Main

A fundamental component of neural computation is how populations of neurons encode and transmit information to downstream brain areas1,2. Each cortical area communicates with many areas and contains a heterogeneous population of neurons that project to distinct downstream targets3,4,5,6. For the transmission of information between brain areas, the relevant neural codes are likely formed by populations of neurons that communicate with the same downstream target area, allowing their activity to be read out collectively. However, in most studies of neural population codes, populations have been analyzed without the knowledge of whether the cells project to the same target. It is, therefore, an open question of what principles underlie coding in populations of neurons that project to the same target area.

The information encoded in a population of neurons is affected by correlations between the activity of different neurons. Experimental and theoretical works have demonstrated how the correlations in activity between pairs of neurons can either enhance the population’s information, due to synergistic neuron–neuron correlations, or increase redundancy between neurons, which establishes robust transmission but limits the information encoded7. Most of this understanding arises from considerations of typical or average pairwise correlation values in populations. However, recent work reports that pairwise correlations in large populations can take on additional network structures, such as hubs of redundant or synergistic interactions8,9,10. Also, theoretical studies propose that a network-level structure of pairwise correlations may enhance neural population information11. Notably, whether projection pathways have network-level structures that enhance the information encoded in the projection pathway or aid their transmission to other brain areas has not been studied.

We studied the population codes in projection pathways between cortical areas in the posterior parietal cortex (PPC). PPC is a sensory-motor interface involved in decision-making tasks, including those involved in navigation12,13,14,15,16,17,18. PPC has heterogeneous activity profiles, including cells encoding various sensory modalities, locomotor movements and cognitive signals, such as spatial and choice information12,19,20,21,22. PPC is densely interconnected with cortical and subcortical regions in a network containing retrosplenial cortex (RSC) and anterior cingulate cortex (ACC)23. In addition, population codes in PPC contain correlations between neurons that benefit behavior24,25,26. Here we study PPC in a flexible navigation-based decision-making task because navigation decisions require the coordination of multiple brain areas to integrate signals across areas and also because PPC activity is necessary for mice to solve navigation decision tasks12,27,28,29.

We developed statistical multivariate modeling methods to investigate the population codes in cells sending axonal projections to the same target. We discovered that, in PPC neurons projecting to the same target, pairwise correlations are stronger and arranged into a specialized network structure of interactions. This structure consists of pools of neurons with enriched within-pool and reduced across-pool information-enhancing (IE) interactions, with respect to a randomly structured network. This structure enhances the amount of information about the mouse’s choice encoded by the population, with proportionally larger contributions for larger population sizes. Remarkably, this IE structure is only present in populations of cells projecting to the same target, and not in neighboring populations with unidentified outputs. Such structure is present when mice make correct choices, but not when they make incorrect choices. We propose that specialized network structures in PPC populations, which comprise an output pathway, enhance signal propagation in a manner that may facilitate accurate decision-making.

Results

A delayed match-to-sample task that isolates components of flexible navigation decisions

We developed a delayed match-to-sample task using navigation in a virtual reality T-maze (Fig. 1a)27. The T-stem contained a black or white sample cue followed by a delay maze segment with identical visual patterns on every trial. When mice passed a specific location, a test cue was revealed as a white tower in the left T-arm and a black tower in the right T-arm, or vice versa. The sample cue and test cue were chosen randomly and independently in each trial, and the two types of each cue defined four trial types. Mice received rewards when they turned into the T-arm whose color matched the sample cue. Thus, mice combined a memory of the sample cue with the test cue identity to choose a turn direction at the T-intersection. After training, mice performed this task with approximately 80% accuracy (Extended Data Fig. 1a,b). Incorrect trials occurred interleaved with correct trials at a relatively constant rate throughout the session, suggesting that errors were due to inaccurate decision-making rather than disengagement (Extended Data Fig. 1c).

