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J Neurophysiol 97: 795-805, 2007. First published November 15, 2006; doi:10.1152/jn.00652.2006
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A Population Coding Account for Systematic Variation in Saccadic Dead Time

Department of Experimental Psychology, University of Bristol, Bristol, United Kingdom

Submitted 22 June 2006; accepted in final form 9 November 2006

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
During movement programming, there is a point in time at which the movement system is committed to executing an action with certain parameters even though new information may render this action obsolete. For saccades programmed to a visual target this period is termed the dead time. Using a double-step paradigm, we examined potential variability in the dead time with variations in overall saccade latency and spatiotemporal configuration of two sequential targets. In experiment 1, we varied overall saccade latency by manipulating the presence or absence of a central fixation point. Despite a large and robust gap effect, decreasing the saccade latency in this way did not alter the dead time. In experiment 2, we varied the separation between the two targets. The dead time increased with separation up to a point and then leveled off. A stochastic accumulator model of the oculomotor decision mechanism accounts comprehensively for our findings. The model predicts a gap effect through changes in baseline activity without producing variations in the dead time. Variations in dead time with separation between the two target locations are a natural consequence of the population coding assumption in the model.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Interacting with a dynamic visual environment requires that observers are able to rapidly adjust their behavior to reflect the changing visual input. A fundamental component of such flexible behavior is to orient to parts of the visual scene that have become important and to orient away from regions that are no longer relevant. The double-step paradigm can be used to investigate the ability of the saccadic eye movement system to rapidly change the goal of the saccadic eye movement in response to dynamic visual input (Westheimer 1954). In this paradigm, the observer is presented with a peripheral target, which jumps to another location in the visual field. The task of the observer is to fixate the peripheral target. When the position of the target changes, the saccade program has to be adjusted to reflect this change.

In a typical double-step trial, a first target appears at, for example, 15°, and is followed after a variable interstimulus interval by a second target at, for example, 30°, on the same axis and hemifield (Becker and Jürgens 1979; Lisberger et al. 1975). The amplitudes of the resulting saccades are systematically related to the interval between the onset of the saccade and that of the second target (termed D). For small values of D (less than 80 ms), the saccade tends to be directed to the first target, and for larger values of D (more than 130 ms), the saccade tends to be directed toward the second target. At smaller separations between the two targets, intermediate values of D give rise to a gradual transition with saccades frequently landing in between the two targets. The relation between first saccade amplitude and D is described by an amplitude transition function.

The amplitude transition function can be used to define a so-called saccadic dead time (SDT). This is the point in time at which new visual information (i.e., a 2nd target step) can no longer change the upcoming movement. The typical amplitude transition function pattern described in the preceding text suggests that this "point of no return" lies 80 ms prior to movement onset (Findlay and Harris 1984). The 80-ms estimate is often related to afferent delays in the transmission of visual information to the oculomotor system and efferent delays between the issue of oculomotor commands and the actual movement of the eye muscles (Becker 1991). However, note that our definition of the SDT here is a functional one: it is the last moment at which a planned movement can be altered. As such, in our conception of the dead time decisional components play a role in addition to afferent and efferent delays (Becker 1991) (see modeling section in Model of the oculomotor decision mechanism). Importantly, the SDT is often assumed to remain relatively constant (Beutter et al. 2003; Findlay and Harris 1984; Ludwig et al. 2005; Van Loon et al. 2002). For example, in models of eye-movement control in reading, the SDT corresponds to what is often termed a "nonlabile" stage of saccade programming (Reichle et al. 1998). Such models may incorporate some variability in the duration of this stage, but the SDT distribution is thought to remain constant across experimental conditions (but see Engbert et al. 2005), meaning that in these models the SDT does not contribute toward systematic latency differences.

However, inspection of published amplitude transition functions suggests that the assumption of a constant dead time may not be correct. For instance, in Becker and Jürgens (1979), it is clear that the SDT depends on the spatial configuration of the two target steps (e.g., 2 steps in the same direction vs. 2 steps in the opposite direction; see their Fig. 5 and Table 1). In the present study, we aimed to 1) examine the dependence of the SDT on saccade latency; 2) parametrically investigate the dependence of the SDT on the spatial configuration of the two steps; 3) develop a model of the oculomotor decision mechanism that accounts for variability in basic saccade latency and direction, and naturally captures the double step findings.

View larger version (10K): [in this window] [in a new window]   FIG. 1. Experimental methods. A: schematic illustration of the sequence of displays in the gap and overlap conditions of experiment 1. B: direction of the first saccade as a function of interstimulus interval. Data from observer 3, upward saccades, gap condition. Smooth curves are cumulative Gaussian fits. C: same data but as a function of D. The dead time estimates are derived from the cumulative Gaussian fits to these functions.

