Neural signals in the mammalian cortex are notoriously noisy. They manifest as a sequence of action potentials (spikes) that approximate non-stationary Poisson point processes. Therefore, to characterize the signal produced by a neuron, electrophysiologists typically combine the spike times from many repetitions or trials relative to the time of an event (e.g., stimulus onset) to yield the average firing rate of the neuron as a function of time. Such trial-averaged firing rates are the main staple of systems neuroscience (Here and throughout, trial average and across-trial average refer to the mean of signal values, over all specified trials at the same time, ${\displaystyle t}$, relative to a trial event (e.g., motion onset)). They are the source of knowledge about spatial selectivity (e.g., receptive fields), feature selectivity (e.g., direction of motion, faces vs. other objects), and even cognitive signals associated with working memory, anticipation, attention, motor planning, and decision-making. But there is an important limitation.

Trial averages suppress signals that vary independently across trials. In many cognitive tasks, such as difficult decisions, the variable component of the signal is the most interesting because it is this component that is thought to explain the variable choice and response time. This variability is thought to arise from a decision process that accumulates noisy evidence in favor of the alternatives and terminates when the accumulated evidence for one alternative, termed the decision variable (DV), reaches a terminating bound. The DV is stochastic because the integral of noisy samples of evidence is biased Brownian motion (or drift-diffusion) and this leads to a stochastic choice and response time on each decision. However, the stochastic part of this signal is suppressed by averaging across trials. We will use the term drift-diffusion because it is the expression most commonly applied in models of decision-making (Ratcliff and Rouder, 1998; Gold and Shadlen, 2007), and we will consider the noise part—that is, diffusion—as the signal of interest.

In the setting of difficult perceptual decisions, studied here, bounded drift-diffusion reconciles the relationship between decision speed and accuracy. It also explains the trial-averaged firing rates of neurons in the lateral intraparietal area (LIP) that represent the action used by monkeys to indicate their choice. These firing rate averages show motion-dependent, ramping activity that reflects the direction and strength of motion, consistent with the drift component of drift-diffusion (Roitman and Shadlen, 2002). Up to now, however, the diffusion component has not been observed, owing to averaging.

There is thus a missing link between the mathematical characterization of the decision process and its realization in neural circuits, leaving open the possibility that drift-diffusion dynamics do not underlie LIP activity (e.g., Latimer et al., 2015), or emerge only at the level of the population, without explicit representation by single neurons. We reasoned that these and other alternatives to drift-diffusion could be adjudicated if it were possible to resolve the DV giving rise to a single decision.

This stratagem is now feasible, owing to the development of high-density Neuropixels probes, which are capable of recording from deep sulci in the primate brain. Here we provide the first direct evidence for a drift-diffusion process underlying single decisions. We recorded simultaneously from up to 203 neurons in area LIP while monkeys made perceptual decisions about the direction of dynamic random dot motion (Newsome et al., 1989; Gold and Shadlen, 2007). Using a variety of dimensionality reduction techniques, we show that a drift-diffusion signal can be detected in such populations on individual trials. Moreover, this signal satisfies the criteria for a DV that controls the choice and reaction time (RT). Notably, the signal of interest is dominated by a small subpopulation of neurons with response fields that overlap one of the choice targets, consistent with earlier single-neuron studies (e.g., Shadlen and Newsome, 1996; Roitman and Shadlen, 2002; Churchland et al., 2011; Gold and Shadlen, 2007).

Two monkeys made perceptual decisions, reported by an eye movement, about the net direction of dynamic random dot motion (Figure 1a). We measured the speed and accuracy of these decisions as a function of motion strength (Figure 1b, circles). The choice probabilities and the distribution of RTs are well described (Figure 1b, traces) by a bounded drift-diffusion model (Figure 1c). On 50% of the trials, a brief (100 ms) pulse of weak leftward or rightward motion was presented at a random time. The influence of these pulses on choice and RT further supports the assertion that the choices and RTs arose through a process of integration of noisy samples of evidence to a stopping bound (Figure 1—figure supplement 1; Stine et al., 2020; Stine et al., 2023; Hyafil et al., 2023).

