**Abstract**:

**Abstract**
**This study analyzes how human participants combine saccadic and pursuit gaze movements when they track an oscillating target moving along a randomly oriented straight line with the head free to move. We found that to track the moving target appropriately, participants triggered more saccades with increasing target oscillation frequency to compensate for imperfect tracking gains. Our sinusoidal paradigm allowed us to show that saccade amplitude was better correlated with internal estimates of position and velocity error at saccade onset than with those parameters 100 ms before saccade onset as head-restrained studies have shown. An analysis of saccadic onset time revealed that most of the saccades were triggered when the target was accelerating. Finally, we found that most saccades were triggered when small position errors were combined with large velocity errors at saccade onset. This could explain why saccade amplitude was better correlated with velocity error than with position error. Therefore, our results indicate that the triggering mechanism of head-unrestrained catch-up saccades combines position and velocity error at saccade onset to program and correct saccade amplitude rather than using sensory information 100 ms before saccade onset.**

^{1}Hz; peak to peak displacement amplitude: [40..60]°) for a random duration ([3000..3750] ms). With this range of parameters, 1.8 to 4.5 cycles of motion were presented to the subject. Around the end of the red target motion, a second green target was briefly presented (flash target, duration: 10 ms) at a random position inside a virtual annulus around current pursuit target position (inner radius of 15°; outer radius of 30°). The trial ended with a red fixation at the center of the screen for 500 ms. The second part of the protocol started at the flash presentation or at pursuit target extinction, whichever occurred first and lasted up to trial end. A trial lasted 6 s, and we had a 0.5-s intertrial interval. Thus a block lasted 162 s, and we allowed subjects to pause for ∼30–45 s between two blocks. Therefore a recording session never lasted more than 30 min. Finally, no subject did more than one recording session per day to avoid fatigue.

^{2}to 500°/s

^{2}. All the trials were aligned with respect to the onset of the pursuit target movement.

*L*) as a phase in a cycle of target motion (

_{ON}*ϕ*) using:

_{ON}*f*in Equation 1 represents target oscillation frequency. Following this conversion, each saccade had a phase onset inside a [0..1] range within a cycle.

_{t}*PE*

_{100}) and the retinal slip (

*RS*

_{100}) 100 ms before saccade onset as in de Brouwer, Missal et al. (2002). Additionally, due to the predictive nature of our protocol, we computed the acceleration error (

*AE*

_{100}) as the difference between the target and gaze accelerations 100 ms before saccade onset.

*PPE*), the predicted velocity error (

_{ON}*PVE*), and the predicted acceleration error (

_{ON}*PAE*) at the onset of the gaze shift as the difference between the target position, velocity, and acceleration and gaze position, velocity and acceleration. We qualified these variables as “predicted” in opposition to the “sensory” variables of the previous paragraph because they could not come from a direct reading of a sensory input (because of internal delays). We postulate that they must be generated by an internal model. This point is developed in the Discussion.

_{ON}*PE*

_{100}could correspond to either a lag (target moving to the right, with target velocity > 0) or an advance of the gaze with respect to the target (target moving to the left, with target velocity < 0). The same kind of reasoning can be applied to interpret retinal slips and acceleration errors. Therefore, the sign of

*PE*

_{100},

*RS*

_{100},

*AE*

_{100},

*PPE*,

_{ON}*PVE*, and

_{ON}*PAE*was corrected by the sign of target velocity at the specified time (100 ms before saccade onset or at saccade onset). Following this normalization, a positive (negative) value of

_{ON}*PE*

_{100}or

*PPE*always corresponds to gaze behind (ahead of) the target. Similarly, a positive (negative) value of

_{ON}*RS*

_{100}or

*PVE*always corresponds to gaze moving slower (faster) than the target while a positive (negative) value of

_{ON}*AE*

_{100}or

*PAE*always corresponds to gaze accelerating (decelerating) with respect to the target.