View larger version (15K): [in this window] [in a new window]   FIG. 2. Direction transition functions from experiment 1. —, cumulative Gaussian fits to the data from the overlap condition; - - -, fits to the data from the gap condition.

View larger version (7K): [in this window] [in a new window]   FIG. 3. Illustrative direction transition functions and mean dead time estimates from experiment 2. A: direction transition function at 30° separation from observer 3, upward saccades. B: direction transition function from the same observer at 90° separation. C: mean dead time across observers as a function of separation. Error bars are 1 SE.

View larger version (15K): [in this window] [in a new window]   FIG. 4. Illustration of the stochastic accumulator model. Top: bank of units coding saccade directions. As a result of visual stimulation these units receive an input that spreads to neighboring units as illustrated by the Gaussian profiles above the array. Bottom: example accumulation patterns for 2 units in a single-step (left) and double-step (right) trial. One unit, x90, is centered on the location of the 1st target, and its activity is shown (black traces). A competing unit, xi, codes a direction in between the 2 target locations; its activity is shown (gray traces). Triangle: onset of the 2nd target. For further details, see text.

View larger version (14K): [in this window] [in a new window]   FIG. 5. Model performance under single-step (A and B) and double-step (C–E) conditions. A: cumulative probability distribution of saccade latency from observer 5 (experiment 2, upward saccades). —, model's performance. B: same as A but for saccade direction. C: simulated direction transition function at 30° separation. D: simulated direction transition Function at 90° separation. E: model dead times as a function of separation. , 95% confidence intervals around the mean across observers from experiment 2. Predictions were generated with the following parameters: µstep = 0.0075; step = 0.0018; spread = 21; noise = 0.0059; and = 0.0027.

In experiment 1, we estimated the SDT under an experimental manipulation of saccade latency. Saslow (1967) demonstrated that if the fixation point disappears just before a saccade target appears, saccade latency decreased compared with a condition in which the fixation point remains visible. One component of this gap effect is ocular disengagement that has been shown to operate in the superior colliculus (Dorris and Munoz 1995). Within this structure, cells in the rostral pole are active during periods of stable fixation (Munoz and Wurtz 1993; or for very small eye movements around the fovea, see Gandhi and Keller 1999) and tonically inhibit movement-related cells in the caudal superior colliculus. The absence of a foveal stimulus results in a reduction in activity of the fixation cells, disinhibiting the movement related units. Functionally then, the effect of the gap is a spatially nonspecific increase in the excitability of the oculomotor system. As a result, the system is more ready to respond to incoming sensory information. We examined whether this increased readiness was reflected in the SDT.

In experiment 2, we systematically investigated the influence of the separation between the first and second target on the SDT. Gaze vectors, which represent the saccadic movements of the eye, are coded by large populations of neurons within the superior colliculus and frontal eye fields (Bruce and Goldberg 1985; Lee et al. 1988; Munoz and Wurtz 1995). Increasing the distance between the two target locations will decrease the amount of overlap in the underlying mechanisms coding the movement vectors associated with the two targets. We reasoned that the partial activation of a second saccade program may affect the time it takes the system to modify the first program. In turn, this may influence the SDT.

The experiments reported formed part of a master's thesis by J. W. Mildinhall.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Human observers

In experiment 1, five observers (3 men and 2 women; age range: 23–28 yr) took part as volunteers. In experiment 2, five observers (1 man and 4 women; age range: 19–20) took part as volunteers in return for course credit. All had self-reported normal or corrected-to-normal vision.

Equipment

Displays were presented on a 21-in, gamma-corrected monitor (Eizo FlexScan T965) running at 80 Hz with a 1,024 x 770 pixel resolution. The screen was 57 cm from the chinrest that was used to stabilize head position. Eye movements were monitored with the EyeLinkII (SR Research, Mississauga, Ontario, Canada). This infrared tracking system uses the center of the pupil to sample eye position at 500 Hz with an accuracy of 0.3°. Saccades were detected off-line using velocity and acceleration criteria of 30°/s and 8,000°/s², respectively.

Procedures

The first experiment consisted of 10 blocks of trials over two separate sessions; the second experiment was run in 20 blocks over four sessions. Each block contained 96 trials. At the start of each trial, a fixation point appeared in the center of the screen. The fixation point was a 0.3 x 0.3° black cross. Saccade targets consisted of bright Gaussian patches with a peak luminance of 76 cd/m² presented on a gray background with a luminance of 25 cd/m². The SD of the Gaussian patches was 0.5°. Note that because the targets are different from the central fixation point, strictly speaking we are not dealing with double steps. Nevertheless, we retain the standard nomenclature to emphasize the link with previous double-step studies.

On single-step trials, a target appeared on the vertical meridian at 8° eccentricity, either above or below the central fixation point. On double-step trials, the target was shifted after a variable delay. The interval between the two steps was selected randomly from 25 to 175 ms in steps of 12.5 ms (the duration of a single monitor refresh). The total exposure duration of the target(s) was 750 ms.

