Many common activities, like reading, scanning scenes, or searching for an inconspicuous item in a cluttered environment, entail serial movements of the eyes that shift the gaze from one object to another. Previous studies have shown that the primate brain is capable of programming sequential saccadic eye movements in parallel. Given that the onset of saccades directed to a target are unpredictable in individual trials, what prevents a saccade during parallel programming from being executed in the direction of the second target before execution of another saccade in the direction of the first target remains unclear. Using a computational model, here we demonstrate that sequential saccades inhibit each other and share the brain's limited processing resources (capacity) so that the planning of a saccade in the direction of the first target always finishes first. In this framework, the latency of a saccade increases linearly with the fraction of capacity allocated to the other saccade in the sequence, and exponentially with the duration of capacity sharing. Our study establishes a link between the dual-task paradigm and the ramp-to-threshold model of response time to identify a physiologically viable mechanism that preserves the serial order of saccades without compromising the speed of performance.

*no-step*trials, a green target appeared after a random fixation period between 300 and 800 ms at one among eight possible locations, which were 45° apart from each other at an eccentricity of 12° visual angle from the central fixation spot. In the remaining trials, called

*step*trials, a second target of red color appeared at a randomly chosen location non-adjacent to the green target on the imaginary circular array. These targets were identical in size, shape, and luminance and remained on the screen until the end of the trial. The delay between the appearances of the targets (commonly called the stimulus onset asynchrony or SOA) was randomly chosen from 50, 100, 150, and 200 ms. Subjects were instructed to shift gaze to the location of the green target in no-step trials and generate successive saccades rapidly to the location of the green and red targets in the order they appeared in step trials. No-step and step trials were randomly interleaved. To test the stability of our model, we examined two additional subjects who were not explicitly instructed to generate saccades “as quickly as possible,” unlike other fourteen subjects. Eye movements were recorded using an infrared eye tracker (ISCAN, MA, USA) at 200-Hz sampling rate. TEMPO/VIDEOSYNC (Reflective Computing, MO, USA) software generated stimulus and collected and stored the sampled eye positions in every trial. The beginning and end of each saccade were demarcated offline when the eye velocity crossed a 30°/s threshold. The latency of a saccade was calculated from the delay between the appearance of a target and the onset of the saccade directed to the target. Trials that produced blink-perturbed saccades were not considered for subsequent analyses.

*D*(e.g.,

*D*

_{1}/

*D*

_{2}in Figure 2b). In Equation 1, substituting SOA by (RT1′ − A2 −

*D*) and (A1 + B1) by RT1, we get

*D*and (A2 + B2) by RT2, we get

*D*> 0) of capacity sharing between saccades.

*parallel processing time*) is the time of onset of saccade to the first target relative to the time of appearance of the second target (e.g., PPT

_{1}/PPT

_{2}in Figure 2b), and

*r*(>0) is a constant. Substituting (PPT − A2) by

*D*in the above equations, we get

^{ r×D }) term on the right-hand side of each equation.

*u*

_{1}and

*u*

_{2}are the signals corresponding to saccades in the direction of the first and second targets, respectively (Figure 3, left). The rate of increase of

*u*

_{1}and

*u*

_{2}remain fixed in a trial but vary across trials. In light of previous findings that while the likelihood ratio merely provides an instantaneous evidence in support of a hypothesis between alternatives (Carpenter & Williams, 1995), the log-likelihood ratio serves as a natural currency for making a decision (Bogacz, Usher, Zhang, & McClelland, 2007; Wald & Wolfowitz, 1948), we defined a

*Confidence Index*(CI) as log

*u*

_{1},

*u*

_{2}> 0, the first saccade is more likely to be directed to the first target if

*u*

_{1}=

*u*

_{2}).