_{ON}^{2}), we assumed that the velocity (acceleration) error was integrated to correct saccade amplitude and ensure an accurate saccadic displacement, as shown during head-fixed pursuit (de Brouwer et al., 2001; Schreiber et al., 2006). Therefore, we multiplied the velocity error either by saccade duration

*S*for predicted signals or by saccade duration plus 100 ms (

_{Dur}*S*+ 0.1) for sensory inputs and used those signals to predict saccade amplitude. Similarly, we multiplied acceleration errors by S

_{Dur}^{2}

_{Dur}/2 for predicted signals or by (

*S*+ 0.1)

_{Dur}^{2}/2 for sensory signals. To test our regression method, we generated a linear two-dimensional (2-D) dataset polluted with Gaussian random noise with different variances on each dimension. Then, we applied the orthogonal regression method and the regular least-square method on the same dataset and compared the results. The orthogonal regression always approximated better the slope of the noise-free dataset than the regular least-square method. For example, when the noise on the regressand was 16 times larger than the noise on the regressor, the slope was underestimated by ∼3% with the total least-square and by ∼7% by the regular least-square. This test was used to validate our method.

*p*value of the rank sum tests is given for each comparison. A smaller cvMSE represents a better predictor of the observed behavior. In contrast, we used vaf (variance-accounted-for: the proportion of regressand variance explained by the regression) to express the quality of regular regressions.

*ρ*in first order regressions).

*proportion of saccadic time*”,

*T*

_{sacc}_{,prop}.

*T*

_{sacc}_{,prop}was defined as the ratio of the sum of the duration of all saccades in a trial to the duration of the corresponding trial. A linear regression between target oscillation frequency and proportion of saccadic time resulted in: Equation 2 shows that the proportion of saccadic time doubled on the range of observed frequencies (from 0.126 at 0.6 Hz to 0.279 at 1.2 Hz). Slopes of regression (Equation 2) for individual participants varied from 0.048 ± 0.012 to 0.351 ± 0.014. Therefore, Equation 2 shows that with an increase of target frequency, participants increased the proportion of time during which they executed a saccade, validating our assumption.

*S*) and the position error 100 ms before saccade onset: Equation 3 shows that saccade amplitude is sensitive to a change of position error (represented by the slope of the regression). However, the correlation between

_{A}*PE*

_{100}and

*S*is very weak. Thus the computed slope could not be used trustfully to characterize saccade amplitude as a function of

_{A}*PE*

_{100}.

*S*+ 0.1) in the following regressions:

_{Dur}*PE*

_{100}≪ ≫

*RS*

_{100}:

*p*< 0.001). Therefore, despite the higher correlation between

*RS*

_{100}and saccade amplitude, the quality of the predictor (measured by cvMSE) is worse between

*RS*

_{100}and saccade amplitude than between

*PE*

_{100}and

*S*. When both

_{A}*PE*

_{100}and

*RS*

_{100}are used to model the amplitude of the gaze shift, regression (Equation 5) shows a significant large increase of the quality of the regression with respect to

*RS*

_{100}but a small decrease with respect to

*PE*

_{100}(two-tailed Wilcoxon rank sum test,

*PE*

_{100}≪ ≫ Multi100:

*p*= 0.0105,

*RS*

_{100}≪ ≫ Multi:

*p*< 0.001).

*AE*

_{100}provides information about a change of target velocity that could be used by the CNS to correct saccade amplitude. As for the retinal slip, the integration of the acceleration error would start 100 ms before saccade onset up to saccade offset (resulting in a factor (

*S*+ 0.1)

_{Dur}^{2}/2 in the following regressions). First, we computed a first order regression between

*AE*

_{100}and

*S*: Regression (Equation 6) shows that a first order regression using the acceleration error taken 100 ms before saccade onset was bad predictor of saccade amplitude. Nevertheless, the acceleration term could be used to compensate for the velocity error 100 ms before saccade onset, thus building a better predictor of saccade amplitude. To test this hypothesis, a multiple regression including was

_{A}*AE*

_{100}computed: Regression (Equation 7) shows that the addition of the acceleration error term does not affect significantly cvMSE (two-tailed Wilcoxon rank sum test, MultiAE100 ≪ ≫ Multi100:

*p*= 0.528. MultiAE100 ≪ ≫

*PE*

_{100}:

*p*= 0.029). Therefore, this analysis showed that, contrary to our hypothesis, the addition of an acceleration term to the regression did not increase significantly the quality of the prediction for the multiple regression with

*RS*

_{100}and

*PE*

_{100}.

