Saccade vigor reflects the rise of decision variables during deliberation

Summary

During deliberation, as we quietly consider our options, the neural activities representing the decision variables that reflect the goodness of each option rise in various regions of the cerebral cortex^1–7. If the options are depicted visually, we make saccades, focusing gaze on each option. Do the kinematics of these saccades reflect the state of the decision variables? To test this idea, we engaged human participants in a decision-making task in which they considered two effortful options that required walking across various distance and inclines. As they deliberated, they made saccades between the symbolic representations of their options. These deliberation period saccades had no bearing on the effort they would later expend, yet saccade velocities increased gradually and differentially: the rate of rise was faster for saccades toward the option that they later indicated as their choice. Indeed, the rate of rise encoded the difference in the subjective value of the two options. Importantly, the participants did not reveal their choice at the conclusion of deliberation, but rather waited during a delay period, and finally expressed their choice by making another saccade. Remarkably, vigor for this saccade dropped to baseline and no longer encoded subjective value. Thus, saccade vigor appeared to provide a real-time window to the otherwise hidden process of option evaluation during deliberation.

eTOC

During deliberation, primates make saccades between options, and the neural activities representing the goodness of each option rise in various cortical regions. Korbisch et. al. demonstrate that saccade velocities also rise during deliberation: the rate of rise and the final difference in peak velocity reflects the difference in subjective value.

Results

Imagine a robot moving its camera to gather information from its environment and is considering two possible actions. The choice that this robot makes, and even its decision-making algorithm, may be indistinguishable from a biological brain that consumes the same visual information. However, in one respect our actions might differ: during deliberation, our brain may move our eyes in a way that reflects the goodness that we assign to each option, thus providing a real-time window to how we made our choice.

Our reasoning for this conjecture is that during perceptual or value-based decision-making, our options are often presented visually, and as we deliberate, neurons in the posterior parietal cortex ^1–3, the frontal eye field ^4,5, and the motor cortex ^6,7, increase their activities to encode the real-time evidence that has been accumulated regarding the goodness of each option. Some of these regions project to the superior colliculus, which in turn directs the machinery that moves our gaze. Thus, it seems possible that during deliberation, as we look back and forth between our options, saccade vigor may provide a real-time window to the process of thinking that underlies decision-making.

To evaluate this possibility, we considered decisions that involved effortful options: people deliberated options that involved walking across various inclines and distances (Figure 1A). As they quietly thought about their options, they made saccades between the symbolic representations, but they did not reveal their choice at the end of deliberation. Rather, they waited during a delay period, and then indicated their choice by making one final saccade (Figure 1B). In this way, we separated movements that were made for the purpose of information acquisition (deliberation period saccades), movements that revealed the result of deliberation (decision period saccade), and movements that executed the choice (walking).

Figure 1. Saccade vigor during deliberation.

A) Outline of experimental protocol. The experiment was comprised of three parts: familiarization, which involved repeated exposure to walking at various inclines on a treadmill; decision-making, where participants’ eye movements were recorded as choices were made; and realization, when a random subset of previous choices were realized on the treadmill. B) Phases of a single trial. Participants were given time to deliberate their choices (maximum of 4.5 s), after which, following a delay period, made a saccade to indicate that choice. On each trial, one choice, defined as the reference option, was always either 4%/6 min or 5%/5.5 min C) Example trajectory of eye movements over the course of deliberation. The two graphics representing the movement options were placed at approximately 25% and 75% of the horizontal distance of the computer monitor, a 15-degree visual angle separation on average across participants. Once a choice was in mind, participants pressed the keyboard’s spacebar. Each dot represents a gaze localization sampled at 1000 Hz. D) Example participant S0 demonstrated choice sensitivity to both incline and duration differences. Grey regions represented 95% CI for logistic fit. Note that the nonmonotonic increase results from distribution of choice options. E) Saccade vigor profiles for example participant. Left Sub-selection of saccades which horizontally traversed the computer monitor (see Figure 1C). Peak velocity was greater towards the option ultimately chosen during the decision phase (preferred option). Right Average velocity traces for participant S0 for saccades which traversed the monitor between choice option icons (15±1°). F) Left Saccade vigor during the deliberation period for participant S0. Right Saccade vigor across all participants. G) Left The difference in vigor between the two options increased over the course of deliberation. Right Highlighting the difference in vigor for only the final two saccades. Individual participant vigor differences are represented as colored points. Error bars are SEM. See also Figure S1.