*Degree of Concurrency*(DoC) as the duration of overlap between the planning stages of consecutive saccades in a step trial divided by the average duration of saccade planning in no-step trials, i.e., DoC =

*D*= B1, and DoC ≤ 0 implies serial processing of saccades. Replacing

*D*by

*n*(>0) is a proportionality constant. We assumed that

*r*

_{1}is the rate of saccade planning to a singleton target in a no-step trial. Because no-step and step trials were randomly interleaved, subjects were expected to start planning a saccade to the first target at the same rate (

*r*

_{1}) in the step trial unaware that a second target would be forthcoming. Replacing the duration (B1) of saccade planning in the no-step trial by

*r*(= 0.0033 ms

^{−1}) in Equations 7 and 8 from the fit (Figure 2d). Because

*r*is related to the rate (

*r*

_{1}) of planning the saccade in isolation (no-step trial). Hence e

^{ r×D }∝

^{ r×D }in Equations 7 and 8 by

*m*(>0) is a proportionality constant, we get

*D*are

*D*, the respective saccade latency increases by

*u*

_{1}) and second (

*u*

_{2}) targets is reduced at every time step by the amount given on the right-hand side of Equations 14 and 15, respectively. When the rate of accumulation of information to generate a saccade to the second target is considerably higher than that to the first target, exertion of an inhibitory control may be a means to prevent the signal rising to generate the second saccade from reaching threshold before the first saccade is generated. The purpose of such inhibitory control is to dampen the ascent of the signal thereby elongating the planning stage of the saccade to the second target. This will cause the duration (

*D*) of capacity sharing to increase, which in turn will increase the latency of the saccade to the first target (RT1′). Such an increase in RT1′ will demand further slowdown of planning the saccade to the second target, and the interaction between planning stages will continue until the signal rising to generate a saccade to the first target reaches the threshold. To test the idea that both mutual inhibition and capacity sharing are critical to preserve the order of execution of consecutive saccades, we designed a model using Matlab Simulink software with an embedded S-Function written in C language, which consists of two processing stages, an accumulation stage and an attenuation stage (Figure 3).

*t*are denoted by

*u*

_{1,t }{= (

*r*

_{1}×

*t*)}, and

*u*

_{2,t }{=

*r*

_{2}× (

*t*− SOA)}, where

*r*

_{1}and

*r*

_{2}are independently sampled from two sets of normally distributed numbers with means that correspond to the mean rate of planning saccades to the first and second targets; (4) the variance of the rate of planning saccades in the direction of two targets are the same. (5) Each of the two attenuators at the next stage receives signals

*u*

_{1,t }and

*u*

_{2,t }from both accumulators at time

*t*+ 1, where these input signals inhibit each other and generate output signals

*X*

_{1,t+1}and

*X*

_{2,t+1}, respectively. In the absence of any interaction, the output from an attenuator merely reflects the activity of its corresponding accumulator. (7) The strength of inhibition exerted by each decision signal on another depends on the proportion of capacity they share. (8) Because the firing rate of a neuron cannot be negative, our model prevents

*X*

_{1}and

*X*

_{2}from falling below zero. (9) The interaction continues until either

*X*

_{1}or

*X*

_{2}reaches the threshold normalized to 1 to elicit a saccade in the direction of the corresponding target. During simulation of the model,

*X*

_{1,t }and

*X*

_{2,t }were computed as

*m*and

*n*) of the model that are independent of any idiosyncratic behavior of individual subjects. No-step trials that yielded identical (±10

^{−4}ms

^{−1}) reciprocals of saccade latencies were grouped together to estimate the mean (

*μ*) and standard deviation (

*σ*) of rate of saccade processing (visual processing and motor planning) from a fit by a Gaussian function of the form

*f*(

*x*) =

*μ*(= 0.005) and

*σ*(= 0.00095) were the coefficients of the fit (Figure 2c). In this analysis, we did not include trials that produced saccades of latency shorter than 100 ms. The goodness of fit in terms of

*R*

^{2}(= 0.97), which was close to the maximum (= 1.0), suggested that the Linear Approach to Threshold with Ergodic Rate (LATER) model that assumes a Gaussian distribution of the rate of saccade processing can emulate saccade latencies in no-step trials (Reddi & Carpenter, 2000). A two-sample Kolmogorov–Smirnov test showed that the distribution of the reciprocal of saccade latency in 2404 correct no-step trials was indifferent (

*P*= 0.178) from that of equal number of random samples from a normal distribution with mean (±

*SD*) of (0.005 ± 0.00095) ms

^{−1}.

