We next evaluate the hypothesis that the drift-diffusion signal, Sx⁢(t), is the DV that controls the choice and response time. We have identified several coding directions that produce candidate DVs, and as we will see below, there are also other coding directions of interest that can be derived from the population. Additionally, PCA indicates that the dimensionality of the data is low, but greater than 1 (participation ratio = 4.4 ± 1.3; Mazzucato et al., 2016; Gao et al., 2017). Therefore, one might wonder whether it is sensible to assume that the DV can be approximated by a scalar measure arising from a single coding direction as opposed to a higher dimensional representation. Two decoding exercises are adduced to support the assumption.

We constructed a logistic decoder of choice using each neuron’s spike counts in 50 ms bins between 100 and 500 ms after motion onset. As shown in Figure 4a, this What-decoder (orange) predicts choice as accurately as a decoder of simulated data from a drift-diffusion model (black) using parameters derived from fits to the monkeys’ choice and RT data (see ‘Methods’). The simulation establishes a rough estimate of the decoding accuracy that can be achieved, given the stochastic nature of the choice, were we granted access to the drift-diffusion signal that actually determines the decision. In this analysis, the decoder can use a different vector of weights at each point in time (time-dependent coding directions; see Peixoto et al., 2021). However, if the representation of the DV in LIP is one-dimensional, then a decoder trained at one time should perform well when tested at a different time. The red curve in Figure 4a shows the performance of a What-decoder with a fixed training-time (450 ms after motion onset; red arrow). This decoder performs nearly as well as the decoder trained at each time bin. The heatmap (Figure 4b) generalizes this observation. It shows two main features for all times 300<t<500 ms (dashed box). First, unsurprisingly, for a What choice decoder trained on data at one time t=x, the predictions improve as the testing time advances (the decoding accuracy increases along any vertical) as more evidence is accrued. Second, and more importantly, decoders tested at time t=y perform similarly, independent of when they were trained (there is little variation in decoding accuracy along any horizontal). This observation suggests that a single vector of weights may suffice to decode the choice from the population response.

The second decoder is trained to predict whether a saccade to the contralateral choice target will be initiated in the next 150 ms. This When-decoder is trained by logistic regression to predict a binary output: 1 at all time-points that are within 150 ms of an upcoming saccade and 0 elsewhere (Figure 4—figure supplement 1). We validated the When-decoder by computing the area under an ROC (AUC) using the held-out (odd) trials (mean AUC over all time points: 0.84), but this is tangential to our goal. Although the When-decoder was trained only to predict the time of saccades, our rationale for developing this decoder was to test whether the When coding direction can be used to predict the choice. The green trace in Figure 4a shows the accuracy of SWhen⁢(t) to predict the choice. The performance is almost identical to the choice decoder, despite being trained on a temporal feature of trials ending in the same left choice. This feat is explained by the similarity of signals produced by the When- and other coding directions. Note the similarity of the trial-averaged SWhen signals displayed in Figure 4c to those in Figure 2 (see also Figure 4—figure supplement 2, right). Indeed, the cosine similarity between the When and Ramp coding directions is 0.67±0.03 (Figure 4d). In light of this, it is not surprising that the weighting vectors derived from both the What- and When-decoders also render single-trial drift-diffusion traces that resemble each other and those rendered by other coding directions (Figure 4e). Together these analyses support the assertion that the DV is likely to be captured by a single dimension, consistent with Ganguli et al., 2008.

If the one-dimensional signals, Sx⁢(t), approximate the DV, they should explain the variability of choice and RT for trials sharing the same direction and motion strength. Specifically, (i) early samples of Sx⁢(t) should be predictive of choice and correlate inversely with the RT on trials that result in contraversive (leftward) choices, (ii) later samples ought to predict choice better and correlate more strongly (negatively) with RT than earlier samples, and (iii) later samples should contain the information present in the earlier samples and thus mediate (i.e., reduce the leverage) of the earlier samples on choice and RT. Each of these predictions is borne out by the data.

The analyses depicted in Figure 5 allow us to visualize the influence of the single-trial signals, Sx⁢(t), on the choice and RT on that trial. We focus on the early epoch of evidence accumulation (200–550 ms after random dot motion onset) and restrict the analyses to decisions with RT≥670 ms and coherence≤6.4%. The RT restriction eliminates 17% of the eligible trials. Larger values of Sx⁢(t) are associated with a larger probability of a left (contraversive) choice and a shorter RT, hence negative correlation between Sx⁢(t) and RT. We use the term, leverage, to describe the strength of both of these associations. The leverage on choice (Figure 5a, black traces) is the contribution of Sx⁢(t) to the log odds of a left choice, after accounting for the motion strength and direction (i.e., the coefficients, β1⁢(t) in Equation 8). The leverage on RT (Figure 5b) is the Pearson correlation between Sx⁢(t) and the RT on that trial, after accounting for the effect of motion strength and direction on Sx and RT (see ‘Methods’). The leverage is evident from the earliest sign of evidence accumulation, 200 ms after motion onset, and its magnitude increases as a function of time, as evidence accrues (Figure 5, top). The filled circle to the right of the traces in each graph shows the leverage of Sx at t=550 ms, which is 120 ms before any of the included trials have terminated. Both observations are consistent with the hypothesis that Sx represents the integral of noisy evidence used to form and terminate the decision. Two control analyses demonstrate that the degree of leverage on choice and RT do not arise by chance: (i) random coding directions in state space produce negligible leverage (Figure 5—figure supplement 1, top), and (ii) breaking the trial-by-trial correspondence between neural activity and behavior eliminates all leverage (see Reviewer Figure 1 in reply to peer review).