Saccade vigor increased during deliberation

During a typical trial, the participants made saccades from one symbol to the other (Figure 1C). This specific participant had a propensity towards choosing options with lower relative inclines and durations (Figure 1D). Intriguingly, during the deliberation period the saccade velocities reflected this preference. Peak velocity as a function of amplitude was greater when the saccade was towards the option that was ultimately chosen (Figure 1E, left panel; see Figure S1 for additional participant). To quantify this effect we defined saccade vigor as the ratio between the actual peak velocity and the participant’s expected peak velocity, normalized by amplitude ^8,9.

For this participant, vigor near the end of the deliberation period appeared larger if the movement was directed towards the ultimately chosen option (preferred) (Figure 1F, left panel). Across all participants, saccade vigor during deliberation gradually increased by 4.93±1.013% (LMM, p=0.00107). The rate of increase was faster for saccades directed towards the preferred option (β=6.726e-3 ± 6.66e-3 [±95% C.I.], p=0.0478) (Figure 1F, right panel). When we considered the entire deliberation period, saccade vigor towards the preferred option was significantly greater (β=1.29e-2±5.29e-3, p=1.84e-6) (Figure 1G, left panel). This effect of preference on vigor was particularly prominent in the final two saccades of the deliberation period (t₍₂₁₎=−3.165 p=0.00467; Figure 1G, right panel).

The rise in saccade vigor resembled variables that are commonly employed in models of decision-making ^10–13. When the decision is easy and the deliberation period is brief, the decision-variable toward the preferred option rises faster than when the decision is hard. Does saccade vigor follow this property?

Indeed, saccade vigor increased more rapidly when the deliberation period was short (Figure 2A, left subplot, time × short/long interaction; β=−1.95e-2±8.66e-3, p=1.03e-5). Remarkably, the faster rate of increase in vigor was present in the short deliberation trials even when we considered only those saccades that were made toward the reference option (Figure 2A, right subplot; β=−2.33e-2±1.18e-2, p=1.2e-4). Similarly, in short deliberation trials the saccades that were made toward the preferred option exhibited a rise in vigor that was faster than compared to long deliberation trials (Figure 2B, left subplot; β=−2.19e-2±7.43e-3, p=7.6e-9). Even saccades to the nonpreferred option exhibited a rise in vigor that was marginally faster on shorter trials than longer ones (Figure 2B, right subplot; β=−2.41e-2±2.25e-2, p=0.036).

Figure 2. Rate of rise in vigor reflects preference.

A) Rate of change in vigor over the course of deliberation differed between short and long deliberations. Short deliberations were defined as trials shorter than participants’ median time, and long deliberations were greater than this time. Saccade vigor data are plotted against time normalized as percentage of participant median deliberation. Left Data averaged across all saccades. Right Data averaged only for saccades directed towards the reference option. Even though the absolute value did not change, rate of vigor change when directed towards the reference significantly differed between short and long deliberations. B) Data are presented separated by preference (whether directed towards option ultimately chosen or no). Rate of rise in vigor also differed between short and long for preferred saccades (Right), and to a lesser extent for nonpreferred saccades (Left). Linear trendlines represent regression analysis of aggregate data. Error bars and shaded regions represent SEM.

In summary, saccade vigor increased during deliberation. The rate of rise was faster for saccades that were directed toward the preferred option and was faster in trials in which participants ended the deliberation earlier.

Saccade vigor indicated degree of preference

Is saccade vigor a binary measure of preference, or a continuous indicator of degree of preference? To answer this question, we tried to quantify the subjective cost, or the utility, that each participant assigned to each option. This required fitting a model to the choices that the participants had made.

We began by checking whether the choices were rational: did the participants prefer the option that was less effortful? Indeed, as the difference between the incline of the alternative option increased with respect to the reference, the participants were more likely to choose the option with the lower incline (GLMM, β=0.713±0.0879, p<2e-16, Figure 3A). Similarly, the participants were more likely to choose the option with lower duration (β=1.38±0.096, p<2e-16, Figure 3A). Next, in order to quantify the subjective cost that each participant assigned to an option, we calculated the expected metabolic cost ^14,15 in units of Joules for a person of mass m to walk a duration t on an incline g with average velocity v:

$$e=\left(1.5m+m\left(1.5v^2+0.35vg\right)\right)t$$ (1)

Figure 3. Choices and deliberation time.

