Smooth pursuit eye movements add motion to the retinal image. To compensate, the visual system can combine estimates of pursuit velocity and retinal motion to recover motion with respect to the head. Little attention has been paid to the temporal characteristics of this compensation process. Here, we describe how the latency difference between the eye movement signal and the retinal signal can be measured for motion perception during sinusoidal pursuit. In two experiments, observers compared the peak velocity of a motion stimulus presented in pursuit and fixation intervals. Both the pursuit target and the motion stimulus moved with a sinusoidal profile. The phase and amplitude of the motion stimulus were varied systematically in different conditions, along with the amplitude of pursuit. The latency difference between the eye movement signal and the retinal signal was measured by fitting the standard linear model and a non-linear variant to the observed velocity matches. We found that the eye movement signal lagged the retinal signal by a small amount. The non-linear model fitted the velocity matches better than the linear one and this difference increased with pursuit amplitude. The results support previous claims that the visual system estimates eye movement velocity and retinal velocity in a non-linear fashion and that the latency difference between the two signals is small.

**E**and

**R**(indicated by

*f*and

*g*in the figure). According to the standard linear model, the perceived head-centered motion

*r*and

*e*are the gains of the retinal and eye movement signals, respectively. Previous studies have shown that many examples of motion perception during smooth pursuit are well approximated by this model. In particular, the linear model is able to quantify how the Aubert–Fleischl phenomenon (moving objects appear slower when pursued) and the Filehne illusion (stationary objects appear to move during pursuit) vary with eye velocity (Freeman, 2001). It also describes perceived motion direction during pursuit quite well (Souman et al., 2005a). However, the linear model is less able to describe other instances of velocity perception, such as general velocity matching tasks in which observers are asked to estimate the speed of stimuli that are neither stationary nor moving at the speed of the pursuit target (Freeman, 2001; Turano & Massof, 2001). In these cases, models with non-linear transducers fit the data better, although the improvement is quite small (Souman, Hooge, & Wertheim, 2006). A specific example of a non-linear model is discussed in more detail below.

**R**and eye velocity

**E**with gains

*r*and

*e,*respectively ( Equation 1). If

**R**and

**E**vary in time and the signals have different latencies, this will affect the perceived velocity

*t*) at time

*t*. As we used sinusoidal movements of one single frequency

*f*in our experiment, these latencies translate into phase shifts:

*R*and

*E*now represent movement amplitudes,

*φ*is the phase of the retinal image motion with respect to the pursuit target,

*ρ*is the phase shift of the retinal signal,

*θ*is the phase of the eye movement with respect to the pursuit target, and

*ɛ*represents the phase shift of the eye movement signal. Figure 2 illustrates this equation in a phasor plot. Sinusoidal motion is represented as a vector, with the angle with respect to the positive

*x*-axis indicating phase and the distance to the origin representing amplitude. Adding sinusoids of the same frequency is equivalent to adding vectors in the phasor plot. By simple trigonometry, the amplitude

_{ p}in the pursuit interval equals the perceived amplitude during fixation

_{ f}. This gives:

*f*and

*p*refer to fixation interval and pursuit interval, respectively. Dividing both sides by

*r*

^{2}gives

*e*/

*r*and the phase difference

*ɛ*−

*ρ*. Note that the individual gains

*e*and

*r*and the individual phases

*ɛ*and

*ρ*cannot be resolved (see Freeman, 2001; Freeman & Banks, 1998; Souman et al., 2006).

*r*and

*e*. Replacing the linear relationships in Equation 4 by these non-linear transducers of motion amplitude and solving for the retinal amplitude during fixation at the point of subjective equality gives

_{p}and

*r*and

*e*and the phase difference

*ɛ*−

*ρ*. As with the linear model, the individual transducer functions (Equations 6 and 7) can only be estimated up to an arbitrary scale factor (for further discussion, see Freeman, 2001).

*e*/

*r,*with the phase difference between the signals set to zero (

*ɛ*−

*ρ*= 0). We distinguish between cases with accurate pursuit ( Figures 3A– 3B) and with a 15° pursuit lag ( Figures 3C– 3D). In Figure 3A, the amplitude matches are shown as a function of the retinal phase in the pursuit interval, with retinal amplitude held constant at 1°. This corresponds to one set of conditions in our experiment. The linear model predicts that the squared amplitude matches should lie on a sinusoid. Veridical amplitude matches would equal the head-centered motion amplitude of the motion on the screen and fall on the dotted line. This corresponds to a gain ratio

*e*/

*r*= 1. Reducing the gain ratio produces similarly shaped curves, but with smaller amplitudes and different offsets from zero. If an observer completely fails to compensate for the effects of the eye movements at all (i.e.,

*e*/

*r*= 0), the amplitude matches will be equivalent to the retinal motion amplitude (horizontal dashed line).