Importantly, the leverage at earlier times is mediated by the later sample at t=550 ms. The blue traces in all graphs show the remaining leverage once this later sample is allowed to explain the choice and RT—by including, respectively, an additional term in the logistic regression (Equation 9) and calculating the partial correlation, conditional on Sx⁢(t=0.55). We assessed statistical significance of the mediation statistics, ξCh and ξRT (Equations 7 and 10) in each session for the three signals shown in Figure 5 using a bootstrap procedure (see ‘Methods’, Equation 10). Mediation is significant in 47 of the 48 comparisons (all p<0.023, median p<10−317). The one non-significant comparison is ξCh for STincon in session 2 (p=0.73). The mediation is significant when this comparison is included in the combined data (p<10−9, paired samples t-test). The stark decrease in leverage is consistent with one-dimensional diffusion in which later values of the signal contain the information in the earlier samples plus what has accrued in the interim. Had we recorded from all the neurons that represent the DV, we would expect the mediation to be complete (e.g., partial correlation = 0). However, our recorded population is only a fraction of the entire population. Indeed, the observed degree of mediation is similar to values obtained from simulations of weakly correlated, noisy neurons (Figure 5—figure supplement 1, bottom).

There is one additional noteworthy observation in Figure 5 that highlights the importance of the Tincon neurons. The top and middle rows (Sramp and SPC1) contain a second, open symbol, which is simply a copy of the filled symbol from the bottom row (STincon). The yellow traces show significant cross-mediation of Sramp and SPC1 by the sample, STincon⁢(t=0.55) (all p<0.05; median p<10−268; bootstrap as above). This signal, carried by 9–21% of the neurons, mediates signals produced by the full population of 54–203 neurons nearly as strongly as Sramp and SPC1 mediate themselves. The observation suggests that minimal leverage is gained by sophisticated analyses of the full NSS compared to a simple average of Tincon neurons. This is both reassuring and disquieting: reassuring because the Tincon neurons compose the dominant projection from LIP to the portions of the superior colliculus (SC) and the frontal eye field involved in the generation of saccades toward the contralateral choice target (Paré and Wurtz, 1997; Ferraina et al., 2002); disquieting because the functional relevance of these neurons is not revealed by the other coding directions. The weights assigned to the Tincon neurons span all percentiles (mean IQR: 49–96; mean 71st percentile, A⁢U⁢C=0.74±0.05) in the ramp coding direction. They contribute disproportionately to PC1 and the What- and When- decoders but not enough to stand out based on their weights. Indeed, the ability to predict that a neuron is Tincon from its weight or percentile is remarkably poor (Figure 5—figure supplement 2).

These observations support the idea that the single-trial signals, Sramp, SPC1, and STincon, approximate the DV used by the monkey to make its decision. In Figure 5—figure supplement 3, we show that the SWhat and SWhen coding directions achieve qualitatively similar results. Moreover, a late sample from Sx⁢(t) mediates the earlier correlation with RT and choice of signals rendered by other coding directions, Sy⁢(t), at earlier times. Such cross-mediation is consistent with the high degree of cosine similarity between the coding directions (Figure 4d). The observation suggests that the DV is a prominent signal in LIP, discoverable by a variety of strategies, and consistent with the idea that it is one-dimensional. In Figure 4—figure supplement 4, we show that linear and nonlinear decoders achieve similar performance, which argues against a non-linear embedding of the DV in the population activity.

Up to now, we have focused our analyses on resolving the DV on single trials, paying little attention to how it is computed or to other signals that may be present in the LIP population. The drift-diffusion signal approximates the accumulation, or integral, of the noisy momentary evidence—a signal approximating the difference in the firing rates of direction-selective (DS) neurons with opposing direction preferences (e.g, in area MT; Britten et al., 1996). DS neurons, with properties similar to neurons in MT, have also been identified in area LIP (Freedman and Assad, 2006; Shushruth et al., 2018; Fanini and Assad, 2009; Bollimunta and Ditterich, 2012), where they are proposed to play a role in motion categorization (Freedman and Assad, 2011). We hypothesize that such neurons might participate in routing information from DS neurons in MT/MST to those in LIP that contain a choice target in their response fields.

We identified such DS neurons using a passive motion viewing task (Figure 6a and b, left). Neurons preferring leftward or rightward motion constitute 5–10% of the neurons in our sample populations Table 1. Figure 6 shows the average firing rates of 51 leftward-preferring neurons (Minleft, Figure 6a) and 26 rightward-preferring neurons (Minright, Figure 6b) under passive motion viewing and decision-making. The separation of the two traces in the passive viewing task is guaranteed because we used this task to identify the DS neurons. It is notable, however, that the DS is first evident about 100 ms after the onset of random dot motion, and this latency is also apparent in mean firing rates grouped by signed coherence during decision making (Figure 6a and b right). The activity of DS neurons is modulated by both the direction and strength of motion. However, unlike the Tin neurons, the traces associated with different motion strengths are mostly parallel to one another and do not reach a common level of activity before the saccadic eye movement (i.e., they do not signal decision termination).

In addition to their shorter onset latency, the direction-selectivity of Min neurons precedes the choice-selectivity of Tincon neurons by ∼100 ms (Figure 6c). The responses bear similarity to DS neurons in area MT. Such neurons are known to exhibit choice-dependent activity insofar as they furnish the noisy evidence that is integrated to form the decision (Britten et al., 1996; Shadlen et al., 1996). We computed putative single trial direction signals by averaging the responses from the left- and right-preferring DS neurons, respectively. The resulting signals, SMinleft⁢(t) and SMinright⁢(t), have weak leverage on choice, but the leverage does not increase as a function of time (Figure 6d, left). This is what would be expected if the Min neurons represent the noisy momentary evidence as opposed to the accumulation thereof (Mazurek et al., 2003). We failed to detect a correlation between RT and either SMin signal (Figure 6d, right). This is surprising, but it could be explained by lack of power—a combination of small numbers of Min neurons, narrow sample windows (50 ms boxcar) and the focus on the long RT trials. Indeed, we found a weak but statistically significant negative correlation between RT and the difference in leftward vs. rightward signals, averaged over the epoch 0.1≤t≤0.4s from motion onset (p=0.0004;H0 : ρ ≥ 0, see ‘Methods’).

