The causal pathway for perceptual decisions in the primate brain is still debated. The lateral intraparietal cortex (LIP) has been a contender as a causal decision-making locus, as it demonstrates accumulation of perceptual evidence (Shadlen and Newsome, 2001; Huk and Shadlen, 2005; Gold and Shadlen, 2007; Kiani and Shadlen, 2009; Hanks et al., 2014) and because stimulating LIP neurons can lead to more quickly reaching a decision threshold (Hanks et al., 2006). Yet a recent study has shown that inactivating large sections of LIP has little effect on decision-making (Katz et al., 2016), suggesting that other areas with which LIP is closely associated may redundantly play a causal role (Hanks and Summerfield, 2017). Such redundancy is also suggested by a study in analogous brain regions in rats (Erlich et al., 2015).

Here, we use data from one of these closely related areas, area 8Ar in dorsolateral prefrontal cortex. Area 8Ar, like LIP, carries information about planned saccades in direction discrimination tasks (Kim and Shadlen, 1999; Constantinidis and Goldman-Rakic, 2002; Hussar and Pasternak, 2009; Kiani et al., 2014, 2015). The accuracy, latency, and confidence in the decision are influenced by noise in the input (experimentally controlled in the RDM task by introducing random motion in the stimulus that varies the percentage of dots moving coherently in one direction), which has measurable effects in both prefrontal cortex (Kim and Shadlen, 1999) and LIP (Shadlen and Newsome, 2001). This sensitivity to the strength of the input suggests that these brain areas do not only represent the decision once made, but are also involved in the decision making process, accumulating information about sensory input.

At the most abstract level, decision-making can be fitted using a variety of continuous or discrete one-dimensional random walks or diffusion models with fixed or variable thresholds (Gold and Shadlen, 2007). A number of closely related simple neural network models, extended to include lateral inhibition or recurrent activation, recover several features of these empirical findings (Gold and Shadlen, 2007; Ratcliff and McKoon, 2008), including timescales in the decision process that are much slower than those of individual neurons. More detailed mechanistic models have successfully reproduced important aspects of the observed decision-making process using individual spiking neurons and also emphasize the emergence of slowly acquired information at the neuronal level using the terminology of recurrent “reverberation” (Wang, 2002). (See Discussion for more details about existing models.)

Most of these models assume a single phase in which information is accumulated. They do not, however, explicitly consider the collective properties of this accumulation—is the information about the decision localized in individual neurons or encoded at the population level? Furthermore, is accumulated information shared or transmitted across the population of neurons? The observation that neuronal behavior is qualitatively different before and after the go cue (see Figures 1, 2) minimally suggests that there are two processes taking place, rather than just one accumulation phase.

Existing decision-making literature tends to neglect neural behavior after the go cue, treating it as “choice execution.” We argue that post-go-cue behavior is an extension of decision making at the system level and view the process between the go cue and the saccade as essential to collective decision-making—“reading out” the information that is, before this point, only available by pooling information from many cells.

First, we quantify how much information about the decision is encoded in individual neural firing rates. We find substantial heterogeneity over neurons and as a function of time. As shown in Figure 2A and described in the figure legend, we can group neurons into three classes based on when their firing patterns encode information about the decision.

Figure 2B compares histograms of mutual information at the time of the go cue and 500 ms later, demonstrating that while no individual neuron encodes more than 0.1 bits of information about the decision at the time of the go cue, many individual neurons encode a substantial amount of information during and after the saccade indicating the decision. This observation motivates splitting the process into two distinct temporal periods, Phase I (pre go cue) and Phase II (post-go cue) (Figure 1).

We next assess whether information about the decision is encoded collectively at the whole population level or within a subpopulation and how this quantity compares to the information encoded at the individual neuron level.

An encoding based on Linear Discriminant Analysis (LDA) allows us to verify that the population encodes more information than any single neural unit by producing a lower bound on the mutual information encoded jointly by the entire population (see Methods). As shown in Figure 3A, this collective encoding is able at the time of the go cue to predict the decision on more than 80% of out-of-sample trials, corresponding to a collective mutual information (CMI) reaching about 0.5 bits (Figure 11A). Shortly after the go cue, CMI rises to nearly 1 bit, with nearly perfect out-of-sample predictions.