*m*(= 17.62) and

*n*(= 4) were optimized by minimizing the difference between expected saccade latencies and simulated saccade latencies in 1000 step trials at SOA = 50 ms, using “Least Square Method” in “Parameter Estimation” tool of Simulink (The Mathworks) software. We derived the expected latencies of saccades directed to the first and second targets using SP (= 0.655),

*r*(= 0.0033 ms

^{−1}), RT1 (= 208 ms), and RT2 (= 318 ms), as obtained from exponential fits (Figure 2d), and

*D*

*D*) of overlap between planning stages of consecutive saccades was equal to the difference between the duration of planning a saccade to the first target and SOA. The rate (

*r*

_{1}) of planning a saccade to the first target varied from trial to trial and was calculated, first, by subtracting 70 ms from the expected duration of saccade processing, then calculating the reciprocal of the difference. The reciprocal of a value sampled from a normal distribution with mean (= 0.005 ms

^{−1}) and standard deviation (= 0.00095 ms

^{−1}) provided the expected duration of saccade processing. The same value of

*r*

_{1}was used in the model as the rate of increase of

*u*

_{1}. The rate (

*r*

_{2}) of increase of

*u*

_{2}in each simulated step trial was sampled from a distribution of mean (=

^{−1}) = 0.004 ms

^{−1}and standard deviation = 0.00095 ms

^{−1}.

*t*= 0 when an integrator started accumulating information. Another integrator started accumulating information after SOA. A pair of attenuators individually received output from both integrators and attenuated them. When the output signals of attenuators reached the threshold, the simulator registered the onset of saccades in the corresponding directions. The interval between the time when an integrator started accumulation and the corresponding attenuated signal reached the threshold was considered as the duration of the planning stage of a saccade. We calculated saccade latency by adding a visual processing delay of 70 ms to the duration of the planning stage and an additional 50 ms as the duration of saccade execution to calculate the time of saccade end.

Parameter | Value |
---|---|

μ _{1} (mean 1st saccade processing rate) | 0.005 ms^{−1} |

σ (standard deviation of 1st saccade processing rate) | 0.00095 ms^{−1} |

r _{1} (rate of 1st saccade planning) | r _{1} = 1 1 R − 70 , R was sampled from a normal distribution with mean (=μ _{1}) and standard deviation (=σ) |

μ _{2} (mean rate of 2nd saccade planning) | 0.004 ms^{−1} |

r _{2} (rate of 2nd saccade planning) | Sampled from a normal distribution with mean (= μ _{2}) and standard deviation (=σ) |

SP (sharing proportion) | 0.655 |

m | 17.62 |

n | 4 |

*SD*) skewness, first: 1.4 ± 0.68; second: 0.83 ± 0.87] as shown in Figure 1b (thin lines), two trends were observed across the population of subjects. First, the duration of fixation between consecutive saccades or intersaccadic interval (ISI) decreased with increasing parallel processing time (PPT) or the duration of overlap between processing stages of saccades [mean (±

*SD*) linear regression slope = −0.29 (±0.23),

*P*< 0.05]. Second, the latency of consecutive saccades increased as the stimulus onset asynchrony (SOA) decreased [mean (±

*SD*) linear regression slope, first: −0.15 ± 0.21; second: −0.59 ± 0.36;

*P*< 0.05]. In the aggregated data across subjects, a decrease in ISI down to a limit at longer PPTs (Figure 1c) and an increase in saccade latency with decreasing SOA (Figure 1d) indicated a capacity-limited mechanism of saccade programming.

_{2}(>SOA

_{1}), processing stages of saccades overlap for a relatively shorter period of time (PPT

_{2}< PPT

_{1}). Because the ongoing preparation of a saccade does not prevent preparation of the subsequent saccade when each saccade in a sequence uses independent pool of resources (Figure 2a, left), latencies of consecutive saccades should remain invariant across all parallel processing times (PPTs) as shown in Figure 2a (right). In the capacity-limited condition, planning stages of consecutive saccades share capacity for an interval of time denoted by

*D*

_{1}, when the second target appears after SOA

_{1}(Figure 2b, top). As a result, B1 increases to B1′, and B2 increases to B2′. Accordingly, RT1 increases to RT1′ (= A1 + B1′) and RT2 increases to RT2′ (= A2 + B2′). When the second target appears after SOA

_{2}(>SOA

_{1}), the processing stages of saccades overlap for a relatively shorter period of time (PPT

_{2}< PPT

_{1}), and capacity is shared for a shorter duration (

*D*

_{2}<

*D*

_{1}; Figure 2b, bottom). As a result, RT1 increases to RT1″ (= A1 + B1″) and RT2 increases to RT2″ (= A2 + B2″), where RT1 < RT1″ < RT1′ and RT2 < RT2″ < RT2′. The capacity-limited model predicts that the latency of saccades directed to the first and second targets will increase with PPT (Figure 2b, right).