We considered the hypothesis that these DS signals are integrated by the Tincon neurons to form the DV. The heatmap in Figure 6e supports this hypothesis. On each trial, we formed the ordered pairs, {x,y}, where x=S~Minleft(tx)−S~Minright(tx) and y=S~Tincon(ty). The tilde in these expressions indicates the use of standardized residual values, for each signed motion strength. The heatmap shows the correlation of these quantities across trials. If the hypothesis were true, the correlations should be positive for ty>tx when tx>100 ms and ty>200 ms, and if the operation approximates integration, the level of correlation should be consistent at all lags, ty-tx>100 ms. The correlations are significant in the epoch of interest, and they differ significantly from the average correlations in the rest of the graph (i.e., tx<100, ty<200, or ty<tx, p<0.0001 permutation test). Although correlative, the observation is consistent with the idea that evidence integration occurs within area LIP, rather than inherited from another brain area (Zhang et al., 2022; Bollimunta and Ditterich, 2012).

Two adult male rhesus monkeys (Macaca mulatta, Primate Products) were used in the experiments. All training, surgery, and experimental procedures complied with guidelines from the National Institutes of Health and were approved by the Institutional Animal Care and Use Committee at Columbia University (protocols AAAN4900 and AC-AAAW4454). A head post and two recording chambers were implanted under general anesthesia using sterile surgical procedures (for additional details, see So and Shadlen, 2022). One recording chamber allowed access to area LIP in the right hemisphere. The other was placed on the midline, allowing access to the SC. Those recordings are described in Stine et al., 2023. Here we report only on the neural recordings from LIP, focusing on the epoch of decision formation.

The monkeys were trained to interact with visual stimuli presented on a CRT video monitor (Vision Master 1451, Iiyama; viewing distance 57 cm; frame rate 75 Hz) using the psychophysics toolbox (Brainard, 1997; Pelli, 1997; Kleiner et al., 2007). Task events were controlled by Rex software (Hays et al., 1982). The monkeys were trained to control their gaze and make saccadic eye movements to peripheral targets to receive a liquid reward (juice). The direction of gaze was monitored by an infrared camera (EyeLink 1000; SR Research, Ottawa, Canada; 1 kHz sampling rate). The tasks involve stages separated by random delays, distributed as truncated exponential distributions(1)f(t)={αλe−t−tminλtmin≤t≤tmax0otherwise

where t_min and t_max define the range, λ is the time constant, and α is chosen to ensure the total probability is unity. Below, we provide the range (tmin to tmax) and the exponential parameter λ for all variable delays. Note that because of truncation, the expectation 𝔼⁢(t)<tmin+λ.

In the main task (Figure 1a), the monkey must decide the net direction of random dot motion and indicate its decision when ready by making a saccadic eye movement to the corresponding choice target. After acquiring a central fixation point and a random delay (0.25–0.7 s, λ= 0.15), two red choice targets (diameter 1 dva) appear in the left and right visual fields. The random dot motion is then displayed after a random delay (0.25–0.7 s, λ= 0.4 s) and continues until the monkey breaks fixation. The dots are confined to a circular aperture (diameter 5 dva; degrees visual angle) centered on the fixation point (dot density 16.7 dots⋅dva^-2s^-1). The direction and strength of motion are determined pseudorandomly from ±{0,3.2,6.4,12.6,25.6,51.2}% coherence. The sign of the coherence indicates direction (positive for leftward, which is contraversive with respect to the recorded hemisphere). The absolute value of coherence determines the probability that a dot plotted on frame n will be displaced by Δ⁢x on frame n+3 (Δt=40ms), as opposed to randomly replaced, where Δ⁢x=575 dva, consistent with 5 dva⋅s^-1 speed of apparent motion (see also Roitman and Shadlen, 2002). The monkey is rewarded for making a saccadic eye movement to the appropriate choice target. On trials with 0% coherent motion, either saccadic choice is rewarded with probability 1/2. Errors are punished by extending the intertrial interval by up to 3 s (see Stine et al., 2023, for additional details). On approximately half of the trials, a 100 ms pulse of weak motion (±3.2% or 4.0% coherence for monkeys J and M, respectively) is added to the random dot motion stimulus at a random time (0.1–0.8 s, λ= 0.4) relative to motion onset (similar to Kiani et al., 2008). Monkey M performed 9684 trials (five sessions); monkey J performed 8142 trials (three sessions). The data are also analyzed in a companion paper that focuses on the termination of the decision (Stine et al., 2023).

In the visually instructed delayed saccade task (Hikosaka and Wurtz, 1983), one target is displayed at a pseudorandom location in the visual field. After a variable delay (monkey M: 0.4–1.1 s, λ= 0.3; monkey J: 0.5–1.5 s, λ= 0.2), the fixation point is extinguished, signaling ‘go’. The monkey is rewarded for making a saccade to within ±2.5 dva of the location of the target. In a memory-guided variant of the task (Gnadt and Andersen, 1988; Funahashi et al., 1989), the target is flashed briefly (200 ms) and the monkey is required to make a saccade to the remembered target location when the fixation point is extinguished. These tasks provide a rough characterization of the neural response fields during the visual, perisaccadic and delay epochs. Neurons are designated Tincon if they exhibit spatially selective activity at the location of the response target in the visual hemifield contralateral to the recorded hemisphere. This determination is made before analyzing the activity in the random dot motion task. We refer to the unweighted mean firing rate as STincon. Neurons are designated Tinips if they exhibit spatially selective activity at the location of the response target in the visual hemifield ipsilateral to the recorded hemisphere. These analyses were conducted post hoc, after spike sorting.

The passive motion-viewing task is identical to the main task, except there are no choice targets and only the strongest motion strength (±51.2% coherence) is displayed for 500 ms (1 s on a small fraction of trials in session 1). The direction is left or right, determined randomly on each trial (Pleft=12). The monkey is rewarded for maintaining fixation until the random dot motion is extinguished.

