Interacting with a dynamic environment calls for close coordination between the timing and direction of motor behaviors. Accurate motor behavior requires the system to predict where the target for action will be, both when action planning is complete and when the action is executed. In the current study, we investigate the time course of velocity information accrual in the period leading up to a saccade toward a moving object. In two experiments, observers were asked to generate saccades to one of two moving targets. Experiment 1 looks at the accuracy of saccades to targets that have trial-by-trial variations in velocity. We show that the pattern of errors in saccade landing position is best explained by proposing that trial-by-trial target velocity is taken into account in saccade planning. In Experiment 2, target velocity stepped up or down after a variable interval after the movement cue. The extent to which the movement endpoint reflects pre- or post-step velocity can be used to identify the temporal velocity integration window; we show that the system takes a temporally blurred snapshot of target velocity centered ∼200 ms before saccade onset. This estimate is used to generate a dynamically updated prediction of the target's likely future location.

^{2}). If either was above threshold, the eye movement is classified as a saccade; a distance threshold of 0.1 deg was used to delay the onset of the saccade until the eye had moved significantly. At the end of the saccade, the signal was terminated if both the velocity and acceleration values were below the criterion thresholds, or if the inter-sample movement was below the distance threshold. This latter component of the algorithm was designed to prevent saccades being “stretched” as a result of erroneous endpoint localization (for a more complete description, see Stampe, 1993). Visual inspection of the eye traces confirmed that the parsing algorithm reliably identified the saccade start and endpoints.

^{2}. The target patch was signaled by increasing its peak luminance to 82 cd/m

^{2}. The background luminance was 20.5 cd/m

^{2}. The standard deviation of the patches was 0.32 deg of visual angle.

*k*seconds before saccade termination, and that a constant offset of

*d*degrees is applied to the position grab to determine ideal landing position, then the error (

*ɛ*) between target position and saccade landing position is

*v*

_{t}is the speed of the target. The fixed offset model therefore predicts a linear relationship between

*v*

_{t}and

*ɛ*with slope −

*k*and intercept

*d*. Complete velocity dependence would show as a flat function aligned with the abscissa, or perhaps parallel to it in the presence of a general bias to under- or overshoot. The dashed lines in Figure 2 illustrate the predictions of a reasonable fixed-offset model in which the observer samples the “final” target position at 100 ms before saccade initiation (e.g., Becker & Jürgens, 1979; de Brouwer et al., 2002; Heywood & Churcher, 1981) and applies a fixed offset taking into account the average velocity experienced over the course of the experiment (15 deg/s) and the duration of the saccade (derived from the average duration for each observer across all trials).

*p*= 0.48), it is clear that amplitude did. As such, it is likely that saccades to the faster targets were larger in amplitude because participants are less likely to “catch” the moving target when it was near the center of the screen. If so, then the corresponding undershoot will also tend to be larger.

*D*) for one observer. For clarity,

*D*= 500 ms indicates that the velocity step occurred 500 ms before saccade onset.

*n*, we can quantify to what extent the saccade was driven by the pre- and post-step velocities. We calculate two errors (see de Brouwer et al., 2002 for a similar approach):

*ɛ*

_{n}

^{old}(the difference between saccade landing position and the point where the target would have been if it had not changed speed) and

*ɛ*

_{n}

^{new}(the difference between saccade landing position and the actual position of the target when the saccade landed). The pre-step error can be calculated as

*x*

_{n}is the horizontal component of each observed saccade landing position from the point at which the velocity change occurred,

*τ*

_{n}is the saccadic duration, and

*v*

^{old}is the original target velocity. We assume that

*x*

_{n}is a random sample from a Gaussian distribution with unknown mean (

_{n}) and variance (

*σ*

_{n}

^{2}). We also assume that

*τ*

_{n}is drawn from a Gaussian distribution with unknown mean (

_{n}) and variance (

*ς*

_{n}

^{2}). We make no initial assumptions about the manner in which the variance in

*τ*

_{n}and

*x*

_{n}varies as a function of

*D*. In Figure 4A, the predicted landing position on the basis of the pre-step speed is shown by the dashed lines, which are based on the average saccade duration. These lines are shown for illustration purposes only; in actual fact, these functions will be slightly different from one trial to the next, depending on the saccade duration on that particular trial. However, these functions demonstrate the basic pattern of results: early in time the saccade landing positions follow the pre-step speed, but over time they fall more in line with the new velocity. In a similar manner to Equation 2, the post-step error can be expressed as

*v*

^{new}is the target velocity after the speed change.

