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Journal of Neurophysiology logoLink to Journal of Neurophysiology
. 2015 Jul 29;114(3):1850–1861. doi: 10.1152/jn.00395.2015

Distinct dynamics of ramping activity in the frontal cortex and caudate nucleus in monkeys

Long Ding 1,✉
PMCID: PMC4575978  PMID: 26224774

Abstract

The prefronto-striatal network is involved in many cognitive functions, including perceptual decision making and reward-modulated behaviors. For well-trained subjects, neural responses frequently show similar patterns in the prefrontal cortex and striatum, making it difficult to tease apart distinct regional contributions. Here I show that, despite similar mean firing rate patterns, prefrontal and striatal responses differ in other temporal dynamics for both perceptual and reward-based tasks. Compared with simulation results, the temporal dynamics of prefrontal activity are consistent with an accumulation of sensory evidence used to solve a perceptual task but not with an accumulation of reward context-related information used for the development of a reward bias. In contrast, the dynamics of striatal activity is consistent with an accumulation of reward context-related information and with an accumulation of sensory evidence during early stimulus viewing. These results suggest that prefrontal and striatal neurons may have specialized functions for different tasks even with similar average activity.

Keywords: basal ganglia, decision making, reward, perception, temporal accumulation

the prefrontal cortex-striatum network has been studied extensively for its roles in cognition. Anatomically, the prefrontal cortex projects heavily to the striatum (particularly the caudate nucleus) as part of a circuit that loops through the basal ganglia and thalamus, then back to the prefrontal cortex (Alexander et al. 1986; Parent and Hazrati 1995). Given their anatomical interconnections, it is not surprising that the prefrontal cortex and caudate nucleus have been implicated in similar cognitive functions, including perceptual decision making (Ding and Gold 2013; Gold and Shadlen 2007), categorization (Miller and Buschman 2007; Seger and Miller 2010), reward-based reinforcement learning (Dayan and Daw 2008; Doya 2000; Frank et al. 2009), oculomotor control (Bruce and Goldberg 1985; Goldberg and Bushnell 1981; Hikosaka et al. 2000), and time estimation (Meck et al. 2008). These similarities extend to many common patterns of task-related modulation of single-unit activity in these brain regions, especially in well-trained animals (Antzoulatos and Miller 2011; Ding and Gold 2010, 2012; Ding and Hikosaka 2006; Kobayashi et al. 2007; Pasupathy and Miller 2005; Seo et al. 2012).

For example, in monkeys performing an asymmetric reward saccade task (Fig. 1A), some neurons in the dorsolateral prefrontal cortex, frontal eye field (FEF), and caudate nucleus exhibit context-dependent “ramping” activity (i.e., a gradual increase in average spike rate over several hundred milliseconds) before reward-predicting cues (Fig. 1C; Ding and Hikosaka 2006; Itoh et al. 2003; Kobayashi et al. 2007; Lauwereyns et al. 2002). This activity has been hypothesized to contribute to reward-biased saccade latencies. Similarly, in monkeys performing a visual motion direction discrimination task (Fig. 1B), some FEF and caudate neurons show ramping activity while the monkey is viewing the stimulus and forming its perceptual decision (Fig. 1D; Ding and Gold 2010, 2012; Kim and Shadlen 1999). Such activity has been hypothesized to reflect the accumulation of sensory evidence to form the decision.

Fig. 1.

Fig. 1.

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Task designs and average responses of caudate and frontal eye field (FEF) neurons. A: asymmetric reward task. During the fixation period, the visual environment was the same, but reward context alternated between blocks of trials. See methods for more task details. B: motion discrimination task. During the motion period, motion strength (%coherence) was randomly selected for each trial. Only correct trials were rewarded, with equal reward size for both choices. See methods for more task details. C: average firing rates of caudate (n = 38) and FEF (n = 50) neurons showed similar reward context-modulated ramping activity during the pretarget fixation period of the asymmetric reward task. Vertical dashed lines denote the epoch used for analyses of temporal dynamics. D: average firing rates of caudate (n = 43) and FEF (n = 29) neurons showed similar coherence and choice-modulated ramping activity during the motion viewing period of the motion discrimination task. Note that the average caudate activity was less truncated in time here than in the original report (Ding and Gold 2010). Gray scale indicates coherence levels. Vertical dashed lines denote the epoch used for analyses of temporal dynamics.

One interpretation of the similarity between FEF and caudate neural responses under these conditions is that task-relevant signals are computed in cortex and then simply relayed to the basal ganglia to inform action selection or other functions. In fact, a number of network models of the prefrontal-basal ganglia pathway have adopted this assumption (Bogacz et al. 2006; Cavanagh et al. 2011; Lo and Wang 2006; Rao 2010). Alternatively, the cortex and basal ganglia may make different computational contributions to different tasks but in a manner that is not readily apparent in simple measures of neural activity like average spike rate. Here my aim was to distinguish these possibilities in the contexts of reward-biased behavior and perceptual decision making, which require the brain to interpret different types of information, and of apparently similar buildup activity that relates directly to behavioral performance on these tasks. Using variance of conditional expectation (VarCE; Churchland et al. 2011) and temporal correlations, I show that these brain regions can exhibit quite different, task-dependent temporal dynamics, suggesting that informational flow in the fronto-striatal network is flexibly controlled to meet diverse task demands.

METHODS

Data source and inclusion.

Two previously published data sets are used in this study. All training, surgery, and experimental procedures were in accordance with the National Institutes of Health (NIH) Guide for the Care and Use of Laboratory Animals and were approved by Institutional Animal Care and Use Committees at the National Eye Institute/NIH and the University of Pennsylvania, respectively. One set of data was obtained from two monkeys performing an asymmetric reward saccade task (Fig. 1, A and C; Ding and Hikosaka 2006). For this task, the monkey initiated a trial by looking at a visual fixation point presented at the center of the screen. One second after the monkey acquired fixation, a visual target was flashed at one of two peripheral locations and the monkey was required to make a delayed saccade to that location after the disappearance of the fixation point. Reward context was alternated in blocks such that in one block of trials one target location was paired with a large reward while the other location was paired with a small reward. Analyses were limited to activity on correct trials during a 700-ms epoch before target presentation (Fig. 1C; the first 300 ms of the fixation period was excluded to avoid contamination of visual/saccade-related responses while fixation was acquired). Neurons with average activity in the epoch that differentiated between reward contexts were included for further analyses (n = 50 and 38 for FEF and caudate, respectively). The reward context that was associated with higher activity during the pretarget period is designated as the “preferred” reward context, and the other context is designated as the “null” reward context.