Interestingly, at the same time that the collective information jumps to its maximum value, there is a switch in the distributed nature of the encoding: many individual units become highly informative in Phase II, providing redundant information. This contrasts with Phase I, in which much more information is contained at the population level than in any single unit. To quantify the redundancy of the encoding, we ask how many units need to be included in the LDA encoding in order to reach 95% of the maximal collective predictive power. As shown in Figure 4, about 20 units need to be included for peak performance in Phase I, but this drops sharply to 1 or 2 at the time of the saccade. (See Figure 13 for a version of this plot aligned by saccade time, showing explicitly that this decrease in distributedness starts before the saccade.)

Figure 5 demonstrates that all information about the decision is encoded in class H units in Phase I, and that this changes to include class M units in Phase II. In addition, information is acquired over a longer timescale in Phase I than in Phase II. Table 1 summarizes the observed properties of the three classes.

We also find information at the population level specifically about the input stimulus, but it is small compared to the information about the decision, and is significant only during Phase I. This can be seen in Figure 11C: at its peak during Phase I, we estimate that the LDA encoding provides only about 0.02 bits of information per trial about whether the coherence of the input is strong or weak (out of a possible 1.0 bit; see Appendix). Though this value is similar to the uncertainty in our estimate of the mutual information, we can be confident that it is nonzero by noting that the LDA encoding can predict the coherence of out-of-sample trials significantly better than chance (Figure 6A).

We explore the relationship between timescales of information accumulation, memory, and informational redundancy using a simple dynamical model. We start simply by representing individual neurons as having a state that (1) is persistent on the timescale of tens of ms, (2) transiently affects the states of other neurons via a firing rate that saturates as a function of the current state, and (3) is subject to random noise. The model consists of N homogeneous, all-to-all coupled neurons with individual firing rates r_i = tanhx_i, whose dynamics are governed by:

Here, s represents an input signal with magnitude proportional to the coherence of the visual stimulus and sign indicating the dominant direction of motion, c is the strength of positive interactions between every pair of neurons, τ = 10 ms sets the timescale of decay for a single neuron, and the final term represents noise with variance Γ² [drawn from a Gaussian distribution for simplicity: 〈ξ(t0)ξ(t1)〉=δ(t1-t0)N(0,Γ2)]. See Methods and Appendix for additional details motivating this form for the dynamics. We assume that the N neurons are fully responsible for making the decision, and that the experiment measures some subset of these units (N_measured < N). As shown in Figures 3, 4, 6, 8, (see also Figures 11, 12) we find that this model can capture both the integration and storage of the decision during Phase I and the consensus formation and propagation of the decision during Phase II.

Our simple dynamic rate model model produces behavior that is critically dependent on the degree of recurrent excitation, controlled by c̄=c(N-1), and the amount of neural noise Γ (Figure 7). For fixed Γ, there exists a critical value of recurrent excitation c̄* such that for c̄<c̄*, the only stable attractor is at r→=r→0=0→, while for c̄>c̄*, two stable attractors, r→+ and r→-, emerge symmetrically on two sides of r→0 (see Appendix).¹ Bistability is required for persistent activity that remembers the decision during the delay period. The informative direction in rate space lies along the vector v→=r→+-r→-, which in this simple model weights all units equally and with the same sign.

The degree of recurrent excitation c̄ also controls two other aspects of decision-making: (1) the timescale τ_decision over which information is accumulated and (2) the redundancy of the encoding of the decision.

First, the relevant timescale for motion along the decision direction v→ is inversely related to the distance from the instability threshold c̄* (see Figure 7C): e.g., without noise (Γ = 0), c̄*=1, and τdecision=τ/(c̄-c̄*). This is the phenomenon of “critical slowing down,” which slows motion along v→ when the system is near the threshold.² Hence, units that forget on a short timescale can still contribute to aggregate-level behavior that integrates over longer timescales. This emergent timescale has been recognized in “reverberation” models (Wang, 2002) as an essential feature of distributed decision-making.

Secondly, sufficiently close to the threshold, noisy individual cells are only weakly constrained to have a similar state as the others; whereas summing over many cells can reliably predict the behavior of the whole, individual cells do not encode much information. This corresponds to the “synergistic” state of Phase I. As recurrent excitation increases, consensus is more strongly enforced, leading to individuals containing more information. This corresponds to the “redundant” state of Phase II. This dependence is demonstrated in Figure 7B: the number of units one needs to measure to obtain maximal information is largest near the transition point.