*RS*

_{100}or

*AE*

_{100}than by

*PE*

_{100}(the population correlation is significantly different than zero only for

*RS*

_{100}and

*AE*

_{100}), confirming our observation when all the saccades were pooled together. At the subject level, table 2 shows that the quality of the regression increases between single regressions with

*PE*

_{100}and multiple regressions with

*PE*

_{100}and

*RS*

_{100}for five out of eight subjects. Seven subjects showed a statistically significant increase of the quality of the fit when we compared the single regression with

*PE*

_{100}to the multiple regression with

*PE*

_{100},

*RS*

_{100}, and

*AE*

_{100}. Only two subjects showed a statistically significant increase of the quality of the fit when we compared both multiple regressions. Finally, we did not see any statistically significant increase of the quality of the fits at the population level. This shows that the sensory signals, measured 100 ms before saccade onset poorly represent saccade amplitude.

*S*and

_{A}*PPE*,

_{ON}*PVE*, or

_{ON}S_{Dur}*PAE*(S

_{ON}^{2}

_{Dur}/2) as independent variables. We also computed a second-order regression between saccade amplitude and both

*PPE*and

_{ON}*PVE*. Finally, we computed a multiple regression that includes also the acceleration error at saccade onset (

_{ON}S_{Dur}*PAE*) to account for the sinusoidal target movement. The linear regressions resulted in: cvMSE of first-order regressions (Equations 8 and 9) clearly show that the variability of the amplitude of a saccade in our paradigm was better explained by predicted velocity error than by predicted position error (two-tailed Wilcoxon rank sum test, PPE ≪ ≫ PVE:

_{ON}*p*< 0.001). Additionally, two-tailed Wilcoxon rank sum test between cvMSE of simple regression (Equation 12) and multiple regression (Equation 11) confirmed that the addition of retinal slip significantly increased the quality of the fit (two-tailed Wilcoxon rank sum test, PPE ≪ ≫ Multi:

*p*< 0.001). Two-tailed Wilcoxon rank sum test between cvMSE of simple regression (Equation 9) and multiple regression (Equation 11) did not show a statistical difference between the two regressions (two-tailed Wilcoxon rank sum test, PVE ≪ ≫ Multi:

*p*= 0.143). The addition of the acceleration term in regression (Equation 12) generated a significantly better predictor than Equation 9 and Equation 11 (two-tailed Wilcoxon rank sum test, PVE ≪ ≫ MultiAE:

*p*< 0.001, Multi ≪ ≫ MultiAE:

*p*< 0.001).

*p*< 0.001. MultiAE ≪ ≫ PE

_{100}:

*p*< 0.001). This analysis shows that the amplitude of a saccade is better explained by a regression using position, velocity and acceleration errors at saccade onset than any model with errors 100 ms before saccade onset.

*PPE*,

_{ON}*PVE*, and

_{ON}*PAE*) and saccade amplitude. This was also the case at the population level, the correlation with

_{ON}*PVE*being the highest.

_{ON}S_{Dur}*PPE*and

_{ON}*PVE*or

_{ON}*PPE*,

_{ON}*PVE*, and

_{ON}*PAE*; this increase remained statistically significant at the population level. In addition, all the subjects showed a statistically significant increase of the quality of the regressions between the multiple regressions, which was also statistically significant at the population level.

_{ON}*N*= 50) for saccade latency bins of 50 ms. cvMSE was computed as in Equation 12 with

*PPE*,

_{ON}*PVE*, and

_{ON}*PAE*as independent variables. Then we computed the mean cvMSE during the first second following the onset of target motion (pursuit initiation) and between 2 and 3 s following the onset of target motion (steady-state pursuit). We observed a significant increase of the quality (decrease of cvMSE) of the model between the two time bins (two-tailed Wilcoxon rank sum test,

_{ON}*p*< 0.001). Finally, we checked if this difference was observed for different frequency ranges of target motion. To do so, we divided the dataset into target frequency bins of 0.1 Hz. Then we repeated the bootstrap analysis of cvMSE. Finally, we compared the quality of the fit across the different frequency bins. Table 5 shows the results of these tests inside the different frequency bins. There was a significant increase of the fit quality for all comparisons.