In experiment 1, the central fixation point was extinguished 200 ms prior to the onset of the first target on half the trials. On the other half of the trials, the fixation point remained visible throughout the trial. The second target, if present, appeared at an angular separation of 45° from the first target, either to the left or to the right, at 8° eccentricity. The sequence of events on double-step trials in experiment 1 is illustrated in Fig. 1A. In experiment 2, the central fixation point was visible throughout the trial, but the angular separation between the two targets was varied from 30 to 120° in steps of 30°. Again, the second target could appear either to the left or to the right of the first target location.

Each block started with a nine-point calibration procedure. Participants fixated a black cross (identical to the fixation point) at each of the nine points in random order on a 3 x 3 grid. In each trial, observers were instructed to saccade to the final target position. Within a block, 2/3 of the trials were single-step trials, and one-third of the trials were double-step trials. Inclusion of a large number of single-step trials is necessary to induce observers to actually program a movement to the first target location. Single- and double-step trials were randomly intermixed.

Data analysis

Trials in which the initial eye position deviated by >1° from the fixation point were rejected. Trials in which saccades were thought to be anticipatory (latency <100 ms) were rejected. Trials in which the initial saccade amplitude was <4° or directed to the wrong half of the screen were also eliminated.

The critical variable in this study is the estimated SDT. Because all targets appeared equidistant from fixation, for each experimental condition, we related the angle of the first saccade to D (Aslin and Shea 1987). We will refer to these functions as direction transition functions. Each first saccade landing position was transformed so that the first target location corresponded to a 90° angle (a vertical, upward saccade), and the second target corresponded to a rightward displacement with respect to the first step. This choice is of course arbitrary but allows for pooling of the data across directions and direct comparisons across conditions.

The procedure to estimate the SDT is illustrated in Fig. 1, B and C. B shows the angle of each first saccade as a function of the delay between the two target steps. In this figure, the second target step corresponds to an angle of 45°. Thus at short interstimulus intervals, the observer frequently made a saccade directly to the second target; however, at long interstimulus intervals, the first saccade generally landed at the first target location. C shows the same data but plotted as a function of D, the interval between the onset of the first saccade and that of the second target. Thus at short values of D, the change in target position could not be incorporated in the movement plan, and the observer fixated the first target location. At larger values of D, a change in the movement plan is possible, and so the observer fixated the second target location. This pattern is similar to that described for saccade amplitudes in the INTRODUCTION and to that for direction reported in the literature (Aslin and Shea 1987; Findlay and Harris 1984).

The smooth curves through the data points in Fig. 1, B and C, are cumulative Gaussians with parameters µ and with upper and lower asymptotes constrained to align with the angles of the two target centers. Because these functions are fitted to data from individual trials, we minimized the sum of the absolute residuals (a standard least-squares fit would give too much weight to outliers). The SDT is defined as the last moment before saccade onset at which the movement to the first target could still be altered. We derived this estimate from the parameters of the cumulative Gaussian fit to the direction transition function (i.e., the function illustrated in C): SDT = µ – 1.65. The multiplier in this equation corresponds to the Z score associated with cumulative probability of 0.95. The SDT estimate is illustrated by the dotted drop-line in Fig. 1C.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Experiment 1

Eight percent of trials were excluded from the analysis because of improper fixation at the start of the trial, too short an amplitude, incorrect direction, and anticipatory latency. The following analyses are based on all remaining trials. The mean saccade latencies were analyzed with a repeated-measures ANOVA with the following factors: saccade direction (up, down), gap (gap, overlap), and step (single, double). There were significant main effects of gap [F(1,3) = 33.0, P < 0.01] and step [F(1,3) = 29.3, P < 0.01]. The main effect of step is unsurprising as this analysis includes the saccades that were directly aimed at the second target, and their latencies include the interstimulus interval. Latency was found to vary monotonically (approximately sigmoidally) as a function of direction with saccades directed to intermediate locations having intermediate latencies. No other main effects or interactions were significant. The robust and large gap effect was found for each observer, in both single- and double-step conditions. The gap effect, expressed as the difference between mean saccade latency, ranged from 49 to 111 ms across the five observers. On average this was a 28% decrease in latency.

Inspection of saccade direction as a function of interstimulus interval (as in Fig. 1B) shows that in the gap condition a very small interstimulus interval (less than 60 ms) is required to ensure that the saccade is directed to the second target. In the overlap condition (not shown), it appeared that interstimulus intervals of up to 120 ms can still result in a saccade toward the second target. The critical question is whether this difference between the two fixation point conditions still exists once the data are aligned on saccade onset (i.e., expressed as a function of D). Figure 2 shows the direction transition functions for every individual observer. The data points from the gap and overlap conditions are shown in the same panels. Data from the upward saccade conditions are shown in the top row; data from the downward conditions are shown in the bottom row. Note that in most of the conditions, the direction transition functions are smooth, continuous functions with a gradual transition between initial to final angle responses. However, it is clear that the data points and the curves from the gap and overlap conditions essentially lie on top of each other.