The combination of low redundancy and slow dynamics therefore suggests that Phase I should correspond to c̄ only slightly greater than c̄*, and large redundancy in Phase II suggests a larger c̄. Indeed, as displayed in Figures 3, 4, 8, the simple rate model reproduces the qualitative behavior of the system by changing the single parameter c̄ between Phase I and II, with c̄II>c̄I>c̄*. Specifically, the increased redundancy of Phase II is associated with a faster timescale. Together, these results imply a dual coding theory for collective decision-making through critical slowing down, summarized in Figure 9.

Additionally, the model explains how neurons responsible for the decision can encode a relatively small amount of information about the input stimulus coherence. As the attractors representing the decision do not depend on the coherence, information is only contained in the speed with which those attractors are approached. This speed is in turn related to the magnitude of the input current (s), which can be small if N is sufficiently large to magnify a small signal.

Using a simple linear relationship between stimulus coherence and input s produces a good fit to the observed psychometric function, shown in Figure 8. The figure also demonstrates the model's prediction (as in Wang, 2002) that using the same large recurrent excitation c̄ in both Phase I and Phase II would lead to a faster decision process, which would integrate the input stimulus over a shorter time and therefore produce poorer performance in trials with more ambiguous, less coherent visual stimuli.

In prior studies the representational status of areas 8Ar and LIP has remained ambiguous, and is often described as partly sensory and partly motor. We find that whereas spiking activity in 8Ar cells is strongly predictive of saccadic eye-movements, there is little residual information concerning the visual stimuli (Figures 3, 11). In other words, these cells are primarily predictive of motor behavior and not sensory input, and in this informational sense are almost purely motor.

Yet this does not rule out the measured neurons being part of a group of similar cells that are collectively fully responsible for the decision. In the rate model, the simulated cells are fully responsible for the decision but measuring a subset of the cells reveals only a small amount of information about the input signal—and even this information is quickly lost once the system reaches an attractor state representing the decision. Thus, because we have data on only a small fraction of all neurons in these areas, it is feasible that 8Ar neurons as a whole could be solely responsible for mapping sensory data onto the decision.

Many previous studies that attempt to model the perceptual decision-making system (e.g., Wang, 2002) and neural computation more generally (see Lisman, 2015 for a review) have focused on interactions within and between two distinct groups of neurons: the pyramidal neurons and the inhibitory interneurons.

These neural classes are typically identified from electrode data by differences in firing rate, spiking waveform, burstiness, and refractory period (e.g., Csicsvari et al., 1999; Diba et al., 2014). Pyramidal cells are typically more informative about (selective with respect to) sensory input than interneurons (e.g., Diba et al., 2014). Additionally, sensory selective cells and/or cells identified as pyramidal have been shown in many contexts to fire more independently of one another: pyramidal compared to parvalbumin-expressing interneurons in the visual cortex (Hofer et al., 2011); visually responsive neurons in V1 (Ecker et al., 2010); excitatory vs. inhibitory neurons in the prefrontal cortex (Constantinidis and Goldman-Rakic, 2002); pyramidal vs. interneurons in rat hippocampus (Diba et al., 2014). This suggests classifying our Class H (and perhaps Class M) units as pyramidal cells and Class L as interneurons.

In Table 2, we compare the model of Equation (1) to related binary decision-making models that vary in their level of detail. The equations shown here use a single population, but an equivalent formulation can be found in each case consisting of two competing populations (Usher and McClelland, 2001; Gold and Shadlen, 2007). We do not include models using discrete neural firing rate states (e.g., Latimer et al., 2015), which we expect to have equivalent behavior at the collective level.

The simplest models are most analytically tractable, abstracting away both the distributed nature of the computation and mechanisms for saturation and persistent memory, beginning with the simplest drift diffusion model (Bogacz et al., 2006). These models assume that a separate mechanism makes and stores the decision once a stochastic process reaches a given threshold. This simple process successfully explains the integration of sensory information, the distribution of reaction times, and accuracy as a function of input coherence.

However, models that do not include decay or saturation of firing rates cannot explain the loss of information about the coherence of the input late in the trial (Figure 6A). Diffusive models constrained to stay within specified bounds could potentially alleviate this problem, as could so-called ‘leaky integrators’ with a state that decays back to a starting value (Kiani et al., 2008). Such models have been ruled out because they predict persistent susceptibility to sensory evidence presented late in the trial, which is not seen in experiments (Kiani et al., 2008).

