce the ellipse orientations we found.

*H*=

*R*+

*E*). The most obvious dimensions to use are therefore

*R*and

*E*. However, two recent studies suggest that observers do not rely on these motion cues but rather the relative motion (between pursued target and background) and the motion of the pursued target itself. Using a speed discrimination task, Freeman, Champion, Sumnall, and Snowden (2009) showed that observers do not have direct access to retinal motion when making discrimination judgments during pursuit—instead, observers use the relative motion between pursuit target and background object, even when feedback concerning absolute retinal motion was explicitly provided. In the case of the pursued target, Welchman, Harris, and Brenner (2009) showed that observers summed eye velocity information with retinal slip information when discriminating the motion-in-depth of a target tracked by a vergence eye movement (we have found similar evidence in unpublished investigations of speed and direction discrimination for pursued stimuli moving in the fronto-parallel plane).

*Rel*) and pursuit target motion (

*T*). We do not mean this to imply that retinal motion and eye velocity are ignored by the observer—rather, retinal motion and eye velocity are incorporated into the estimates of relative motion and target velocity. Importantly, relative motion by itself does not tell the observer how a background object is moving with respect to the head—it simply informs the observer how two objects are moving with respect to one another. To determine the head-centered velocity of the background stimulus, the observer must add relative motion to velocity of the pursuit target (

*H*=

*Rel*+

*T*). The current paper therefore asks whether the visual system contains relatively low-level mechanisms explicitly tuned to

*H*, or whether it is inferred by some more circuitous route.

*T*

_{s},

*Rel*

_{s}) and a test stimulus (

*T*

_{t},

*Rel*

_{t}), where

*T*

_{t}=

*T*

_{s}+ Δ

*T*and

*Rel*

_{t}=

*Rel*

_{s}+ Δ

*Rel*. The variation in head-centered velocity of the background object is therefore Δ

*H*= Δ

*T*+ Δ

*Rel*, such that a line of constant Δ

*H*has slope of −1 (the negative diagonal in Figure 1 defines Δ

*H*= 0). Suppose we are able to obtain thresholds for discriminating test from standard for the set of directions

*θ*. The blue ellipse labeled “components” in Figure 1 describes the expected threshold contour if observers base judgments on individual components rather than their combination (the figure assumes that sensitivity to relative motion is greater than pursuit target motion, which is why the major axis of the “components” ellipse is horizontal). Along the cardinal axes, only one motion component conveys any useful information, so in these directions thresholds are limited by one component alone (dotted lines). In all other directions, however, useful information is conveyed by both components, so observers may gain a statistical advantage due to probability summation (e.g., Alais & Burr, 2004).

*T*and

*Rel*to yield

*H*. Ideally, if observers based their judgments on head-centered velocity alone, the threshold contour would consist of two lines parallel to the negative diagonal. In this case, observers would find it particularly difficult to differentiate any pair of stimuli that lie along lines of constant Δ

*H*because these form head-centered “metamers”. In practice, however, for relatively extreme values of Δ

*T*and Δ

*Rel*, observers are likely to be able to differentiate standard and test on the basis of individual components (see Hillis et al., 2002). Hence, the resulting thresholds will produce a closed contour oriented with respect to the negative diagonal.

^{2}. The random dot pattern appeared in an annulus window with inner and outer radii of 1° and 8°, respectively. The movement of the window was yoked to the pursuit target. The target, dot pattern, and window moved horizontally in all conditions investigated.

*T*

_{s},

*Rel*

_{s}) = (+4, +4)°/s, with the dot stimulus moving at 8°/s on the screen. In the “opposite” condition, (

*T*

_{s},

*Rel*

_{s}) = (+4, −4)°/s. The dot stimulus in this case was always stationary on the screen.