*D*in Equations 3 and 4 by (PPT − A2), where we considered an average visual delay of 70 ms in the frontal eye field as the duration of visual processing (A2) of the second target. From the linear fit (

*R*

^{2}= 0.94; green dotted line) of the data by RT1′ = (1 − SP) × (PPT − A2) + RT1, we obtained RT1 = 209.1 ms and SP = 0.349. To account for the non-linearity in the plot, we also fitted RT1′ by RT1′ = {e

^{ r×(PPT−A2)}} × (1 − SP) × (PPT − A2) + RT1, where

*r*> 0. From the exponential fit (

*R*

^{2}= 0.99; green solid line), we obtained RT1 = 208.5 ms, SP = 0.655, and

*r*= 0.0033 ms

^{−1}. Note that in correct no-step trials, the mean (±

*SD*) saccade latency across the population of subjects was 205 (±40) ms. Subsequently, we fitted the plot of RT2′ versus PPT by RT2′ = SP × (PPT − A2) + RT2, using SP (= 0.349) as obtained from the linear fit of RT1′, and by RT2′ = {e

^{ r×(PPT−A2)}} × SP × (PPT − A2) + RT2, using SP (= 0.655) and

*r*(= 0.0033 ms

^{−1}) as obtained from the exponential fit of RT1′. We obtained RT2 = 398 ms and 318.4 ms from the linear (red dotted line) and exponential (red solid line) fits, respectively. The exponential function fitted the data better than the linear function (

*R*

^{2}: linear = 0.52, exponential = 0.93) and closely approximated the average (±

*SD*) latency (328 ± 95 ms) of saccades to the second target that appeared maximum 70 ms prior to the onset of the first saccades (i.e., 0 < PPT ≤ 70).

*SD*) of 0.005 (±0.00095) ms

^{−1}. Simulation of 2000 step trials at the stimulus onset asynchrony (SOA) of 50 and 200 ms each showed that saccades targeted to the second target were executed first in 17.35% and 9% trials, respectively.

*SD*) simulated latency (first: 213 ± 58 ms; second: 373 ± 112 ms) was not different from the corresponding mean (±

*SD*) observed latency (first: 225 ± 65 ms; second: 371 ± 112 ms). On average (±

*SD*), a total of 336 (±64) correct step trials contributed data to calculate the mean saccade latency at each SOA. The mean (±

*SD*) observed latency of the first and second saccades significantly (

*P*< 0.001) increased from 219 (±63) ms to 252 (±71) ms and from 342 (±90) ms to 445 (±121) ms, respectively, as SOA decreased from 200 to 50 ms. Similarly, the mean (±

*SD*) simulated latency of the first and second saccades significantly (

*P*< 0.001) increased from 207 (±41) ms to 232 (±82) ms and from 343 (±76) ms to 438 (±147) ms, respectively, as SOA decreased from 200 to 50 ms (Figure 4b). Step trials with identical (±10 ms) PPTs were grouped together to plot the mean ISI against the mean PPT for each bin spanning 20 ms (Figure 4c). Each data point was weighted by the reciprocal of the standard deviation of the corresponding mean ISI for the fit by a function of the form

*f*(PPT) =

*a*× e

^{−b×PPT}+

*c*. Slopes of the exponential fits (

*R*

^{2}: simulation = 0.92, experiment = 0.75) reached above −0.1 at PPT of 134 ms (observed) and 157 ms (simulated) and leveled off {ISI

_{min}= lim

_{PPT→∞}

*f*(PPT) =

*c*} at ISI of 196 ms (observed) and 190 ms (simulated). Simulated saccades directed to the second target started at least 67 ms and on average (±

*SD*) 237 (±93) ms after the end of saccades directed to the first target, indicating that the order of sequential saccades was always maintained.

*P*< 0.001 (KR),

*P*= 0.21 (KV)]. Saccades directed to the second target were progressively postponed as SOA decreased [regression slope: −0.63 (KR), −0.80 (KV); correlation coefficient: −0.97 (KR), −0.99 (KV);

*P*< 0.05]. In contrast, none of the subjects postponed saccades directed to the first target when SOA decreased.