A) Participants showed consistent sensitivities to incline and duration. Shaded region represents 95% CI of logistic fit. B) Capture rate model predictions for participant choices (dotted line). Solid line is the mean across participants. Dots are individual participants. C) Deliberation time varied with capture rate difference. Across participants, average deliberation time was longer for smaller capture rate differences. Error bars are SEM.

In this equation, all parameters are known thus there are no parameters to fit. Next, we estimated the utility of an option via the difference between the expected reward and energetic cost, divided by duration of the walk:

$$J=\frac{\alpha -e}{t}$$ (2)

The above expression represents the capture rate or rate of net reward ¹⁶, a measure of goodness of the option often used to describe foraging behavior and decision making. The capture rate correctly predicted 86.6±1.2% (mean±SEM) of choices that the participants made (Figure 3B). Alternative models were tested based on linear combinations of incline and duration or based on effort cost alone. All alternative models performed either similarly to capture rate or worse (see Methods, Table S1). As no model clearly outperformed the rest, we moved forward with the model that explained choices with the most superior performance (capture rate, Eq. 2).

Equipped with this quantitative description of utility for each option (Eq. 2), we tested it with the behavioral metric of deliberation time ^17–19. We found that deliberation time varied robustly with the capture rate difference of the two options (GLMM, β = −6.032±2.116, p=2.4e-8; Figure 3C). Deliberation was longest for capture rate differences near 0 J/kg·s and decreased as options differed in value; thus, the greater the difference in the capture rate of the two options, the more quickly participants were able to arrive at their decision.

Individual fixation durations also varied with differences in utility.

Capture rate provided us with a participant-specific measure of the utility of each option, and the difference between the utility of each option provided a measure of the degree of preference. Armed with these quantities, we returned to our original question: does saccade vigor signal which option is preferred, or does it also signal how much one option is preferred over the other?

We examined saccade vigor on easier trials compared to more difficult ones, based on capture rate difference, to determine whether the saccades revealed degree of preference. In easy trials, saccade vigor appeared to increase at a faster rate than in the hard trials (Figure 4A). Across all participants, saccade vigor increased during deliberation (main effect; β=2.026e-2±7.434e-3, p=3.43e-5). However, the rate of increase in vigor (the slope of the relationship with time), was influenced by the capture rate differential of the two options (interaction term |rate| × time; β=3.281e-4±2.38e-4, p=0.00688). Thus, the greater the difference in the utility of the two options, i.e., the easier the decision, the greater the change in saccade vigor during deliberation (Figure 4B, left). This effect was particularly strong for saccades directed towards the preferred option (Figure 4B, right panel; |rate| × time × preference; β=1.361e-3 ± 1.159e-3, p=0.0214). Thus, the greater the degree of preference, the greater the rate at which saccade vigor increased over the deliberation period.

Figure 4. Saccade vigor during deliberation reflects degree of preference.

A) Participant S0 showed a difference in vigor rate increase during deliberation dependent on capture rate difference. Left For trials with the smallest capture rate differences (i.e. more difficult choices), vigor change was relatively flat. Right Those trials with greater differences in capture rate had steeper rates of vigor increase. B) Left For all participants, the rate of saccade vigor increase (1/s) over the course of deliberation significantly increased with absolute capture rate difference. Right This effect is greater towards preferred options. The smallest and largest 50% of capture rate differences per participant were aggregated and the average rate of change for both preferred and nonpreferred saccades were calculated. Points are the per-subject calculated slopes. C) Saccade vigor towards reference and alternative options significantly varied with capture rate difference. As the relative utility of an option increased, vigor towards that option increased. D) The difference in vigor towards the reference option and alternative option increased with capture rate difference. E) Reaction time changed with capture rate difference. With increasing capture rate difference, reaction times decreased (p=0.002). Participants’ reaction times were normalized to median values. F) Vigor of the decision saccade was significantly less than the preceding final saccade in the deliberation phase. G) Participants who deliberated more quickly made relatively faster saccades. Error bars are SEM, shaded region are 95% CI.