Table 1 lists the SDT estimates derived from the direction transition functions for each observer. Each of these estimates is based on data from 44 to 80 trials (range across observers and conditions). A reliable SDT could not be extracted for the downward saccades in the overlap condition for observer 3. As a result, this observer's SDT estimates were based on the gap and overlap data pooled together. The estimates were entered in a repeated-measures ANOVA with saccade direction (up, down) and gap (gap, overlap) as factors. Although the SDT estimates tended to be somewhat shorter (by 9 ms), when the first target appeared in the lower visual field, this effect was not reliable. In addition, as suggested by Fig. 2, there was no effect of the fixation point manipulation. Despite an overall reduction in mean saccade latency of 68 ms as a result of the gap manipulation there was only a 1-ms difference in estimates of the SDT between these two conditions.

The gap manipulation clearly had a powerful effect on the latencies of the first saccades but did not affect the SDT. The reduction in saccade latency is consistent with the increase in oculomotor readiness that results from ocular disengagement (Dorris and Munoz 1995). Nevertheless the SDT appears to remain fixed. Thus in spite of the overall increase in excitability within the oculomotor system, overriding an existing movement plan costs the same amount of time. As a consequence, these results suggest that the variability in saccade latency caused by the fixation point manipulation does not stem from, or result in, a reduction in the SDT. If at least part of the SDT reflects afferent and efferent delays in the conduction of neural signals, the results suggest that such delays are the same in gap and overlap conditions. This, in turn, suggests that the pathway underlying saccade generation under both conditions is one and the same.

Experiment 2

The rejection criteria resulted in the exclusion of 2.5% of the trials. The mean latencies of the first saccades from only the double-step trials were entered into a repeated-measures ANOVA with saccade direction (up, down) and separation (30, 60, 90, 120°) as factors. The main effect of separation was significant [F(3,12) = 15.1, P < 0.001]: saccade latency increased with separation. The main effect of direction was also significant [F(1,4) = 10.2, P < 0.05]: upward saccades were initiated faster. There was a significant interaction between direction and separation [F(3,12) = 10.6, P < 0.001], which appeared to be due to a particularly pronounced directional effect at a separation of 90°. The latency advantage for upward saccades has been reported before (e.g., Honda and Findlay 1992; Ludwig and Gilchrist 2003) but is not always reliable (as in experiment 1).

As in the first experiment, SDT estimates were derived from the direction transition functions for each combination of observer, saccade direction, and angular separation. Figure 3 illustrates two of these functions from one observer. Note again the presence of a substantial number of saccades directed to an intermediate position at the short separation, creating a continuous direction transition function (A). When the separation is larger (B), saccades were directed either to the first or to the second target but tended not to fall in between the two.

Table 2 lists the SDT estimates for each individual observer. Each direction transition function was based on 51–79 saccade endpoints (range across observers and conditions). Again it appears that the SDTs were somewhat shorter for trials with a downward first step. In addition, the function appears nonmonotonic for upward first steps but increases monotonically for downward first steps. Despite these trends, a repeated-measures ANOVA showed that only the main effect of angular separation was significant [F(3,12) = 19.2, P < 0.001]. This effect is illustrated in Fig. 3C, which shows the mean SDT as a function of separation. The SDT increases with separation up to a point (90°) and remained level for the largest separation. The SDT difference between 90 and 120° was not reliable. Note that the SDT estimates at a separation of 45° from experiment 1 are consistent with the illustrated function.

Model of the oculomotor decision mechanism

Traditional models of double-step behavior incorporate a decision mechanism that is subject to random temporal variations (Becker and Jürgens 1979). Specifically, these temporal variations are assumed to follow a distribution with the same shape as the observed saccade latencies. This variability is thought to be directly related to the incidence of initial or final angle responses (i.e., saccades directed to the 1st or 2nd target, respectively). In this sense, there clearly is a strong link between saccade latency and landing position, even though the computations underlying both variables are distinct within these models. However, the mechanism that causes latency variability is ad hoc and has little explanatory power: variability in saccade latency is accounted for by simply placing exactly the same kind of variability inside a black box somewhere in the system.

Sequential sampling models are very successful in accounting for reaction time and choice behavior in situations with a limited number of response alternatives (Brown and Heathcote 2005; Luce 1986; Ratcliff and Rouder 1998; Ratcliff and Smith 2004; Smith 1995; Usher and McClelland 2001). This class of models has also been used to account for saccade latency variability (Carpenter 1981; Carpenter and Williams 1995). The fundamental idea is that saccade generation can be regarded of as a process of accumulating activity up to some critical threshold. When the activity exceeds this threshold, a saccade is generated. Saccade latency variability is accounted for by assuming the rate of accumulation varies within and/or between trials. Critically though, this variation is not of the same type as that of the saccade latencies themselves but of an a priori more plausible Gaussian nature (Carpenter 1981). Recently, Ray et al. (2004) have used this type of accumulator model to predict patterns of intersaccadic intervals in double-step saccade sequences.