We fit a neurally inspired variant of the drift-diffusion model (Figure 1c) to the choice-RT data from each session. The model constructs the decision process as a race between two accumulators: one accumulating evidence for left and against right (e.g., left minus right) and one accumulating evidence for right and against left (e.g., right minus left). The decision is determined by the accumulator that first exceeds its positive decision bound, at which point the decision is terminated. The races are negatively correlated with one another, owing to the common source of noisy evidence. We assume they share half the variance, ρ=-0.5≈-0.71, but the results are robust to a wide range of reasonable values. The decision bounds are allowed to collapse linearly as a function of time, such that(2)B(t)=B0−αt,α≥0

We used the method of images (van den Berg et al., 2016; Shan et al., 2019) to compute the probability density of the accumulated evidence for each accumulator (which both start at zero at t=0) as a function of time (t) using a time step of 1 ms. The decision time distributions rendered by the model were convolved with a Gaussian distribution of the non-decision times, tnd, which combines sensory and motor delays, to generate the predicted RT distributions. The model has six parameters: κ,B0,α,μnd,σnd, and C0, where κ determines the scaling of motion strength to drift rate, C0 implements bias in units of signed coherence (Hanks et al., 2011), μnd is the mean non-decision time, and σnd is its standard deviation (Table 2). Additional details about the model and the fitting procedure are described in van den Berg et al., 2016.

We fit the race model described above to the combined behavioral data across all sessions (separately for each monkeys) and used the best-fitting parameters for monkey M (see Table 2) to simulate a total of 60,000 trials representing all signed coherences of the motion discrimination task. Each simulated trial yields a time series for two DVs, one for each accumulator in the race. We assume that the model-derived non-decision time (tnd=317 ms; Figure 1b) comprises visual and motor processing times at the beginning and end of the decision: 200 ms from motion onset to the beginning of evidence integration, and the remaining 117ms after termination. The latter approximates the variability observed in the saccadic latencies in the delayed saccade task and is simulated using a normal distribution, 𝒩⁢(μ,σ), where μ=117 ms and σ=39 ms (Stine et al., 2023). In this variable time period between decision termination and the response (saccade), the simulated DVs were assigned the values they had attained at the start of this epoch. For all analyses that employ these simulations, we use the DV for the left-choice accumulator because the neural recordings were from LIP in the right hemisphere.

We used prototype ‘alpha’ version Neuropixels1.0-NHP45 probes (IMEC/HHMI-Janelia) to record the activity of multiple isolated single units from the ventral subdivision of area LIP (LIP_v; Lewis and Van Essen, 2000). We used anatomical MRI to identify LIP_v and confirmed its physiological hallmarks with single-neuron recordings (Thomas Recording GmbH) before proceeding to multi-neuron recordings. Neuropixels probes enable recording from 384 out of 4416 total electrical contacts distributed along the 45-mm-long shank. All data presented here were recorded using the 384 contacts closest to the tip of the probe (Bank 0), spanning 3.84 mm. Reference and ground signals were directly connected to each other and to the monkey’s head post. A total of 1084 neurons were recorded over eight sessions (54–203 neurons per session) (Table 1).

The Neuropixels 1.0-NHP45 probe uses a standard Neuropixels 1.0 headstage and is connected via the standard Neuropixels1.0 5m cable to the PCI eXtensions for Instrumentation (PXIe) hardware (PXIe-1071 chassis and PXI-6141 and PXIe-8381 I/O modules, National Instruments). Raw data were acquired using the SpikeGLX software (http://billkarsh.github.io/SpikeGLX/), and single units were identified offline using the Kilosort 2.0 algorithm (Pachitariu et al., 2016; Pachitariu et al., 2020), followed by manual curation using Phy (https://github.com/cortex-lab/phy).

The spike times from each neuron are represented as delta functions of discrete time, si,n⁢(t), on each trial i and each neuron n (d⁢t=1 ms). The weighted sum of these si,n⁢(t) gives rise to the single-trial population signals:(3)Six⁢(t)=∑nwn⁢si,n⁢(t)

where the superscript, x, identifies the method or source that establishes the weights—that is, the coding direction in NSS or the neuron type contributing to a pooled average (e.g., Tincon). For visualization, the single-trial signals are smoothed by convolution with a truncated Gaussian using the MATLAB function, gausswin (width =80 ms, width factor =1.5, σ≈26 ms). Unless otherwise specified, all other analyses employ a 50 ms boxcar (rectangular) filter; values plotted at time t include data from t-24⁢ to ⁢t+25 ms.

We used several methods to define coding directions in the NSS defined by the population of neurons in each session. For PCA and choice decoding, we standardized the single-trial firing rates for each neuron using the mean and standard deviation of its firing rate at in the epoch 200≤t≤600 ms after motion onset. This practice led to the exclusion of two neurons (session 1) that did not produce any spikes in the normalization window. Those neurons were assigned zero weight.

Neurons were classified post hoc as Tin by visual-inspection of spatial heatmaps of neural activity acquired in the delayed saccade task. We inspected activity in the visual, delay, and perisaccadic epochs of the task. The distribution of target locations was guided by the spatial selectivity of simultaneously recorded neurons in the SC (see Stine et al., 2023, for details). Briefly, after identifying the location of the SC response fields, we randomly presented saccade targets within this location and seven other, equally spaced locations at the same eccentricity. In monkey J, we also included 1–3 additional eccentricities, spanning 5–16 degrees. Neurons were classified as Tin if they displayed a clear, spatially selective response in at least one epoch to one of the two locations occupied by the choice targets in the main task. Neurons that switched their spatial selectivity in different epochs were not classified as Tin. The classification was conducted before the analyses of activity in the motion discrimination task. The procedure was meant to mimic those used in earlier single-neuron studies of LIP (e.g., Roitman and Shadlen, 2002) in which the location of the choice targets was determined online by the qualitative spatial selectivity of the neuron under study. The Tincon neurons in the present study were highly selective for either the contralateral or ipsilateral choice target used in the RDM task (AUC=0.89±0.01 for 97% of neurons, Wilcoxon rank-sum test). Given the sparse sampling of saccade target locations, we are unable to supply a quantitative estimate of the center and spatial extent of the RFs. We next describe the methods to establish the coding directions.