We next considered saccades towards the reference or alternative options, rather than towards the preferred or nonpreferred options. This has the effect of making changes in relative utility linear. We found the average vigor of saccades towards the reference increased with increasing relative capture rate (β=4.785e-3±2.017e-3, p=3.36e-6; Figure 4C). Saccade vigor towards the alternative option decreased with decreasing relative utility (β=−2.38e-3±1.435, p=0.0011). Thus, when one option had a larger capture rate than the other, saccade vigor towards that option was also greater.

We further examined whether the difference in saccade vigor between the reference and alternative options reflected the underlying difference in their utilities, i.e., whether the difference in vigor reflected an individual’s degree of preference for one option over the other. Indeed, we found a significant relationship between the relative vigor and relative utility (ρ=0.774, p=0.02422; Figure 4D).

In summary, saccade vigor increased as participants deliberated between two effortful options. As the choice became easier, vigor increased at a faster rate. Near the end of the deliberation period, the differential in vigor for saccades directed toward the preferred option versus the other option tracked the difference utility. Thus, the rate of increase in saccade vigor and relative vigor of the final saccades appeared to reveal the degree of underlying preference.

Reaction time of the decision saccade, but not its vigor, varied with preference

Once participants indicated their deliberation had concluded (press of a key), they fixated on a centrally-located stimulus. The later appearance of two circles on either side of the screen cued them to make a saccade and indicate their choice. The time it took to initiate this saccade, i.e., reaction time, was correlated with the capture rate difference between the options (β=−1.054±0.657, p=0.00168). The greater the capture rate difference, the shorter the reaction time (Figure 4E).

However, saccade vigor dropped significantly from the end of the deliberation phase to the decision phase (β = 0.046±0.02, p=0.000162; Figure 4F), and increasingly so over the course of the experiment (GLMM, phase × trial interaction, β = −0.03399±0.012, p=6.45e-8). Furthermore, unlike the saccades during the deliberation period, the vigor of the decision saccade did not vary with capture rate difference (β=0.0699± 0.223, p=0.538).

Thus, saccade vigor rose during deliberation but dropped back to near baseline when deliberation ended. Saccade vigor during deliberation was sensitive to the difference in the utilities of the two options, but not when the saccade signaled the choice. The easier the decision, the shorter the reaction-time of this decision saccade.

Individuals who had greater saccade vigor tended to have shorter deliberation periods

Participants who had greater than average saccade vigor allocated significantly less time for deliberation (t₍₂₀₎=−3.068, ρ=−0.566, p=0.00607; Figure 4G). In other words, participants who tended to make faster saccades also tended to deliberate for a shorter period.

Discussion

Thinking is rarely subject to measurement because nothing is revealed until we act. Decision-making offers a framework for examining the process of thinking because it dissociates planning from execution: we think about what we might do, evaluate our options, and reach a decision, all the while enjoying the freedom to ignore the immediacy of the chosen action briefly, or even indefinitely. Here, our results show that when the options are presented as visual symbols, the thought processes that consider those options leave their impressions on saccade vigor.

But how might this come about? Vigor of saccades is closely linked to activities of neurons in the superior colliculus ²⁰, which in turn receives information regarding the utility of the visual stimulus from the frontal eye field, parietal cortex, and the basal ganglia. This information is conveyed via a balance of excitation from the cerebral cortex, and inhibition from the basal ganglia. When the stimulus has a greater subjective value, the inhibition that substantia nigra reticulata (SNr) neurons impose on the superior colliculus is suppressed ²¹, while the excitation that the frontal eye field ²² and the parietal cortex neurons ²³ impose is increased. Neurons in these regions show ramping activity during the deliberation period, reflecting accumulation of evidence regarding the goodness of choosing one option or the other ^1,12. Perhaps it is this ramping activity that regulates the burst in the superior colliculus neurons, which in turn alters saccade vigor.

Critically, we found that saccade vigor was not always a reflection of the decision variables. Despite the fact that the decision variables must have been maintained during the delay period, saccade vigor dropped following the end of deliberation. That is, saccade vigor reflected the putative decision variables only when the eyes were engaged in acquiring information relevant for deliberation.