The other set of data was obtained from two monkeys performing a visual motion direction discrimination task (Fig. 1, B and D; Ding and Gold 2010, 2012). For this task, the monkey viewed a random-dot motion stimulus centered on the screen and indicated its decision about the global motion direction of the stimulus by making a saccade toward the corresponding target. Motion strength was controlled by varying the percentage of the dots moving in the same direction (coherence). Viewing duration was controlled by the monkey: The motion stimulus was extinguished as soon as the monkey's gaze left a preset fixation window. The same reward was given for correct decisions for both directions. Analyses were limited to activity on correct trials during a motion-viewing epoch (from 250 ms after stimulus onset to 100 ms before saccade onset; Fig. 1D). Neurons with average activity in the epoch that was modulated by choice and stimulus strength were included for further analyses (n = 29 and 43 for FEF and caudate, respectively). The choice that was associated with higher activity during motion viewing is designated as the “preferred” choice, and the other choice is designated as the “null” choice. Note that Fig. 1D differs slightly from Fig. 3 of the original report (Ding and Gold 2010): For the original figure, the curve for each condition was truncated at the median reaction time across all neurons for that condition. For Fig. 1D, average firing rate for each neuron and condition was truncated at the median reaction time for that neuron-condition combination and then averaged across neurons without further truncation, to allow visualization of the mean firing rate trajectories over a longer period.

Fig. 3.

Fig. 3.

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Comparison of temporal correlation between caudate and FEF neurons. A: correlation matrices obtained from pooling each population for the motion discrimination data set. B: first subdiagonal correlation curves from A [i.e., correlation between the ith and (i + 1)th bins; mean ± SE estimated with resampling]. C: histograms of slope values from linear regression of the first subdiagonal correlation as a function of time bins for individual neurons. Filled bars indicate neurons with significant nonzero slope (t-test, P < 0.05). Arrowheads indicate median slope values for each population. D–F: correlation results for the asymmetric reward data set. Same conventions as A–C.

Temporal dynamics measurements.

For each trial, spikes were counted in a 100-ms sliding window with 50-ms steps. A data point plotted at time t was based on spike count in the window [t − 100 ms, t]. Trials were grouped by types (reward contexts or motion strength-choice combinations for the asymmetric reward and motion discrimination data sets, respectively). For each neuron and condition combination, the mean, variance, and Fano factor for spike counts were estimated. Spike count residuals in a time bin were obtained by subtracting the mean spike count in the same time bin over all trials of the same neuron-condition combination.

The VarCE of spike count residuals is the difference between the total variance and the estimated point-process variance, assuming that spike trains are generated with a doubly stochastic process. Following methods described by Churchland et al. (2011), the minimum Fano factor for all conditions of each neuron was denoted as ϕ (Fig. 2A). Point-process variance was estimated as the mean spike count multiplied by ϕ. VarCE was computed by subtracting the point-process variance from the total variance of the spike count residuals across all conditions.

Fig. 2.

Fig. 2.

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Comparison of variance of conditional expectation (VarCE) between caudate (Cd) and FEF neurons. A: illustration of ϕ estimation. Gray traces are Fano factor computed from a neuron for different coherence-choice combinations (solid, preferred choice; dashed, null choice). ϕ denotes the minimum Fano factor for all combinations (horizontal line with arrow), which was used to compute point-process variance for the neuron. B and C: histograms of ϕ estimate for neurons recorded on the motion discrimination (B) and asymmetric reward (C) task. D: VarCE (mean ± SE estimated with resampling) computed from pooling the population data. Dashed lines indicate the start time of the epoch for regression analyses (same as in Fig. 1D). E: histograms of slope values from linear regression of VarCE computed for individual neurons. Filled bars indicate cells with significant nonzero slopes (t-test, P < 0.05). Arrowheads indicate median values for each population. F and G: same conventions as D and E, for the asymmetric reward task. H: median slope values of VarCE estimated with a sliding window of 6 consecutive points (300 ms) for the motion discrimination task. A data point at time t represents the value obtained for the t to t + 300 ms epoch. Asterisk indicates bins with significant regional difference (P < 0.05). I: % of cells showing positive slope values. Asterisk indicates bins with significant regional difference (P < 0.05). J and K: same conventions as H and I, for the asymmetric reward task. L: peak (triangles) and average (circles) median slope values across all subepochs in H and J.

Temporal correlation was estimated by computing a covariance matrix of the spike count residuals in nonoverlapping 100-ms time bins. Correlation coefficients were obtained by normalizing the covariance matrix with values in the diagonal. The correlation coefficient between successive time bins (i.e., the first subdiagonal values of the correlation matrix) was used to track the temporal correlation of neural activity over time.

Statistical analysis.

To test for task-relevant modulation in the average firing rates (FR), linear regressions were performed using

FRi,j=b0+b1Cj+b2Ii

where FRi,j is the value for the ith neuron (Ii) and the jth condition (Cj). For the motion discrimination data, the conditions were the signed coherence (positive and negative for preferred and null choices, respectively). For the asymmetric reward data, the conditions were the preferred and null reward contexts, respectively.

For each region-task combination, two methods were used to test the null hypothesis that the slope value is zero for VarCE/correlation values over time. In the “population-based” method, linear regression coefficients were obtained from VarCE and correlation values measured from spike count residuals pooled from all cells. t-Tests were performed to test against the null hypothesis that the slope was zero. Error bar values shown in figures were bootstrapped by randomly selecting trials (with replacements) from each condition for each neuron and then pooling spike count residuals across cells. In the “single-cell” method, linear regression coefficients were obtained from VarCE/correlation values measured from each neuron. Sign tests were performed to test against the null hypothesis that the median slope across cells was zero. To test for the brain region/task dependence of VarCE and temporal correlation, two-way ANOVAs with interaction were performed on single-cell slope estimates, with brain region and task as factors. For comparisons of VarCE between FEF and caudate neurons within shorter time spans, Wilcoxon rank sum and χ2-tests were used for the slope values and proportion of positive values, respectively.

Simulation.

Spike trains were simulated with doubly stochastic Poisson processes. The time-dependent rate parameter λ(t) for the Poisson processes (rectified at zero) was governed by four models.