We applied linear regression to generate a signal that best approximates a linear ramp, on each trial, i, that terminates with a saccade to the choice target contralateral to the hemisphere of the LIP recordings. The ramps are defined in the epoch spanning the decision time: each ramp begins at fi⁢(t0)=-1, where t0=0.2 s after motion onset, and ends at fi⁢(t1)=1, where t1=tsac-0.05 s (i.e., 50 ms before saccade initiation). The ramps are sampled every 25 ms and concatenated using all eligible trials to construct a long saw-tooth function (see Figure 2—figure supplement 1). The regression solves for the weights assigned to each neuron such that the weighted sum of the activity of all neurons best approximates the saw-tooth. We constructed a time series of standardized neural activity, sampled identically to the saw-tooth. The spike times from each neuron are represented as delta functions (rasters) and convolved with a non-causal 25 ms boxcar filter. The mean and standard deviation of all sampled values of activity were used to standardize the activity for each neuron (i.e., Z-transform). The coefficients derived from the regression establish the vector of weights that define Sramp. The algorithm ensures that the population signal Sramp⁢(t), but not necessarily individual neurons, have amplitudes ranging from approximately -1 to 1.

We employed a lasso linear regression with λ=0.005. The vector of weights assigned across the neurons defines a direction in NSS, Sramp, which we use to render the signal Sramp⁢(t) on single trials by projecting the data onto this direction. To determine the effect of the regularization term in the lasso regression, we recomputed single-trial signals using standard linear regression, without regularization. We then calculated the Pearson correlation between single-trial traces generated by projecting neural data onto the two coding directions (i.e., with and without regularization). The high correlation between single-trial traces (mean⁢r=0.99⁢, across sessions) indicates that the findings are not a result of the regularization applied. Here and elsewhere we compute the mean r using the Fisher-z transform, such that(4)r¯=Zinv⁢(𝔼⁢[Z⁢(rk)])

where Zinv is the inverse Fisher-z.

We applied standard PCA to the firing rate averages for each neuron using all trials sharing the same signed motion coherence in the shorter of two epochs: 200 ms to either 600 ms after motion onset or 100 ms before the median RT for the signed coherence, whichever produces the shorter interval. The results of the PCA indicate that the dimensionality of the data is low, but greater than 1. The participation ratio is 4.4 ± 1.3 (Mazzucato et al., 2016; Gao et al., 2017) and the first three PCs explain 67.1 ± 3.1% of the variance on average (mean ± s.e.m. across sessions). As in all other analyses of neural activity aligned to motion onset, we exclude data in the 100 ms epoch ending at saccade initiation on each trial. We projected the neural data onto the first PC to generate the signal SPC1.

For each experimental session, we trained logistic choice decoders with lasso regularization (λ=0.01) on the population activity in 50 ms time bins spanning the first 500 ms after motion onset and the 300 ms epoch ending at saccade initiation, respectively. Each of the decoders was trained on the even-numbered trials. Decoder accuracy was cross-validated using the activity of held-out, odd trials at the same time point (Figure 4a). For the time bins aligned to motion onset, we also assessed the accuracy of the decoders trained on each of the time bins to predict the choice on time bins on which they were not trained (Figure 4b; King and Dehaene, 2014). We use the decoder trained on the bin centered on t=450 ms to define the What coding direction. We refer to this as the fixed training-time decoder to distinguish it from the standard machine-learning decoder, which assigns a potentially distinct vector of weights at each time point. We applied a similar analysis to simulated data (see ‘Simulated DVs’) to generate the black curve in Figure 4a. Assuming a stochastic drift-diffusion process giving rise to choice and response times, the exercise establishes a rough upper bound on decoder accuracy, were the actual drift-diffusion process known precisely.

This decoder is trained to predict whether a saccade to the left (contralateral) choice target will occur within the next 150 ms. We applied logistic regression with lasso regularization (λ=0.01) to spike counts from each neuron in discrete bins of 25 ms, from 200 ms before motion onset to 50 ms before the saccade. We used only trials ending in a left choice (including errors) and trained the decoder on the even numbered half of those trials. The concatenation of these trials forms a sequence of step functions which are set to 1 if a saccade occurred within 150 ms of the start of the 25 ms time bin and 0 otherwise (Figure 4—figure supplement 1).

The spike counts were also concatenated across these trials to construct column vectors (one per neuron) that match the vector of concatenated step functions. These concatenated vectors, one per neuron, plus an offset (β0), serve as the independent variables of the regression model (one β term per neuron). The proportion of β weights equal to zero, controlled by the lasso parameter, λ, was 0.8±0.02 across sessions. The weights define the SWhen coding direction, which yields single-trial signals, SWhen⁢(t). The When-decoder signal is SWhen⁢(t)+β0. We validated the When-decoder by computing the area under an ROC (AUC) using the held-out (odd-numbered) trials ending in left choices (mean AUC over all time points and sessions: 0.84±0.024, mean ± s.e.).

Our motivation, however, was to ascertain whether the When coding direction also predicts the monkey’s choices on all trials—that is, to perform as a What decoder. To this end, we predicted the choice using the sign of the detrended SWhen(t), formed by subtracting the average of the signal using all trials:(5)Si*⁢(t)=SiWhen⁢(t)-⟨SiWhen⁢(t)⟩i

where ⟨⋯⟩i denotes expectation across all trials contributing values at time t. The choice accuracy isAi(t)={1Si∗(t)>0 & choicei=left1Si∗(t)≤0 & choicei=right0otherwise

The green trace in Figure 4a shows ⟨Ai⁢(t)⟩i.

To combine single-trial data across sessions (e.g., Sx⁢(t)), we first normalize activity within each session as follows. Using all trials ending in the same choice, we construct the trial-averaged activity aligned to both motion onset (0≤tm⁢o⁢t⁢i⁢o⁢n≤0.6 s) and saccade onset (-0.6≤ts⁢a⁢c⁢c≤0 s). This produces four traces. The minimum and maximum values (am⁢i⁢n and am⁢a⁢x) over all four traces establish the range, zero to one, of the normalized signal:(6)s^x(i,t)=sx(i,t)−aminamax−amin

where lowercase s^x⁢(i,t) is the normalized signal across trials i and time t in an individual session.

