The Gaussian shape of reciprocal latency distributions typically found in single saccade tasks supports the idea of a race-to-threshold process underlying the decision when to saccade (R. H. Carpenter & M. L. Williams, 1995). However, second and later saccades in a visual search task revealed decision-rate (=reciprocal latency) distributions that were skewed Gamma-like (E. M. Van Loon, I. T. Hooge, & A. V. Van den Berg, 2002). Here we consider a related family of Beta-prime distributions that follows from strong competition with a signal to stop the sequence, and is described by two parameters: a fixate and saccade threshold. In three saccadic search experiments, we tried to manipulate the two thresholds independently, thereby expecting change in shape and mean of the reciprocal latency distribution. Interestingly, rate distributions for later saccades were significantly better fit by Beta-prime than by Gamma functions. Increases in the distribution’s skew were found with higher display density, but only for second and later saccades. First saccade rate distributions were not altered by the expected target location or by visual information presented prior to the search, but making pre-search saccades did influence both thresholds. The mean rate remained a stereotyped function of ordinal position in the saccade sequence. Our results support strong competition between two decision signals underlying the timing of saccades.

^{−1}) at which the neural signal rises from baseline until threshold. Whereas latencies are typically skewed in their distribution, their reciprocal values are distributed normally (Carpenter & Williams, 1995), fitting the idea of a stochastic decision-rate underlying the timing of saccades.

*s*’ to make a saccade, or to an incremental step toward the threshold ‘

*f*’ to maintain fixated, thus terminating the saccade sequence. Importantly, each decision bit cannot contribute to both races at the same time, hence forming a strong form of competition between the decision to make a saccade or to maintain fixation. Now, the chance

*p*of an incremental step toward the saccade threshold

*s*over the chance of incremental step toward the fixation threshold

*f*is

*r = p/q*. Note,

*q*= 1−

*p*because of our assumption of strong competition, so that

*r*=

*p*/(1−

*p*). The ratio

*r*we associate with the observable decision-rate in our experiments. The ratio

*r*has a probabilistic nature that is described by the probability function Beta-prime: Here the Beta function

*b*(

*s,f*) is used as normalization factor, and the parameters

*s*and

*f*are the thresholds for saccade initiation and maintaining fixation. Note, the observable decision-rate has the dimension [s

^{−1}], whereas the ratio

*r*is dimensionless. Therefore, we need to include a third fit parameter

*τ*[s] to scale the decision-rate distribution along the time axis, but this parameter turns out be close to 1 (Van den Berg & Van Loon, in press, and see Methods).

^{−1}(i.e., fixation durations > 67 ms), grouped into bins of 0.35 [s]

^{−1}, were fitted using Marquardt-Levenberg’s nonlinear fit procedure programmed in Mathematica (Wolfram). A minimum of 50 saccades was required for a fit. Fit results were checked for stable solutions by varying the starting values.

*p*= .5 at the mean rate. Indeed, for each subject, the first saccade data resemble a straight line, but the

*p*(K-S) values (see insets) indicate that the first saccade distribution cannot be distinguished from a Beta-prime or Gamma distribution either. For second and later saccades, the Gaussian function does not fit the data very well, as the plotted data seems to curve, and fast rates (i.e., short latencies) occur more frequently than predicted by a Gaussian distribution. The Gamma function has larger tails, but the Beta-prime function seems best at fitting the data, giving slightly higher p values (e.g., fourth saccades).

*p*(K-S) > 0.05]. The fraction varies considerably across experiments and saccade number. Generally, for first saccades, the fraction of good fits for Gauss, Gamma, and Beta-prime functions is about equal, none of the fractions being consistently greater or smaller over the three experiments. In contrast, for second and later saccades, the fraction of successful Gauss fits is consistently smaller than that for the Gamma function. This confirms the earlier findings by Van Loon et al. (2002). Furthermore, for second and later saccades, the fraction of successful fits for the Beta-prime function was consistently higher than that for the Gamma function in two out of three experiments.