In a model with accumulation (drift-diffusion model, or DD), λ(t) was proportional to the time accumulation of a normally distributed random variable x(t) ∼ N(m,σDD) where m and σDD, the mean and standard deviation of the normal distribution, respectively, were constant values for a given trial. To simulate the asymmetric reward task, the preferred and null reward contexts have different values for m (mPREF > mNULL) and the same σDD. The rate parameter λ(t)=∑0tx(t)+βPREF or NULL, where βPREF or NULL approximated the baseline firing rate for the two reward contexts, respectively. To simulate the motion discrimination task, m varied with signed motion coherence (C), i.e., m = kC, and σDD was set at 1. Accumulation stopped if either of the two choice bounds (+A, −B) was crossed, and the choice was assigned by the first crossed bound (i.e., if +A was reached first, the trial ended with a PREF choice; if −B was reached first, the trial ended with a NULL choice). Trials that did not reach a bound within 4,000 steps (i.e., 4,000 ms) were discarded. As the firing rate dispersion is usually asymmetric between PREF and NULL choices, the rate parameter λ(t) was scaled differently for the two choices: λ(t)=αPREF or NULL∑0tx(t)+βbase, where βbase approximated the baseline firing rate.

In three models that do not implement accumulation but display the same mean and same average variance over the simulated period of the rate parameter as the DD model, λ(t) was proportional to independent samples from normal distributions with time-varying mean: For the “varT” model, x(t) ∼ N(m × t, σDD × t); for the “stdT” model, x(t) ∼ N(m × t, σT × t); and for the “fixedV” model, x(t) ∼ N(m × t, σV). The particular values of σT and σV were set to match the average variance over 700 ms (the duration of the epochs of interest) to that of the “DD” and “varT” models. To simulate the asymmetric reward task, the rate parameter λ(t) = x(t) + βPREF or NULL. Parameters mPREF or NULL, σDD, and βPREF or NULL are the same as in the DD model. To simulate the motion discrimination task, the rate parameter λ(t) = αPREF or NULLx(t) + βbase, and parameters m = kC, σDD, αPREF or NULL, and βbase are the same as in the DD model. Choice outcome, trial truncation, and baseline period were determined by the DD model. For example, if a trial from the DD model was terminated at t with a PREF choice, the corresponding trials for the other three models were simulated with αPREF and truncated at time t.

To approximate the ranges of firing rate for neural data, cell-matched simulations were performed using parameters extracted from neural data. For each neuron on the motion discrimination task, 10 parameters were extracted: 1) the average number of trials across all coherence; 2) the onset time of choice selectivity, used to simulate the sensory delay (Tchoice); 3) the baseline firing rate 100 ms before Tchoice, used to specify βbase; 4) the average slopes of the ramping activity during 200 ms after Tchoice across coherence levels, separately for the preferred and null choices, used to specify αPREF and αNULL, respectively; and 5) A, B, k, and nondecision times obtained by fitting the behavioral psychometric and chronometric functions obtained from the same session to a drift-diffusion model using maximum likelihood methods (Ding and Gold 2010, 2012; Hanks et al. 2006; Palmer et al. 2005).

For each neuron on the asymmetric reward task, five parameters were extracted: the number of trials, the average slope in firing rate over the epoch for the two reward contexts (used as mPREF and mNULL, respectively), and the intercepts (used as βPREF and βNULL, respectively). Because there is no constraint from behavioral data on the asymmetric reward task, multiple values of σDD were tested for how well the simulated VarCE and correlation trajectories approximate the experimental ones (see results).

A likelihood measure was used to assess how well a model captures the temporal dynamics patterns of the experimental data. Specifically, trajectories from 20 iterations of simulations were used to estimate the mean and standard deviation at each time bin. The log likelihood of obtaining the experimental value at a given time bin was computed assuming a normal distribution and summed across bins. As the focus was on the time course of each trajectory, instead of its baseline value, a simulated trajectory was vertically shifted to match its value for the first time bin to that of the experimental trajectory. This first time bin was excluded from likelihood summation.

RESULTS

I analyzed data previously obtained from two monkeys performing an asymmetric reward saccade task (Ding and Hikosaka 2006) and two different monkeys performing a visual motion direction discrimination task (Ding and Gold 2010, 2012). As reported previously, subsets of caudate and FEF neurons showed similar task-relevant modulation of mean firing rates. For the asymmetric reward task, during a “pretarget” period while the monkey waited for the visual target, the visual environment remained constant but the reward context differed between blocks. Both regions exhibited pretarget activity that differentiated between reward contexts (Fig. 1C; n = 50 and 38; linear regressions, test for H0: no dependence on reward context before target appearance, P = 0.0001 and 0.009, respectively, for caudate and FEF). In addition, the average firing rate for the preferred reward context increased with time for both regions.

For the motion discrimination task, during motion stimulus viewing the reward context remained constant but the sensory evidence differed between trials. Neural activity in both regions was modulated by the strength of sensory evidence used to form the direction decision and the decision itself (Fig. 1D; n = 29 and 43; linear regressions, test for H0: no dependence on choice or signed motion coherence during motion viewing, P < 0.0001 for both choice and signed coherence for caudate neurons; P < 0.0001 and P = 0.0004, respectively, for FEF neurons). The similarity was especially strong for the early motion viewing period. Later in the motion viewing period, caudate activity for the PREF choice decreased, whereas FEF activity largely maintained a rising pattern (Ding and Gold 2010, 2012). In addition, the average firing rate for the preferred choice increased with viewing time for both regions. These results from analyzing average firing rate thus appear to support the assumption that, for both tasks, the cortex simply relays asymmetric reward and visual-motion information to the basal ganglia. However, as detailed below, analyses of higher-order statistical structure in the responses challenge this assumption.

VarCE.

The measurement of VarCE assumes that a neuron's spike trains can be generated by a doubly stochastic Poisson process (Churchland et al. 2011). In such a process, the rate parameter for the Poisson process is itself stochastic. VarCE measures the portion of the spike count variance that can be attributed to the stochastic rate parameter and complements more commonly used measures of total spike count variability, such as Fano factor, coefficient of variation, etc. (Brostek et al. 2013). It has been especially useful as a signature of neural computations related to temporal accumulation in the context of decision making (Churchland et al. 2011; de Lafuente et al. 2015; Marcos et al. 2013). Specifically, a doubly stochastic Poisson process can generate different patterns of VarCE and temporal correlation, depending on whether or not the underlying rate parameter is a direct product of temporal accumulation (Churchland et al. 2011): Accumulation results in upward-ramping mean spike count, VarCE, and correlation, whereas time-independent samples with the same upward-ramping mean spike count produce much dampened VarCE and flat correlation. I therefore examined whether and how VarCE for the ramping activity in FEF and caudate neurons differed under the two task conditions and, more specifically, whether the VarCE trajectories follow a upward-ramping pattern indicative of temporal accumulation.

