Identifying Connectivity Distributions from Neural Dynamics Using Flows

Timothy Doyeon Kim; Ulises Pereira-Obilinovic; Yiliu Wang; Eric Shea-Brown; Uygar Sümbül

q-bio.NCMar 27, 2026

Identifying Connectivity Distributions from Neural Dynamics Using Flows

Authors: Timothy Doyeon Kim, Ulises Pereira-Obilinovic, Yiliu Wang, Eric Shea-Brown, Uygar Sümbül

Organizations: Allen Institute, Seattle, WA, USA · University of Washington, Seattle, WA, USA

Abstract

Connectivity structure shapes neural computation, but inferring this structure from population recordings is degenerate: multiple connectivity structures can generate identical dynamics. Recent work uses low-rank recurrent neural networks (lrRNNs) to infer low-dimensional latent dynamics and connectivity from observed activity, enabling a mechanistic interpretation of the dynamics. However, standard approaches for training lrRNNs can recover spurious structures irrelevant to the underlying dynamics. We first characterize the identifiability of connectivity structures in lrRNNs and determine conditions under which a unique solution exists. To find such solutions, we develop an inference framework based on maximum entropy and continuous normalizing flows (CNFs), trained via flow matching. Instead of estimating a single connectivity matrix, our method learns a distribution over connection weights that is maximally unbiased over unidentifiable components while matching the observed dynamics. This approach captures complex yet necessary distributions such as heavy-tailed connectivity found in empirical data. We validate our method on synthetic datasets with connectivity structures that generate multistable attractors, limit cycles, and ring attractors, and demonstrate its applicability in recordings from rat frontal cortex during decision-making. Our framework shifts circuit inference from recovering connectivity to identifying which connectivity structures are computationally required, and which are artifacts of underconstrained inference.

Figures & tables

Figure 1: Connector is a framework that infers minimally structured distribution over the connection weights of the lrRNN trained on neural data. It constructs, based on maximum entropy and continuous normalizing flows (CNFs), a generative model of the connectivity distribution, and can sample new lrRNNs that have mean-field dynamics that match the latent dynamics learned by the original lrRNN.

Figure 2: Identifiability of connectivity distributions. ( A ) Flow field of quadstable attractor dynamics generated from a generalized Hopfield network. Purple lines are example latent trajectories with circles indicating initial conditions. ( B ) Connectivity distribution of the generalized Hopfield network is a mixture of four Gaussians. Here, $x$ -axis represents the first and second components from the samples $\boldsymbol{m}$ drawn from $p(\boldsymbol{m},\boldsymbol{n})$ , and the $y$ -axis represents samples $\boldsymbol{n}$ from $p(\boldsymbol{m},\boldsymbol{n})$ . Color code based on which of the four Gaussians the neuron is drawn from (the neuron’s “cell type”). The same colors are used across B – D based on the ground-truth cell type. ( C ) Ground-truth $\boldsymbol{m}_{i}$ ’s plotted against $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ ’s. ( D ) Connectivity generated from an arbitrary $p(\boldsymbol{n}|\boldsymbol{m})$ that has the same $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}]$ as the ground truth connectivity. Connectivity distributions in B – D are all admissible and generate dynamics nearly identical to the quadstable attractor dynamics in A . ( E ) Connectivity inferred from LINT. Color code based on $4$ -means clustering. ( F ) Connectivity inferred from Connector (our approach). Color code based on $4$ -means clustering. ( G ) The ground-truth and inferred $p(\boldsymbol{n})$ in B , E , F . ( H ) We clustered the neurons based on the learned connectivity using GMM. The 5-fold cross-validated log-likelihood (mean $\pm$ std) was computed to identify the “elbow”.

Figure 3: Comparisons of dynamics under perturbation. We clustered the neurons into four different cell types for each model. We then silenced neurons that belonged to one of the four cell types in the ground-truth ( A ), LINT ( B ), and Connector ( C ).

Figure 4: Connectivity dissimilarity $D(\textrm{True},\textrm{Inferred})$ computed using the measure in Valente et al. (2022) and our measure in Equation ( 11 ). QA: Quadstable Attractors (Figure 2 ), BA: Bistable Attractors (Figure S6 A–F), LC: Limit Cycles (Figure S6 G–J), RA: Ring Attractors (Figure S6 K–N).

