In both regimes, stronger motion was associated with improved choice accuracy and faster RTs, as previously shown (Figure 2A; Roitman and Shadlen, 2002). Behavior during the two regimes revealed a tradeoff between speed and accuracy. In the high accuracy regime, both monkeys exhibited longer RTs and improved performance (Figure 2A). This observation is important because it indicates that the monkeys did not slow down merely by delaying the motor response, but rather, they used the time to accumulate more evidence from the stimulus. Similarly, if the monkeys had failed to pay attention to the early portion of the stimulus, an improvement in accuracy should not have accompanied the longer RTs.

Figure 2

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A bounded accumulation model can explain the behavior in both regimes (Figure 2B). The traditional depiction of this model (Figure 2B) would represent the accumulation of evidence as a random walk between two terminating bounds. We consider a version of this model that is more consistent with neurophysiology (Figure 2C) in which one process is replaced by two competing processes that accumulate moment-by-moment sensory evidence bearing on each alternative (Usher and McClelland, 2001). The choice is dictated by which process first reaches the bound (Figure 2C). This two-process model explains the established neural representations of both saccadic choices in LIP for this task (Mazurek et al., 2003). In the model, RT is determined by a combination of the time taken to reach the bound and a stimulus-independent non-decision time. As explained in ‘Materials and methods’, the models in Figure 2B,C can be treated as mathematically equivalent. For both depictions, the distance from the starting point of evidence accumulation to the bound, termed the ‘excursion’, determines how much evidence must be accumulated in order to commit to a choice. In theory, the excursion may change as a function of decision time, for example, if the bound collapses during the course of decision formation. However, for simplicity, we first consider the restricted set of models with static excursions.

Monkeys traded off speed and accuracy by changing the amount of evidence that was needed to commit to a choice. In particular, SAT was explained by the bounded accumulation model with a change in the decision variable excursion. Overall, the excursion was reduced by 43% ± 2% in the high speed regime compared to the high accuracy regime (monkey D: 58% ± 3%; monkey E: 23% ± 4%). Thus, less accumulated evidence was needed to commit to a choice in the high speed regime compared to the high accuracy regime.

The change in SAT was not explained in a consistent way by the other model parameters. Overall, we observed a 33.4 ± 5.8 ms shorter non-decision time in the high speed regime compared to the accuracy regime, and this effect was not consistent between the two monkeys (monkey D: 3.1 ± 5.2 ms longer in high speed regime; monkey E: 63.7 ± 9.1 ms shorter in high speed regime). Importantly, the RT elongation in the high accuracy regime exceeded any changes observed in non-decision time by a large margin (mean RT elongation, monkey D: 176 ± 4 ms; monkey E: 159 ± 3 ms). Moreover, if changes in RTs were solely a result of altered non-decision time, there would be no associated changes in accuracy. In principle, changes in the SAT could be associated with changes in the signal-to-noise ratio of the sensory evidence, owing perhaps to a change in attentional state, which would lead to a change of sensitivity to motion strength (parameter k of the model). We cannot rule this out, but the model fits provided inconsistent support for such a mechanism in the two monkeys. In the high speed regime, k changed by a factor of 1.36 ± 0.12 and 0.78 ± 0.07, respectively, for monkeys D and E (1.01 ± 0.06 in combined data).

We cannot rule out the possibility that some of the behavioral change was manifested by a competing process in which the monkey simply guessed quickly, so-called ‘fast guesses’ (Ratcliff and Rouder, 1998). A fast guess process could lead to faster and less accurate choices in the high speed regime. However, the level of accuracy for both monkeys at the highest motion strength in the high speed regime suggests that any such fast guess mechanism plays a relatively minor role because fast guesses would reduce accuracy for all stimulus strengths. Together, these considerations show that the change in SAT was achieved primarily via a process that alters the excursion required for the accumulating evidence to support termination.

Traditionally, bounded accumulation models implement this change in excursion by altering the bound height. However, the brain is thought to employ competing accumulator processes for decision making (Figure 2C; Usher and McClelland, 2001; Gold and Shadlen, 2007; Churchland et al., 2008). Accordingly, the brain can achieve a reduction in the excursion of accumulated evidence in two ways: (1) by lowering the bounds for both processes, or (2) by adding a signal to the accumulated evidence for both processes. These can be treated as mathematically equivalent: in both cases the effect of the change in the excursion is to alter the amount of accumulated evidence necessary to terminate the process. However, they imply different neural mechanisms.