Operationally, VarCE is computed as the difference between the total spike count variance and the estimated variance for a Poisson process with the mean rate parameter. Computing VarCE thus requires estimating a proportionality factor that transforms mean firing rate to point-process variance, ϕ (Fig. 2A; also see methods). I found that this factor was not different between FEF and caudate for both tasks (Fig. 2, B and C; Wilcoxon rank sum test, P = 0.863 and 0.142, respectively). Moreover, scaling ϕ artificially by 1.4, 1.2, 0.8, or 0.6 did not qualitatively change the VarCE patterns (data not shown). I therefore consider the estimation of ϕ to be reasonably reliable for identifying potential differences in VarCE between FEF and caudate neurons for the two tasks.

I found several differences in VarCE computed for the caudate and FEF data from the two tasks. First, the overall magnitude of VarCE was greater for the FEF population than for the caudate population, for both tasks (Fig. 2, D and F, compare the ranges on the y-axes). Second, for the motion discrimination task, the VarCE for the caudate population exhibited a sharp early increase, peaked near 400 ms after motion onset, and then gradually decreased until the end of motion viewing (Fig. 2D, left; slope from linear regression with 100-ms time bins: 0.0077 bin−1, t-test for H0: zero slope, P = 0.231). In contrast, the VarCE for the FEF population maintained an increasing trajectory for the majority of the motion viewing period (Fig. 2D, right; slope: 0.029 bin−1, P = 0.0016). Third, for the asymmetric reward task, the VarCE for the caudate population maintained an increasing trajectory throughout most of the fixation period in anticipation of reward-predicting targets (Fig. 2F, left; slope: 0.018 bin−1, P < 0.0001). In contrast, the VarCE for the FEF population followed a decrease-increase-decrease trajectory instead of a monotonic ramp (Fig. 2F, right; slope: −0.0016 bin−1, P = 0.82).

I also observed differences in the VarCE measured from FEF versus caudate data for separate task-specific conditions (Table 1). For the motion discrimination task, the VarCE for FEF neurons had consistently positive slopes (i.e., ramped up) for the preferred choice at all coherence levels and more varied slope values for the null choice. In contrast, the VarCE for caudate neurons did not have reliably nonzero slopes for most conditions. For the asymmetric reward task, the slope in VarCE for FEF neurons was not different from zero for the preferred reward context, whereas the slope for caudate neurons was positive.

Table 1.

Condition-specific results

Choice Coh VarCE Slope VarCE P Corr. Slope Corr. P
Caudate—motion task PREF 0.032 0.025 0.062 0.016 0.333
0.064 0.019 0.189 0.001 0.825
0.128 −0.006 0.702 −0.006 0.581
0.256 0.041 0.038 −0.007 0.798
0.512 0.047 0.322 0.101 0.234
NULL 0.032 0.004 0.540 0.049 0.016
0.064 −0.017 0.066 0.011 0.679
0.128 −0.005 0.112 0.021 0.077
0.256 −0.009 0.204 −0.002 0.924
0.512 −0.030 0.001 −0.022 0.615
FEF—motion task PREF 0.032 0.073 0.030 0.037 0.160
0.064 0.069 0.000 0.047 0.008
0.128 0.112 0.000 0.038 0.001
0.256 0.130 0.011 0.018 0.267
0.512 0.124 0.046 0.033 0.027
NULL 0.032 0.029 0.049 0.020 0.330
0.064 0.018 0.142 0.037 0.135
0.128 −0.009 0.551 0.028 0.052
0.256 −0.020 0.521 0.012 0.707
0.512 −0.125 0.007 −0.031 0.623
Caudate—reward task PREF 0.031 0.000 0.026 0.012
NULL 0.005 0.012 0.021 0.020
FEF—reward task PREF 0.000 0.998 −0.002 0.804
NULL −0.003 0.457 0.003 0.672
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Variance of conditional expectation (VarCE) and temporal correlation were estimated by pooling data from all neurons for each condition. The slope was measured by linear regression with time bin as the regressor.

Coh, motion coherence; FEF, frontal eye field.

P values < 0.05 (t-test) are indicated in bold.

These population results were consistent with linear regression results from the VarCE computed for single neurons from the two brain regions. For the motion discrimination task, the median slope of VarCE as a function of time for individual caudate neurons was 0.0017 bin−1, which was not significantly different from zero (Fig. 2E, left; sign test, P = 0.38). In contrast, the median slope for individual FEF neurons was 0.032 bin−1, which was significantly greater than zero (Fig. 2E, right; P = 0.0011) and greater than those computed for caudate neurons (Wilcoxon rank sum test, P = 0.019). For the asymmetric reward task, the median slope for caudate neurons was 0.0098 bin−1 for caudate neurons, which was significantly greater than zero (Fig. 2G, left; P = 0.0031) and greater than those computed for FEF neurons (P = 0.0007). The median slope for FEF neurons was not different from zero (Fig. 2G, right; P = 0.20). A two-way ANOVA with brain region and task as factors indicated modulation by task (d.f. = 1, F = 5.64, P = 0.019) and by region × task interaction (d.f. = 1, F = 7.73, P = 0.006) but not by brain region alone (F = 1.62, P = 0.206).