*n*attributed to the new velocity is

*ρ*

_{ n}= 0. On the other hand, if the new velocity is fully taken into account, then

*ρ*

_{ n}= 1. If the process is conceived of as a step change (such as our velocity change) falling within a temporal integration kernel

*f*(

*t*), then

*ρ*

_{ n}is the proportion of the area under

*f*(

*t*) that falls after the speed change. A plot of

*ρ*

_{ n}at a range of values of

*D*

_{ n}therefore gives us the integral of the temporal kernel. Figure 4B illustrates these weights in 15 roughly equal bins for illustration purposes only (the actual analyses always involved the data from individual trials). As expected, as

*D*increases, the new velocity is increasingly taken into account. We now only need to fit some reasonable function given the form of the data, and then differentiate to obtain an estimate of the kernel.

*ρ*

_{ n}is a sample.

^{1}Following from Equation 2, we can express the probability distribution from which each

*ɛ*

_{ n}

^{old}is drawn as

*v*

^{new}−

*v*

^{old}) is a constant that we term

*v*

^{Δ}. Obviously, the variability in saccade landing position (

*σ*

_{n}) is factored out of the scaling factor because it affects

*ɛ*

_{ n}

^{new}and

*ɛ*

_{ n}

^{old}equally. The probability distribution of the scaling factor can therefore be expressed as

*ρ*

_{ n}(see Equation 4) gives us the following:

*g*(

*t*) that we translated into Matlab).

*v*

^{Δ}is positive (when the stimulus speeds up) but not when

*v*

^{Δ}is negative. With the latter, we simply multiply both numerator and denominator by −1. Note that the distribution is determined in part by the correlation between the two distributions, which, with some straightforward substitution and reworking of the standard equation for correlation, can be expressed as

_{ n},

*σ*

_{ n},

_{ n}, and

*ς*

_{ n}. Ultimately, it is the variation in mean landing position,

*D*that is of critical interest. The remaining quantities are of less interest but do need to be specified. A complication is that any of these quantities may depend on

*D*, which is implicit in the subscript

*n*. Intuitively, it may seem that with increasing

*D*the amplitudes of the movements are likely to increase: as time elapses since the speed change, the pattern will have moved further and so a larger movement is required in order to capture it. It is well established that endpoint variability tends to increase with movement size (Harris & Wolpert, 1998; van Beers, 2007, 2008). If it is indeed the case that saccade amplitude increases with

*D*, then it would seem plausible that the endpoint variance should increase as well. Likewise, given the well-established relation between movement amplitude and duration (Carpenter, 1988) it follows that saccade duration should increase with

*D*as well, with possible concomitant increase in duration variability. Indeed, there were several weak but significant correlations between

*D*and saccade amplitude, as well as between

*D*and saccade duration.

*σ*

_{ n},

*ς*

_{ n}, and

_{ n}remain constant over

*D*. Moreover, the dependence on

*D*may vary across the step-up and step-down conditions, and we have no a priori expectations about the shape of these dependencies. To capture these relations parametrically would require the introduction of a number of additional free parameters. In order to keep our final model as parsimonious as possible, we chose to sample estimates of

*σ*

_{ n},

*ς*

_{ n}, and

_{ n}directly from the raw data for each observer. To capture any potential variation, we computed amplitude variance, mean duration, and duration variance for values of

*D*ranging from 1 to 500 ms, using a Gaussian kernel.

*D*. Such techniques aim to find a bandwidth that does not unduly smooth the distribution, nor allow an undue amount of high-frequency noise into the distribution (Marron, 1988). We reasoned that as our variables of interest are sampled as a function of

*D*, a bandwidth for the optimal sampling of

*D*itself would also be optimal for the smoothing of

*σ*

_{n},

*ς*

_{n}, and

_{n}as a function of

*D*.

Observer | Kernel widths (ms) | |
---|---|---|

Step up | Step down | |

1 | 27.8 | 26.9 |

2 | 25.8 | 32.2 |

3 | 36.3 | 42.9 |

4 | 32.3 | 29.3 |

5 | 36.8 | 34.7 |

Mean (SD) | 31.9 (4.94) | 33.2 (6.17) |

*τ*

_{ n}and

*x*

_{ n}with this Gaussian smoothing function, in order to produce an estimate of

_{ n},

*ς*

_{ n}, and

*σ*

_{ n}for every value of

*D*, in 1-ms increments. In other words, this allows us to calculate the variance in saccade landing position, the mean saccade duration, and its variability for every saccade

*n*. Convolved values of

_{ n}were given by

*ω*, is the Gaussian smoothing function, sampled at 1-ms intervals and centered on the

*D*value of the

*n*th saccade. The resulting standard deviation in

_{ n}(i.e.,

*ς*

_{n}) is then given by

*σ*

_{ n}as a function of

*D*. Figure 5 shows, for a single observer, the resulting smoothed estimates of

_{ n},

*ς*

_{ n}, and

*σ*

_{ n}as a function of

*D*. Incidentally, we replicated the fits reported below using a common function width of 30.0 ms across all observer/conditions (as this width lies close to the mean across observers), as well as with a range of widths from 20.0 ms to 45.0 ms. In all such circumstances, the replicated fits fell within the 95% confidence intervals reported below.