Fig. 1: Differences in the activity of neurons projecting to distinct cortical targets.

a, Schematic of a delayed match-to-sample task in virtual reality. b, Retrograde virus injections to label PPC neurons projecting to ACC, RSC and contralateral PPC (cPPC). Images are 250 × 350 μm. c, Depth distribution of PPC neurons projecting to different areas. Error bars indicate mean ± s.e.m. over mice. d, Normalized mean deconvolved calcium traces of PPC neurons sorted based on the cross-validated peak time. Vertical gray lines represent onsets of sample cue, delay, test cue, turn into T-arms and reward. Gray dashed lines correspond to 1 s before turn and 0.5 s before reward. Color bar shows normalized mean deconvolved calcium intensity (a.u.). e, Cumulative distribution of the peak activity times for neurons. Compared to nonlabeled population, ***P < 0.001 for ACC-projecting (blue) and RSC-projecting (green) neurons, P = 0.13 for cPPC-projecting (red) neurons; two-sample KS-test. Error bars are s.e.m. computed from bootstrapping. f, Mean ± s.e.m. deconvolved calcium activity of different populations. Nonlabeled neurons are shown in black. g, Mean ± s.e.m. deconvolved calcium activity of example neurons with different encoding properties are shown in different trial conditions. Each trace corresponds to a trial type with a given sample cue and test cue in correct (green choice arrow) or incorrect (red choice arrow) trials. Neurons 1 to 4 encode the sample cue, the test cue, the left–right turn direction (choice) and the reward direction, respectively. Neuron 5 is active in only one of the four trial conditions, both in correct and incorrect trials, and thus encodes multiple task variables. Each subpanel contains four traces, corresponding to the four trial types. Some traces are hidden if they contain deconvolved calcium activity values of zero. Nonlabeled, n = 2,506 cells; cPPC projection, n = 93 cells; RSC projection, n = 127 cells; ACC projection, n = 134 cells. a.u., arbitrary units.

We used two-photon calcium imaging to measure the activity of hundreds of neurons simultaneously in layer 2/3 of PPC. We injected retrograde tracers conjugated to fluorescent dyes of different colors to identify neurons with axonal projections to ACC, RSC and contralateral PPC (Fig. 1b and Extended Data Fig. 1d,g). These areas are major recipients of projections from layer 2/3 PPC neurons4, and the ACC–RSC–PPC network is important for navigation-based decision tasks29,30. The PPC neurons projecting to ACC, RSC and contralateral PPC were intermingled, except that ACC-projecting neurons were enriched in superficial layer 2/3 (Fig. 1c). Neurons projecting to the same area were slightly closer in anatomical space than in unlabeled neurons (Extended Data Fig. 1f). Neurons labeled with multiple retrograde tracers were not observed.

Neurons were transiently active during task trials with different neurons active at different times, and the activity of the population tiled the trial12 (Fig. 1d and Extended Data Fig. 1h). ACC-projecting cells had higher activity early in the trial, while RSC-projecting cells had higher activity later. Contralateral PPC-projecting neurons had more uniform activity across the trial (Fig. 1e,f). These differences in activity levels across the trial suggest that neurons projecting to different targets could contribute to different stages of information processing (Extended Data Fig. 3a). They could encode the sample cue (neuron 1; Fig. 1g), the test cue (neuron 2), the left–right turn direction (choice) (neuron 3) and the combination of the sample cue and test cue that indicates the reward direction (neuron 4). Note that the reward direction (combination of the sample and test cues) and choice are identical on correct trials and opposite on incorrect trials. We also identified neurons that were active only on one of the four trial types in both correct and incorrect trials, thus encoding multiple task variables (neuron 5; Fig. 1g).

Vine copula models to analyze encoding in multivariate neural and behavioral data

To quantify the selectivity of neurons for a task variable, we isolated the contribution of the variable to a neuron’s activity while controlling for other variables that also contribute. This was important because neural activity is modulated by movements of the mouse19,20,31, and a mouse’s movements correlate with task variables (Extended Data Fig. 2a,c). We considered the locomotor movements used to control the virtual environment.