The slope estimates from the entire viewing period so far indicated that the VarCE patterns were different between FEF and caudate neurons for the two tasks: The VarCE patterns for FEF neurons on the motion discrimination task and for caudate neurons on the asymmetric reward task seemed consistent with accumulation processes operating throughout the time epoch of interest, whereas other VarCEs showed nonmonotonic patterns inconsistent with continuous accumulation. To better characterize the temporal characteristics of these latter patterns, I estimated slope values for individual neurons within shorter epochs (300-ms sliding windows). As shown in Fig. 2, H–K, this piecewise analysis provided further support for brain region- and task-dependent differences in VarCE. For the motion discrimination task, during the early motion-viewing period VarCE slopes were similar in FEF and caudate, with similar median slope values (Fig. 2H) and proportion of neurons with positive slope values (Fig. 2I). However, at ∼350 ms after motion onset, the median slope of VarCE and the proportion of neurons with positive slopes were significantly larger for FEF neurons than for caudate neurons (Wilcoxon rank sum test, P = 0.018 and χ2-test, P = 0.046, respectively). Afterward, both regions showed a decreasing trend in median slope value and proportion of positive slopes, although >50% of FEF neurons showed positive slopes throughout the epoch, whereas in caudate a slight majority of neurons had negative slopes by the end of the epoch. These results suggested that VarCE for caudate neurons was consistent with an accumulation process only during early motion viewing. For the asymmetric reward task, the median slope value was consistently positive throughout the epoch for caudate neurons but more varied for FEF neurons (Fig. 2J). Similarly, the proportion of positive slopes was >50% throughout the epoch for caudate neurons but more varied for FEF neurons (Fig. 2K). The largest differences between brain regions were observed at the beginning of the epoch, where caudate neurons had relatively strong positive slopes and FEF neurons had relatively negative slopes (Wilcoxon rank sum test, P = 0.006 and χ2-test, P = 0.0009, respectively). Compared with values for the motion discrimination task, the peak median slope value for the asymmetric reward task was smaller for both regions (Fig. 2L). However, the average value was similar for caudate neurons between tasks (Fig. 2L), whereas the average value was much smaller for FEF neurons for the asymmetric reward task (Fig. 2L), suggesting that caudate neurons may be involved in accumulation-like processes for both tasks, whereas FEF neurons may only be involved for the motion discrimination task.

Collectively, these results revealed task-specific regional differences in VarCE between FEF and caudate neurons. Within the framework of a doubly stochastic Poisson process, VarCE is sensitive to task-relevant computations underlying the rate parameter. The regional differences in VarCE trajectories therefore suggest that, instead of constituting a general relay, FEF and caudate neurons may be involved in distinct computations depending on task context.

Temporal correlation.

The measurement of temporal correlation quantifies the relationship of spike count residuals between two neighboring time bins. Similar to VarCE, a process with temporal accumulation generates correlation that increases with time (Churchland et al. 2011). Unlike VarCE, this measure is agnostic about the underlying spike-generation process and thus is complementary to the analyses presented above.

Similar to the regional differences observed in the VarCE, I observed substantial task-dependent regional differences in temporal correlations measured from ramping activity of caudate and FEF neurons. The first subdiagonal correlation coefficients (correlation in short) reflect spike count correlations between neighboring time bins. As shown in Fig. 3, correlation values tended to be slightly higher for FEF activity than caudate activity for both tasks (brighter pixels for the FEF matrices, Fig. 3, A and D; also compare traces in Fig. 3, B and E). For the motion discrimination task, correlation values tended to ramp up as a function of time within a trial for both caudate (albeit with a slight decrease after ∼600 ms, as seen in Fig. 3B, left; slope of linear regression = 0.017 bin−1, P = 0.044) and FEF (Fig. 3B, right; slope = 0.027 bin−1, P = 0.0083) activity. In contrast, for the asymmetric reward task, correlation values tended to ramp up over time in caudate but not FEF activity (Fig. 3E; caudate slope = 0.025 bin−1, P = 0.0052; FEF slope = −0.0001 bin−1, P = 0.99).

I also observed differences in temporal correlations measured from FEF versus caudate data for separate task-specific conditions (Table 1). For the motion discrimination task, mean correlations ramped up over time (i.e., have positive slopes computed via linear regression) for trials with the preferred choice for FEF, but less consistently for caudate, neurons. In contrast, for the asymmetric reward task, positive slopes were observed for both reward contexts in caudate neurons but not FEF neurons.

These patterns of correlation dynamics as a function of time, measured from the pooled population data, were consistent with results from single-neuron analysis. For the motion discrimination task, the median slope of correlation values versus time across cells was 0.013 bin−1 (sign test, P = 0.0011) for caudate neurons and 0.024 bin−1 (P < 0.0001) for FEF neurons (Fig. 3C). These slope values were larger for FEF neurons than for caudate neurons (Wilcoxon rank sum test, P = 0.033). For the asymmetric reward task, the median slope from single-neuron analysis was 0.013 bin−1 (P = 0.0031) for caudate neurons and −0.0058 bin−1 (P = 0.32) for FEF neurons (Fig. 3F). These slope values were significantly larger for caudate neurons than for FEF neurons (Wilcoxon rank sum test, P = 0.005). A two-way ANOVA with brain region and task as factors indicated modulation by task (d.f. = 1, F = 19.68, P < 0.0001) and by region × task interaction (d.f. = 1, F = 19.38, P < 0.0001) but not by brain region alone (F = 0.0038, P = 0.951). Piecewise estimation of the slope values was not performed because the computation of temporal correlation resulted in too few time samples.

The region- and task-dependent VarCE and correlation patterns were robust with respect to choices of bin size. I repeated the analyses using different bin sizes (60, 80, 100, and 120 ms). As shown in Table 2, for all bin sizes tested median VarCE slope was significantly different from zero only for FEF neurons for the motion discrimination task and for caudate neurons for the asymmetric reward task and not for other region-task combinations. Similarly, for all bin sizes tested, median correlation slope was significantly different from zero for the FEF neurons for the motion discrimination task. A similar pattern of results was evident in the caudate-asymmetric reward combination, except for the 60-ms bin size. For all bin sizes tested, two-way ANOVAs also consistently indicated modulation by the region × task interaction for the slopes of VarCE and correlation and by task for the slopes of correlation.

Table 2.