Figure 5: Connector-inferred connectivity distribution reveals computational cell types and their roles in neural dynamics. ( A ) Dataset from Luo et al. (2025) . The rat listened to a stream of clicks from the left and right speakers and oriented to the side that had more clicks. Neurons from frontal cortex were recorded during this task ( $K_{obs}=240$ ). Figure adapted from Kim et al. (2025) . ( B ) Connectivity distribution inferred with Connector. Each dot denotes a neuron sampled from the distribution ( $K=$ 5,000). ( C ) We clustered the neurons based on the connectivity inferred in B using GMM. The 5-fold cross-validated log-likelihood (mean $\pm$ std) was computed to identify the “elbow”. Connector-inferred connectivity suggests at least 2 clusters, which we plot in pink and green colors in B . ( D ) Flow field (in quiver plot, showing only inside the dotted line—the part traversed by the single-trial latent trajectories—to be consistent with Luo et al. (2025) ) with normalized difference showing relative contributions of cell type A and B to the dynamics. The colored trajectories are trial-averaged latent trajectories grouped by the number of clicks (darker red: more right clicks; darker blue: more left clicks).

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1 : MSE between the ground-truth $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}]$ and our estimate $\hat{\boldsymbol{N}}$ . The regularization coefficient $c=10^{-2}$ for this experiment. For this value of $c$ , our estimate does not converge to the ground-truth $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}]$ as $K$ becomes large.

Figure S2 : MSE between the ground-truth $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}]$ and our estimate $\hat{\boldsymbol{N}}$ . The regularization coefficient $c=10^{-4}$ for this experiment.

Figure S3 : MSE between the ground-truth $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}]$ and our estimate $\hat{\boldsymbol{N}}$ . The regularization coefficient $c=10^{-6}$ for this experiment.

Figure S4 : There are many $p(\boldsymbol{m},\boldsymbol{n})$ ’s that give rise to quadstable attractors, not just the mixture of four Gaussians in Figure 2 B. Notice that $p(\boldsymbol{m})$ learned by this network is unimodal. ( A ) Quadstable attractor dynamics generated from $p(\boldsymbol{m},\boldsymbol{n})$ in B . ( B ) Connectivity distribution of the quadstable attractors in A .

	flow-field similarity to ground truth (MSE)	# of approximate point attractors	latent trajectory similarity to ground truth ( $R^{2}$ )
LINT (Figure 3 B)	0.0894	2	0.695
Connector (Figure 3 C)	0.0097	2	0.929

Appendix

Table S1 : Similarity between inferred flow fields and ground-truth flow field under perturbation.

Figure S5 : The learned $p(\boldsymbol{n}|\boldsymbol{m})$ by LINT depends on its hyperparameters (e.g., learning rate, regularization) and initialization, in contrast to Connector, which learns consistent $p(\boldsymbol{n}|\boldsymbol{m})$ and associated $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}]$ . ( A ) Quadstable attractor dynamics generated from a generalized Hopfield network ( Beiran et al., 2021 ) . (Same as Figure 2 A.) ( B ) Connectivity distribution of the generalized Hopfield network is a mixture of four Gaussians. The conditional covariance of $p(\boldsymbol{n}|\boldsymbol{m})$ is set to be $\boldsymbol{S}=\boldsymbol{I}_{R}$ .(Same as Figure 2 B.) ( C ) Ground-truth $\boldsymbol{m}_{i}$ ’s plotted against $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ ’s. (Same as Figure 2 C.) ( D ) Connector-inferred $\boldsymbol{m}_{i}$ ’s plotted against Connector-inferred $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ ’s. We used the learned $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ here, and set $\boldsymbol{S}=\boldsymbol{I}_{R}$ in the connectivity distribution inferred by Connector in Figure 2 F. ( E ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where network parameters were initialized from $\mathcal{N}(\mu=0,\sigma=2)$ . Learning rate was $0.001$ . No regularization was applied. Color code based on $4$ -means clustering. (Same as Figure 2 E.) ( F ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where orthogonal initialization ( Saxe et al., 2014 ) with gain $=1$ was used. Learning rate was $0.001$ . No regularization was applied. ( G ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where orthogonal initialization with gain $=10$ was used. Learning rate was $0.001$ . No regularization was applied. ( H ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where network parameters were initialized from $\mathcal{U}[-5,5]$ . Learning rate was $0.005$ . No regularization was applied. ( I ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where network parameters were initialized from $\mathcal{U}[-1,1]$ . Learning rate was $0.005$ . No regularization was applied. ( J ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where network parameters were initialized from $\mathcal{U}[-3,3]$ . Learning rate was $0.003$ . Regularization with coefficient $=\textrm{1e-5}$ was applied. ( K ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where network parameters were initialized from $\mathcal{U}[-3,3]$ . Learning rate was $0.001$ . Regularization with coefficient $=\textrm{5e-4}$ was applied. ( L ) Connectivity inferred from LINT ( Valente et al., 2022 ) , where network was initialized with Xavier Uniform ( Glorot and Bengio, 2010 ) with gain $=\textrm{1e-3}$ . Learning rate was $0.001$ . Unlike the models above, we didn’t include the $1/K$ scaling in Equation ( 1 ), and put the $1/K$ scaling factor post-training. No regularization was applied. LINT-inferred connectivities in E - L all generated dynamics nearly identical to the quadstable attractor dynamics in A .