We adapted nonparametric vine copula (NPvC) models to estimate the multivariate dependence among a neuron’s activity, task variables and movement variables (Fig. 2a). This method expresses the multivariate probability densities as the product of a copula, which quantifies the statistical dependencies among all these variables, and of the marginal distributions conditioned on time, task variables and movement variables32,33,34. The mutual information between two variables depends only on the copula and not on the marginal distributions35. Using a sequential probabilistic graphical model called the vine copula32,34, we broke down the complex, data-hungry estimation of the full multivariate dependencies into a sequence of simpler, data-robust bivariate dependencies that were estimated using a nonparametric kernel-based method35 (Fig. 2a). This approach takes into account correlations between all the variables in the multivariate probability, does not make assumptions about the form of the marginal distributions and their dependencies, and is able to capture nonlinear dependencies between variables, thus providing advantages over conventional methods such as generalized linear models (GLMs)24,36,37. By discounting collinearities between task and behavioral variables, this method isolates the contribution of individual variables and improves the estimation of information in neural activity (Extended Data Fig. 4). Using the NPvC, we estimated the expected activity of a neuron for any value of task and movement variables and at any time in the trial (Fig. 2b). The NPvC predicted frame-by-frame held-out neural activity better than a GLM (Fig. 2c)24,36,37.

Fig. 2: Vine copula modeling of neural activity.

a, Schematic of NPvC model of neural activity (r) as a function of time (t), a vector of movement variables (x) with components (({x}_{1},\ldots ,{x}_{n})) and task variable (c). Conditional vine copulas are built between neural activity and all the other variables for each task variable (c). Mixing the vine copula and marginal distributions gives the conditional density function of neural activity and other variables. The vine copula model can be used either to estimate the value of neural activity conditioned over all the other variables (which is the copula fit ({r}_{\mathrm{NPC}})) or to generate samples, or to estimate various conditional entropy and mutual information values. b, Deconvolved calcium activity of two example neurons (black) and the cross-validated predictions of the NPvC model (orange, dashed line) and GLM (pink, solid line). c, Cumulative distribution of FDE across neurons for the GLM and the NPvC model.

We used the NPvC model to estimate the mutual information that could be decoded from a neuron’s activity about each task variable at each time point. This was computed as the information between the actual value of the task variable and the one decoded from the posterior probabilities of task variables computed with the NPvC, conditioned on all other measured variables19,24. In simulations of neural activities modulated either linearly or nonlinearly by movement variables, the NPvC outperformed a GLM in fitting the data with nonlinear dependencies to movement variables (Extended Data Fig. 4 and Supplementary Note ‘Comparison of the performance of NPvC and GLM on simulated neural population data’). The NPvC and GLM both correctly estimated the information conveyed by individual neurons and the information from neuron pairs, conditioned upon movement variables, when the tuning to behavioral variables was linear. When the tuning was nonlinear, the GLM underestimated the neuronal information, whereas the NPvC performed well. Thus, the NPvC provides a more accurate estimate of information and is more robust to nonlinear tuning, consistent with its better fits to the data.

Preferential, but widespread, routing of information

While our focus is on population codes in projection pathways, we first established the information encoded in single neurons and whether that information is specialized in projection pathways. The PPC neurons contained information about each task variable, even after conditioning on the movement variables (Fig. 3a). Sample cue information was high in the sample, delay and test segments. Both sample cue and test cue information were appreciable in the early part of the test segment when the cues needed to be combined to inform a choice (Fig. 3a, left). The PPC neurons thus carried information about the reward direction (combination of the sample and test cues) and the choice, but the choice information was larger, indicating that PPC activity was more related to the turn direction selected by the mouse than the reward direction defined by the cues (Fig. 3a, middle). In addition, PPC neurons contained information about the mouse’s movements (Fig. 3a, right). Individual neurons contained relatively low information, likely due to transient activity patterns, trial-to-trial variability and multiplexing of information about different variables12,19,20,24,26.

Fig. 3: Single-neuron information in labeled projection neurons and nonlabeled cells.

a, Time course of different information components in all PPC neurons. Shading indicates mean ± s.e.m. b, Average single-neuron information in different populations about different task variables during the first 2 s after sample cue onset, delay onset or test cue onset. Error bars indicate mean ± s.e.m. across cells. *P < 0.05, **P < 0.01 and ***P < 0.001, computed using two-sided t test with Holm–Bonferroni correction for multiple comparisons. Nonlabeled, n = 2,506 cells; cPPC projection, n = 93 cells; RSC projection, n = 127 cells; ACC projection, n = 134 cells. rewdir, reward direction.