Statistical results from analyses using different bin sizes

Bin size 100 ms, used in figures
Motion
 Reward
Caudate FEF Caudate FEF
    Median VarCE slope 0.034 0.636 0.197 −0.130
    Sign test P 0.382 0.001 0.003 0.203
    Median Corr. slope 0.129 0.236 0.134 −0.058
    Sign test P 0.001 0.000 0.003 0.322
    ANOVA Region
 Task
 Region × Task
F P F P F P
        Slope of VarCE 0.686 0.409 4.271 0.040 8.737 0.004
        Slope of Corr. 0.678 0.411 15.991 0.000 17.923 0.000
Bin size 120 ms
Motion
 Reward
Caudate FEF Caudate FEF
    Median VarCE slope 0.039 0.778 0.176 −0.133
    Sign test P 0.672 0.000 0.015 0.322
    Median Corr. slope 0.127 0.273 0.154 0.019
    Sign test P 0.003 0.000 0.033 0.322
    ANOVA Region
 Task
 Region × Task
F P F P F P
        Slope of VarCE 1.669 0.198 2.261 0.134 5.405 0.021
        Slope of Corr. 0.232 0.631 6.115 0.014 7.048 0.009
Bin size 80 ms
Motion
 Reward
Caudate FEF Caudate FEF
    Median VarCE slope −0.002 0.517 0.102 −0.038
    Sign test P 1.000 0.000 0.033 0.672
    Median Corr. slope 0.021 0.176 0.071 −0.085
    Sign test P 0.480 0.007 0.033 0.203
    ANOVA Region
 Task
 Region × Task
F P F P F P
        Slope of VarCE 1.264 0.262 1.155 0.284 5.490 0.020
        Slope of Corr. 0.009 0.924 9.951 0.002 10.370 0.001
Bin size 60 ms
Motion
 Reward
Caudate FEF Caudate FEF
    Median VarCE slope 0.000 0.319 0.120 0.007
    Sign test P 1.000 0.000 0.015 0.888
    Median Corr. slope 0.095 0.174 0.043 −0.083
    Sign test P 0.033 0.001 0.119 0.203
    ANOVA Region
 Task
 Region × Task
F P F P F P
        Slope of VarCE 1.415 0.236 0.679 0.411 6.385 0.012
        Slope of Corr. 0.868 0.353 15.093 0.000 7.637 0.006
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Slope values are in s−1. P values < 0.05 are indicated in bold.

Collectively, these results indicate that, despite similar mean firing rate patterns, caudate and FEF spike trains showed distinct VarCE and temporal correlation patterns for the two tasks. The FEF population shows increasing VarCE and temporal correlation during motion viewing on the perceptual decision task but not in anticipation of reward-predictive targets on the asymmetric reward task. In contrast, the caudate population shows increasing VarCE and temporal correlation in anticipation of reward-predictive targets on the asymmetric reward task and during early motion viewing on the perceptual decision task.

Cell-matched simulations.

To examine whether an accumulation-like computation can account for the observed VarCE and correlation time courses in caudate and FEF neurons, I next performed simulations with four doubly stochastic Poisson processes that differ in whether temporal accumulation is directly involved. In the “DD” model, the rate parameter was directly governed by an accumulation-to-bound process. In the other models, the rate parameter was determined by independent samples from stochastic processes with ramping mean values (e.g., reflecting accumulation that occurs elsewhere) and with variance increasing linearly with time (“varT” model) or quadratically with time (“stdT” model) or staying constant (“fixedV” model).

I focused on four questions. First, for the motion discrimination task, given that the ramping activity in FEF and caudate neurons during motion viewing has been postulated to reflect temporal accumulation of visual motion evidence (Ding and Gold 2010, 2012; Kim and Shadlen 1999), can a temporal accumulation process with cell-matched and behaviorally constrained parameters approximate the observed VarCE and correlation patterns? Second, on the asymmetric reward task, although the reward context is known to the monkeys and constant during the pretarget period, the magnitude of reward-context modulation increases with time in cortical activity, caudate activity, and saccade latency (Coe et al. 2002; Ding and Hikosaka 2006, 2007; Kobayashi et al. 2007; Lauwereyns et al. 2002; Takikawa et al. 2002). These observations are consistent with an accumulation-like underlying computation. In relation to our data set, can a temporal accumulation process with cell-matched parameters approximate the observed VarCE and correlation patterns in FEF and/or caudate activity? Third, for each task, can a common process capture the patterns for both FEF and caudate neurons? In other words, are the observed regional differences in VarCE and correlation due to differences in basic firing properties or differences in the computations performed by the two brain regions? Fourth, for each brain region, can a common process capture the VarCE and correlation patterns on both tasks? In other words, can a brain region be involved in a general-purpose computation to meet different task demands?

As detailed in methods, for the motion discrimination data set simulation parameters were fully specified by the experimental data. With these parameters, all four models produced mean firing rates that were modulated by choice and motion coherence (Fig. 4, A and B). Because the DD model used bounded accumulation while the rate parameter for the other models was not bounded, the excursion of the mean firing rate was larger in the non-DD models. As my focus was on comparing experimental and simulation results in terms of the time course of the VarCE and correlation curves and not their absolute average values (the latter of which may be influenced by task-independent factors), I shifted the simulated VarCE and correlation coefficient values vertically to match the corresponding experimentally measured values at the first time bin.

Fig. 4.

Fig. 4.

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Cell-matched simulation results for the motion discrimination data sets. A and B: mean firing rates from the caudate (A) and FEF (B) data sets for the experimental (left) and simulation (4 right panels) data. C: VarCE computed from experimental (black; mean only) and simulation [red, drift-diffusion (DD); blue, variance increasing linearly with time (varT); green, variance increasing quadratically with time (stdT); magenta, variance staying constant (fixedV); means ± SE] data for caudate neurons. The simulated curves were shifted vertically to match the experimental values at the first time bin. D: correlation curves computed from experimental and simulation data. Same conventions as C. The simulated curves were shifted vertically to match the experimental values at the first time bin. E: sum of log likelihood for VarCE and correlation trajectories of different models, for the caudate data set. Values below −10 were truncated. Open bars, sum of log10(LL) for all time bins; filled bars, the sum of log10(LL) for bins 2–8 for VarCE and for bins 2–4 for correlation. F–H: same conventions as C–E, for the FEF data set.