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In addition to the main task, the monkeys performed two control tasks: instructed, delayed saccades to peripheral targets and passive viewing of random dot motion (see ‘Methods’). These control tasks served to identify, post hoc, neurons with response fields that overlap the choice target in the hemifield contralateral to the recording site (${\text{T}}_{\text{in}}^{\text{con}}$), neurons with response fields that overlap the other choice target (${\text{T}}_{\text{in}}^{\text{ips}}$), and neurons with response fields that overlap the random-dot motion stimulus (${\text{M}}_{\text{in}}$; Table 1).

We recorded simultaneously from populations of neurons in area LIP using newly developed macaque Neuropixels probes (Trautmann et al., 2023) while monkeys performed these tasks. The data set comprises eight sessions from two monkeys (1696–2894 trials per session; Table 2). Our primary goal was to identify activity in LIP that relates to the DV, a theoretical quantity that determines the choice and RT on each trial. To achieve this, we formed weighted averages from all neurons in the sample population, including those with response fields that do not overlap a choice target or the motion stimulus. We used several strategies to assign this vector of weights, which we refer to as a coding direction in the neuronal state space (NSS). The projection of the spiking activity from the population of neurons onto the vector of weights gives rise to a scalar function of time, $S^x(t)$, where the superscript $x$ labels the strategy. We focus on such one-dimensional projections because of the long-standing hypothesis that the DV is drift-diffusion, which is a scalar function of time.

We first developed a targeted strategy that would reproduce the well-known coherence-dependent ramping activity evident in the across-trial averages. This strategy applies regression to best approximate 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 on the epoch spanning the decision time: from $t_0=0.2$ s after motion onset to $t_1=0.05$ s before saccade initiation (black lines in Figure 2—figure supplement 1) The epoch is motivated by many previous studies (see Gold and Shadlen, 2007; Shadlen and Kiani, 2013, for reviews). Each ramp begins at $f_i(t_0)=-1$ and ends at $f_i(t_1)=1$. The ramp approximates the expectation—conditional on the choice and response time—of the deterministic components of the drift-diffusion signal, which, in addition to the drift, can incorporate (i) a time-dependent but evidence-independent urgency signal (Churchland et al., 2008; Drugowitsch et al., 2012), and (ii) a dynamic bias signal (Hanks et al., 2011). It can also be viewed as an approximation to firing rates averaged across trials and grouped by contraversive choice and RT quantile (Figure 2—figure supplement 2). Importantly, the fit is not guided by an assumption of an underlying diffusion process. That is, the ramp coding direction is agnostic to the underlying processes whose averages approximate ramps. The weights derived from these regression fits specify a ramp coding direction in the state space defined by the population of neurons in the session. The single-trial signal, $S^{\mathrm{ramp}}(t)$, is rendered as the projection of the population firing rates onto this coding direction.

The left side of Figure 2a shows single-trial activity rendered by this strategy. The right side of the figure shows the averages of the single-trial responses grouped by signed coherence and aligned to motion onset or response time (saccade initiation). These averaged traces exhibit features of the firing rate averages in previous studies of single neurons in LIP (e.g., Roitman and Shadlen, 2002). They begin to diverge as a function of the direction and strength of motion approximately 200 ms after the onset of motion. The traces converge near the time of saccadic response to the contralateral choice target such that the coherence dependence is absent or greatly diminished. Coherence dependence remains evident through the initiation of saccades to the right (ipsilateral) target, consistent with a race architecture—between negatively correlated accumulators—depicted in Figure 1c.