_{ n}with

*D*, where

*σ*

_{ n}and

*ς*

_{ n}jointly determine the scale of the residuals. In the absence of strong a priori expectations on the nature of the integration kernel, we fit

*ρ*

_{ n}with a scaled cumulative Gamma distribution function, shown by the red line in Figure 4B (we did examine fits of a cumulative Gaussian curve, which is symmetrical, but found that the fit was inferior). Note that the function was fit using values of

*D*

_{ n}> 0 only—as negative values of

*D*

_{ n}correspond to instances in which the velocity change occurred after saccade onset, such trials cannot reasonably provide meaningful information about how the velocity change is incorporated into the programming of a saccade. The equation for the Gamma function used is

*k*and

*θ*are shape and scale parameters, respectively. The advantage of this functional form is that it can accommodate both symmetrical and asymmetrical kernels. To accommodate the data, we allow three parameters to vary:

*a*(the upper bound of the curve fit − the lower bound was set to zero),

*k*(shape), and

*θ*(scale). To find a set of best-fitting parameters, we used the Simplex method (Nelder & Mead, 1965) to adjust the model parameters in order to maximize the likelihood of the observed values of

*ρ*

_{n}.

*σ*

_{ n},

*ς*

_{ n}, and

_{ n}for each observer/step condition. For values of

*D*ranging from 1 to 500 in 1-ms increments, we calculated the weightings resulting from each bootstrap parameter set. For each sampled value of

*D*, we then obtained the 95% confidence limits of the bootstrap weighting distributions at that point using the percentile method (Efron & Tibshirani, 1993).

*SEM*= 5.7 ms) before saccade onset. The mean full duration at half-maximum of the kernels is ∼95 ms (

*SEM*= 12.4 ms). After velocity estimation, there is a short gap of some 50–90 ms before saccade onset during which no new speed information is taken into account. Table 2 shows the resulting Gamma fit parameters for all five observers.

Observer | Fit parameters | 95% Confidence intervals | ||||
---|---|---|---|---|---|---|

a | k | θ | a | k | θ | |

1 | 0.94 | 11.92 | 15.96 | 0.90 | 7.67 | 9.62 |

1.00 | 19.26 | 25.92 | ||||

2 | 0.80 | 29.50 | 6.04 | 0.77 | 14.84 | 3.72 |

0.87 | 47.55 | 12.36 | ||||

3 | 0.84 | 13.55 | 15.23 | 0.81 | 8.61 | 8.55 |

0.88 | 24.07 | 24.04 | ||||

4 | 0.81 | 22.76 | 7.34 | 0.77 | 12.82 | 3.68 |

0.85 | 44.98 | 13.24 | ||||

5 | 0.84 | 35.20 | 5.14 | 0.79 | 19.41 | 2.50 |

0.89 | 72.02 | 9.44 |

*D*are simply a sampling of the temporal kernel. If they fail to adequately sample that kernel, then this will simply lead to a lack of adequate constraint in our kernel estimates. It is more likely that the differences in findings reflect the differences in task demands. Tavassoli and Ringach were concerned with pursuit maintenance, which is generally thought to involve the modulation of a continuous movement based on feedback about performance quality and target velocity that minimizes retinal slip (Lencer & Trillenberg, 2008; Lisberger, Morris, & Tychsen, 1987). The feedback involves a combination of retinal (i.e., image displacement) and extra-retinal (i.e., efference copy and proprioception from the ocular muscles) sources of information (e.g., Krauzlis & Lisberger, 1994; Lisberger, Evinger, Johanson, & Fuchs, 1981).

^{1}Specifying the probability distribution is necessary for maximum-likelihood estimation of the kernel parameters. Fits using least squares or least absolute residuals do not require explicit identification of this probability distribution, as a function of

*D*. Indeed, fits using these simpler methods result in similar parameter estimates as the ones reported in Table 2. We specify a more complete model here, because we feel it is important to make explicit what sources of variance contribute to the variation in

*ρ*

_{ n}.