*ɛ*−

*ρ*) affects the predictions. This is more straightforward than the effect of the gain ratio. A phase difference causes the curves to shift horizontally (see Equation 5).

*e*/

*r*produce curves with different heights and slopes. However, the amplitude matches only fall below the retinal amplitude (dashed line) if there is a non-zero phase difference

*ɛ*−

*ρ*between the two signals.

*ɛ*−

*ρ*has a similar effect in both models.

^{2}(the fixation target had a small black hole at its center to improve fixation). The density of the dot patterns was 0.1 dot/deg

^{2}. The bands were 10° high, separated by 10° and covered the entire screen horizontally. Both dot pattern and target could be made to move sinusoidally with an independent amplitude and phase. The frequency of oscillation was 0.5 Hz throughout.

*fminsearch*function. The fitting was performed on the mean squared amplitude matches, averaged across observers. Local minima were avoided by repeating the fitting procedure with different initial values for the parameters.

*n*is the number of observations,

*k*represents the number of free parameters, and

*ɛ*

_{i}are the errors in model predictions. The last term in Equation 9 corrects for the small number of observations (Hurvich & Tsai, 1989; Shono, 2000). Lower values of AIC

_{c}indicate better model performance. Below, we report the differences in AIC

_{c}between the two models, as the absolute values are not interpretable (Burnham & Anderson, 2004).

_{ c}was 1.91 lower for this model than for the linear one). More specifically, the non-linear model was better able to capture the squared amplitude matches for low retinal phases ( Figure 6A) and low retinal amplitudes ( Figure 6B).

*e*/

*r*was close to 0.6, which is similar to previous estimates reported in the literature (Freeman, 2001; Freeman & Banks, 1998; Freeman et al., 2000; Souman et al., 2005a, 2006). The estimated phase difference

*ɛ*−

*ρ*between the two signals was about −12°, indicating that the eye movement signal lagged the retinal signal by about 67 ms. These results are similar to those obtained by Freeman et al. (2000) using a nulling task. For the non-linear model, the ratio of the two power coefficients

*e*/

*r*was 1.45/1.81, which again is very similar to the values reported previously (Freeman, 2001; Souman et al., 2006). The estimated phase difference

*ɛ*−

*ρ*for the non-linear model was −19°, which is slightly larger than that of the linear model, but in the same direction.

Parameter | Linear model | Non-linear model | ||
---|---|---|---|---|

Experiment 1 | Experiment 2 | Experiment 1 | Experiment 2 | |

Ratio e/ r | 0.56 | 0.68 | 0.80 | 0.85 |

Phase difference ɛ − ρ (°) | −11.80 | −6.97 | −18.89 | −6.95 |

_{ c}was lower for the non-linear model by 16.17).

*e*/

*r*for the linear model and the ratio of the two power coefficients for the non-linear model were very similar to those obtained in Experiment 1. Again, both power coefficients were above unity (

*e*= 1.48;

*r*= 1.74). Both models estimated the phase difference

*ɛ*−

*ρ*to be about −7°, indicating that the eye movement signal lagged the retinal signal by about 39 ms.

_{c}showed that the increased goodness-of-fit was not just due to the additional free parameter in the non-linear model. This suggests that the visual system uses non-linear transducers for estimating both retinal velocity and eye velocity. Previous studies have arrived at the same conclusion (Freeman, 2001; Souman et al., 2006; Turano & Massof, 2001).

*e*and

*r*were ∼1.5 and 1.8, respectively. This suggests that the speed transducers as defined in Equations 6 and 7 are expansive. The values are similar to those reported by Freeman (2001) in a study that used constant pursuit and retinal speed. The nature of speed transducers underlying motion perception has received little attention in the literature. In principle, the shape of the speed transducers impacts on magnitude estimates of visual speed and on discrimination performance. The few studies that have employed magnitude estimation tend to report linear or compressive relationships (Kennedy, Hettinger, Harm, Ordy, & Dunlap, 1996; Kennedy, Yessenow, & Wendt, 1972). However, magnitude estimation is susceptible to a number of factors that are not easy to control (Poulton, 1979). Speed discrimination data, on the other hand, cannot be directly related to the transducer functions either. Discrimination thresholds are not only determined by the transducer function but also by the relationship between signal level and noise (Georgeson & Meese, 2006). It is therefore difficult to devise an alternative test for the shape of the non-linear transducers revealed by our experiments.