Information for the sensory-related task variables—sample cue, test cue and their combination that indicates the reward direction—was enriched in ACC-projecting neurons and lowest in contralateral PPC-projecting cells (Fig. 3b). Thus, PPC preferentially transmits sensory information to ACC. In contrast, information about the choice and movements was similar across the projection types, indicating that this information is more uniformly transmitted (Fig. 3b). All three projection types had lower information about the mouse’s movements than the unlabeled cells, suggesting that movement information is enriched in neurons projecting to areas not studied here (Fig. 3b, right). Cells projecting to contralateral PPC often had less information about each variable than the unlabeled cells, suggesting that across-hemisphere communication is less critical for encoding specific task and movement events. RSC-projecting neurons carried the sensory and choice information typical of the PPC population (Extended Data Fig. 3a). Therefore, neurons projecting to different targets differ in their encoding, revealing a specialized routing of signals. However, each projection contains significant information about each variable, showing that PPC also broadcasts its information widely. To understand population coding in projection pathways, we focused on choice information because it is the largest contributor to task-related variables.

Enriched IE pairwise interactions in neurons projecting to the same target

The structure of correlated activity patterns in populations of neurons can impact the transmission and reading out of information7,38. We computed pairwise noise correlations7,38, defined as the correlations in activity for a pair of neurons for a fixed trial type (Methods; Fig. 4a). We focused on the first 2 s after the test cue onset. Remarkably, noise correlations were significantly larger in pairs of neurons projecting to the same target than in unlabeled neurons with unidentified projection patterns, both for signed values (Fig. 4b, left, and Extended Data Fig. 5a) and absolute values (Extended Data Fig. 5b), suggesting that correlations are a key part of coding in output pathways. Noise correlations were higher on correct trials than on incorrect trials, consistent with the possibility that correlations are functionally relevant in guiding behavior25 (Fig. 4b and Extended Data Fig. 5a). We also considered that behavioral variability within a given trial type could contribute to trial-to-trial variability and thus potentially to noise correlations. After using the single-neuron NPvC models to compute partial correlations regressing out the effect of movement variability, noise correlations were lower, confirming that movement variability contributed to traditional noise correlation measures (Fig. 4b, right, and Extended Data Fig. 5a). Even in this case, noise correlations were higher in neurons projecting to the same target than in pairs of unlabeled neurons and higher in correct trials (Fig. 4b, right, and Extended Data Fig. 5a). These patterns of noise correlations were present even for pairs of neurons with similar separation in anatomical distance (Extended Data Fig. 5c).

Fig. 4: Pairwise interactions between pairs of nonlabeled neurons and pairs of neurons projecting to the same target.

a, Schematic of the models to compute pairwise joint probability density functions and conditional joint probability density functions of two neurons with correlated activities (({r}_{1},{r}_{2})) as a function of time (t). A vector of movement variables (x) with components (({x}_{1},\ldots ,{x}_{n})) is represented. Using single-neuron NPvC model outputs, we build different types of pairwise correlation models with or without conditioning over the movement variables. The joint pairwise model is then used to estimate noise correlations or interaction information. b, Left: noise correlations computed for pairs of nonlabeled neurons and pairs of neurons projecting to the same area for correct and incorrect trials. Right: same except for noise correlations conditioned on movement variables. c, Similar to b but for interaction information. d, Average single-neuron choice information in different populations during the first 2 s after the test onset for correct and incorrect trials over all nonlabeled and projection cells. e, Histogram of interaction information divided into pairs of IE (red), IL (blue) and independent pairs (green). In b–d the average is computed over all the simultaneously recorded pairs of nonlabeled or same-target projection cells. Error bars indicate mean ± s.e.m. across all pairs of neurons. *P < 0.05, and ***P < 0.001, t test with two-sided Holm–Bonferroni correction for statistical multiple comparisons. Nonlabeled, n = 145,439 pairs; same projection, n = 1,355 pairs.