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We complemented this regression strategy with principal component analysis (PCA) and use the first PC (PC1), which explains $44\pm 7$% of the variance (mean ± s.e. across sessions) of the activity between 200 and 600 ms from motion onset (see ‘Methods’). This coding direction renders single-trial signals, $S^{\text{PC1}}(t)$ (Figure 2b). In a third strategy, we consider the mean activity of neurons with response fields that overlapped the contralateral choice target (${\text{T}}_{\text{in}}^{\text{con}}$ neurons), which were the focus of previous single-neuron studies (e.g., Shadlen and Newsome, 1996; Platt and Glimcher, 1999; Roitman and Shadlen, 2002). In those studies, the task was modified so that one of the choice targets was placed in the neural response field, whereas here we identify neurons post hoc with response fields that happen to overlap the contralateral choice target. This difference probably accounts for the lower firing rates of the ${\text{T}}_{\text{in}}^{\text{con}}$ neurons studied here. Figure 2c shows single-trial and across-trial averages from these ${\text{T}}_{\text{in}}^{\text{con}}$ neurons. They too render signals, $S_{\text{Tin}}^{\text{con}}(t)$, similar to those derived from the full population. The ${\text{T}}_{\text{in}}^{\text{con}}$ neurons thus furnish a third coding direction defined by a vector of identical positive weights assigned to all ${\text{T}}_{\text{in}}^{\text{con}}$ neurons and 0 for all other neurons in the population. The emboldened single-trial traces in Figure 2 (left) correspond to the same trials rendered by the three coding directions. It is not difficult to tell which are the corresponding traces, an observation that speaks to their similarity, and the same is true for the averages. We will expand on this observation in what follows.

The averages show the deterministic drift component of the hypothesized drift-diffusion process, with the slope varying monotonically with the signed motion strength (Figure 2, right). The rise begins to saturate as a consequence of the putative termination bound—a combination of dropout of trials that are about to terminate and the effect on the distribution of possible diffusion paths imposed by the very existence of a stopping bound. This saturation is evident earlier on trials with stronger motion, hence shorter RT, on average. The positive buildup rate on the 0% coherence motion represents the time-dependent, evidence-independent signal that is thought to reflect the cost of time. It leads to termination even if the evidence is weak, equivalent to collapsing stopping bounds in traditional, symmetric drift-diffusion models (Drugowitsch et al., 2012). Removal of this urgency signal, $u(t)$, from the non-zero coherence traces renders the positive and negative coherence averages symmetric relative to zero on the ordinate (Figure 2—figure supplement 3).

The single-trial responses in Figure 2 do not look like the averages but instead approximate drift-diffusion. We focus on the epoch from 200 to 500 (or 600) ms from motion onset—that is, the first 300 (or 400) ms of the period in which the averages reflect the integration of evidence. Some traces are cut off before the end of the epoch because a saccade occurred 100 ms later on the trial. However, most 0% coherence trials continue beyond 500 ms (median RT $>600$ ms). The single-trial traces do not rise monotonically as a function of time but meander and tend to spread apart from each other vertically. For unbounded diffusion, the variance would increase linearly, but as just mentioned, the existence of an upper stopping bound and the limited range of firing rates (e.g., non-negative) renders the function sublinear at later times (Figure 3a, Figure 3—figure supplement 1). The autocorrelation between an early and a later sample from the same diffusion trace is also clearly specified for unbounded diffusion. The theoretical values shown in Figure 3b and c are the autocorrelations of unbounded diffusion processes that are smoothed identically to the neural signals (see ‘Methods’ and Appendix 1). The autocorrelations in the data follow a strikingly similar pattern. These observations support the assertion that the coherence-dependent (ramp-like) firing rate averages observed in previous studies of area LIP are composed of stochastic drift-diffusion processes on single trials.

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Single-trial drift-diffusion signals approximate the decision variable

We next evaluate the hypothesis that the drift-diffusion signal, $S^x(t)$, 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.

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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 $S^{\text{When}}(t)$ 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 $S^{\text{When}}$ 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 $0.67\pm 0.03$ (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, $S^x(t)$, 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 $S^x(t)$ 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, $S^x(t)$, 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 $\text{RT}\ge 670$ ms and $\text{coherence}\le 6.4\%$. The RT restriction eliminates 17% of the eligible trials. Larger values of $S^x(t)$ are associated with a larger probability of a left (contraversive) choice and a shorter RT, hence negative correlation between $S^x(t)$ 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 $S^x(t)$ to the log odds of a left choice, after accounting for the motion strength and direction (i.e., the coefficients, ${\beta }_1(t)$ in Equation 8). The leverage on RT (Figure 5b) is the Pearson correlation between $S^x(t)$ and the RT on that trial, after accounting for the effect of motion strength and direction on $S^x$ 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 $S^x$ at $t=550$ ms, which is 120 ms before any of the included trials have terminated. Both observations are consistent with the hypothesis that $S^x$ 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).