We computed the cosine similarity between the weight vectors that define coding directions Sramp, SPC1, STincon, SWhat, and SWhen. Mean cosine similarities are portrayed in the heatmap in Figure 4d and also in Figure 4—figure supplement 3, top, where they are accompanied by error bars. We evaluated the null hypothesis that the mean cosine similarity is ≤0 with t-tests. We also performed two control analyses that deploy random coding directions in NSS. For each of the original coding directions, we obtained 1000 random coding directions as random permutations of the weight assignments. The cosine similarities between pairs of such random directions in state space are shown in Figure 4—figure supplement 3, top. The cumulative distribution of cosine similarities under permutation supports p-values less than 1/1000. In a second control analysis, we used random unit vectors as random coding directions (normal distribution with mean 0 and scaled to unit length).

We calculated Pearson correlation to quantify the similarity of the signals generated by pairs of coding directions, x and y. For each trial, i, the detrended signals, S~ix and S~iy, provide ordered pairs, {S~ix⁢(tj),S~iy⁢(tj)} where j indexes successive 50 ms bins between 200 ms after motion onset and 100 ms before saccade initiation. We excluded trials comprising less than four such bins. Each trial gives rise to a correlation coefficient, ri. We report the mean r using Equation 4. The r-values for comparisons across all pairs of coding directions are summarized in Figure 4e, and variability across sessions is portrayed in Figure 4—figure supplement 3, bottom. For each pair of CDs and session, we evaluated the null hypothesis, ℋ0:r¯≤0 (t-test).

We also performed two control analyses that deploy random CDs in NSS. These analyses control for the possibility that the correlations observed in the signals are explained by pairwise correlations between the neurons, regardless of the signals produced by the weighted sums. (i) We generated sets of single-trial traces Srand⁢(t) by projecting the neural responses onto random CDs, defined by permuting the weights of each coding direction (Sramp, SPC1, STincon, SWhat, and SWhen). For each pair of CDs, we compute within-trial correlations between ordered pairs of trials using the same method applied to the original signals. We repeat this process for a total of 1000 random permutations per pair of CDs, per session. (ii) We sample a pair of random weight vectors from standard normal distributions. Each weight vector has a dimension equal to the number of recorded neurons in the session. The weight vectors are normalized to sum to 1. We generate random CDs using these weights and compute within-trial correlations using the same method applied to the original signals. We repeat this process 1000 times per session. For both analyses, we evaluated the null hypothesis that the observed correlations are not greater than those produced by the random projections (t-tests using the Fisher-z transformed correlations). Mean ± stdev of the mean r-values between ordered pairs for both control analyses are summarized in Figure 4—figure supplement 3, bottom.

The leverage of single-trial signals, Sx⁢(t), on choice and RT was assessed using the earliest 300 ms epoch of putative integration (0.2<t<0.5 s from motion onset), restricting analyses to trials with RTs outside this range (0.67<RT<2 s). The single-trial signals are smoothed with a 50 ms boxcar filter and detrended by subtracting the mean S⁢(t) for trials sharing the same motion strength and direction (i.e., signed coherence). The RTs are also expressed as residuals relative to the mean RT, using trials sharing the same signed coherence and choice. We include trials with |coh|≤0.064 that result in choices of the left response target in this analysis. Including trials of |coh|≤0.128 produced comparable results. The leverage on RT is the Pearson correlation between the residual signals S~⁢(t) at each time 0.2<t≤0.5, and RT~ on that trial, where the tilde indicates residual. Correlations were computed per session and then averaged across sessions (Equation 4). We also show the correlation at t = 0.55 s, using the ordered pairs, {S~(t=0.55),RT~}. We quantify mediation of the leverage of earlier samples by the later sample of Sx by computing partial correlations {S~(t),RT~∣S~(t=0.55)}, also notated RS~⁢(t),R⁢T~∣S~⁢(0.55) in Figure 5. We show this mediation at all time points. We also report a mediation statistic (ξRT) using the time point 200 ms after the beginning of putative integration (i.e., S⁢(t=0.4)):(7)ξ RT=1−RS~(0.4),  RT~∣S~(0.55)2RS~(0.4),  RT~2

The rationale for using the 400 ms time point is (i) to allow the process to have achieved enough leverage on RT so that a reduction is meaningful and (ii) to preserve a substantial gap between this time and the sample at t=0.55 s (e.g., to avoid autocorrelations imposed by smoothing). The rare cases in which there was no negative correlation between S⁢(t=0.4) and RT were excluded from this summary statistic, ξRT, because no mediation is possible (Session 2: Sramp & SPC1). When combining values of ξRT across sessions, we rectify any RS~(t),RT~∣S~(0.55)>0 to zero. This occurs rarely when the mediated correlation is near zero, typically at early times.

We compute the leverage on choice, ξCh, using trials with |coh|≤0.064 and the same time points as for ξRT. Instead of an R-squared measure, we based ξCh on coefficients derived from logistic regression:(8)Pcon(t)=11+e−(β0coh+β1(t)S(t)),0.2≤t≤0.5.

where β0coh is a set of constants that accounts for the proportion of contralateral choices at each signed motion strength and β1⁢(t) is the simple leverage of S⁢(t) on choice, analogous to simple correlation. The regression analysis was performed separately for each session. The coefficients β1⁢(t) were divided by their standard error and then averaged across sessions. This normalization step was implemented to control for potential variation in the magnitude of S⁢(t) (and therefore of β1⁢(t)) across sessions. Analogous to partial correlation, we include the later time point and fit(9)Pcon(t)=11+e−(β0coh+β1∗(t)S(t)+β2S(0.55))

where β1*⁢(t) is the amount of leverage at time t given the S⁢(0.55). The regression coefficients β1*⁢(t) were averaged across sessions after dividing them by the standard error of the β1⁢(t) coefficients obtained from Equation 8. That is, the same normalization factors were used for the mediated and unmediated leverage. The summary statistic for choice mediation is defined by(10)ξCh⁢(t)={1β1*⁢(t)≤01-β1*⁢(t)β1⁢(t)β1*⁢(t)>0undefinedβ1⁢(t)≤0⁢(i.e., mediation is not possible)

For both types of mediation, we also test whether the earlier S⁢(t) is mediated by the Tincon neurons by substituting STincon⁢(0.55) for S⁢(0.55) in Equation 9 and in the expression for partial correlations.