*T*

_{t},

*Rel*

_{t}) = (

*T*

_{s}+ Δ

*T*,

*Rel*

_{s}+ Δ

*Rel*), where the increments Δ

*T*=

*gT*

_{s}cos

*θ*and Δ

*Rel*=

*g Rel*

_{s}sin

*θ*. The parameter

*g*defines the step size, and

*θ*is the direction of the discrimination task in

*T*–

*Rel*space (see Figure 1). Italics denote speed, emphasizing that test stimuli could not flip phase through 180° for any given

*θ*. The increments Δ

*T*and Δ

*Rel*were controlled by a 3-down 1-up staircase. Along any direction

*θ*, the ratio of Δ

*T*to Δ

*Rel*was constant, with the staircase changing the distance between the test and standard in steps

*D*=

*g*(

*T*

_{s}

^{2}cos

^{2}

*θ*+

*Rel*

_{s}

^{2}sin

^{2}

*θ*)

^{1/2}. Staircases were terminated after 9 reversals, with the step size before the first reversal set to

*g*= 0.2 and all subsequent step sizes set to

*g*= 0.1. Sixteen directions in

*T*–

*Rel*space were investigated (

*θ*= 0° to 337.5° in increments of 22.5°). Within one experimental session, 4 of these different directions were investigated, each assigned one staircase. Staircases were randomly interleaved. In total, three replications of each staircase were completed. Staircases for “same” and “opposite” were blocked, with observers S1–S3 completing “same” blocks first, and S4 and S5 completing “opposite” blocks first.

*T*

_{s}, −

*Rel*

_{s}). By definition, this also flips the test, such that both test and standard rotate 180° about the origin in Figure 1. For the “same” condition, trials therefore alternated between the upper right and lower left quadrants; for the “opposite” condition, trials alternated between upper left and lower right quadrants. Data were collapsed within these quadrant pairings.

*D*(as defined above), which corresponds to length along a direction

*θ*. Error rates from each direction condition

*θ*were concatenated with those from the condition

*θ*+ 180° (for the latter

*D*=

*D** − 1). A Gaussian was then fit to the data using maximum likelihood minimization, with standard deviation and a lapse rate parameter free to vary and mean fixed at

*D*= 0. Lapse rate was constrained to be less than 6% (Wichmann & Hill, 2001). Threshold values were defined as the standard deviation of the Gaussian. We also estimated 95% confidence limits by bootstrapping 999 thresholds. In some cases, the lower and upper confidence intervals are unequal because the distribution of standard deviations was sometimes asymmetric. Trials were excluded on the basis of eye movements if a saccade was detected.

*SE*= 6.0°),

*t*(4) = 2.61,

*p*< 0.05, one-tailed; “opposite” = 17.0° (

*SE*= 4.4°),

*t*(4) = 3.81,

*p*< 0.01, one-tailed). Hence our results lie somewhere in between the predictions based on the use of individual cues (ellipses oriented along cardinal) and the prediction based on the use of head-centered cues (oriented along negative diagonal). Such a pattern of results suggest that discrimination was based on a mixture of two strategies; on some trials, observers combined components and based their judgments on head-centered motion, whereas on other trials they used the individual components. Below we present a model that simulates such a strategy and is able to produce ellipse orientations like those found here.

*SE*= 10.9°), “opposite” = 157.6° (

*SE*= 9.4°),

*t*(4) = 0.004,

*p*= 0.95, two-tailed). Hence, the lack of asymmetry between “same” and “opposite” conditions contrasts with work in other areas that report an anisotropy (e.g., motion smear: Tong et al., 2006; though see Morvan & Wexler, 2009). We note that we have previously failed to find this asymmetry in analogous experiments on retinal speed discrimination during pursuit and have discussed this finding in more detail elsewhere (Freeman et al., 2009).

*θ*. For the “opposite” condition, eye movements were quite accurate—average retinal slip is close to 0. For the “same” condition, pursuit tended to be faster than required. The influence of the direction of background motion on pursuit is well documented and explains the differences found here (Lindner & Ilg, 2006; Spering, Gegenfurtner, & Kerzel, 2006; Yee, Daniels, Jones, Baloh, & Honrubia, 1983). Closer inspection also revealed an influence of interval order. The lower panel of Figure 3 shows that eye speed decreased from intervals one to three in the “opposite” condition, whereas eye speed remained more or less the same in the “same” condition. These differences may reflect an influence of background motion on pursuit interacting with certain judgment strategies based on appearance. For instance, if the first two intervals appeared the same, observers could have decided that the final interval was the odd one out before it was displayed. In this case, the final interval could be ignored, perhaps leading to lower eye speeds.