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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 $S^x(t=0.55)$. We assessed statistical significance of the mediation statistics, ${\xi }^{\text{Ch}}$ and ${\xi }^{\text{RT}}$ (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 ${\displaystyle \mathrm{p}<{10}^{-317}}$). The one non-significant comparison is ${\xi }^{\text{Ch}}$ for $S_{\text{Tin}}^{\text{con}}$ in session 2 (p=0.73). The mediation is significant when this comparison is included in the combined data (${\displaystyle \mathrm{p}<{10}^{-9}}$, 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 ${\text{T}}_{\text{in}}^{\text{con}}$ neurons. The top and middle rows ($S^{\mathrm{ramp}}$ and $S^{\text{PC1}}$) contain a second, open symbol, which is simply a copy of the filled symbol from the bottom row ($S_{\text{Tin}}^{\text{con}}$). The yellow traces show significant cross-mediation of $S^{\mathrm{ramp}}$ and $S^{\text{PC1}}$ by the sample, $S_{\text{Tin}}^{\text{con}}(t=0.55)$ (all ${\displaystyle \mathrm{p}<0.05}$; median ${\displaystyle \mathrm{p}<{10}^{-268}}$; 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 $S^{\mathrm{ramp}}$ and $S^{\text{PC1}}$ mediate themselves. The observation suggests that minimal leverage is gained by sophisticated analyses of the full NSS compared to a simple average of ${\text{T}}_{\text{in}}^{\text{con}}$ neurons. This is both reassuring and disquieting: reassuring because the ${\text{T}}_{\text{in}}^{\text{con}}$ 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 ${\text{T}}_{\text{in}}^{\text{con}}$ neurons span all percentiles (mean IQR: 49–96; mean 71st percentile, $AUC=0.74\pm 0.05$) 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 ${\text{T}}_{\text{in}}^{\text{con}}$ from its weight or percentile is remarkably poor (Figure 5—figure supplement 2).

These observations support the idea that the single-trial signals, $S^{\mathrm{ramp}}$, $S^{\text{PC1}}$, and $S_{\text{Tin}}^{\text{con}}$, approximate the DV used by the monkey to make its decision. In Figure 5—figure supplement 3, we show that the $S^{\text{What}}$ and $S^{\text{When}}$ coding directions achieve qualitatively similar results. Moreover, a late sample from $S^x(t)$ mediates the earlier correlation with RT and choice of signals rendered by other coding directions, $S^y(t)$, 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.

Activity of direction-selective neurons in area LIP resembles momentary evidence

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 (${\text{M}}_{\text{in}}^{\text{left}}$, Figure 6a) and 26 rightward-preferring neurons (${\text{M}}_{\text{in}}^{\text{right}}$, 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 ${\text{T}}_{\text{in}}$ 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).

Figure 6

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In addition to their shorter onset latency, the direction-selectivity of ${\text{M}}_{\text{in}}$ neurons precedes the choice-selectivity of ${\text{T}}_{\text{in}}^{\text{con}}$ 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, $S_{\text{Min}}^{\text{left}}(t)$ and $S_{\text{Min}}^{\text{right}}(t)$, 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 ${\text{M}}_{\text{in}}$ 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 $S_{\text{Min}}$ signal (Figure 6d, right). This is surprising, but it could be explained by lack of power—a combination of small numbers of ${\text{M}}_{\text{in}}$ 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 ${\displaystyle 0.1\le t\le 0.4s}$ from motion onset (${\displaystyle \mathrm{p}=0.0004;{\mathcal{H}}_0}$ : ρ ≥ 0, see ‘Methods’).