Our design contrasted with previous studies in which the act of making a saccade was itself associated with a reward or punishment ^24–27. Here, saccades were a natural part of acquiring information. The fact that the options indicated future expenditure of effort, rather than acquisition of reward, suggests that saccade vigor may vary positively with subjective value regard less of whether that value is associated with reward or effort.

We found that individuals who tended to make saccades with greater vigor also tended to make decisions faster. This builds on previous findings where individuals who made faster saccades were also less willing to wait for reward ²⁸, those who reached with greater vigor tended to exhibit a greater cost of time ²⁹, and those who preferred to run to a target (rather than walk), tended to run with greater vigor ³⁰. This potential link between individual differences in decision-making and movement control remains a fascinating area of further study.

Participants displayed a slight choice bias for the reference option, which was also present in saccade vigor. The bias may be related to exposure-dependent effects on subjective value assignment, which could be potentially explained by familiarity possessing intrinsic value³¹. Several studies have shown a relationship between increased gaze-duration and propensity for selection ^32–36. Here, participants were exposed to one of the two reference images on every trial, and thus experienced greater exposure to the reference options. Krajbich et al. proposed that attention, measured as fixation duration, biases choice ³⁷. Indeed, we observed that participants tended to gaze for longer durations and were more likely to look last at the reference option. Thus, the greater exposure coupled with the greater gaze duration within a trial may have contributed to the bias toward the reference options.

In summary, during deliberation, the vigor of eye movements increased, but at a greater rate toward the option that had the greater utility. The difference in the rate of rise in vigor reflected the degree of preference. Intriguingly, this relationship between vigor and utility was present only during deliberation when the eyes were actively engaged in acquiring information. The link disappeared when the deliberation period ended, a delay period ensued, and another saccade indicated the choice. People who had faster saccades also made faster decisions, implying that saccade vigor may also be predictive of decision-making behavior at a trait-like level ⁹.

STAR Methods

RESOURCE AVAILABILITY

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the lead contact, Alaa Ahmed (alaa@colorado.edu)

Materials Availability

This study did not generate new unique reagents

Data and Code Availability

De-identified human movement data, choice data, and original code have been deposited at OSF and are publicly available as of the date of publication. DOIs are listed in the key resources table. Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Key resources table.

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
Raw and analyzed data	This paper	DOI 10.17605/OSF.IO/YBE4U

EXPERIMENTAL MODEL AND SUBJECT DETAILS

Human participants

Participants were recruited from the University of Colorado Boulder (n=22; 8 Female; age: 23.71±3.07, mean ± s.d.) with no known neurological or motor deficits. Each participant signed a consent form approved by the CU Boulder Internal Review Board and was paid $10/hr for participation in the study. Two participants wore necessary corrective eyeglasses.

METHOD DETAILS

Experimental design

Participants performed a decision-making task in which they considered walking various inclinations and durations, each represented by a visual symbol on a computer monitor. A total of 360 decisions were presented per participant, with 180 unique decisions repeated twice, counterbalanced by left-right presentation order on-screen.

Participants were first familiarized to the various potential inclines before any choices were presented. Half of the participants were familiarized to inclines of 0, 2, 4, 6, 8, and 10%, while the other half experienced inclines of 0, 1, 3, 5, 7, and 9% each for 3 minutes apiece at a pace of 1.4 m/s. A brief reset period of 30s was allowed between each exposure. Familiarization order was randomized per participant. During the familiarization procedure, a graphic representing the current incline was presented to the participant. This same graphic was then later used during the choice-selection period to represent the incline on-screen.

After the familiarization period, the choice-selection period commenced. Eye-movements were recorded using an SR Research Eyelink 1000 Plus sampling at 1000 Hz. Standard 9-point calibration was performed before any data collection, and tracking was routinely verified over the course of the experiment. Choices were always between either one of two reference pairs (4%/6min or 5%/5.5min) and an alternative option of incline and duration varying between 0 to 10% and 0.5 to 10 min respectively. 90 total alternative options were presented, each consisting of a different incline and duration as compared to the reference option presented for that trial. After every 90 decisions, participants were given a short rest of 10 minutes to reduce eye strain. Immediately following this break, a 9-point calibration was again performed to account for any shift in participant posture. Participants were informed that of the selected choice pairs, four were randomly selected and will be performed after all decisions were made. We further instructed participants that the choice of movement duration held no bearing on the overall time spent for the experimental protocol.