We recorded neural activity from 70 neurons while the monkeys performed the task (35 in the high speed regime and 35 in the high accuracy regime). In each recording session, we screened neurons based on whether they exhibited spatial selectivity during a delayed saccade task and persistent delay activity during a memory-guided saccade task. For the motion discrimination task, one of the choice targets was placed in the RF of each recorded neuron; the other choice target was placed in the opposite hemifield.

In both speed-accuracy regimes, LIP neurons exhibited signatures of evidence accumulation. These features are illustrated in Figure 3, and they are similar to previous observations (Roitman and Shadlen, 2002; Churchland et al., 2008). Initially, the responses were elevated in the presence of the choice targets, one of which was in the neuron's RF. For the first 200 ms after motion onset, there was no relationship between stimulus strength and firing rate (Figure 3A,B; Equation 1, p>0.1 for both regimes in each monkey). Onset of the random dot motion (outside the RF) then induced a transient depression in the firing rate that recovered over a 200-ms period. This pattern was similar for speed and accuracy regimes, but the average firing rate in this epoch was 15% ± 1% lower in the high accuracy regime (monkey D: 24% ± 2% lower in the high accuracy regime; monkey E: 8% ± 2% lower in the high accuracy regime; p<0.01 in all cases), as discussed in detail below.

Figure 3

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After the transient depression in the firing rate, the average responses began to display characteristic ramping activity that depended on stimulus strength and direction (Figure 3A,B). To characterize this relationship, we estimated the ‘buildup rate’ for each motion strength (‘Materials and methods’). Stronger motion in the T_in direction led to larger buildup rates, and stronger motion in the T_out direction led to smaller buildup rates (Figure 3C; Equation 2, p<0.01 for each monkey in both regimes). This is consistent with the bounded accumulation framework, which posits that the drift rate of the decision variable should depend on the strength of the evidence. Interestingly, larger buildup rates were observed in the high speed regime (Figure 3C; Equation 3, p<0.05 for each monkey). Importantly, however, there was not a significant difference between the slope of the relationship between buildup rate and motion strength between the two regimes (Equation 3, p>0.1 for each monkey). This suggests that the sensory evidence influences the decision variable in the same way in both speed-accuracy regimes, consistent with the conclusions drawn from the fits to behavior.

In both speed-accuracy regimes, LIP neurons also reflected a signature of decision commitment. Shortly before saccade initiation for T_in choices, rather than reflecting stimulus strength, responses attained a common level of activity (Figure 4A; Equation 1, p>0.3 for coherence effect 75 ms before the saccade for each monkey in both regimes). The coalescence of T_in responses is also evident when responses are grouped by RT rather than stimulus strength (Figure 4B,C; Equation 4, p>0.2 for RT effect 75 ms before the saccade for each monkey in both regimes). This overall pattern is consistent with the idea that LIP responses support the presence of a decision bound which when reached, terminates further deliberation and triggers commitment to a choice.

Figure 4

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Interestingly, the level of this decision bound was undifferentiated by the speed-accuracy regime. Recall that the bounded evidence-accumulation model explained the behavioral data with a change in the excursion of the decision variable in the two regimes. The brain could implement a change in excursion by altering the threshold level of LIP firing rate necessary to commit to a choice. However, we did not observe significant differences between the firing rates in the two regimes within 200 ms of the saccade for either monkey (Figure 5A; F test, Equation 5, p>0.5 for each monkey).

Figure 5

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Alternatively, the excursion could be modified by means of an evidence-independent signal that adds directly to the accumulated evidence. We looked for such a signal by averaging responses in each speed-accuracy regime across all stimulus strengths and both directions to extract the component of the response that was not dependent on the evidence. This evidence-independent component was significantly elevated in the high speed regime relative to the high accuracy regime (Figure 5A). In one monkey, this difference began from before motion onset and persisted throughout the period of evidence accumulation (p<0.001). For the other monkey, a significant difference emerged 100 ms after motion onset (p<0.01) and persisted at all time points thereafter (p<0.001). Thus, for both monkeys, LIP neurons undergo a smaller excursion in firing rate from the start to the end of the decision in the high speed regime compared to the high accuracy regime. It appears that an evidence-independent signal effectively pushes the neural representation of accumulated evidence closer to a threshold.