Noisy accumulation of activity up to a relatively fixed threshold, and its relation to saccade latency, has been shown to occur in the frontal eye fields (Hanes and Schall 1996) and superior colliculus (Ratcliff et al. 2003). Moreover, it has been demonstrated that both within the colliculus (McPeek and Keller 2002; McPeek et al. 2003; Port and Wurtz 2003) and the frontal eye fields (Bichot et al. 2001), multiple sites associated with different potential saccade targets may be activated simultaneously. In the following text, we develop a simple stochastic accumulator model that incorporates these principles to account for variability in both saccade latency and direction.

Model description

We make the following key assumptions within the model: 1) direction is coded by a bank of units, each of which corresponds to a unique direction; 2) visual input (i.e., the onset of a target) results in an increase in activity in the unit centered on the target location, and the activity is shared with neighboring units in a graded fashion (population coding: Lee et al. 1988; McIllwain 1986; see also Ratcliff 1981 for an example of this principle in a more cognitive model of perceptual matching); 3) each unit accumulates evidence in favor of the target being in its response field (Carpenter and Williams 1995; Ratcliff et al. 2003). Thus the rate of accumulation is greatest in the unit coding the target center and drops off with increasing distance from this unit. 4) The activity of each unit is subject to passive decay or leakage (Brown and Heathcote 2005; Usher and McClelland 2001). 5) If the activity of a unit exceeds some critical threshold, a saccade is generated to the location coded by that particular unit with the latency determined by the time that the threshold was reached plus a constant efferent delay. 6) The rate of accumulation varies across target onsets and within a trial. 7) The within-trial noise is independent over time and space (i.e., across units).

These assumptions are illustrated in Fig. 4. The top half of the figure shows an array of units coding different, neighboring, directions. The onset of the first target provides the greatest input into the unit coding a saccade directed to the center of the target (x₉₀: for the purpose of illustration this is assumed to be a vertical, upward saccade), but neighboring units also receive a positive input. The bottom half of the figure shows the accumulation of activity over time for two units. On a single target trial (left), unit x₉₀ receives the largest input, which translates into the fastest rate of accumulation (black solid trace). The activity starts to rise after a constant, afferent delay, and when the threshold is reached, a saccade is generated after an additional constant, efferent delay. A neighboring unit (x_i: shown in gray) still receives some positive input and will therefore accrue some activation toward the threshold. As a result of within-trial noise, the unit that wins the race to threshold does not always have to be the one that received the greatest input. In this way, the model accounts for variability in saccade direction.

Figure 4, bottom right, illustrates the pattern of activity in a typical double-step trial. The activity is shown for the same two units that were displayed in the single-step illustration. The appearance of the first target leads to an increase in activity of unit x₉₀ in much the same way as on a single-step trial (black trace). In this example, the interstimulus interval is 100 ms (marked by the triangle on the abscissa), but the second target only affects the activity after an additional afferent delay. At that point, after 150 ms, unit x_i (gray trace) is activated more strongly than before when it received a small amount of activation associated with the first target. Unit x₉₀ will now only be activated to the extent that it is involved in coding the second target location. However, in this example, this activation is not enough to outweigh the passive decay. As a result, the activity tends to decrease somewhat after the second target's activity has reached the motor map. This decrease reduces the unit's chances of winning the race. Indeed, it is soon overtaken by unit x_i, and with the aid of the within-trial noise, this unit reaches threshold first. As a consequence, a saccade to an intermediate location is generated (i.e., global effect saccade) (Findlay 1982; Ottes et al. 1984).

Model simulations

Each unit x_i codes one direction, with i = 0–359° in integer steps. A visual input is delivered to the unit centered on the target location, x_m (where m = 90 or 270° in the experiments reported in this paper but only 90° in these simulations). Afferent and efferent delays were constants set to 50 and 25 ms, respectively, on the basis of neural recording data (reviewed in Becker 1991). The afferent delay is denoted d. Baseline and threshold activity levels were arbitrarily set to 0 and 1 respectively. The activity of each unit depends on: positive activation as a result of target onset, noise, and passive decay. Each of these factors will be described in the following text.

The mean accumulation rate of the unit centered on the target location, R_m, is assumed to be a monotonic function of the strength of the visual input: R_m = f(input). However, accumulation rate varies between trials, perhaps reflecting internal noise in the visual response to the stimulus (Smith 1995). As a result of this noise, the rate of accumulation varies in a Gaussian manner, depending on parameters µ_step and _step. Carpenter (1981) has shown that a Gaussian-distributed accumulation rate results in the typical long-tailed latency distribution. The input is shared with neighboring units as follows

(1)

Thus the mean accumulation rate of unit x_i is a function of the input to center unit x_m and the distance between units x_i and x_m. The larger the distance between the center unit and the neighboring unit, the smaller the positive activation will be. One can think of each unit accumulating a quantity that represents the likelihood of the target being in its response field (Gold and Shadlen 2001).