*S*corresponds to the head displacement during the gaze saccade and

_{A,H}*S*corresponds to the amplitude of the gaze saccade. We extracted percentiles 20 (range for the subjects: 0.22–0.55) and 80 (range for the subjects: 0.44–0.89) of each subject's HC distribution. Then, we computed regressions (Equations 7 and 12) separately for small HC (HC smaller than percentile 20) and large HC (larger than percentile 80) for each subject. We compared the quality of the fit for the corresponding conditions (small HC: MultiAE100 ≪ ≫ MultiAE and large HC: MultiAE100 ≪ ≫ MultiAE). The amplitude of the saccades was always better explained by a regression using the parameters sampled at saccade onset (corresponding to regression Equation 12) than by a regression using the parameters sampled at 100 ms before saccade onset (corresponding to regression Equation 7). This analysis demonstrates that the difference between our study and the previous study of de Brouwer, Missal et al. (2002) is not explained by the release of the head since it is valid for the whole range of head contribution.

_{A}*PPE*and

_{ON}*PVE*(see Methods) as a function of the saccadic onset phase in Figure 3. Solid black lines in Figure 3 represent a typical frequency-normalized position signal (arbitrary amplitude). Thin dotted lines represent the mean over 0.025 phase onset bins of either

_{ON}*PPE*(upper row in Figure 3) or

_{ON}*PVE*(lower row in Figure 3) for all target frequencies pooled together. It must be noted that the steps observed in the upper row of Figure 3 appeared because of the sign change of the target velocity that influenced the computation of predicted position error (see Methods). Figure 3 shows that gaze position lagged the target during a large part of the movement (

_{ON}*PPE*> 0). However, inside the [0.25..0.325] and [0.75..0.85] ranges, gaze was, on average, in phase or leading the target at saccade onset (

_{ON}*PPE*≤ 0). The lower row in Figure 3 shows that gaze moved on average more slowly than the target during the movement at saccade onset (

_{ON}*PVE*> 0). This is supplementary evidence that the head-unrestrained tracking gain is inferior to one, as reported by Daye et al. (2012). To quantify this observation, the horizontal thick black lines under each row represent the time when

_{ON}*PPE*or

_{ON}*PVE*were significantly higher than zero (dashed black lines, one-sided

_{ON}*t*test,

*P*< 0.05) or significantly smaller than zero (solid black lines, one-sided

*t*test,

*p*< 0.05). Importantly, the bottom row of Figure 3 shows that

*PVE*was never statistically smaller than zero (there is no horizontal solid black lines under the bottom row in Figure 3). In contrast, the predicted position error (upper row of Figure 3) shows that there were parts of the cycle for which gaze was on average in advance with respect to the target.

_{ON}*PPE*and

_{ON}*PVE*to target oscillation frequency. Thin solid color lines in Figure 3 represent the mean over 0.025 phase bins of either

_{ON}*PPE*(upper row in Figure 3) or

_{ON}*PVE*(lower row in Figure 3) for target oscillation frequency bins of 0.1 Hz (centered at either 0.65, 0.75, 0.85, 0.95, 1.05, or 1.15 Hz). One can clearly see that target oscillation frequency does not modulate

_{ON}*PPE*(color lines are superimposed in upper row of Figure 3) but has a profound effect on

_{ON}*PVE*(color lines are distinct in lower row of Figure 3). To validate this observation, we computed the time during which

_{ON}*PPE*for two successive frequency bins (0.65 Hz with 0.75 Hz, 0.75 Hz with 0.85 Hz, etc.) were statistically different (

_{ON}*t*test,

*p*< 0.05). The same procedure was applied to compare

*PVE*for two successive frequency bins. Thick horizontal dashed colored lines under the upper row of Figure 3 represent the times during which

_{ON}*PPE*for two successive frequency bins (represented by color components of the dashed lines) were statistically different. Similarly, thick horizontal dashed colored lines under the bottom row of Figure 3 represent the times during which

_{ON}*PVE*for two successive frequency bins (represented by color components of the dashed lines) were statistically different. Therefore, these analyses show that the distribution of predicted position error was not modulated by target oscillation frequency (no clear pattern emerges from the frequency comparison) while the predicted velocity error increased with increasing target oscillation frequency.