This conclusion receives additional support from an analysis of individual neurons. Across all recorded neurons, there was a significant negative correlation between the estimated evidence-independent neural response and the mean RTs for the corresponding behavioral sessions (Figure 5B, r = −0.57, p<0.01). Of course, this correlation is driven to a large extent by the changes in the mean values between the two regimes. Nonetheless, there is variation within each regime, and for neurons in each regime individually, the correlation was significant (r = −0.40 and −0.63 for high speed and high accuracy regimes, respectively; p<0.05 in both cases).

Beyond correlation, these changes in the evidence-independent neural signal explained the changes observed in the behavior between the two speed-accuracy regimes. One way to appreciate this is to fit the behavior in both regimes with the bounded accumulation model such that the only difference between the two regimes is established by the measured evidence-independent neural signals. Notice that in both regimes, the evidence-independent signal increased over time (Figure 5A), consistent with urgency signals reported previously (Ditterich, 2006; Churchland et al., 2008; Cisek et al., 2009). For each monkey, we approximated the evidence-independent signals in each regime with hyperbolic functions (Figure 6A), and used these neurally-derived ‘urgency functions’ to control the model's excursion for each regime. Importantly, the change in excursion between the two regimes is not fit to the behavioral data; it is measured from the neural data. In this way, the only free parameters in the model are the three that are shared between the two regimes (‘Materials and methods’). This neurally-constrained model provided a satisfying account of the magnitude of the changes in performance and RT for the two regimes (R² = 0.99 for monkey D; R² = 0.97 for monkey E; Figure 6B).

Figure 6

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In traditional diffusion models, like the one depicted in Figure 2B, the neurally-derived, evidence-independent signal would be represented by a time-dependent change in the height of the bound (Ditterich, 2006; Drugowitsch et al., 2012). In the high speed regime, the starting point of the bound is nearer to the origin, and the rate of collapse is more severe. In the race-like architecture (Figure 2C), a change in bound height is equivalent to a change in the starting point of both accumulations, and a collapsing bound is equivalent to a time-dependent, monotonically increasing signal that is also added to both accumulators. The neural responses are consistent with the idea that both components comprise a singular, evidence-independent signal that adds to the evidence-dependent activity in LIP. Nonetheless, one might ask how well each component alone could explain the behavioral changes between the two speed-accuracy regimes. To test this, we fit the behavioral data using only the baseline or only the time-dependent component of the neurally-derived, evidence-independent signal. The model comparison, summarized in Table 1, suggests that the time-dependent component of the signal is more important, although both components contribute to the fits in Figure 6D, especially for monkey D. This dissection is far less important than the rather remarkable observation that a pair of signals derived from the physiology in two speed-accuracy regimes and scaled identically, can explain the dramatic differences in the behavior.

Two rhesus monkeys (Macaca mulatta) were trained to perform a motion discrimination task. Visual stimuli were presented on a computer monitor (75 Hz frame rate) using the Psychophysics Toolbox for Matlab (Brainard, 1997). Trials began with the appearance of a single dot that the monkey was required to fixate. To minimize effects of anticipation on behavior and neural activity, random intervals between trial events were implemented using truncated exponential distributions, t_min + exprnd(µ), where exprnd is a random sample from an exponential distribution with mean = µ. If the generated interval was longer than the maximum allowed time for the epoch (t_max), the interval was resampled to ensure it was less than t_max. After a variable delay (t_min = 300 ms, µ = 300 ms, t_max = 1 s), two bright red choice targets appeared at an equal distance from the fixation point and 180° apart. After another variable delay (t_min = 300 ms, µ = 300 ms, t_max = 1.5 s), the random dot motion stimulus appeared in an aperture 5° in diameter that was centered at the fixation point. The dynamic random dot stimulus consists of sets of random dots, which are shown for one video frame and then updated 40 ms later (e.g., dots in frame 1 were updated in frame 4; dots in frame 2 were updated in frame 5, and so forth). To update, each dot is either replaced at a random location, or displaced by 14.4 min arc (i.e., 6.0 deg s⁻¹) with probability C (expressed as a percent), termed the motion coherence or motion strength. The dot density was 16.7 dots deg⁻² s⁻¹.