Instead, the loss of information about coherence and the loss of susceptibility can be parsimoniously explained if the decision process involves approaching stable attractors that store the decision during the delay period. Including feedback is one way to produce multiple attractors. In the simplest case, this leads to the Ornstein-Uhlenbeck (OU) model (Busemeyer and Townsend, 1993; Bogacz et al., 2006), which produces multiple attractors when combining positive feedback with saturation or boundedness (Zhang and Bogacz, 2010).

None of these low-dimensional models, however, address the issue of how the computation is distributed over multiple cells. The noisiness of individual neurons likely necessitates larger populations of neurons in order to accumulate persistent information over longer timescales.

Existing models that incorporate collections of cells include those with populations of spiking neurons (Wang, 2002; Lo and Wang, 2006; Wimmer et al., 2015) or of neurons with continuous states governed by noisy dynamics (Wong and Wang, 2006). As in our model, these studies have emphasized the importance of recurrent interactions to slow the effective timescale using the phenomenon of “reverberation,” essentially equivalent to critical slowing down. One proposed model also includes separate neurons that act as a switch governing the transition from information accumulation to decision commitment (Lo and Wang, 2006). This is distinct from the single-population story we present here, instead hypothesizing that information accumulation and consensus happen in different brain regions. Generally, these studies have not explicitly addressed the distributedness of neural information, but we expect the models may display similar phenomena with respect to population dynamics, coding, and redundancy. It will be useful in future work to confirm this.

The model presented in Equation (1) captures the details necessary to describe the distributed nature of the computation but abstracts away most details of neurobiology. It can be viewed as a distributed implementation of an Ornstein-Uhlenbeck model (Busemeyer and Townsend, 1993; Bogacz et al., 2006; Zhang and Bogacz, 2010) that includes biologically realistic saturation of firing rates.

The results we find here in the specific case of the random dots task in prefrontal cortex hint at more general design principles for decision-making and collective computation. For instance, much attention has been paid to the idea that it may be beneficial for collective information processing systems to exist near a symmetry breaking transition, or critical point (Langton, 1990; Mora and Bialek, 2011; Plenz and Niebur, 2014). We can come to a similar but more subtle conclusion in our dynamic model: a system near a transition (our Phase I) is indeed successful in producing distributed, collective states that remain more sensitive to exogenous inputs (as has been emphasized in “criticality” research; e.g., Shew et al., 2009), but reaching consensus (in Phase II) requires moving away from the transition toward a collectively boring but useful “frozen” state. Locating, managing, and controlling dynamics with respect to such transitions and collective states is certainly important for not only a brain involved in cognition and learning, but in controlling collective behavior across many biological and social systems.

Mutual information is the information shared by any two streams of data A and B. This can be thought of in information theoretic terms as the average number of bits that are revealed about B upon measuring A. For instance, with A representing a firing rate and B the behavioral output (a choice of either left or right), the mutual information I(A, B) represents how much information the neuron's rate provides about the decision variable. The mutual information between A and B is the sum of the marginal entropies (the maximum potential information content of the paired system) minus the joint entropy (the amount of noise in the paired system); see Appendix.

In these analyses, we employ rates of neural firing averaged over time bins of length 200 ms (assuming rate encoding and ignoring details about the precise timing of spikes) in order to calculate dependencies among neural and behavioral states using mutual information. Entropies are estimated using the NSB method (Nemenman et al., 2004, 2002). We consider a mutual information to be significant if it is greater than 0.01 bits, roughly the resolution of entropy estimation for the amount of data we have, which is also estimated using the NSB method.

To combine neural firing rates into a collective encoding, we use LDA. Given neural rate data from many trials and classification of those trials as left decisions and right decisions, LDA attempts to find the linear combination of neural rates that is most informative of the class (left or right).

LDA makes the simplifying assumption that data from each class is produced by a multidimensional Gaussian specified by the observed mean μ→α and covariance matrix C_α for each class α. In this case, projecting any given data vector r→ along the LDA vector, defined as

produces a number designed to be informative about the class from which the data vector came (with maximal performance guaranteed when C₁ = C₂). That is, the LDA vector v→ provides a weighting of individual neural rates that, when summed, approximately maximally separates the two classes.