We considered the hypothesis that these DS signals are integrated by the ${\displaystyle {\mathrm{T}}_{\mathrm{i}\mathrm{n}}^{\mathrm{c}\mathrm{o}\mathrm{n}}}$ neurons to form the DV. The heatmap in Figure 6e supports this hypothesis. On each trial, we formed the ordered pairs, $\{x,y\}$, where ${\displaystyle x={\tilde{S}}_{\mathrm{M}\mathrm{i}\mathrm{n}}^{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}(t_x)-{\tilde{S}}_{\mathrm{M}\mathrm{i}\mathrm{n}}^{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}(t_x)}$ and ${\displaystyle y={\tilde{S}}_{\mathrm{T}\mathrm{i}\mathrm{n}}^{\mathrm{c}\mathrm{o}\mathrm{n}}(t_y)}$. 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 $t_y>t_x$ when $t_x>100$ ms and $t_y>200$ ms, and if the operation approximates integration, the level of correlation should be consistent at all lags, $t_y-t_x>100$ 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., $t_x<100$, $t_y<200$, or ${\displaystyle t_y<t_x}$, 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).

We have observed a neural representation of the stochastic process that gives rise to a single decision. This is the elusive drift-diffusion signal that has long been thought to determine the variable choice and response time in the perceptual task studied here. The signal was elusive because it is the integral of noisy momentary evidence, hence stochastic, and undetectable in the firing rates when they are computed as averages over trials. The averages preserve the ramp-like drift component, leaving open the possibility that the averages are composed of other stochastic processes (e.g., Latimer et al., 2015; Cisek et al., 2009). By providing access to populations of neurons in LIP, macaque Neuropixels probes (Trautmann et al., 2023) allowed us to resolve, for the first time, the evolution of LIP activity during a single decision.

The present findings establish that the ramp-like averages arise from drift-diffusion on single trials, and this drift-diffusion signal approximates the DV that arbitrates the choice and RT on that trial. We used a variety of strategies to assign a weight to each neuron in the population such that the vector of weights defines a coding direction in NSS. The weighted averages render population firing rate signals on single trials. Our experience is that any method of assigning the weights that captures a known feature of evidence accumulation (or its termination in a saccadic choice) reveals drift-diffusion on single trials, and this also holds for data-driven, hypothesis-free methods such as PCA. This is because the actual dimensionality of the DV is effectively one—a scalar function of time that connects the visual evidence to a saccadic choice (Ganguli et al., 2008). Thus a weighting established by training a decoder at time $t=\tau$ to predict the monkey’s choice performs nearly as well when tested at times other than the time the decoder was trained on (i.e., $t\ne \tau$; Figure 4).

The different strategies for deriving coding directions lead to different weight assignments, but the coding directions are linearly dependent (Figure 4d). They produce traces, $S(t)$, that are similar (Figure 4e) and suggestive of drift-diffusion. Traces accompanying trials with the same motion coherence meander and spread apart at a rate similar to diffusion (i.e., standard deviation proportional to $\sqrt{t}$), and they exhibit a pattern of autocorrelation, as a function of time and lag, consistent with diffusion (Figure 3). The calculations applied in the present study improve upon previous applications (e.g., Churchland et al., 2011; de Lafuente et al., 2015; Shushruth et al., 2018) by incorporating the contribution of the smoothing to the autocorrelations. The departures from theory are explained by the fact that the accumulations are bounded. The upper bound and the fact that spike rates must be non-negative (a de facto lower reflecting bound) limits the spread of the single-trial traces.