Depending on the well-characterized relationships between signal and noise correlations, noise correlations can either reduce or enhance the information in neural populations7,38. To quantify how much a neuron pair’s noise correlations increase (IE) or decrease (information-limiting (IL)) information about a task variable, for each pair of neurons, we computed the interaction information (Fig. 4a). This was defined as the mutual information between the actual value of the task variable and the value decoded from the pair’s activity using NvPCs trained with noise correlations minus the information between the variable’s actual value and the value decoded from the pair’s activity using NvPCs trained blind to noise correlations. The former refers to the actual information carried by the pair, and the latter refers to independent information.

Interaction information was on average positive, and thus noise correlations were on average IE (Fig. 4c). Remarkably, interaction information on correct trials was larger in pairs of neurons projecting to the same area compared to pairs of unlabeled cells (Fig. 4c). Furthermore, interaction information was higher on correct trials than on incorrect trials. For pairs of neurons projecting to the same target, interaction information was even closer to zero on incorrect trials (Fig. 4c). In contrast, in single neurons, choice information was similar between correct and incorrect trials (Fig. 4d), suggesting a specific role of neuron–neuron interactions for generating correct behavioral choices. Similar results were obtained when comparing decoding performance (fraction correct) of NvPC decoders trained with or without correlations (Extended Data Fig. 5g). Therefore, pairwise interactions differ in populations projecting to the same target relative to surrounding neurons, enhance the information in an output pathway and may aid accurate decisions.

Interestingly, there was a wide range of interaction information values across pairs of neurons, with extended and almost symmetric tails for pairs of neurons with IE (significantly positive) and IL (significantly negative) interaction information (Fig. 4e and Extended Data Fig. 5f). Pairs of neurons for which the sign of signal correlations (that is, similarity of choice selectivity of the neurons) was the same as the sign of noise correlations had IL interactions, whereas pairs with opposite signs for signal and noise correlations had IE interactions38,39,40 (Extended Data Fig. 5k,l). For both IE and IL pairs, interaction information was higher in absolute value in correct trials compared to error trials, consistent with higher noise correlations resulting in greater levels of both types of interactions41 (Extended Data Fig. 5e). The IE interactions had higher magnitude interaction information in pairs of neurons with the same projection target compared to unlabeled pairs, whereas IL interactions had similar magnitudes of interaction information between same projection and unlabeled pairs (Extended Data Fig. 5e). Thus, PPC contains a rich mix of IE and IL pairs, with IE interactions having a preferential contribution to output pathways.

We also computed the interaction information for individual projection targets and the sample cue and test cue (Extended Data Fig. 5d). The RSC-projecting pairs had similar interaction information about the mouse’s choice compared to unlabeled pairs, consistent with RSC-projecting neurons being most like the unlabeled population. Test cue information had similar but weaker results to choice information. For sample cue information, interaction information was not different between cells projecting to the same target and unlabeled pairs. Thus, there exists a possible specificity of interactions between neurons with respect to different task variables and projection pathways.

The network structure of pairwise interactions

Two networks with the same set of IL and IE interactions can differ in how these interactions are organized within the network (Fig. 5a,b). For example, the same set of interaction pairs can be distributed either randomly (Fig. 5a, top) or structured as clusters containing enriched IE or IL interactions (Fig. 5a, middle and bottom).

Fig. 5: The presence of network structure of interaction information in populations projecting to the same target.