_{ON}*PPE*(red markers) in Figure 4E or

_{ON}*PVE*(blue markers) in Figure 4D. To represent the relationships between

_{ON}*PPE*and

_{ON}*PVE*independently of the bins' width, we first normalized the area under the histogram represented in Figure 2 to obtain the distribution represented in Figure 4A. Then we separated the data into two pools: one grouped positive values of

_{ON}*PPE*(gaze lagged behind the target) and the corresponding

_{ON}*PVE*(filled markers). The second group pooled the negative values of

_{ON}*PPE*(gaze led the target) and the corresponding

_{ON}*PVE*(open markers). Finally, to build Figure 4D, we plotted the mean

_{ON}*PVE*at a relative onset time (e.g.,

_{ON}*Y*

_{1}in Figure 4B) as a function of the normalized number of saccades (e.g., X in Figure 4A) at the same relative onset time. The same procedure was used to build Figure 4E (e.g., using X and

*Y*

_{2}).

*PPE*for the positive and the negative values of

_{ON}*PPE*is shown in Figure 4E. A similar trend of the number of saccades as a function of

_{ON}*PVE*is represented in Figure 4D. With an increase of the velocity error, there was an increase of the number of saccades. Surprisingly, the increase of

_{ON}*PVE*was accompanied by a decrease of the

_{ON}*PPE*. Therefore with a small velocity error, the saccadic system triggered few saccades and tolerated a larger position error. On the other hand, a larger number of saccades were triggered with a small position error and a large velocity error. We fitted an exponential function on

_{ON}*PVE*and

_{ON}*PPE*for the two pools of data (positive and negative values of

_{ON}*PPE*). The four fits resulted in:

_{ON}*PPE*. Dashed lines represent the fits (Equations 16–17) corresponding to the negative values of

_{ON}*PPE*. The fits quantitatively confirm our interpretation above: The number of triggered saccades increased with the retinal slip, even if the position error decreased. This analysis shows that velocity error at saccade onset was the major parameter to trigger saccades in our head-unrestrained paradigm. It also confirms that saccade amplitude is better correlated with velocity error than with position error.

_{ON}*S*≥ 5°. The 5, 50, and 95 percentiles of the resulting HC distribution are respectively equal to 0.142, 0.465, and 0.873. This shows that the head contributed on average for about 50% of the gaze saccadic displacement in our experiment.

_{A}*PPE*was computed similarly to

_{OFF}*PPE*using the sign of

_{ON}*Ṫ*to correct for the target velocity at the offset of the saccade.

_{OFF}*PVE*than

_{ON}*PPE*is not surprising. In head-restrained conditions, de Brouwer, Missal et al. (2002) showed that catch-up saccade amplitude was better described by a regression on

_{ON}*PE*

_{100}and

*RS*

_{100}than by a single regression on either

*PE*

_{100}or

*RS*

_{100}. Becker and Jürgens (1979) showed that the last time a change of target position can influence saccade amplitude was around 100 ms before saccade onset. Therefore, de Brouwer, Missal et al. (2002), de Brouwer et al. (2001), and de Brouwer, Yuksel et al. (2002) proposed that

*PE*

_{100}and

*RS*

_{100}correspond to sensory inputs delayed by 100 ms. A decade before Becker and Jürgens (1979), Barmack (1970) showed that a step of target velocity occurring 50 ms before saccade onset still modulates the amplitude of catch-up saccades. All those results pointed towards an influence of the retinal slip when programming gaze saccade amplitude in head-restrained conditions but with different latencies between a stimulus change and a modification of the behavior.

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