The single-trial signals, $S_i^x(t)$, approximate the DV that gives rise to the choice and decision time on trial $i$ (Figure 5, Figure 5—figure supplement 3). Support for this assertion is obtained using a conservative assay, which quantifies the leverage of the first 300 ms of the signal’s evolution on decision outcomes—choice and RT—occurring at least 670 ms after motion onset. Naturally, the signals do not explain all the variance of these outcomes. The sample size is limited to $N$ randomly selected, often weakly correlated neurons. The sample size and correlation are especially limiting for the ${\text{T}}_{\text{in}}^{\text{con}}$ neurons ($𝔼(r)=0.067\pm 0.0036$). Control analyses show that the degree of leverage on behavior and mediation of these relationships by later activity is on par with that obtained from simulated, weakly correlated neurons (Figure 5—figure supplement 1). In addition, because they are identified post hoc, many have response fields that barely overlap the choice target. Presumably, that is why their responses are weak compared to previous single-neuron studies in which the choice targets were centered in the response field by the experimenter. Yet even this noisy signal, $S_{\text{Tin}}^{\text{con}}$, mediates signals produced by coding directions using the entire population (Figure 5).

The ${\text{T}}_{\text{in}}^{\text{con}}$ neurons were the first to be identified as a plausible candidate neural representation of the DV, based on firing rate averages (Shadlen and Newsome, 1996; Platt and Glimcher, 1999). This neural type is also representative of the LIP projection to the region of the SC that represents the saccadic vector required to center the gaze on the choice target. (Paré and Wurtz, 1997). In a companion study by Stine et al., 2023 we show that the SC is responsible for cessation of integration in LIP. Features of the drift-diffusion signal from the ${\text{T}}_{\text{in}}^{\text{con}}$ neurons are correlated with bursting events in corresponding populations of ${\text{T}}_{\text{in}}^{\text{con}}$ neurons in the SC, including the final saccadic burst that ends the decision with a saccade to the contralateral choice target. Stine et al., 2023 also show that inactivation of ${\text{T}}_{\text{in}}^{\text{con}}$ neurons in the SC has little effect on $S_{\text{Tin}}^{\text{con}}$ signals in LIP.

Previous studies of LIP using the random dot motion task focused primarily on the ${\text{T}}_{\text{in}}^{\text{con}}$ neurons (cf. Meister et al., 2013). It was thus unknown whether and how other neurons contribute to the decision process. The Neuropixels probes used in the present study yield a large and unbiased sample of neurons. Many of these neurons have response fields that overlap one of the two choice targets, but the majority have response fields that overlap neither the choice targets nor the random dot motion. Our screening procedures (delayed saccades and passive motion viewing tasks) do not supply a quantitative estimate of their spatial distribution. It is worth noting that neurons with response fields that overlap neither of the two choice targets were assigned nonzero weights by the What- and When-decoders, and yet, removal of the task-related neurons that represent the choice targets and motion (i.e., ${\text{T}}_{\text{in}}$ and ${\text{M}}_{\text{in}}$) decreases decoding accuracy more substantially than removing all but the ${\text{T}}_{\text{in}}$ neurons (Figure 4—figure supplement 5). The accuracy the decoder achieves is likely explained by neurons with weak responses that simply failed to meet our criterion for inclusion in the ${\text{T}}_{\text{in}}$ and ${\text{M}}_{\text{in}}$ categories (e.g., neurons with response fields that barely overlap the choice targets or RDM). Some neurons outside these groups might reflect normalization signals from the ${\text{T}}_{\text{in}}$ and ${\text{M}}_{\text{in}}$ neurons (Shushruth et al., 2018; Carandini and Heeger, 2011), imbuing broad, decision-related co-variability across the population. It thus seems possible that higher dimensional tasks (e.g., four choices instead of two) could decrease correlations among groups of neurons with different response fields.