Because the mediation statistics, ξRT and ξCh, are, by their definition, non-negative, we assess statistical significance by bootstrapping. For each session, we construct 1000 surrogate data sets equal in size to the original data by sampling with replacement. The standard deviation of the leverage and mediation values at each time approximates the standard error. We compare the distribution of the mediation statistics, ζ, to their distribution under the null hypothesis, ℋ0, that the values arise by chance, instantiated by breaking the correspondence with the trial giving rise to the later sample, Sx⁢(t=0.55⁢s). The permutation maintains correspondence in signed motion coherence. We compare the distributions of mediation from the bootstrap and ℋ0 using the Wilcoxon rank-sum test.

To test whether the observed leverage of neural activity on choice and RT is achieved by projections onto arbitrary coding directions, we generated random weight vectors by permuting the weights associated with the first PC for each session. We projected activity onto this random coding direction, applied the mediation analyses described above to this signal, and repeated this process 1000 times to produce a null distribution at each time point. The reported p-values represent the probability that the observed leverage was generated from this null distribution.

We performed a similar analysis to test whether the observed leverage depends on the trial-to-trial correspondence between neural activity and behavior. Here, the null distribution at each time point was generated by randomly permuting the trial indices associated with the neural activity and those associated with the behavioral measures.

Finally, we used the simulated data from the racing accumulator model to test the degree of leverage and mediation expected had we known the ground-truth DV on each trial (Figure 5—figure supplement 1). Because the simulated DV is noiseless (and the process is Markovian), the mediation is expected to be complete for all time points tested in the analyses. We therefore took two steps to make the simulated data more comparable to neural data: (i) we sub-sampled the simulated data to match the number of trials in each session. (ii) We generated N noisy instantiations of the signal for each of the sub-sampled, simulated trials, where N is the number of Tincon neurons in each session. The added noise is independent across time points and weakly correlated across all N×(N-1)/2 neuron pairs (r≈0.09). We then applied the mediation analyses to the mean of these signals and repeated this process 1000 times.

We performed three control analyses to determine whether the results of the mediation analysis were specific, meaningful, and comparable to the results obtained for the DV of a race model. To showcase that the results are specific, we generated random weight vectors by permuting the weights of the PC1 coding direction. We repeated this procedure 1000 times per session and projected the data along these directions in NSS to generate Srand. We then computed the mediation analyses detailed above on these signals and determined significance by comparing the leverage on choice and correlation with RT of Sramp, SPC1, and STincon to the null distributions of Srand at each time point.

To estimate an upper limit for the degree of possible leverage and mediation, we simulated 60,000 trials using the race model that best fits the behavioral data of monkey M (see ‘Simulated DVs’). For any noise-free representation of a Markovian integration process, the leverage of an early sample of the DV on behavior would be mediated completely by later activity as the latter sample by definition encompasses all variability captured by the earlier sample. We, therefore, took two steps to make the simulated DVs more comparable to real neural data. (i) For each session, we first subsampled the simulated data to match the each session. (ii) To evaluate a DV approximated from the activity of nTincon neurons per session rather than the true DV represented by the entire population, we then generated n noisy instantiations of the signal for each simulated trial. The added noise is independent across time points and weakly correlated across neurons (r≈0.09). We then computed the measured DV Ssim as the mean activity of these n simulated neurons. We repeated this procedure 1000 times per session. Figure 5—figure supplement 1, bottom, displays the mean and standard deviation across permutations of the leverage of Ssim⁢(t) on behavior. The simulation results highlight that we would not expect the mediation of the leverage on behavior by a later sample to be complete (i.e., zero mediated leverage for all t < 0.55).

The mean pairwise correlation between Tincon neurons, reported in ‘Results’, is based on all pairs of simultaneously recorded Tincon neurons in each session and all trials with RT>0.5 s. For each neuron and each trial, we compute the time-averaged activity over the epoch 0.2≤t≤0.4. These scalar values are converted to residuals by subtracting the mean (for each neuron) across all trials sharing the same signed motion coherence. The residuals from all eligible trials are concatenated for each neuron to support the calculation of N×(N-1) Pearson r values, where N is the number of Tincon neurons in the session. The mean correlation for all pairs of Tincon neurons across all sessions is computed using Equation 4.

We identified DS neurons (Min neurons) using the passive motion-viewing task (described above). We classified a neuron as Min if it satisfies two criteria. The first criterion is a short-latency response to the onset of random dot motion, which we defined as a fivefold increase in firing rate relative to baseline in the first 80 ms following motion onset and a greater increase in the rate of rise in activity in the same 80 ms epoch, compared to any rise in activity in the 200 ms preceding motion onset. The second criterion is direction selectivity. We calculated the area under the ROC (AUC) comparing leftward versus rightward for two separate epochs: (i) 0.15≤t<0.3 s and (ii) 0.3≤t<0.5 s. Neurons were determined to be DS if the AUC in either epoch exceeded 0.6. We excluded one neuron from this analysis because it switched its direction preference in the two epochs. We also excluded neurons that had previously been classified as Tin.

In total, 6 of the 152 Tincon neurons fall into this group: three per monkey; at most two in a session. Removal of these neurons has negligible effects on the findings as pairs of STincon constructed with and without removal are strongly correlated (r = 0.9867).