*Rel*

_{ i },

*T*

_{ i }, and

*H*

_{ i }, where

*H*

_{ i }=

*Rel*

_{ i }+

*T*

_{ i }and

*i*= interval 1, 2, or 3.

*Rel*

_{ i }and

*T*

_{ i }were corrupted by Gaussian noise with a mean of 0 and a standard deviation

*σ*. The precision of

*H*

_{ i }was therefore assumed to be fully determined by noise at the input stage (i.e.,

*σ*

_{Rel}and

*σ*

_{T}). For the “combination strategy,” the odd interval was taken as the

*H*

_{ i }that was most different from the mean of the other two intervals. For the “individual-components strategy,” an odd interval was identified separately for

*Rel*

_{ i }and

*T*

_{ i }, using the method described for

*H*

_{ i }. This potentially yields two different candidate odd intervals on each trial, one determined by

*Rel*

_{ i }and one determined by

*T*

_{ i }. In these cases, the signal corresponding to the “most different” odd one out was chosen.

*k*” to determine the probability on each trial of using the “combination strategy” or “individual-components strategy.” The parameter

*k*therefore set the weighting or mixture between strategies. For instance, with

*k*= 0, the model's choice was determined entirely by the individual-components strategy. Conversely, with

*k*= 1, the model's choice was determined entirely by the combination strategy. With

*k*= 0.5, the probability of using either strategy was the same on each trial. In the latter case, the resulting threshold therefore comprised a mixture of judgments based on combination and individual-component strategies. To determine discrimination-ellipse orientation as a function of

*k*, a series of simulations was run for a range of directions

*θ*through

*Rel*–T space. Each simulation sampled the underlying psychometric function by running 10,000 trials at 7 equally spaced steps along the given direction

*θ*. Step size “

*g*” was set to 0.5 and values of Δ

*Rel*and Δ

*T*were calculated as described in the Methods section. Thresholds and ellipse orientation were then derived using the fitting procedures also described in the Methods section.

*k*between strategies. We investigated five different levels of relative noise between

*Rel*and

*T*, corresponding to the five lines in the figure (dashed lines represent the same ratio

*σ*

_{Rel}:

*σ*

_{T}as the solid lines, but with a factor of 10 decrease in noise). Figure 4B provides examples of the ellipses returned by the model at three of these levels of relative noise. We did not investigate a full range of noise values—the simulations shown in Figure 4 are simply meant to demonstrate “proof of principle.” In Figure 4A, the red lines show the results for

*σ*

_{Rel}<

*σ*

_{T}and the green lines

*σ*

_{Rel}>

*σ*

_{T}(corresponding to the upper and lower rows, respectively, in Figure 4B). When

*k*= 0, the model always uses the individual-components strategy and so the ellipse's major axis is oriented parallel to the least precise motion cue (see first column of Figure 4B for examples). When

*k*= 1, the model makes choices based on the combination strategy and so the thresholds lie parallel to the oblique. The “ellipse” in this case is not closed (see Figure 4B, end column). Between these values of

*k*, the orientation of the ellipse rotates away from the cardinal axis toward the oblique. The mean deviation from cardinal across observers was 16.4°, suggesting that they used the combination strategy between 10% and 20% of the time (this assumes that the noises present in our observers are within the range used in the simulations).

*σ*

_{Rel}=

*σ*

_{T}. Using individual components in this case (i.e.,

*k*= 0) produces a circle because the underlying input signals are equally precise. Hence orientation is undefined at this value of

*k*. As

*k*increases, the circle becomes stretched along the negative oblique (as shown in Figure 4B). Thus, for cases where input noises are equal, the defining feature of mixing the two strategies is a change in shape but not orientation.

*T*axis for most observers (S1–S4), the motion signals associated with the pursuit target are less precise than those associated with the relative motion of the background.