LDA simultaneously provides a framework for predicting the output given rate data. Looking at any given set of rates r→, the LDA's estimate of the relative log-likelihood of left vs. right can be written as:

Thus, positive L(r→) corresponds to a (maximum likelihood) prediction that r→ came from a left trial, and negative to prediction of a right trial.

Equation (1) describes a dynamic Hopfield network (Hopfield, 1984) that includes Gaussian noise on neuron states and is restricted to uniform positive interactions among all cells.

We imagine that we measure some subset N_measured of N neurons that are collectively responsible for the decision, with no input from other areas of the brain except for a temporary signal s, which is proportional to the signed coherence of the visual stimulus.

Although this simplified picture does not include spiking, models that include spiking have been shown to produce similar behavior (Wang, 2002). Our model can be mapped onto a model with separate populations that are positively selective for leftward and rightward stimuli by assuming that (1) the variables r_i represent the difference from a typical firing rate, (2) oppositely selective cells are mapped onto r_i with opposite sign while simultaneously changing the sign of s_i and the sign of c_ij when i and j are oppositely selective cells, corresponding to mutual inhibition between opposite populations.

Experimental data were provided by Roozbeh Kiani and William Newsome. Data were gathered in accordance with the recommendations of the National Institutes of Health Guides for the Care and Use of Laboratory Animals. The protocol was approved by the Stanford University Animal Care and Use Committee (IACUC number 9720). The data is a subset of that described in Kiani et al. (2014, 2015). Details of the stimulus and eye monitoring method are explained in those references. Neural spiking data consists of spike times measured at a resolution of 1/30 ms for 169 neural units using a 4 mm × 4 mm multielectrode array (see Figure 10). Data includes 1,778 trials taken from one animal in one recording session, with the signed coherence of the stimulus for each trial chosen at random from the set {−0.32, −0.16, −0.08, −0.04, −0.02, −0.01, 0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32}. Other task data for each trial include the direction of the decision saccade as well as the times of the onset of the stimulus, end of stimulus, go cue, and response saccade.

In a continuous-time model of neural activity (Hopfield, 1984; Beer, 1995), the time derivative of the state x_i of synapse i is the sum of external input s, leak current proportional to x_i, and input from other neurons in the system. Input from each other neuron is assumed to be proportional to its current firing rate r_j, which is a sigmoidal function of its current state: r_j = g(x_j). This produces:

where the timescale of the cell returning to equilibrium in the absence of other signals is set by τ_i. For simplicity, we assume complete homogeneity, with all-to-all synaptic connections of strength c, constant individual timescales τ_i = τ, and g_i(x_i) = r_i = tanh(x_i):

Adding a Gaussian noise term ξ, with 〈ξ(t0)ξ(t1)〉=δ(t1-t0)N(0,Γ2), we obtain Equation (1):

The timescale τ is set at 10 ms to match the order of magnitude of the characteristic timescales of synaptic receptors (Moreno-Bote and Parga, 2010).

Importantly, we expect the qualitative features of the relationship between timescales and redundancy to be insensitive to the exact form of the dynamics. These features are produced near any collective transition displaying critical slowing down, as described below.

As the strength c of recurrent excitatory interactions increases, there is a transition from one stable state at r_i = 0 to two stable states at ri=±r*. Without noise (Γ = 0) and assuming identical individuals r_i = r, the dynamics becomes one-dimensional:

As can be seen by measuring the stability of the x = 0 state (taking a derivative of Equation 8 with respect to x), this transition (bifurcation) is controlled by the degree of recurrent excitation c̄=c(N-1), with the transition occurring at c̄=c̄*=1. Above the instability transition and neglecting noise, the final state retains indefinitely a perfect memory of the sign of the input s. Furthermore, the characteristic timescale of moving away from x = 0, which sets the timescale for the decision process, is inversely related to the distance from this transition: τdecision=τ/(c̄-1). When noise is added (Γ > 0), the attractors are destabilized, but we expect equivalent stability to be regained by moving to larger c̄. This dependence of c̄* on Γ is demonstrated in Figure 7.

The remaining parameters of the model can be set by matching the qualitative characteristics of informational dynamics observed in the data (see Figures 3, 4). First, the degree of recurrent excitation is set to roughly match the timescales with which information increases in Phases I and II: with no noise, c̄I=1.1 corresponds to accumulation on the scale of 100 ms in Phase I, and c̄=1.5 to a faster timescale of 20 ms in Phase II.