Figure S6 : Connector accurately infers the connectivity distributions of bistable attractors ( A – D ), limit cycle ( G – J ), and approximate ring attractor ( K – N ). $K=$ 1,000. ( A ) Ground-truth bistable attractor dynamics generated from an lrRNN with a Gaussian connectivity distribution ( Beiran et al., 2021 ) . ( B ) Ground-truth connectivity distribution of the bistable attractors. The conditional covariance of $p(\boldsymbol{n}|\boldsymbol{m})$ is set to be $\boldsymbol{S}=\boldsymbol{I}_{R}$ . ( C ) Ground-truth bistable attractors $\boldsymbol{m}_{i}$ ’s plotted against $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ ’s. ( D ) Connectivity of bistable attractors inferred from Connector. ( E ) Connectivity of bistable attractors inferred from LINT ( Valente et al., 2022 ) . No regularization was applied. ( F ) Connectivity of bistable attractors inferred from LINT ( Valente et al., 2022 ) . Regularization with coefficient $=\textrm{1e-5}$ was applied (regularization coefficient higher than or equal to 1e-4 did not converge). ( G ) Ground-truth limit cycle dynamics generated from an lrRNN with a Gaussian connectivity distribution ( Beiran et al., 2021 ) . ( H ) Ground-truth connectivity distribution of the limit cycle. The conditional covariance of $p(\boldsymbol{n}|\boldsymbol{m})$ is set to be $\boldsymbol{S}=\boldsymbol{I}_{R}$ . ( I ) Ground-truth limit cycle $\boldsymbol{m}_{i}$ ’s plotted against $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ ’s. ( J ), Connectivity of limit cycle inferred from Connector. ( K ), Ground-truth approximate ring attractor dynamics generated from an lrRNN with a Gaussian connectivity distribution ( Beiran et al., 2021 ) . We have an approximate ring due to the finite size of the network ( $K=$ 1,000). ( L ), Ground-truth connectivity distribution of the approximate ring attractor. The conditional covariance of $p(\boldsymbol{n}|\boldsymbol{m})$ is set to be $\boldsymbol{S}=\boldsymbol{I}_{R}$ . ( M ) Ground-truth approximate ring attractor $\boldsymbol{m}_{i}$ ’s plotted against $\mathbb{E}[\boldsymbol{n}|\boldsymbol{m}_{i}]$ ’s. ( N ) Connectivity of approximate ring attractor inferred from Connector.

Figure S7 : Connector accurately infers connectivity distributions of bistable attractors generated from a Student- $t$ distribution and a log-normal distribution. The ground-truth networks had $K_{obs}=$ 1,000. The samples from the Student- $t$ distribution were generated by using the same mean and covariance as in Figure S6 B, but with the number of degrees of freedom $\nu=3$ . The samples from the log-normal distribution were generated by exponentiating the samples in Figure S6 B. The ground-truth and Connector-sampled flow fields are similar to each other; however, Connector’s $p(\boldsymbol{n}|\boldsymbol{m})$ is Gaussian, and therefore does not match the ground-truth $p(\boldsymbol{n}|\boldsymbol{m})$ , which is a Student- $t$ or a log-normal.

Figure S8 : Identifiability in the presence of external inputs. ( A ) lrRNN trained on context-dependent decision-making task in Dubreuil et al. (2022) . Adapted from Figure 4b of Dubreuil et al. (2022) . ( B ) Latent trajectories of the ground-truth lrRNN (blue) and latent trajectories inferred from the neural activity of the ground-truth lrRNN (orange and red). Orange lines are from the latent trajectories inferred by LINT ( Valente et al., 2022 ) , and red lines are from the latent trajectories generated from an lrRNN sampled from the connectivity distribution inferred by Connector. The inferred latent trajectories matched the ground-truth trajectories well, both for LINT and Connector. ( C – D ) Despite both LINT and Connector-generated networks having similar latent trajectories, their connectivities can be different. In particular, $\boldsymbol{n}$ given $\boldsymbol{m},\boldsymbol{b}$ can be different between the ground truth and the inferred networks due to the degeneracy (3) discussed in Section 3 . We empirically find this mismatch. In C , notice the mismatch in the histograms approximating $p(\boldsymbol{n})$ for the ground-truth and LINT-inferred networks, with the LINT-inferred network having a sharp peak around the origin. In D , we show the histograms approximating $p(\boldsymbol{n})$ for the ground-truth and Connector-based networks. Here for the Connector-inferred connectivity distribution, we set $\boldsymbol{S}=\boldsymbol{I}_{R}$ . The histograms approximating $p(\boldsymbol{n})$ are similar between the ground truth and Connector, more so than between the ground truth and LINT.