The activity of unit x_i is also subject to random perturbations over time during the accumulation of activity toward the threshold. Thus at each time step the activity of unit x_i is altered by noise, denoted , drawn from a zero-mean Gaussian distribution with SD _noise. This noise is independent and identically distributed across space and time. Importantly, we imposed a lower bound on the activity in that it was not allowed to go below 0 (rectification) (Usher and McClelland 2001). From Eqs. 2A–2C it is clear that without such a bound negative activity could, as a result of decay, turn into positive activation. This is computationally undesirable, and neurally implausible.

Finally, each unit's activity decays passively at a rate that depends on its activity level so far (the activity is multiplied by –). Over time, such decay reduces activity levels back to baseline. As such, this mechanism ensures that activity cannot grow without bound. As shown in Fig. 4, decay reduces the activity of those units that have a high level of activity and no longer receive an input. Without this mechanism, units centered on the first target location would almost always reach threshold first.

This model has five free parameters: µ_step, _step, _spread, _noise, and , and its dynamics are governed by the following equations (Ratcliff and Smith 2004; Usher and McClelland 2001)

(2A)

(2B)

(2C)

The subscripted numbers 1 and 2 in Eqs. 2B and 2C indicate the inputs associated with the two targets. The symbol denotes the time scale and was set to 1 ms in these simulations. To model single-step trials, only Eqs. 2A and 2B are relevant.

These stochastic, dynamic equations describe the change in activity in unit x_i, during a small time step dt. Before the visual input has reached the motor map (Eq. 2A), units integrate noise, and their activity will be near the baseline level (see activity plots in Fig. 4). Once the afferent delay has passed, units close to the one centered on the (1st) target location receive a positive input from the stimulus in addition to the noise (Eq. 2B). On double-step trials, units receive a second input to the extent that they are involved in coding the second target location. The positive activation is received after an additional afferent delay (Eq. 2C). All units have their activity reduced by a constant factor, , at every instance of time. The dynamic equations were approximated through simulations with a time step of 1 ms. By exploring the parameter space, we aimed to have the model produce realistic single-step distributions of saccade latency and direction, produce SDT estimates within the range of that observed empirically, display variations in SDT with step separation, and maintain a constant SDT despite large latency differences.

Figure 5 shows the model performance for a given set of parameters (given in the legend). A and B illustrate the predicted single-step latency and direction cumulative probability distributions (solid lines). These predictions were generated through simulation of 1,000 trials (a larger number of simulated trials did not noticeably improve the stability of the predicted pattern). The data points in both panels illustrate the empirically observed cumulative distributions for one observer (observer 5, from experiment 2, upward saccades only). In the latency domain (A), the model adequately captures the skewed right-hand tail of the distributions (as is evident from the shallower slope in the upper quantile). The observed direction variability has been shifted to align the median with 90° (the model does not produce the small and idiosyncratic directional biases displayed by human observers), and the model provides a reasonable fit to the directional variability (B). Thus a small set of assumptions and parameters is all that is needed to characterize variability in both saccade latency and direction under single-step conditions. We now turn to the model behavior under double-step conditions.

Examples of the model's direction transition functions are shown in Fig. 5, C and D. These data were generated with the same set of parameters that was used to simulate the single-step data. Each of these functions was based on simulation of 1,200 trials, 75 at each interstimulus interval (ranging from 25 to 175, in 10-ms steps). As in the empirical data (Fig. 3), global effect saccades occur when the separation between the two targets is small (Fig. 5C) but not when the separation is increased (Fig. 5D). The gray solid lines in these panels show the fits of a cumulative Gaussian to the simulated data. From these fits, SDT estimates can be derived as described in METHODS. The mean SDT estimates derived from 20 runs of this simple model are plotted in panel E (solid line). Multiple double-step runs were simulated because the SDT estimates at larger separations were more variable compared with the estimates at shorter separations. This variability is not surprising given that these estimates are derived from the more discontinuous direction transition functions. These transition functions lack the intermediate data points (i.e., global effect saccades) to tightly constrain the cumulative Gaussian fit. The gray shaded region in E represents the 95% confidence region around the mean SDT estimates from experiment 2. Clearly the model produces SDT estimates within the empirically observed range. In addition, as a result of the population coding assumption, the model naturally predicts the initial increase in SDT with separation, and the subsequent leveling off with further increases in step separation.