The fact that the raw averages from a small number of weakly correlated ${\text{T}}_{\text{in}}$ neurons furnish a DV on par with that furnished by the full population underscores the importance of this functional class. The role of the ${\text{M}}_{\text{in}}$ neurons is less well understood. Freedman and colleagues described direction selective neurons in LIP, similar to our ${\text{M}}_{\text{in}}$ neurons (Freedman and Assad, 2011; Fanini and Assad, 2009; Sarma et al., 2016). They showed that the neurons represent both the direction of motion and the decision in their task. In contrast, we do not observe the evolution of the decision (i.e., DV) by the ${\text{M}}_{\text{in}}$ neurons (Figure 6). The latency of the direction and coherence-dependent signal as well as its dynamics resemble properties of DS neurons in area MT. The delayed correlation between ${\text{M}}_{\text{in}}$ and ${\text{T}}_{\text{in}}$ responses evokes the intriguing possibility that ${\text{M}}_{\text{in}}$ neurons supply the momentary evidence, which is integrated within LIP itself (Zhang et al., 2022). Future experiments that better optimize the yield of ${\text{M}}_{\text{in}}$ neurons will be informative, and direct, causal support will require perturbations of functionally identified ${\text{M}}_{\text{in}}$ neurons, which is not yet feasible. A natural question is why LIP would contain a copy of the DS signals that are already present in area MT. We suspect it simplifies the routing of momentary evidence from neurons in MT/MST to the appropriate ${\text{T}}_{\text{in}}$ neurons. This interpretation leads to the prediction that DS ${\text{M}}_{\text{in}}$ neurons would be absent in LIP of monkeys that are naïve to saccadic decisions informed by random dot motion, as has been observed in the SC (Horwitz et al., 2004). Further, when motion is not the feature that informs the saccadic response—for example, in a color categorization task (e.g., Kang et al., 2021)—LIP might contain a representation of momentary evidence for color (Toth and Assad, 2002; Sereno and Maunsell, 1998).

The capacity to record from many neurons simultaneously invites characterization of the population in NSS, in which the activity of each neuron defines a dimension. Often, population activity is confined to a low-dimensional subspace or manifold within the NSS (Vyas et al., 2020). An ever-more-popular viewpoint is that representations within these subspaces are emergent properties of the population—that is, distributed, rather than coded directly by single neurons—a dichotomy that has its roots in Barlow’s neuron doctrine (as updated in Barlow, 1994). Indeed, it is tempting to conclude that the drift-diffusion signal in LIP is similarly emergent based on our NSS analyses—the identified subspaces (i.e., coding directions) combine neurons with highly diverse activity profiles. In contrast, grouping neurons by the location of their spatial response field reveals a direct coding scheme: ${\text{T}}_{\text{in}}$ neurons directly represent the accumulated evidence for making a particular saccade and ${\text{M}}_{\text{in}}$ neurons represent the momentary evidence. We argue that this explanation is more parsimonious and, importantly, more principled. Grouping neurons based on spatial selectivity rests on the principle that neurons with similar RFs have similar projections, which is the basis for topographic maps in the visual and oculomotor systems (Schall, 1995; Silver and Kastner, 2009; Kremkow et al., 2016; Felleman and Van Essen, 1991). In contrast, there are no principles that guide the grouping of neurons in state space analyses, as the idea is that they may comprise as many dimensions as there are neurons that happen to be sampled by the recording device.

The present finding invites both hope and caution. It may be useful to consider a counterfactual state of the scientific literature that lacks knowledge of the properties of LIP ${\text{T}}_{\text{in}}$ neurons—a world without Gnadt and Andersen, 1988 and no knowledge of LIP neurons with spatially selective persistent activity. In this world we have no reason to entertain the hypothesis that decisions would involve neurons that represent the choice targets. We do know about DS neurons in area MT and their causal role in decisions about the direction of random dot motion (Salzman et al., 1992; Fetsch et al., 2018; Ditterich et al., 2003; Liu and Pack, 2017). We also know that drift-diffusion models explain the choice-response time behavior. Guided by no particular hypothesis, we obtain population neural recordings in the random dot motion task. We do not perform the saccade and passive viewing control experiments. What might we learn from such a dataset? We might apply PCA and/or train a choice decoder or possibly a When-decoder. If so, we could discover the drift-diffusion signal and we might also infer that the dimensionality of the signal is low. However, we would not discover the ${\text{T}}_{\text{in}}$ neurons without a hypothesis and a test thereof. We might notice that the coding directions that reveal drift-diffusion often render a response at the onset of the choice targets as well as increased activity at the time of saccades to the contralateral choice target. These facts might lead us to hypothesize that the population might contain neurons with visual receptive fields and some relationship to saccadic eye movements. We might then query individual neurons, post hoc, for these features, and ask if they render the drift-diffusion signal too. The inferences could then be tested experimentally by including simple delayed saccades in the next experiment. The hope in this counterfactual is that data-driven, hypothesis-free methods can inspire hypotheses about the mechanism. The caution is to avoid the natural tendency to stop before the hypotheses and tests, thus accepting as an endpoint the characterization of population dynamics in high dimensions or a lower dimensional manifold. If LIP is representative, these mathematically accurate characterizations may fail to illuminate the neurobiological parsimony.