*p*(K-S) is a measure of goodness of fit that is corrected for sample size, we observed that

*p*(K-S) values could still decrease when adding more samples. This observation may implicate that the beta-prime model still does not fully capture all aspects of saccade timing in visual search. Most importantly, it means that goodness-of-fit comparisons should be restricted to equally sized data. In that light, the increased fraction of successful fits with higher ordinal saccade number (Figure 3, Experiment I) might also be attributed to the reduced number of saccades (insets, Figure 2). However, we can validly compare the relative goodness of each fit per data set. For later saccade data, Beta-prime fitted better than Gamma in 77% of all fits, the goodness of fits

*p*(K-S) being significantly different (

*p*

_{140}< .001 in a two-sided paired

*t*test). For first saccade data,

*p*(K-S) values were not significantly different between Beta-prime and Gauss (

*p*

_{41}< .44, paired t test). Overall, the Beta-prime seems best suited for describing the rate data. For this reason, in the following we look only at the Beta-prime parameters.

*τ*to be close to 1 for each experimental condition, ordinal number in the saccade sequence, and subject (

*τ*= 1.05 s ± 0.06 SD over all fits). When fitting with a fixed parameter (

*τ*= 1 s)

*p*values turned out to be somewhat higher (25%). Including a third scale parameter also resulted in lower fit quality for the Gauss and Gamma function, so we assume this is a general effect of overfitting, by an enhanced chance of finding a local minimum in the residuals. Using a fixed scale factor

*τ*= 1 affected the Beta-prime parameters only marginally, resulting in about 10% smaller (

*s+f*) values and 3% smaller ratios s/(

*f*−1). Therefore, effectively, the rate distri-butions can be well described by a Beta-prime function that takes only two free parameters

*s*and

*f*. All reported fit parameters and goodness of fit have been obtained using this constant

*τ*of 1 s.

*s*and

*f*, we associate the threshold for saccade initiation and hold fixation, respectively. In our results, however, we will report the ratio

*s*/(

*f*−1) and the sum (

*s+f*) instead, because these terms more directly describe the shape of the Beta-prime distribution. Mathematically, the ratio of the thresholds

*s*/(

*f*− 1) equals the mean of the Beta-prime distribution, thus a higher ratio implies a higher mean rate (i.e., shorter latencies). Furthermore, the sum of the thresholds (

*s+f*) reflects the asymmetry in the distribution [i.e., a lower sum of thresholds (

*s+f*) will broaden the Beta-prime distribution and skew it toward shorter rates, i.e, longer latencies]. These relations will be explained below.

*f*. This means that for a saccade to occur, the chance

*p*of an incremental step to-ward the saccade threshold must be proportionally (about four times) larger than the chance

*q*for a step toward the fixation threshold. Because we identify the ratio

*p/q*with our observed decision-rate r, the latter must on average be proportional (about a factor four) with the ratio of decision thresholds.

*s*/(

*f*−1) (i.e., mean decision-rate). Lowering both thresholds will bring the decision in favor of a saccade more under the influence of spontaneous activity. Thus, the rate distribution will broaden. Moreover, this broadening will be asymmetrical about the mean decision-rate. Given that the fixation threshold is 4 times lower than the saccade threshold, a proportional lowering of fixation and saccade thresholds will increase the chance of reaching the fixation threshold more than the chance of reaching saccade threshold. To keep the mean decision-rate constant, relatively many fast decision-rates will need to occur, hence the tendency for the Beta-prime distribution to have more tail at higher rates when the threshold sum is lowered.

*F*

_{2,33}= 13,

*p*< .001). We also found that the average number of saccades per trial (4.4, 3.9, and 3.7 saccades for Lo, Mi, and Hi, respectively) decreased significantly with denser displays (

*F*

_{2,6474}= 300,

*p*< .001 in an ANOVA across subjects and target eccentricities).

*s*and

*f*that follow from the Beta-prime fits. To compare the variability over individuals and experiments, we plotted the sum (

*s+f*) and the ratio

*s*/(

*f*−1) of the thresh-olds for the different experiments, split by subject, for first or later saccades (Figure 9a–9d). Generally, considerable differences were found between individuals, but individual data showed consistent levels over time.

*s*and

*f*(Figure 5a). This general skew in rate distributions for later saccades confirms earlier observations by Van Loon et al. (2002), but extends them to longer sequences (Van Loon et al. analysis was confined to the first four saccades in a sequence). As follows from Figure 5, the ratio of thresholds (or mean rate) for first saccades was markedly lower than for later saccades. Interestingly, second saccades had clearly increased rates compared to the other saccades.