We estimated the latency of DS responses using the CUSUM method (Ellaway, 1978; Lorteije et al., 2015). We employed a receiver operating characteristic (ROC) analysis to estimate the selectivity of each Min neuron to motion direction. The AUC reflects the separation of the distributions of spike counts (100–400 ms after motion onset) on single trials of leftward and rightward motion, respectively. We included only correct trials with response times greater than 450 ms and motion strengths above 10% coherence. For each neuron with AUC>0.6, we computed the difference in spike counts (25 ms bins) between correct trials featuring leftward and rightward motion. Subsequently, we accumulated these differences over time, following the CUSUM method. The resulting difference is approximately zero before the onset of direction selectivity and then either increases or decreases monotonically, depending on the preferred motion direction. To identify the transition between these two regimes, we fit a dog leg function to the cumulative sum of spikes: a flat line starting at t0=0 followed by a linearly increasing component beginning at t1>t0. The time of the end of the flat portion (between 0 and 500 ms from motion onset) of the fit was taken as the latency. Estimating latencies based on cumulative sums of spikes helps mitigate the effect of neuronal noise. The fitting step reduces the effect of the number of trials on latency estimates compared to traditional methods that rely on t-tests in moving windows.

The analysis of the correlations shown in Figure 6e is based on the spike counts of the Minleft, Minright, and Tincon neurons calculated in 25 ms windows. We computed residuals by subtracting from each trial and time bin its average over trials of the same signed coherence. The spike count residuals were then z-scored independently for each time bin and session. Trials from different sessions were concatenated, and the baseline activity—last 100 ms before motion onset—was subtracted from each trial. We refer to the resulting signals as S~Minleft, S~Minright, and S~Tincon. Trials with response time less than 0.55 s were discarded, and the correlations between the difference, S~Minleft−S~Minright, and S~Tincon ,were calculated for all pairs of time steps between 0 and 500 ms (Figure 6e). Statistical significance was assessed using permutation tests, as follows. Two regions of interest (ROIs) were defined based on the time from stimulus onset for the Min (x) and Tin (y) dimensions. The first region of interest, ROI1, is characterized by tx>100, ty>200, and ty>tx. According to our hypothesis that the Min neurons represent the momentary evidence integrated by Tincon neurons, we anticipated high correlations in this region. The second region of interest, ROI2, is defined by tx>100, ty>200, and ty<tx. If contrary to our hypothesis Min and Tin signals were influencing each other bidirectionally, we would expect high correlations in this region. We calculated the difference in correlations between these two groups, ⟨ρROI1⟩-⟨ρROI2⟩, where the expectation is over the time bins within each region of interest. This difference was compared to those obtained after randomly shuffling the order of the trials for one of the dimensions before calculating the pairwise correlations (Nshuffles=200). We assess significance with a z-test given the mean and standard deviation of the values obtained under shuffling. The analysis was repeated with an alternative ROI2 defined by tx<100 and ty<200, representing the times before direction selectivity is present in at least one of the two dimensions.

To assess the leverage of M in signals on choice and RT (Figure 6d), we performed the same logistic regression and pairwise correlation analyses as in Figure 5, substituting the Minleft and Minright for Sx. The leverage on choice is not mediated by a later sample of either Min signal (ξCh≤9.6%; not shown), and there is negligible leverage on RT to mediate. We suspect the failure to detect leverage of Min is explained by a lack of power, owing to the focus on long RT trials, narrow sample windows (50 ms boxcar), and the small number of Minleft and M in right neurons. We support this suspicion with a simpler correlation analysis using the difference of the Min signals (standardized as in the previous paragraph):(11)ψk=mean(S~Minleft(t)−S~Minright(t))

on the interval 0.1≤t≤0.4 s from motion onset, on each trial, k, including trials with contraversive choices and RT≥500 ms. We calculated the Pearson correlation coefficient between ψk and RT. Response times were z-scored independently for each signed motion strength and session. We evaluated the null hypothesis that the correlation coefficient is non-negative. The reported p-value is based on a one-tailed t-statistic.

The analyses in Figure 3 compare the variance and autocorrelation of the single-trial signals, Sramp⁢(t), to those expected from unbounded drift-diffusion. To mitigate the effect of the bound, we focus on the earliest epoch of putative integration (200–506 ms after motion onset; six 51 ms counting windows) and the weakest motion strengths (|coh|≤3.2%). The single-trial signals are detrended by the mean across trials sharing the same signed motion coherence and baseline corrected by subtraction of Siramp⁢(t=0.2) from all time points on each trial i.

The variance as a function of time and the autocorrelation as a function of time and lag are well specified for the cumulative sum of discrete iid random samples, but the autocorrelation is affected by the boxcar filter we applied to render the signals. We incorporated the correction in our characterization of unbounded diffusion. The derivation is summarized in Appendix 1, and we provide MATLAB code in the GitHub repository. The theoretical values shown in Figure 3 assume a 1 kHz sampling rate and standard Wiener process (i.e., samples drawn from a normal distribution with {μ=0,σ=d⁢t}). The evolution of variance would be a line from 0 to 1 over the first second of integration. The key prediction, shown in Figure 3a, is that the variance of mean single-trial signals, Si⁢(t), over the epoch 26± 25 ms should double in the epoch (26+51)±25 ms, and triple in the epoch (26+2×51)±25 ms), and so on for each successive non-overlapping running mean. We therefore use arbitrary units, normalized to the measured variance of the first point. We do not know the variance of the drift-diffusion signal that S⁢(t) is thought to approximate, but we assume it can be decomposed—by the law of total variance—to a component given by drift-diffusion and components associated with spiking and other nuisance factors. We therefore search for a scalar non-negative factor ϕ≤1 that multiplies all terms in the diagonal of the empirical covariance matrix (i.e., the variance) before normalizing to produce the autocorrelation matrix. We search for the value of ϕ that minimizes the sum of squares between Fisher-z transformed correlation coefficients in the theoretical and empirical autocorrelation matrices (Figure 3b and c). Standard errors of the variance and autocorrelations in Figure 3a and c are estimated by a bootstrap procedure respecting the composition of motion strength and direction (the s.e. is standard deviation of each variance and autocorrelation term across 500 repetitions of the procedure.