Having found a set of parameters that produces realistic behavior under the manipulations of experiment 2, we now examine the model behavior under simulated gap/overlap conditions (i.e., as in experiment 1). In these simulations, the parameters were fixed at the same values as those that generated Fig. 5. The separation between the two steps was 45°, just as in experiment 1. The gap condition was simulated by increasing the baseline activity level (set to 0.25). The consequences of this manipulation are illustrated in Fig. 6. A and B show activity profiles for a set of two units under double-step conditions in the same way as in Fig. 4. The black solid line represents a saccade program associated with the first target; the gray line represents a saccade program associated with the second target. The simulated interstimulus interval between the two steps is 100 ms (indicated by the triangles in A and B). In A (overlap condition), baseline activity is set to 0. On the onset of the second step, the first saccade program has not accumulated enough information to warrant saccade initiation, and it is overtaken by the second saccade program. A saccade directly to the second target is generated. In B, the baseline activity is increased to 0.25 (gap condition). The same activity profiles are shown as in A, but because of the increase in the baseline, the initial increase in activity associated with the first target is now sufficient to reach the threshold first. A saccade to the first target results, with a shorter latency (see vertical dotted lines in A and B).

View larger version (18K): [in this window] [in a new window]   FIG. 6. Simulated fixation point manipulation. A: activity profiles of 2 units centered on the 1st (black) and 2nd (gray) target locations. Conventions as in Fig. 4. Vertical dotted line, latency with which the threshold was reached; triangle, onset of the 2nd step. B: same activity profiles as in A but with elevated baseline activity. Under these conditions, there is not enough time for the 2nd saccade program to overtake the 1st. C: saccade direction as a function of interstimulus interval. Solid gray line, cumulative Gaussian fit to the simulated overlap condition data; dashed gray line, fit to the simulated gap condition data. D: direction transition functions for the gap and overlap conditions. Data are the same as in C but now aligned on saccade onset. These simulations were generated with the same parameters used to generate Fig. 5.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
During movement programming, there comes a point in time at which the system is committed to executing a certain movement. Changing the visual input at various intervals prior to movement onset can be used to probe for the last moment at which a planned movement can be altered. The double-step paradigm is an example of using this approach to quantify the dead time of the saccadic system. In the present study, we showed that the SDT remains constant despite large variations in overall saccade latency induced by a fixation point manipulation and varies substantially with the separation between the two target locations.

The SDT depends, at least partly, on delays in conveying sensory information and motor commands. The finding of a constant SDT in experiment 1 suggests that saccades undergap and overlap conditions depend on the same afferent and efferent pathways. In addition, we have no a priori reason to believe that these delays vary across the visual field in such a way as to produce the pattern of variation in SDT with separation that we obtained in experiment 2. As a result, we aimed to account for these findings in a way that does not invoke differences in the afferent and efferent delays. The functional dead time also includes a decisional component, and we have attempted to provide a possible characterization of this component in the form a stochastic accumulator model.

Our model consists of a one-dimensional motor map with units coding different saccade directions. This framework can be considered an extension of Carpenter's LATER model to multiple, independent units coding multiple possible saccade programs. In addition, it incorporates within-trial noise to help account for variability in saccade direction and also saccade latency. The presence of this particular kind of noise is a critical feature of many sequential sampling models of this type (Ratcliff and Smith 2004; Smith and Ratcliff 2004).

The model very effectively accounts for the distribution of saccade latencies and directions from the single-step trials using a small set of common parameters. Only five free parameters are needed: the spread of activation (i.e., the extent of population coding), the mean activity increase at the center of the target location, between-trial noise in the accumulation rate, within-trial noise, and a decay coefficient. These parameters also suffice to generate direction transition functions that have the appropriate shape. Global effect saccades are generated when the separation between the two target steps is small. With increasing separation, the functions become more and more discontinuous. This pattern is well established in the double-step literature (Aslin and Shea 1987) and is also shown in Fig. 3.

This model accounts well for the data observed in the current experiments. It accounts for the gap effect on saccade latency and its lack of influence on the SDT by modeling the gap manipulation as a change in the baseline level of activity (Dorris and Munoz 1995; Dorris et al. 1997). In addition, it accounts for the variation in SDT with separation as a natural result of population coding within the motor map. Importantly, the model captures these empirical effects while, possibly unrealistically, assuming constant (across trials and experimental conditions) afferent and efferent delays.

The model is clearly limited in some respects. First, there are a number of issues concerning neural plausibility. Although neurally inspired, the model cannot be directly identified with any particular neural motor map, for instance, in the frontal eye fields or superior colliculus. Nor should the activity of the units within the model be identified with the activity of single neurons. Clearly a saccade is not triggered when a single neuron reaches some threshold level of activity (Goossens and Van Opstal 2006; Lee et al. 1988), and the assumption of independent noise across units is also inconsistent with real neural coding (Shadlen and Newsome 1998). In addition, movement endpoint variability is the result of noise at various stages, including perceptual localization, saccade target selection, and movement execution. In essence, our model collapses the noise arising from all these different processes into one mechanism. The current model is fundamentally a functional model that embodies a set of neurally plausible principles (population coding, noisy build-up of activity, leakage, and rectification).