The choice-selection period of the experiment consisted of two distinct phases: the deliberation phase and the decision phase. During the deliberation phase, two graphics representing the choice pairs were presented on-screen at approximately 25% and 75% of the screen width (960-pixel separation, visual angle deviation of ~15°). Participants were given 4.5 seconds to freely deliberate between the two options, and once a preferred choice was in mind, the space bar was pressed to advance to the next phase. If participants had failed to press the spacebar within the allotted timeframe, the trial timed-out and the choice-pairs were repeated at a random later point in the experiment. Should a participant fail to decide within the allotted time, the decision pair was recycled into a random future index for re-selection.

Next, the participants fixated a small cross at the center of the screen for a minimum of 1s (the average fixation duration was 1668 ± 70 ms, mean ± SEM). After this fixation period, the decision phase began. During this phase, two identical grey circles appeared on-screen at the same locations as the previously presented graphics. Participants then were to saccade towards the side of the screen within 1.5 seconds that corresponded to their preferred movement option in the deliberation phase. Failure to initiate a saccade within the allotted time was considered a failure to make a decision, and the trial was repeated later, and the selection defaulted to the more metabolically costly of the two. If there was a saccade that ended within one of the two circles on-screen, a final screen with the previous corresponding choice pair graphic was displayed to provide feedback and decision confirmation to the participant.

Importantly, the separation of the deliberation phase from the readout of the decision differentiates this study from our previous in Reppert et al.⁸. In that work, participants indicated their choice with a button press, and were free to continue gazing at the images after they made their decision. As such, saccade vigor over the course of a trial can include saccades made both before and after the decision. Here we separated saccades made during deliberation from the saccades made after the decision (see Figure 4F).

After all choices were made, a random sample of four previous choices made were given to participants to then perform. Before doing so, participants waited 15 minutes to account for any variation in time selection between participants. Thus, the total duration of the protocol was constant. Finally, participants walked on the same treadmill and performed the incline-duration movement pairs randomly selected, again at the same set velocity of 1.4 m/s.

QUANTIFICATION AND STATISTICAL ANALYSIS

Saccade vigor

Eye movement data was converted first from pixel location data to visual angle degrees for both x- and y-coordinates. Instantaneous velocity was calculated via second order Savitzky-Golay filter with a window size of 21 msec. Data was then processed for missing or physiologically impossible saccade velocities. First, samples greater than a threshold of 850 deg/second were identified as “high velocity” samples and the median velocity of all other samples was then calculated. The contiguous timespans including these “high velocity” samples and all other samples in which velocity was greater than the median value calculated were excised from further analysis as part of the denoising procedure. Lastly, all samples with a calculated saccade velocity of zero or NaN were also removed from consideration (<1% of all samples). Saccades were automatically identified via a post-hoc adaptive velocity threshold algorithm ³⁸. Blinks were identified by continuous samples with missing x- and y- screen coordinate position data. Remaining data was then characterized as fixation periods, where continuous time indices uninterrupted by a saccade or blink taken as indicative of a single fixation event.

As individuals differed in overall saccade velocity and behavior ^28,39, a within-participant measure of saccade vigor was calculated similar to Reppert et. al. (2015). For each participant, the amplitude-velocity relationship was fitted via the following hyperbolic function:

$$v_n={\alpha }_n\left(1-\frac{1}{1+{\beta }_{n}x}\right)$$ (3)

This functional form was chosen as previous work has shown that a hyperbolic relationship is a generally good fit for saccade data ²⁸. Nonlinear Least Squares curve fitting was performed per participant (via curve_fit function in the Python scipy.optimize package) for both nasal and temporal saccades. Both α and β were free parameters for the curve fit, with x the given saccade amplitude, and v the saccade peak velocity for participant n. Given the fit parameters α and β, the expected peak saccade velocity was calculated. Dividing the measured peak saccade velocity by the expected velocity defined the within-participant vigor metric (referred to as vigor), with values >1 indicative of more vigorous saccades than typical, and values <1 being less vigorous.

For visualization of the rate of change in saccade vigor over the course of a deliberation period, in each trial with at least two saccades, we calculated the rate of increase in saccade vigor (ΔVigor/ΔTime) and calculated its z-score per participant (See Figure 4B, left panel).