*m*(= 17.62) and

*n*(= 4), we compared the simulated data with the data recorded from individual subjects. We measured the sharing proportion (SP), the mean rate of processing saccades in the direction of the first target (

*μ*

_{1}), the mean rate of planning saccades in the direction of the second target (

*μ*

_{2}), and the variance (

*σ*) in the mean rate of saccade processing for each subject following the techniques described in the Finding parameters of the model section. Values of these parameters are shown in Table 2. Both subjects allocated about 18% (i.e., 1−SP) of total capacity for planning saccades to the second target in comparison to 34.5% in previous cases when other subjects were encouraged to speed up saccades to the second target. Figures 5a and 5b (right panels) show the plots of the mean simulated ISI against the mean simulated PPT (top) in ten bins, each spanning 15 ms on the abscissa, and the mean RT of the simulated first and second saccades against SOA (bottom) for subjects KR and KV, respectively. Our model successfully simulated the behavior of subject KR who exhibited some degree of parallel processing of sequential saccades and subject KV as well who apparently processed consecutive saccades in series [regression slope: −0.27 (KR), −0.09 (KV); correlation coefficient: −0.09 (KR), −0.03 (KV);

*P*< 0.05 (KR),

*P*= 0.49 (KV)]. Saccades directed to the second target were progressively postponed as SOA decreased [regression slope: −0.60 (KR), −0.82 (KV); correlation coefficient: −0.99;

*P*< 0.05]. In contrast, the simulated latency, like the observed latency, of saccades directed to the first target did not show correlation with SOA.

Subject | Parameter | |||
---|---|---|---|---|

SP | μ _{1} (ms^{−1}) | μ _{2} (ms^{−1}) | σ (ms^{−1}) | |

KR | 0.8155 | 0.0056 | 0.0030 | 0.0012 |

KV | 0.8179 | 0.0061 | 0.0044 | 0.0013 |

*m*= 0), while other parameters of the model are kept fixed (Figure 6b). Note that the order of saccade execution is switched in less than 50% of total simulated trials even when two targets appear simultaneously (SOA = 0 ms), because the average rate of planning saccades to the second target was naturally slower than that to the first target (Table 1). The attenuated signals

*X*

_{1}and

*X*

_{2}in a simulated step trial and that averaged across the trials are shown in Figures 6c and 6d, respectively. The order of saccades in a sequence may be switched if a saccade to the second target is planned independently (

*m*= 0) at a relatively faster speed. Our model suggests that the order of consecutive saccades is maintained by mutual inhibition (

*m*> 0) between decision signals rising to a threshold to generate saccades, irrespective of the speed of their rise (Figure 6c). Figure 6d shows the output of each attenuator averaged across 1000 trials and aligned on the beginning of the simulated first saccades, which suggests that the rate of presaccadic increase in the activity of neurons related to saccade planning is slower when the planning stage of a saccade temporally overlaps with that of another saccade, in comparison to when a saccade is planned in isolation. A steady rise of the decision signal corresponding to the first saccade slowly halts after the appearance of the second target until ∼100 ms prior to the saccade onset due to an inhibitory interaction.

*μ*

_{1,KR}= 0.0056 and

*μ*

_{1,KV}= 0.0061 compared to

*μ*

_{1,Population}= 0.005) effectively reduced the duration of capacity sharing. Thus, the magnitude of postponement of first saccades decreased as well, resulting in a putative processing bottleneck.

_{1}/SOA

_{2}in Figure 2b, top), planning stages of consecutive saccades overlap (SOA + A2) after the onset of the first target and planning of both saccades continue simultaneously until the onset of a saccade. Suppose that the fraction of capacity allocated for planning a saccade in the direction of the first target, denoted by the

*sharing proportion*, is SP (0 < SP < 1). The remaining capacity (1 − SP) is assumed to be allocated for planning another saccade in the direction of the second target. A saccade in the direction of the first target is planned with full capacity (SP = 1) for a period of (SOA + A2 − A1) until planning of a saccade to the second target begins. During this period, the oculomotor system finishes {(SOA + A2 − A1) × SP} or (SOA + A2 − A1) part of planning of a saccade directed to the first target. The remaining part of the planning, which is {B1 − (SOA + A2 − A1)}, takes time equal to

*D*=

*D*× (1 − SP)] or

*Onderzoekingen gedaan in het Physiologisch Laboratorium der Utrechtsche Hoogeschool,*Tweede reeks, 1868–1869, II, 92–120)