a, Schematics of a random network (top), a network with clustered interactions (middle) and a network with modular structure (bottom). b, Sketch of how networks with the identical probability of IE and IL interactions can be organized randomly or in clusters of IE or IL pairs. Red and (+) indicate IE pairs. Blue and (−) indicate IL pairs. c, Relative triplet probability with respect to a random network for nonlabeled and same-target populations during correct (left) and incorrect (right) trials. The random network has the same distribution of pairwise interactions, except that they are shuffled between neurons. d, Global cluster coefficient of IE and IL subnetworks relative to a random network. e, Schematic of a two-pool network model. f, Schematic of the space of two-pool networks is quantified in terms of the probability of IE pairs in pool 1 (x axis) and pool 2 (y axis) minus the probability of these pairs in a random network. Red indicates IE pools and blue indicates IL pools. g, Schematic of some examples of symmetric and asymmetric networks corresponding to different points in the two-pool network space along the diagonal (red dashed line) or antidiagonal (blue dashed line) axes. h, Left: the probabilities of different triplets computed for different two-pool networks minus the triplet probabilities in a random network, computed analytically as derived in Supplementary Note ‘Analytical calculation of triplet probabilities in the two-pool model of network structure’ for two-pool networks. Right: the triplet probabilities for three example model networks sampled from the space of two-pool networks. i, At each point in the 2D space of two-pool networks, comparison of the triplet probabilities for the model network and the empirical data. The similarity index ranges from 1 for similar networks to 0 for completely different networks. Yellow circles correspond to networks that are more similar to data for nonlabeled (left) and same-target projection networks (right). The four dashed contours correspond to two-pool networks with similar value of each of the four triplet probabilities to the one obtained from data (from Fig. 5c for correct trials). j, Using pools of neurons defined based on their choice selectivity to left or right choices, the average interaction information within the pool and between the pools, relative to a random network. Pink indicates same-target projections. Black indicates nonlabeled neurons. Statistical tests are performed over all the pairs of neurons in each pool. k, Similar to j but for the probability of IE pairs within or between pools, relative to a random network. Error bars indicate mean ± s.e.m. estimated using bootstrapping. *P < 0.05, **P < 0.01 and ***P < 0.001, two-sided t test with Holm–Bonferroni correction for statistical multiple comparisons. Nonlabeled, n = 1,080,382 triplets; same projection, n = 30,204 triplets.

Graph-theoretic measures can identify structured arrangements of pairwise links in a network. The simplest motifs in a graph beyond pairs are interaction triplets42,43. A network with IL clusters has a larger number of ‘−,−,−’ triplets compared to a random network, and a network with IE clusters has more ‘+,+,+’ triplets compared to a random network, where ‘−’ and ‘+’ indicate IL and IE pairwise interaction links, respectively (Fig. 5a,b). For interactions of choice information, we computed the difference in the probabilities of ‘−,−,−’, ‘−,−,+’, ‘−,+,+’ and ‘+,+,+’ triplets between our data and an unstructured network obtained by randomly shuffling the position of pairwise interactions within the network, without changing the set of interaction values. In the unlabeled population, triplet probabilities were like those in a random network, indicating that the pairwise interactions are not structured (Fig. 5c). However, in populations of neurons projecting to the same target, there was an enrichment of ‘+,+,+’ and ‘−,−,+’ triplets and fewer ‘−,−,−’ and ‘−,+,+’ triplets compared to a random network. Notably, this structure was present only on correct trials and not when mice made incorrect choices (Fig. 5c). The network structure was present in ACC-projecting, RSC-projecting and contralateral PPC-projecting populations for choice information but was less apparent for sample cue and test cue information (Extended Data Fig. 6a,b).

We then used a graph global clustering coefficient42,43 to measure IL or IE clusters. This coefficient compares the frequency of specific triplets in real data to a shuffled network, measured as the ratio between the number of specific closed triplets and the number of all triplets, normalized to the same quantity computed from shuffled networks. On correct trials, the clustering coefficient obtained from data was larger compared to a shuffled network for IE interactions, meaning that these interactions were clustered. In contrast, for IL interactions, the clustering coefficient was smaller than for a shuffled network, indicating that they were set apart (Fig. 5d). This clustering was not found in unlabeled neurons and when mice made incorrect choices (Fig. 5d). Thus, pairwise interactions are clustered in the network of neurons projecting to the same target.

To exemplify how the network’s topology relates to the triplet distributions, we considered a simple two-pool model in which the network structure can be varied parametrically, while keeping the overall set of values for IE and IL interactions constant (Fig. 5e). Each pool was parameterized by the difference in probability of IE pairwise interactions within the pool relative to a random network. Thus, the range of possible network models resides in a two-dimensional (2D) space consisting of the enrichment of IE interactions in pool 1 along one axis and in pool 2 along the second axis (Fig. [5f](https://www.nature.com/ar