Decision making

Participant choices were modelled via general linear mixed models (GLMM) with a logistic link function. Choice response, SelRef was coded as a binomial response variable, either 0 for selection of the alternative, or 1 for selection of the reference option; as such, a binomial distribution with logistic link function was used. A typical Gaussian linear mixed-effects model could not be used to ensure that the fitted values ranged from 0 to 1. With the logistic link function, for each choice, the log-odds that participants would select the reference option was estimated. Thus, the model description that for a given choice i, participant j will choose the reference option is of the form:

$$\begin{array}{l}SelRe{f}_{ij}~Binomial\left({\theta }_{ij}\right)\\ E\left[SelR{e}_{ij}\right]={\theta }_{ij}\\ \mathrm{logit}\left({\theta }_{ij}\right)={\beta }_0+\mathit{\beta }{\mathit{X}}_{\mathit{i}}+{\beta }_{0_j}+{\mathit{\zeta }}_{\mathit{j}}{\mathit{W}}_{\mathit{i}}\\ {\mathit{\zeta }}_{\mathit{j}}~N\left(0,{\sigma }^2\right)\end{array}$$ (4)

β₀ is the fixed effect intercept, β the fixed effect coefficients, β_0j the participant-specific intercept, ζ_j the random effect coefficients, X_i the fixed effect parameters for choice i, and W_i the random effect parameters for choice i. Regression coefficients and their confidence intervals were estimated via R package lme4 ⁴⁰. Random effect coefficients were assumed normally distributed with mean 0 and covariance matrix σ²_.

To fit the capture rate model of choice preference, a nonlinear mixed effects model was used to fit both the participant-specific free parameter α (Eq 2) and regression coefficient. Distribution of fitted α values across participants followed a log-normal distribution.

$$\begin{gathered} {\alpha }_j=\exp \left({\gamma }_0+{\gamma }_j\right) \\ \mathrm{\Delta }{J}_{ij}=\frac{{\alpha }_j-e_{ref,i}}{Du{r}_{ref,i}}-\frac{{\alpha }_j-e_{alt,i}}{Du{r}_{alt,i}} \\ logit\left({\theta }_{ij}\right)=\beta \mathrm{\Delta }{J}_{ij} \\ {\gamma }_j~N\left(0,{\sigma }^2\right) \end{gathered}$$ (5)

Behavioral metrics

Aside from the choice selection, saccade velocity, and other velocity related metrics (e.g. peak velocity, vigor, and trial average vigor acceleration), other measures known to correlate to choice-difficulty, primarily deliberation time and reaction time, were also collected. Participants were provided a maximum of 4.5 seconds from the time of stimulus onset to indicate a choice via keypress. Deliberation time was defined as the time spent from stimulus onset to choice-indication via keypress. If no indication were made, the trial was considered timed out, the choice recycled for a later selection at a random later point in the experiment. Participants were then advanced to the next trial. After successful choice-indication, and the brief fixation period, the decision phase began. Reaction time during this phase was defined as the time from stimulus onset (presentation of two grey circles) to saccade initiation as identified by the algorithm utilized. Should participants time out during the decision phase, the choice was also recycled for later selection. All trials during which participants timed-out either during the deliberation or decision phase were excluded from analysis unless otherwise specified (17.41±20.53 trials; mean±s.d. across participants). Trial number was accounted for as fixed factor within behavioral regression models (reaction time, deliberation time, and saccade vigor).

To compare saccade vigor across participants, we fit a hyperbolic function for all saccades, producing a population-averaged expected velocity as a function of amplitude ⁹. We next defined the vigor of each saccade for each participant with respect to this expected value, and then averaged the vigor of all saccades for each participant. The result was the participant’s average saccade vigor with respect to the population mean.

Statistical analysis specifications

All statistical analyses were performed using the R programming language, specifically either the lmer or hglm package for regression analyses. Specifics on regression predictor variables and diagnostics are included in supplementary material. All t-tests performed were two-tailed. Appropriate diagnostics were performed on all regression analysis, ensuring that assumptions regarding homoskedasticity, random effects distributions, and residual distributions were met.