*F*

_{2,57}= 6.5,

*p*= .003 in an ANOVA with data pooled over the four subjects and saccade 2 up to 6).

*s*/(

*f*−1)] for either first or later saccades when density increased. This lack of a decreased mean rate seems to contrast with reports of longer fixation duration when the number of stimulus elements per fixation in-creases (i.e., Mackworth, 1976; Moffitt, 1980) or when the number of distractors increases (Walker et al., 1997). How-ever, our stimulus may have been different in that the line elements do not merely act as distractors but also help to locate the target line. Thus, a possible effect of more distractors may have been balanced by an increased target saliency.

*p*= .25). The probability for a target at 6, 12, and 18-deg eccentricity thus was 25% each. In a second block, foveal targets were presented in half of the trials (

*p*= .5). In that case, the probability of a target at eccentricities 6, 12, or 18 deg was 16.7% each. A block was completed in sessions of 240 trials to obtain a minimum of 480 trials per subject.

*F*

_{1,28}= 1.0,

*p*= .3) (see Figure 4). Also, the mean sequence length was constant (3.5 saccades) in both conditions (

*F*

_{2,4300}= 3.0,

*p*= .4). The potential appearance of a foveal target did have an effect. The performance with 50% foveal targets was significantly lower than the middle density condition of Experiment I (

*F*

_{1,16}= 10,

*p*= .006 across three subjects). Remarkably, when the target was presented centrally, subjects often did make a saccade sequence (in 92 and 83% of the trials with 25 and 50% foveal targets, respectively, averaged over subjects). The sequence length in that case clearly decreased with higher foveal target probability (3.7 and 2.8 saccades, respectively). For these central targets, the performance was 0.7 and did not change significantly with higher foveal target probability (

*F*

_{1,8}= 1.3,

*p*= .3).

*p*= .09; two-tailed t test). Also, compared to Experiment I (Mi density condition, three subjects) no significant difference was found between the sum or ratio of thresholds, neither for first saccades (

*F*

_{2,6}= 0.17,

*p*=.8 and

*p*=.6, respectively) nor for later saccades (

*F*

_{2,38}= 0.6,

*p*= .5 and

*p*= .9, respectively).

*F*

_{1,58}= 29,

*p*< .001 across five subjects (see Figure 4). Subjects reported that the saccadic task during the pre-search display hampered subsequent search more than the fixation task. No effect of refreshing the line positions was evident. Moreover, whether a line display or just a fixation point was presented prior to the search display did not significantly alter performance (

*F*

_{2,6}= 1.2,

*p*=.36, comparison of the two fixation conditions of Experiment II with the middle density condition of Experiment I across three subjects). The mean sequence length for the fixation conditions (2.2 saccades) was about a saccade shorter than in Experiments I and II, probably as a result of the reduced search time. Moreover, the sequence was significantly shorter for the saccade conditions (2.0 saccades) (

*F*

_{1,7174}0, < 001). Furthermore, a small increase in sequence length was found when refreshing lines, but only in the fixation condition.

*F*

_{1,16}= 0.34,

*p*= .57, data pooled over saccade conditions). In contrast, making pre-search saccades significantly decreased the threshold sum for first saccades (Figure 8a,

*F*

_{1,16}= 4.68,

*p*= .046, data pooled over refresh conditions). Closer examination (Figure 9a) showed that this effect was caused by two subjects (EP and JB). The other three subjects showed no effect. Experimental conditions did not systematically influence the sum of thresholds in later saccades (Figure 8a), or the ratio of thresholds in first and later saccades (Figure 8b).

*F*

_{1,7}= .21,

*p*= .7, data from three subjects). Thus, the timing of the first saccade did not depend on the search task.

*F*

_{1,4}= 493,

*p*= .8, data from three subjects). Thus, both findings suggest that the prior visual history has no influence on first saccade rate distribution.

*F*

_{1,58}= 29,

*p*< .001, data from five subjects, pooled over target eccentricities and refresh conditions).

*F*

_{1,2five}= 4.7,

*p*= .04, data from three subjects pooled over target eccentricities).

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