For each trial, there were two possible directions of motion, differing by 180°. For one monkey (monkey E), the directional axis of motion corresponded to that formed by the choice targets. For the second monkey (monkey D), the directional axis of motion always corresponded to the horizontal meridian, but the choice target configuration was as described above, identical to the other monkey. Motion strength (the percentage of coherently moving dots) was chosen randomly from a set C ∈ {0%, 3.2%, 6.4%, 12.8%, 25.6%, 51.2%}. The monkey's task was to determine the direction of coherent motion, which it indicated by making a saccade to the appropriate choice target (e.g., rightward target for rightward motion, leftward target for leftward motion). The monkey received a liquid reward for all correct choices, and on a random half of the trials in which 0% coherence motion was displayed. The monkeys received no reward for the incorrect choices.

The monkeys were trained to perform a RT version of the motion discrimination task in two different speed-accuracy regimes: a faster speed (hence, lower accuracy) or higher accuracy (hence, slower speed) one. Each regime was maintained for multiple consecutive daily sessions. For both monkeys, we started with the faster speed regime first, followed by the higher accuracy regime, and then switched back to the faster speed regime, with details as follows. The monkeys were initially trained on a version of the task that used an experimenter-controlled duration for the visual stimulus. Both monkeys achieved stable psychometric thresholds in this version of the task before proceeding to the RT version. In the RT version of the task, the color of the fixation point was changed from red to blue and the monkey was allowed to respond any time after motion onset. Previous training experience revealed a natural tendency of monkeys to respond with short latencies. We exploited this tendency by providing only a small incentive for accuracy; a time out penalty for incorrect choices was added to the standard inter-trial interval to discourage fast errors:

where t_ITI is the inter-trial interval, t₁ is a constant, typically 1 s, and t_R is the RT. With this approach, the monkeys established a stable speed-accuracy regime with well-behaved psychometric and chronometric functions. We refer to this as the high speed (low accuracy) regime.

After collecting data from LIP neurons in multiple consecutive daily sessions in this regime, each monkey was switched to a higher accuracy regime. This was achieved through the conditional delay of reward. Specifically, reward was always given for correct choices, but it was delayed relative to motion onset. The duration of this delay was 600–800 ms. So, if the delay was 800 ms and the monkey responded correctly in 500 ms, it would have to wait an additional 300 ms before delivery of the reward. This eliminated any reward-based incentive to respond faster than the minimum delay to reward, encouraging longer deliberation times that would improve accuracy. After multiple daily sessions with this reward structure, the monkeys achieved a new stable speed-accuracy set point involving longer RTs and higher accuracy. Data were collected from LIP neurons in multiple consecutive daily sessions in this regime.

Finally, the monkeys were moved back to the high speed regime. We performed a variety of manipulations to bring the monkeys to a similar speed-accuracy regime as observed in the initial high speed sessions. At first, the monkeys preserved their higher accuracy set point even after removal of the conditional delay of reward for multiple daily sessions. After the additional removal of the time out penalty for incorrect choices, one monkey (monkey E) slowly reached a faster speed set point over the course of 15 sessions. The other monkey (monkey D) required an additional manipulation. For 50% of trials, the motion stimulus was extinguished after 600 ms, so no further sensory information was available. After three sessions with this design, the monkey reached a faster speed set point. The task structure then reverted back to allowing the motion stimulus to be displayed indefinitely, and the monkey maintained the faster speed set point. After stable behavior was achieved in this regime, data were collected from 12 more LIP neurons (7 for monkey D, 5 for monkey E). For both monkeys, the behavior in these final fast speed sessions was similar to the earlier fast speed sessions that were collected before the high accuracy sessions (mean RT difference <35 ms and mean accuracy difference <2.5%). The main effects on neural responses were similar as well, so data were combined across the two sets of high speed sessions.

Behavioral data were fit using a bounded evidence-accumulation model, also known as bounded drift-diffusion (Figure 2B). The model explains choice and decision time as an accumulation of momentary evidence to an upper bound or a lower bound, corresponding to the two direction choices. The momentary evidence gathered in each time step is drawn from a Gaussian distribution with unit variance for 1 s and mean µ determined by a linear transform of the motion strength: µ = kC, where C is the motion strength and k is a free parameter that scales the motion strength appropriately. The process terminates when the accumulated evidence, termed the decision variable, reaches ±B, the upper and lower decision bounds. The bound reached first by the accumulated evidence determines the choice, and the decision time is determined by how long it takes to reach that bound. RT is a combination of this decision time with an additional non-decision residual time, t_nd, which accounts for visual and motion delays and other factors that are independent of the decision process. The basic model (fits in Figure 2) uses three free parameters: k, B, and the mean t_nd ($\overline{t_{nd}}$).