Second, because the dynamics is only slightly perturbed by the input (Figure 11C), we next set s = 0 and match the behavior of information redundancy (Figure 4A) solely by varying the total number of model neurons N, the number of measured neurons N_measured, and the amount of noise Γ. As N becomes larger, we expect that a larger Γ will produce the same dynamics and variance along the decision dimension v→, while corresponding to larger noise and thus less information carried by individuals.³ We find that N = 500, N_measured = 5, and Γ = 0.16 reproduces the qualitative features of the two phases. We do not attempt here to estimate the number of neurons truly involved in the decision-making process; this will require a more sophisticated model than presented here. Specifically, this model does not capture the strong heterogeneity that exists even within the defined cell classes, which will be important to estimating the actual number of required cells.

Finally, we set the form of the external input s representing transient sensory information. We assume that each cell receives input that is proportional to the signed coherence of the visual stimulus (with sign determined by the dominant direction of motion): s = s_cζ for coherence ζ. The signal is applied for the duration of the visual stimulus, 810 ms, after which it is set to zero. The proportionality constant s_c is set by roughly matching the proportion of correct decisions as a function of coherence (Figure 8). The values of all model parameters are listed in Table 3.

We emphasize that there are very likely other sets of parameters that equally well match the qualitative features of the data, so the values of the individual parameters are not meant to represent best choices to be used in other contexts. For instance, moving along the transition line in Figure 7 produces largely equivalent behavior, demonstrating that distance from the point of instability is most important. We use the model not to infer specific parameters for individual level behavior, but rather (1) to show that it is possible to reproduce the behavior seen in both temporal phases using a simple model, and (2) to demonstrate the close relationships among distance from instability, timescale of accumulation, and redundancy of representation.

The model simulation recapitulates the timing of trials. In Phase I, with c̄=c̄I and initial conditions r_i = 0 ∀i, the stimulus is presented for 810 ms, followed by an 810 ms delay period⁴ during which s = 0. The signed coherence takes values {−0.32, −0.16, −0.08, −0.04, −0.02, −0.01, 0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32}, with each signed coherence simulated 140 times for a total of 1820 simulation trials.⁵ Phase II consists of two more 810 ms time periods with increased c̄=c̄II, with initial conditions given by the final timestep of Phase I. Integration is performed using straightforward Euler timesteps, with 500 timesteps per 810 ms time period. Decreasing the step size by a factor of two does not qualitatively affect the results.

The mutual information is a standard measure of dependence in information theory. Calculating mutual information between random variables A and B begins with estimating the probabilities of all possible pairs of states of A (with states a₁, a₂, …) and B (with states b₁, b₂, …). Call these probabilities P(a₁, b₁), P(a₁, b₂), P(a₂, b₁), …. The marginal entropy S(B) measures the number of bits⁶ necessary to communicate the state of B without knowing the state of A. This marginal entropy S(B)=−∑jP(bj)logP(bj), where P(bj)=∑iP(ai,bj) sums over the unknown state of A. Similarly, S(A)=−∑iP(ai)logP(ai) with P(ai)=∑jP(ai,bj). Finally, the joint entropy measures the number of bits necessary to describe the combined state of A and B: S(A,B)=−∑i∑jP(ai,bj)logP(ai,bj), and the mutual information is given by the difference of the total possible entropy of the paired data and its actual entropy:

Intuitively, I(A, B) measures the reduction of uncertainty about the state of B when given the state of A, and vice versa.

Estimating the mutual information between the collective firing pattern of the measured neurons and a behavioral variable requires estimating the entropy of each (Equation 9). The entropy of the neural firing is difficult to calculate because it has many dimensions and thus many possible states. ⁷

If we hypothesize a specific encoding, however, we can use the Data Processing Inequality to produce a lower bound on the mutual information. LDA constitutes such a hypothesis; by transforming the neural rate data r→ at a given time into a single continuous variable v=v→·r→ (with v→ given by Equation 2), the mutual information of v with the variable in question provides a lower bound on the true mutual information.

In Figure 11, to estimate the entropy of the LDA projection v used to calculate the lower bound on the CMI, we bin values for v into N_B equally-spaced bins (with N_B≪N_s) and use the NSB method (Nemenman et al., 2004, 2002) on this reduced space. We choose N_B = 30 by increasing N_B until the mutual informations shown in Figure 11 saturate.