Overall, the model with accumulation (DD) was better at capturing the patterns of VarCE and correlation than the models without accumulation (varT, stdT, and fixedV) (Fig. 4, C–H). VarCE trajectories from the DD model were similar to experimental curves during the early motion viewing period for caudate neurons (Fig. 4C) and throughout the motion period for FEF neurons (Fig. 4F). Only the DD model produced the rising temporal correlation patterns seen in experimental results, while the other models produced flat trajectories for both regions (Fig. 4, D and G). To quantify these visual impressions, I computed the log10 likelihood (LL) of observing the experimental results for the four models (Fig. 4, E and H; values were truncated at −10 for better visualization). I considered a positive sum of log(LL) as indication of a good match and a >1 difference in the mean log(LL) between two models as strong evidence for one model prevailing over the other (Kass and Raftery 1995). On the basis of these criteria, for the full motion viewing epoch, the DD model provided good approximation for FEF neurons (Fig. 4H) but not caudate neurons (Fig. 4E), although the match was significantly improved if only the early motion viewing epoch was considered (Fig. 4E). For either the full or early motion viewing epoch, the DD model consistently outperformed other models for both brain regions: The differences in mean log(LL) between the DD model and the other three models were all >1.6. Scaling down the mean firing rate excursion of the non-DD models, by applying a single scale factor for all neurons, resulted in a slight decrease in VarCE and no significant change in the time course of correlation (data not shown), which worsened the match to experimental results for these models. These results suggest that the temporal dynamics of FEF and caudate activity is more consistent with an underlying computation that accumulates motion evidence. Furthermore, because the DD model used parameters that were fitted to behavioral data and produced continuously increasing VarCE for both FEF and caudate data sets, the simulation results suggest that the plateau-decrease VarCE pattern for caudate neurons during late motion viewing was likely not an artifact of early decision termination in those recording sessions but rather reflected disengagement of those neurons from an ongoing, behaviorally relevant accumulation process.

For simulations of the motion discrimination data set, the relationship between mean firing rate and σDD was constrained by behavioral performance. The asymmetric-reward data set, however, only specified the simulation parameters that are related to the mean firing rates, leaving σDD as an unconstrained guess parameter in simulations. As expected, using the cell-matched parameters, all models produced qualitatively similar reward context-modulated ramping mean firing rates for both brain regions (Fig. 5, A and B; data not shown for non-DD models). The accompanying VarCE and correlation patterns, on the other hand, depended on the particular values of σDD and the model identity. I therefore performed simulations using different σDD values to examine whether any σDD-model combination can approximate the observed VarCE and correlation patterns.

Fig. 5.

Fig. 5.

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Cell-matched simulation results for the asymmetric reward data sets. A and B: mean firing rates from the caudate (A) and FEF (B) data sets for the experimental (left) and simulation data (right; DD model only). C: simulated VarCE and correlation with the DD model, σDD = 0.02 and caudate neuron parameters. The simulated curves were shifted vertically to match the experimental values at the first time bin. D–F: same conventions as C. G and H: sum of log likelihood of both VarCE and correlation trajectories as a function of σDD for different models. G: the caudate data set. H: the FEF data sets. Values were truncated at −150. Positive values are indicated with filled symbols. Arrows indicate values that correspond to examples in C–F.

Figure 5, C–F, illustrate the simulated VarCE and correlation curves with the DD and varT models using two σDD values. With σDD = 0.02, the DD model generated VarCE and correlation curves that were similar to the experimental curves for caudate neurons (Fig. 5C) but failed to do so for FEF neurons (Fig. 5E). With σDD = 0.04, the varT model generated a VarCE curve that was similar to the experimental curve for caudate neurons but with a flat correlation curve that deviated far from the experimental curve (Fig. 5D). With the same σDD value, the varT model failed to match VarCE and correlation curves to the corresponding experimental curves for FEF neurons (Fig. 5F), although it was better than the DD model in Fig. 5E. In my sample of σDD values, only the DD model with σDD values in the range of ∼0.016–0.024 produced positive sum log10LL values for caudate activity (Fig. 5G). No combination produced positive sum log10LL values for FEF activity, although the non-DD models outperformed the DD model at most σDD values (Fig. 5H). These results suggest that, for the asymmetric reward task, a temporal accumulation process can account for the observed VarCE and correlation patterns in caudate neurons but not for FEF neurons.

Combining all simulation results, for the motion discrimination task temporal accumulation can capture the patterns of VarCE and correlation for FEF neurons during the full motion viewing epoch and for caudate neurons during the early motion viewing epoch. For the asymmetric reward task, none of the models simulated here can account for the VarCE and correlation patterns of both FEF and caudate neurons, suggesting that the two regions have different functional roles on this task. When considered across tasks, the temporal dynamics of caudate activity is consistent with a general-purpose accumulation-like computation for both tasks, whereas the temporal dynamics of FEF activity does not reflect similar underlying computations on the two tasks, despite similar task-relevant ramping mean firing rates. These results further suggest that the regional differences in VarCE and correlation are more likely due to task-specific presence/absence of temporal accumulation-like underlying computations than differences in basic firing properties.

DISCUSSION

To better understand functional differences between the prefrontal cortex and striatum, I analyzed spike count properties of subsets of neurons previously recorded in monkeys performing two decision tasks: a motion discrimination task used extensively for studying perceptual decisions and an asymmetric reward task used extensively for studying reward-biased behaviors. These neurons showed gradually increasing (ramping up) mean firing rates for the preferred condition (preferred choice or reward context). Despite the similarities in mean firing rate, my analyses of spike count variability and temporal correlation patterns revealed task-specific regional differences. The FEF population showed increasing VarCE and temporal correlation throughout the motion viewing period on the perceptual decision task but not in anticipation of reward-predictive targets on the asymmetric reward task. In contrast, the caudate population showed increasing VarCE and temporal correlation in anticipation of reward-predictive targets on the asymmetric reward task and during early motion viewing on the perceptual decision task. Compared with simulation results of models with and without temporal accumulation, these patterns suggest that FEF and caudate neurons are more involved in temporal accumulation of visual motion information to reach a perceptual decision and temporal accumulation of reward context-related information to modulate saccade behavior, respectively.

Before discussing the functional implications of these results, I note a potential caveat. As acknowledged above, the partition of spike count variability for VarCE estimation relies on the assumptions that spike trains can be characterized as a doubly stochastic process and the appropriate portion of point-process variance can be reliably estimated (Churchland et al. 2011). These assumptions have been examined and applied in analyses of cortical activity (Churchland et al. 2011; Geisler and Albrecht 1995; Goris et al. 2014; Lee and Seo 2011; Nawrot et al. 2008). In contrast, the validity of these assumptions for striatal activity remains to be more rigorously tested. Nevertheless, several considerations suggest that the observed regional differences are robust and interpretable. First, corticostriatal and striatal neurons show similar firing properties in anesthetized rats, in terms of the dependence on state transitions of membrane potential, the shape of interspike interval distribution, and spike train variability (Stern et al. 1997). Consistent with these findings from intracellular neuron recordings, estimates of ϕ in this study, which were based on extracellular neuron recordings and controlled the point-process variance estimation, did not differ between FEF and caudate neurons. In addition, doubly stochastic processes with cell-specific parameters can capture the temporal patterns of VarCE and correlation of caudate neurons during early motion viewing on the perceptual task and before target onset on the asymmetric reward task. These results suggest that the first assumption is a reasonable approximation for the purpose of this study. Second, artificially scaling ϕ did not change the patterns of region- and task-specific difference, suggesting that these patterns are robust with respect to the precision of estimating the portion of point-process variance. Third, the differences in VarCE were also accompanied by congruent differences in correlation patterns. As the measurement of correlation did not rely on the validity of the above assumptions, these results thus gave me confidence that the observed regional differences reflect functionally important modulation of spike count statistics.