This model can be described in a mathematically equivalent way in which one process is replaced by two competing processes with perfectly anti-correlated accumulations of moment-by-moment sensory evidence bearing on each alternative (Figure 2C; Usher and McClelland, 2001). In this version, the choice is dictated by which process first reaches the bound. A signal that adds an equivalent amount to both processes would be equivalent to a reduction of the bound of both processes by the same value. We refer to the distance between the bound and starting point of each process as the ‘excursion’. In a single process version of the model, the excursion is controlled entirely by the heights of the decision bounds. A symmetric reduction of the magnitude of each bound is mathematically equivalent to an equally-sized additive signal in the two-process version of the model.

To determine how the model accounts for the difference in behavior between the speed and accuracy regimes, we allowed all three parameters of the model to vary between the two regimes. As is well known, the simplified model, described above, cannot explain mean RT on error trials or the full shapes of the RT distributions for this task (Ratcliff, 1978; Ratcliff and Rouder, 1998; Ditterich, 2006). We therefore maximized the likelihood of the choice proportions (binomial error) and the mean and SD of the RT from correct choices (Gaussian error). Each model's predictions for these values were derived from the analytic solutions for the boundary-crossing probability and first-passage times for a bounded drift-diffusion process (Palmer et al., 2005).

We recorded from 70 well isolated single neurons in the lateral intraparietal area (LIP) of two monkeys while they performed the reaction time (RT) motion discrimination task. In monkey D, 17 neurons were recorded in the high speed regime and 15 in the high accuracy regime. In monkey E, 18 neurons were recorded in the high speed regime and 20 in the high accuracy regime. Standard methods were used for extracellular recording of action potentials from single neurons (e.g., Roitman and Shadlen, 2002).

Neurons were selected using anatomical and physiological criteria. Registration of structural MRI scans to a standard cortical atlas (Caret software; Van Essen et al., 2001) was used to identify LIP and to direct the placement of recording electrodes. Registration of recording sites to structural MRI suggests that our neural recordings were obtained from the ventral subdivision of area LIP (LIPv; Lewis and Van Essen, 2000). Once the proper anatomic location was identified, we recorded from neurons during the motion task if they met two criteria: (1) spatially selective activity during visually-guided delayed saccade trials at both the time of visual onset and the time of the saccade, and (2) memory activity during the delay period for memory-guided saccade trials (Gnadt and Andersen, 1988). Both criteria were assessed qualitatively by the experimenter after finding a neuron with an isolated spike waveform.

In the visually-guided delayed saccade task, each monkey maintained its gaze at a central fixation point while a target was displayed at a peripheral location. The monkey was required to maintain fixation for a delay taken from a truncated exponential as described above (t_min = 300 ms, µ = 600 ms, t_max = 2 s). After this delay, the fixation point was extinguished, instructing the monkey to make a saccade to the location of the target. This task was used to find the response field (RF) of each neuron—that is, the region of space where a target would elicit an increased response for the neuron throughout this task. In the memory-guided saccade task, the peripheral target was flashed briefly (150 ms) rather than maintained. Thus, the monkey was required to remember its location during the delay period. After the delay and the instruction to go, the monkey was required to make a saccade to the remembered location of the flashed target. When collecting neural data while the monkey performed the direction discrimination task, one target was placed in the center of the recorded neuron's RF and the other target was placed in the opposite hemifield. We refer to the target in the neuron's RF and its associated motion as T_in; we refer to the other target and its associated motion as T_out.

All training, surgery and experimental procedures were in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals and were approved by the University of Washington Animal Care Committee.

Population responses for motion discrimination trials were calculated by combining normalized responses across all neurons. Responses for each neuron were normalized by division by the mean peri-saccadic response (0–50 ms prior to saccade initiation) measured during a block of visually-guided delayed saccade trials. The spikes from each trial were aligned with respect to two trial events. Spikes were aligned to motion onset and saccade initiation, respectively, for analyses of the beginning and end of the decision process. When aligning to motion onset, the time period within 100 ms before saccade initiation was excluded to reduce contamination of the averages with peri-saccadic activity. Similarly, when aligning to the saccade, the time period within 200 ms after motion onset was excluded to avoid contamination by the dip in firing rate that follows motion onset. Response averages (running means) were calculated after applying a symmetric (non-causal) boxcar filter to the spike trains (boxcar width = 50 ms), appropriately normalized.