Second, the model is obviously limited in that it is restricted to only one dimension, that of saccade direction. As a result, it cannot account for any interactions between saccade direction and amplitude. Previous authors have reported that intermediate angle saccades tend to have shorter amplitudes than saccades directed to either the first or second target (Aslin and Shea 1987; Findlay and Harris 1984). We replicated this finding in the current study. Figure 7 plots the saccade amplitude as a function of saccade direction in experiment 1. The first target appeared at an angle of 90°; the second step was at 45°. For each observer, we computed a smoothed function using a Gaussian kernel with a SD of 9°, moving in 1° steps. The solid black line shows the averaged function across observers; the gray shaded region represents the 95% confidence interval around this function. The horizontal dashed line shows the predicted behavior if observers' saccades had the "correct" amplitude, regardless of direction: the average amplitude would lie on the equal eccentricity arc connecting the two targets in the display. The curved dotted line shows the predicted amplitudes if observers landed on the shortest path between the two targets (in display coordinates). Aslin and Shea (1987) found that intermediate angle saccades fell close to this straight path. Our empirical findings are also more in line with this behavior. Each observer showed the dipped function that resembled the plotted average (note the relatively tight confidence limits for the intermediate directions compared with the initial or final angle). Superimposed on this pattern is a general tendency to undershoot the required amplitude.

View larger version (18K): [in this window] [in a new window]   FIG. 7. Kernel estimate of saccade amplitude as a function of direction. The function was computed for each observer, using a Gaussian kernel with a SD of 9°. The width of the kernel was determined using a cross-validation procedure, minimizing the sum of squared errors between iteratively omitted data points and the kernel estimates for these missing data points (Hastie et al. 2001). —, average across 5 observers from experiment 1 (pooled over gap/overlap and 1st target position). Shaded region, 95% confidence interval around this average.

However, the actual saccade amplitude may be controlled by subsequent processes that occur during the saccade (Goossens and Van Opstal 2006; Optican and Quaia 2002; Quaia et al. 1999). For instance, suppose an 8° saccade is programmed to an intermediate location. The saccade is initiated, but the directional error is noted further downstream, and an on-line corrective mechanism truncates the saccade in favor of a subsequent movement to the second target. One would expect this correction to occur more frequently for intermediate angle responses, resulting in the kind of pattern displayed in Fig. 7. In this conception, saccade direction and amplitude are coded together in motor maps found throughout the saccadic system but may become decoupled as a result of internal feedback mechanisms monitoring the progression of the saccade.

The presence of this kind of corrective mechanism has been invoked to account for the curvature of saccade trajectories (McSorley et al. 2004; Quaia et al. 1999, 2000). Strongly curved saccades may be regarded as part of a continuum that also includes sequences of two saccades that are typically observed under double-step conditions (Ottes et al. 1984) or in visual search (McPeek et al. 2000). As such, a model account for the amplitude and direction interaction needs to incorporate the appropriate mechanisms that govern the dynamics of saccades themselves. Such models of saccade execution exist (e.g., Arai and Keller 2005; Optican and Quaia 2002; Quaia et al. 1999) but are themselves restricted in that they do not (aim to) account for the processes of target selection and timing of saccade initiation. Our model may be viewed as one that specifies the initially selected endpoint that constitutes the input into the saccade generating circuitry captured by other models.

Although clearly not a complete model of saccade generation, the current model offers a compelling account of the empirical data reported in this study. As a result, despite the limitations outlined in the preceding text, the model can be used to generate behavioral and neurophysiological predictions. Specifying the decision units with in the model makes explicit the complex interplay of factors involved in determining where to look under double-step conditions: interstimulus interval, strength of the signal at the first target location, signal strength at the second target location, the noisy build-up of activity associated with either of these saccade programs, the amount of overlap in the coding of the two target locations, and the rate of passive decay or leakage. From this perspective, the SDT is the minimum time needed for units coding directions away from the first target step to reliably win the race to threshold. Thus conditions that affect the accumulation rate, spread of activity, noise, and/or decay can be expected to affect the saccadic dead time.

	GRANTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
This work was supported by an equipment grant from the Wellcome Trust to I. D. Gilchrist.

	ACKNOWLEDGMENTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
We thank R. Baddeley for a suggestion to use a kernel estimate of saccade amplitude. We also thank the reviewers for thoughtful comments.

	FOOTNOTES

 
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Address for reprint requests and other correspondence: C.J.H. Ludwig, Dept. of Experimental Psychology, University of Bristol, 12a Priory Rd., Bristol BS8 1TU, UK (E-mail: c.ludwig{at}bristol.ac.uk)

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