For statistical analysis of deliberation times and reaction times, GLMMs were fitted using Gamma distributed errors with the canonical inverse link function to account for the data’s highly skewed variance structure and to control for heteroskedasticity. Analysis of saccade vigor and rate of vigor change in both phases of the experiment were done via LMM with Gaussian distributed errors.

To compare vigor towards the reference and alternative, we calculated the average per-participant difference in vigor between saccades directed towards the two options if made as part of the final two saccades per trial, binned by capture rate difference (bin count = 8, 1298±871 saccades per bin, mean±s.d.). We then measured the Spearman rank correlation between these mean differences and utility.

Alternative models of decision variables

Other candidate decision variables were considered and compared against each other to determine their ability to explain choice (Table S1).

In addition to the capture rate model (Eq 5) we considered a simple additive model with independent sensitivities to duration and incline (Eq 7):

$$\begin{gathered} logit\left({\theta }_{ij}\right)=\left({\beta }_I+{\beta }_{Ij}\right)\mathrm{\Delta }Inclin{e}_{ij}+\left({\beta }_D+{\beta }_{Dj}\right)\mathrm{\Delta }Duratio{n}_{ij} \\ {\beta }_{Ij}~N\left(0,{\sigma }_I^2\right) \\ {\beta }_{Dj}~N\left(0,{\sigma }_D^2\right) \end{gathered}$$ (7)

As the ζ parameters above, participants’ individual sensitivities to incline and duration (β_Ij, β_Dj) were assumed to be normally distributed across the population with a mean of zero and estimated variances σ_I² and σ_D².

It is possible that participants based their decisions solely on the effort cost of the options. To consider this we tested two additional models that evaluated the goodness of the option based on its effort cost alone with either linear perception or nonlinear perception of time. In comparison to the capture rate model, we found both alternates markedly worse in predicting participant choice (ΔAIC=1576 and 1202 respectively).

We also compared the alternative decision variables’ predictive power for saccade vigor. We found that the capture rate model and additive model correlated significantly with the difference in saccade vigor at the end of the trial (p = 8e-4 and p = 3.3e-6, respectively). The total effort models and had worse AIC scores and did not significantly correlate with the difference in saccade vigor (p=0.21 and p=0.19, respectively). Similarly, only capture rate and the additive model significantly correlated with the rate of vigor increase within a trial (p = 0.003 and p = 0.006, respectively), while both effort models did not predict vigor rate (p = 0.4 and p = 0.07, respectively). All four decision variables exhibited a significant correlation between deliberation time and reaction time (p’s < 0.01), and all failed to explain decision saccade vigor.

Considering both choices and the behavioral data (deliberation time, reaction time and vigor), we found that the capture rate and simple additive model overall performed similarly, with the effort models less descriptive of either choices or saccade vigor.

Using the simplified choice model (Eq 7) we tested whether individual sensitivity to option duration was correlated with average vigor (See Figure 4G). We found the correlation to be insignificant (t20 = 0.20475, p = 0.8398)

Control studies

The low-level properties of a visual stimulus can bias the decisions that participants make. For example, when utilities of the stimuli are similar, people are biased toward choosing the brighter stimulus ⁴¹. During the deliberation period, the symbols that represented the effortful options had differing luminosity. Salience is greater for stimuli that have greater luminosity, thus raising the concern that differences in vigor may be related to luminosity. However, we designed our task so that luminosity was anti-correlated with capture rate (t₆₆₀₆=17.9, p<2.2e-16). That is, the more effortful options were represented by more luminous symbols. Yet, saccade vigor was greater toward the less effortful options. Beyond that association, saccade vigor was not affected (GLMM, β=−1.259e-2±2.04e-2, p=0.227).

We asked whether the relation between faster participant-specific vigor and faster deliberation time could be a result of simply making faster saccades (Figure 4G). We found that this difference in vigor could not account for the range of deliberation times measured over the experiment, as the maximum per-saccade difference in duration (from the slowest to the fastest) amounted to approximately 15 milliseconds, whereas the range in average deliberation times was approximately 2 seconds.

Highlights.

• Saccade vigor increases during deliberation, reflecting decision variables.

• Saccade vigor indicates degree of subjective preference.

• Rate of change in vigor increases with difference in the value of the options.

• Individuals’ relative vigor correlates with average deliberation time.