For plotting purposes (Figure 3A,B), motion strengths were binned into five groups: strong T_in (51.2% coh favoring T_in), medium T_in (25.6, 12.8, and 6.4% coh favoring T_in), weak (0 and 3.2% coh), medium T_out (25.6, 12.8, and 6.4% coh favoring T_out), and strong T_out (51.2% coh favoring T_out). All statistical tests were performed using the actual motion strengths; they were not influenced by the groupings of the trials. When plotting responses aligned to stimulus end (Figure 4B,C), trials were sorted into quintiles based on RT. This was done separately for the responses for each neuron and then combined across corresponding quintiles.

To estimate buildup rate during decision formation (Figure 3C), we applied weighted regression to the mean firing rates for each motion strength and direction. The weighting is based on the time-dependent covariance of the running mean (i.e., the sample covariance of the boxcar filtered trials, divided by the number of trials), thereby correcting for temporal correlation induced by filtering and in the LIP responses themselves (Churchland et al., 2011). The chosen analysis window was on the initial buildup, beginning 200 ms after motion onset and ending at the point in time when 50% of the trials were still included in the averages for each motion strength in the two speed-accuracy regimes, taking into account the RTs associated with each motion strength in each regime. The value of 200 ms is based on the established latency between stimulus motion (near the fovea) and the emergence of signals in LIP neurons that represent accumulation of evidence in favor of peripheral choice targets in the RFs of the neurons (Roitman and Shadlen, 2002; Huk and Shadlen, 2005; Kiani et al., 2008; Churchland et al., 2011), and it is supported in the current study by the time of the earliest significant effects of stimulus strength on LIP responses. We chose an ending point that equated the attrition of trials from each of the averages—that is, the fraction of trials that contribute to the averages. Attrition of trials associated with the fastest RTs biases the estimate of the buildup rate toward shallower slope because the faster rise times drop out of the average. An alternative approach would be to estimate the buildup rate using identical time intervals in both regimes, but Monte Carlo simulations revealed that it would result in a larger underestimation in the high speed regime than using an approach that based the endpoint on a matched level of trial attrition (results not shown). The latter approach does not eliminate the bias entirely, but it minimizes the difference in the bias of the buildup rate estimates in the two regimes.

For the single neuron analysis (Figure 5B), we derived estimates of the evidence-independent response for each neuron and the mean RT for the corresponding behavioral session. Because of the noise inherent in measurements from single neurons, we estimated the evidence-independent signal by averaging responses across all motion strengths and choices. For the same reasons as described above, for each neuron we fit a line to the responses in an interval beginning 200 ms after stimulus onset and ending at a point in time when 50% of the trials for highest motion strength contributed to the average, thereby ensuring inclusion of at least 50% of trials at all time points for all motion strengths. From the linear fits, we obtained the firing rate 300 ms after motion onset (i.e., reflecting 100 ms of processing time in LIP) and divided this by the mean response 75 ms (using 50 ms bin width) before saccades to T_in. This yields an estimate of the urgency signal in units of ‘fraction of neural bound’ for each neuron. The scatter plot (Figure 5B) displays the relationship between this statistic and the mean RT for each of the 70 experiments. We report standard correlation coefficients and report significance based on the Fisher z-transformation.

Statistical comparisons of neural measures were based on regression analyses using weighted least squares, which respect the standard error for point estimates and their correlation, when applicable. To test the effect of motion strength on firing rate during a specified epoch for each speed-accuracy regime, we fit the model:

where y_nfr is the mean normalized firing rate for each motion strength during the measured epoch, C is the motion strength, and for all equations, β_i are fitted regression coefficients. The null hypothesis is that motion strength does not affect firing rate (H₀: β₁ = 0). To test the effect of motion strength on the buildup rate of the neural responses for each speed-accuracy regime, we fit the model:

where y_bu is the buildup rate at each motion strength (see above) and C is the motion strength. The null hypothesis is that motion strength does not affect buildup rate (H₀: β₁ = 0). To test whether motion strength influenced the buildup rate differently between the two speed-accuracy regimes, we fit the expanded model:

where y_bu is the buildup rate at each motion strength (see above), C is the motion strength, and I_SA is an indicator variable for the speed-accuracy regime (1 for speed, 0 for accuracy). The null hypothesis that the regime does not affect the offset of the buildup rate is: H₀: β₂ = 0. The null hypothesis that the regime does not affect the relationship between buildup rate and motion strength is: H₀: β₃ = 0.