We consider a discrete time (sampling interval $dt$) Wiener process with independent random increments ${ϵ}_k$ on time step $k$ that are zero-mean noise with variance ${\sigma }^2({ϵ}_k)=dt$ (i.e., unit variance per second). The accumulated evidence (i.e., decision variable, DV) on time step $p$ is

For such a Wiener process,

As the increments ${ϵ}_k$ are independent across time, ${\displaystyle \text{Cov}({ϵ}_j,{ϵ}_k)}$ is 0 for $j\ne k$ and $dt$ for $j=k$

We define the mean DV over a window of $\pm n$ points as

We consider two time points $p<q$ with window $n$ such that there is no overlap and hence $p+n<q-n$

Given that ${\displaystyle \sum _{j=1}^n\sum _{k=1}^{n}min(j,k)=\sum _{j=1}^n{j}^2=n(n+1)(2n+1)/6}$

Therefore, the correlation

In contrast for the point estimates at $p$ and $q$

It is useful to re-express the above two equations in terms of actual time $t_p$ and $t_q$ and window size $t_n$. Substituting for $p$, $q$, and $n$ with $t_p/dt$, $t_q/dt$ and $t_n/dt$

Matlab code for all analyses and graphs are available at GitHub (copy archived at Steinemann, 2024). The data are deposited at Zenodo.

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Author details

1. Natalie Steinemann

   Zuckerman Mind Brain and Behavior Institute, Columbia University, New York, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Validation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   Contributed equally with

   Gabriel M Stine

   Competing interests

   No competing interests declared

2. Gabriel M Stine

   Zuckerman Mind Brain and Behavior Institute, Columbia University, New York, United States

   Present address

   Department of Brain and Cognitive Science, MIT, Cambridge, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing – original draft, Writing – review and editing

   Contributed equally with

   Natalie Steinemann

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4906-0461

3. Eric Trautmann

   Zuckerman Mind Brain and Behavior Institute, Columbia University, New York, United States

   Contribution

   Resources, Supervision, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

4. Ariel Zylberberg

   1. Zuckerman Mind Brain and Behavior Institute, Columbia University, New York, United States
   2. Howard Hughes Medical Institute, Chevy Chase, United States
   3. Department of Neuroscience, Columbia University, New York, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Supervision, Validation, Visualization, Methodology, Writing – original draft, Writing – review and editing

   Contributed equally with

   Daniel M Wolpert and Michael N Shadlen

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-2572-4748

5. Daniel M Wolpert

   1. Zuckerman Mind Brain and Behavior Institute, Columbia University, New York, United States
   2. Howard Hughes Medical Institute, Chevy Chase, United States
   3. Kavli Institute, Columbia University, New York, United States

   Contribution

   Conceptualization, Software, Formal analysis, Visualization, Methodology, Writing – original draft, Writing – review and editing

   Contributed equally with

   Ariel Zylberberg and Michael N Shadlen

   Competing interests

   consultant to CTRL- Labs Inc, in the Reality Labs Division of Meta. This entity did not support or influence this work

   "This ORCID iD identifies the author of this article:" 0000-0003-2011-2790

6. Michael N Shadlen

   1. Zuckerman Mind Brain and Behavior Institute, Columbia University, New York, United States
   2. Howard Hughes Medical Institute, Chevy Chase, United States
   3. Department of Neuroscience, Columbia University, New York, United States
   4. Kavli Institute, Columbia University, New York, United States

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   Contributed equally with

   Ariel Zylberberg and Daniel M Wolpert

   For correspondence

   shadlen@columbia.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-2002-2210

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