Neural rates are calculated using a bin width of 200 ms for measuring information about the decision (Figure 3) and 500 ms for measuring information about the coherence of the stimulus (Figure 6). “Best unit in Phase I” refers to the unit with largest average mutual information in 200 ms time bins in Phase I (in the 2000 ms before the go cue). Data is split randomly into an equal number of in- and out-of-sample trials; we plot the resulting mean (line) and standard deviation (shaded area) over 20 of these random partitions in the case of predicting the output decision, and 100 partitions in the case of predicting motion coherence.

To test the amount of information present about coherence of the input, we split the trials into “strong coherence” trials (coherence value equal to 0.08, 0.16, or 0.32; 820 trials; relative frequency 0.461) and “weak coherence” trials (coherence value equal to 0, 0.01, 0.02, or 0.04; 958 trials; relative frequency 0.539). Mutual information and out-of-sample prediction are measured with respect to the binary classification of strong vs. weak, leading to a maximum possible mutual information of −0.461log0.461−0.539log0.539 = 0.996 bits. Previous studies have found that variation in firing rates is opposite in sign depending on the output direction (Shadlen and Newsome, 2001). For this reason, without knowing the eventual output, differences in firing rates due to stimulus coherence tend to be invisible using the linear LDA encoding. The coherence LDA calculations and predictions are thus performed separately in the two output direction conditions and then averaged; this corresponds to the task of predicting the coherence of the signal given both neural firing rates and the eventual output direction.

In Figure 11, we plot the mutual information measures corresponding to the predictive power plotted in Figure 3.

To measure the number of units needed to reach a certain predictive performance, we add neurons one at a time ordered by their individual mutual information with the output at the given time (calculated with bin width 200 ms). Plotted lines indicate means and shaded regions indicate standard deviations over 20 realizations of in-sample and out-of-sample partitioning, as described above with regard to Figure 3.

Though it is intuitive, our measure of the number of individual units needed to reach the collective performance may not be ideal for future experiments in that we expect it to become uninformative for large N when individual neural behavior is heterogeneous. If there is some small fraction of neurons that are individually very informative (as was found in Shadlen and Newsome, 2001), then the number of units needed to reach the collective performance will always approach 1 as N becomes large. Alternatively, we can use an information theoretic measure of redundancy that also takes into account the number of very informative units. We use here the relative redundancy:

estimating the collective and individual mutual informations in the same way as in Figure 11. This redundancy is plotted as a function of trial time in Figure 12. The story is the same as that told by Figure 4: Units become more redundant in the information they encode about the output near the time of the saccade in Phase II.

Finally, we would like to test directly the extent to which the change in redundancy happens before or after the saccade. Aligning by the saccade time instead of the go cue, and using a smaller time window of 100 ms for finer temporal resolution, produces the right half of Figure 13. We see that the abrupt increase in collective information and decrease in distributedness begins before the saccade. This rules out an interpretation in which the change is caused only by visual feedback coming from the execution of the eye movement. The datapoint at t = 0 corresponds to a time window from 50 ms before to 50 ms after the saccade, during which it is unlikely that any visual feedback signal has reached the measured neurons. Even ignoring this datapoint, however, the number of units needed to saturate predictive power decreases significantly before the saccade: Saturation using rates in the 100 ms prior to the go cue requires 23 ± 5 units, whereas the 100 ms prior to the saccade requires 11 ± 2 units (and a 100 ms window 250 ms after the saccade requires 2.8 ± 0.4 units, in each case averaged over 20 in- and out-of-sample shuffles, as in Figure 4).

Besides avoiding premature saturation, another argument for decreased interactions during decision-making comes from a “wisdom of the crowd” argument, in which noise in individual decisions is best removed from an average by having individuals cast independent votes. We do not focus on this explanation because the magnitude of the effect in general depends on the specifics of the interactions, and for some cases (e.g., each individual moves their opinion closer to the average of individuals it interacts with) has no effect on the accuracy of the decision.

For fixed means and variances, it is true that having noise correlations that are the same sign as signal correlations leads to worse performance (as in Jeanne et al., 2013, originally explored in Abbott and Dayan, 1999). Yet this is not easily connected to optimizing interactions for decision-making, because changing interactions in a dynamical context does not typically leave means and variances fixed.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.