To test whether firing rate depended on RT during a particular epoch for each speed-accuracy regime, we binned trials into RT quintiles for each individual neuron. Then, we combined responses across each RT quintile for each speed-accuracy regime and fit the model:

where y_nfr is the normalized firing rate for each RT quintile and T_RT is the mean RT across all neurons for trials that fall into the corresponding quintile. The null hypothesis that RT does not affect firing rate is: H₀: β₁ = 0.

To compare mean responses derived from particular epochs in the two regimes, we used either a t test or weighted regression to detrend the responses. For example, the difference in mean firing rate in the 200 ms epoch preceding the saccade is captured by the following equation,

where y is the running mean normalized firing rate. The time-dependent terms achieve detrending. The null hypothesis, H₀: β₂ = 0, is evaluated by an F statistic using weighted regression that incorporates the covariance of the running sample means. For both monkeys, β₃ was not significantly different from zero, and removal of this interaction term had no effect on the conclusions.

To make neurally-constrained fits of the behavioral data, we measured the change in neural excursion in the two speed-accuracy regimes, as follows. We fit the evidence-independent neural responses (urgency signals derived from responses combined across motion strength and choice) from 200 ms after motion onset (the start of decision-related ramping activity) to the time point beyond which fewer than 50% of the trials for the highest motion strength contributed to the averages (same reasons as explained above). This fit was based on the following hyperbolic function:

where y_nfr is the measured population neural response for each speed-accuracy regime. The parameter corresponds to the neural response at decision time t = 0 (200 ms after motion onset). r₀+r_∞ corresponds to the asymptotic value of the urgency signal. The parameter t_1/2 is a rate parameter that controls how fast the urgency signal approaches its asymptote. We measured the difference between this urgency signal (which begins at r₀ at t = 0) and r_b, the neural activity at the time of putative decision commitment (75 ms before the saccade [Gold and Shadlen, 2007]), to determine the neurally-measured excursion as a function of time for each speed-accuracy regime. Thus, the neurally-measured excursion is described by the following equation:

where the subscript i indexes each of the two regimes.

The pair of neural excursion functions derived from the two speed-accuracy regimes can be used to constrain the excursion in the bounded accumulation model. In the model, the excursion is determined by the height of the decision bound, so we convert the time-dependent excursion E(t) to a time-dependent bound B(t). E(t) is in units of normalized neural firing rate and the model's B(t) is in units of the standard deviation of a Wiener process (see Palmer et al., 2005; Shadlen et al., 2006). To achieve the conversion, we apply an identical scale factor, α, to the urgency functions from the two regimes. The bound is thus described by

where the subscript i indexes each of the two regimes. The bounded accumulation model was then fit to the choice proportions and the mean RTs from correct choices in both regimes with three free parameters that are all shared: α as described above, and k and $\overline{t_{nd}}$ as described previously. Because there is not an analytic solution for drift-diffusion with time-varying bounds, we solved for mean RTs and accuracy numerically (i.e., propagating the probability distributions of accumulated evidence to the absorption bounds). Importantly, the only differences in the model between the two regimes are the parameters in Equation 7 that are directly estimated from the neural responses. Thus, the neurally-constrained fitting procedure allows us to assess whether the measured differences in neural activity can account for the behavioral changes associated with SAT without any additional free parameters that are different between the two regimes.

We also compared these fits to neurally constrained fits employing only the static or the dynamic component of E(t): $B_i\left(t\right)=\alpha E_i\left(0\right)$ and $B_i\left(t\right)=\alpha \left[E_i\left(t\right)-E_i\left(0\right)\right]$, respectively. These models have the same degrees of freedom as the full model (i.e., α, k, $\overline{t_{nd}}$). Comparisons were based on the Bayes Information Criterion (BIC). Models with lower BIC values are preferred over alternatives with higher BIC values.

The funder had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank D Wolpert for helpful advice, C Fetsch, N Odean, Shushruth, and L Woloszyn for comments, and A Boulet and K Ahl for technical assistance.

Animal experimentation: This study was performed in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. All of the animals were handled according to approved institutional animal care and use committee (IACUC) protocols (#2896-01) of the University of Washington. All surgery was performed under isoflurane anesthesia and analgesics to avoid pain and distress.

© 2014, Hanks et al.

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