What can we infer from these task-dependent patterns of VarCE and temporal correlation? In line with a previous report, simulations based on an accumulation model predicted a gradual increase of VarCE and temporal correlation (Churchland et al. 2011). For the motion discrimination task, although both caudate and FEF activity show patterns of mean firing rate consistent with motion evidence accumulation (Ding and Gold 2010, 2012; Kim and Shadlen 1999), FEF activity was more robust in exhibiting the predicted patterns of VarCE and temporal correlation, similar to LIP activity (Churchland et al. 2011), and caudate activity exhibited rising patterns only during early motion viewing. These observations suggest that cortical neurons are more intimately involved in the temporal accumulation of motion evidence to form the final decision. During the early accumulation phase, the rate of rise in VarCE of caudate activity was approximated by the DD model, but the rise in correlation was less than predicted. The varT and stdT models, with ramping mean and independent samples that may be thought of as products of pooling, produced increasing VarCE and flat correlation. These results thus suggest a pooling-relay scheme, where caudate neurons inherit accumulated signals from cortical areas through convergent corticostriatal projections (Bogacz and Gurney 2007; Lo and Wang 2006; Rao 2010; Zheng and Wilson 2002). During the late motion viewing phase, mean firing rate, VarCE, and correlation all deviate from predictions of continued accumulation. Why and how caudate neurons transition out of the accumulation process remain to be examined. It is interesting to speculate whether these caudate neurons participate in an initial assessment of trial difficulty for online adjustment of decision formation and how the transition times of these caudate neurons influence behavioral performance.

For the asymmetric reward task, despite a similar prevalence of caudate and FEF neurons demonstrating reward-context modulation of activity before target presentation (Ding and Hikosaka 2006), caudate neurons showed the rising VarCE and temporal correlation patterns as predicted by the DD model, while the nonmonotonic time courses for FEF neurons do not conform to predictions of any of the models studied here. These observed regional differences are inconsistent with a simple FEF-caudate feedforward relay scheme as for the motion discrimination task. It remains to be tested whether a DD model incorporating information multiplexing as observed in other high-order cortical neurons could approximate the temporal dynamics of FEF activity (Meister et al. 2013; Park et al. 2014). A systematic application of temporal dynamics analyses to other brain regions involved in reward-biased behavior may provide insights into the underlying functional circuitry.

It has been proposed that, similar to sensory evidence, the value of different stimuli may also be accumulated over time to support value-based decisions in human subjects (Krajbich et al. 2010, 2012; Towal et al. 2013). My results further extend the idea that accumulation is a general brain computation that can be applied to a broad range of information. The rising mean firing rates, VarCE, and correlation in caudate activity before target onset suggest that, even in the absence of stimuli, an accumulation-like computation is involved to support asymmetric reward-biased behavior. Furthermore, because all of these characteristics in caudate activity can be well approximated by the DD model, which accumulates a noisy, reward context-related quantity, the accumulation-like computation may operate on an internal reward context-related signal. This interpretation may appear puzzling: For the asymmetric reward task, reward context was altered in blocks of trials such that the current reward context was known to the monkeys and constant during the pretarget period. In other words, unlike for the motion discrimination or value-based decision tasks, where accumulation is needed to identify the stimulus or the higher-valued option, here the monkeys do not need to accumulate information to determine the identity of current reward context. It is thus unclear why an accumulation-like computation is required. I speculate that the accumulation-like computation is a part of the process that incorporates the identity of current reward context into a behavioral bias. For example, biases in saccade latency have been consistently observed on asymmetric reward saccade tasks. For the same saccade target, latency is shorter if the larger reward is expected (Lauwereyns et al. 2002). Consistent with the idea of temporal accumulation, the magnitude of such a bias depends on the duration of the pretarget period, with larger bias corresponding to longer pretarget duration (Ding and Hikosaka 2007). The behavioral benefits of the time-dependent bias remain to be identified, but caudate neurons may accumulate a noisy neural representation of the fixed reward context information to contribute to the saccade latency bias. Alternatively, caudate neurons may contribute to time estimation, which has also been proposed to use accumulation as the underlying computation (Merchant et al. 2013; Treisman 1963), and the pretarget activity reflects reward-context modulation of the accumulated timing information. Under the present task conditions, these two schemes are mathematically equivalent. It awaits further investigation how the reward context-dependent accumulation process relates to other accumulation-like computations for time estimation and/or preparation for self-initiated movement (Schurger et al. 2012).

In summary, focusing on neurons with similar task-relevant modulation in mean firing rates, analyses of spike count variability and temporal correlations suggest that the subpopulations of neurons in FEF and the caudate nucleus have different functional roles in the accumulation of reward context and sensory information. Because caudate neurons appear to be directly involved in the accumulation of reward context information and to inherit signals reflecting accumulated sensory evidence from FEF, they may serve strategic roles in forming decisions that must appropriately combine sensory and reward context information. Combined with differences in mean firing rate patterns observed in the larger population of the two regions, these results suggest that, instead of serving as a simple relay, the inner workings of the fronto-striatal network are flexibly controlled to meet diverse task demands. Future efforts to understand computations underlying the fronto-striatal interaction and the functional roles of the larger cortico-basal ganglia network will likely benefit from analyses of VarCE, correlations, and other response properties beyond mean firing rates.

GRANTS

This work was supported by National Eye Institute Grant 1R01 EY-022411.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: L.D. conception and design of research; L.D. performed experiments; L.D. analyzed data; L.D. interpreted results of experiments; L.D. prepared figures; L.D. drafted manuscript; L.D. edited and revised manuscript; L.D. approved final version of manuscript.

ACKNOWLEDGMENTS

I thank Anne Churchland and Michael Shadlen for generous sharing of source code and tips for the VarCE analysis and Joshua Gold, Yin Li, and Takahiro Doi for providing valuable comments and suggestions.

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