Figure S9 : We trained LINT ( Valente et al., 2022 ) on the neural activity of the lrRNN that performs a context-dependent decision-making task (lrRNN data obtained from Dubreuil et al. (2022) ). Consistent with our analyses in Appendix C , $\boldsymbol{M}$ and $\boldsymbol{B}$ in LINT are linearly identifiable, and this leads to the ground-truth $p(\boldsymbol{m},\boldsymbol{b})$ and our inferred $p(\boldsymbol{m},\boldsymbol{b})$ matching each other well (after the transformation in Equation ( 5 ) to match the inferred connectivity to the ground-truth connectivity).

Figure S10 : Same as Figure S9 but for a transformer-based LVM. This LVM is similar to NDT ( Ye and Pandarinath, 2021 ) , but with mapping from the latent $\boldsymbol{z}_{t}$ to neural activity following Equation ( 45 ). Our analyses in Appendix C suggest that $\boldsymbol{M}$ and $\boldsymbol{B}$ in LVMs are not identifiable in general. Consistent with this result, we find that the ground-truth $p(\boldsymbol{m},\boldsymbol{b})$ and the $p(\boldsymbol{m},\boldsymbol{b})$ inferred from the transformer-based LVM do not match well (even after linear transformation as in Figure S9 ), despite the model producing neural activity patterns that are highly similar to the ground truth ( $R^{2}=0.99$ ).

Figure S11 : Extended analyses for Figure 5 . ( A ) Flow field inferred by FINDR ( Kim et al., 2025 ) . ( B ) Flow field obtained from lrRNN trained via the distillation approach in Appendix G.4 . ( C ) Flow field from a new lrRNN sampled from the connectivity distribution inferred by Connector ( $K=$ 5,000). ( D ) Silhouette scores for the connectivity of lrRNN in B (orange; baseline), and silhouette scores for the connectivity of lrRNN in C (green; Connector). ( E ) Flow field from cell type A in the network generated in C (Equation ( 86 )). ( F ) Flow field from cell type B in the network generated in C (Equation ( 86 )). ( G ) Same as Figure 5 D. The normalized difference index is computed with Equation ( 89 ) in Section F.1 . ( H ) Instead of using C , if we use the connectivity from B to cluster neurons and compute the normalized difference index (Equation ( 89 )), cell types A and B did not partition the state space in a way similar to Figure 5 D. ( I ) Instead of using the normalized difference index in Equation ( 89 ), we used Equation ( 90 ) and Equation ( 91 ) for the left and right panels, respectively. We found that cell types A and B partition the state space in a way similar to Figure 5 D using these indices. ( J ) Instead of using $k$ -means clustering as in Figure 5 B, cell types A and B were clustered using GMM. The turning points of the trial-averaged latent trajectories coincide with changes in the relative contributions of the two cell types identified from GMM, similar to what we find in Figure 5 D. ( K ) In Figure 5 B, the samples $\boldsymbol{n}_{i}$ were drawn from $p(\boldsymbol{n}|\boldsymbol{m},\boldsymbol{d})=\mathcal{N}(\boldsymbol{\mu}(\boldsymbol{m},\boldsymbol{d}),\boldsymbol{S})$ , assuming that $\boldsymbol{S}=\boldsymbol{0}$ . Here, we let $\boldsymbol{S}$ be 1.5 times the standard deviation of $\{\boldsymbol{\mu}(\boldsymbol{m}_{i},\boldsymbol{d}_{i})\}^{K}_{i=1}$ . We then performed clustering on $\{\boldsymbol{m}_{i},\boldsymbol{n}_{i},\boldsymbol{d}_{i}\}_{i=1}^{K}$ and found that cell types A and B partition the state space in a way similar to Figure 5 D.

Figure S12 : Sampling new neurons from connectivity distribution inferred from data in Figure 5 . We visualize the task-relevant component of neural activity $\boldsymbol{M}\boldsymbol{z}_{t}$ in Equation ( 93 ). FINDR was trained on $K_{obs}=240$ neurons, and the resulting task-relevant component inferred from FINDR is shown as the orange density distribution. For Connector, we sampled $K=5{,}000$ neurons, and the corresponding task-relevant component is shown as the blue density distribution. The plots show snapshots of the task-relevant component of neural activity at $t=1\mathrm{s}$ from stimulus onset, separately for leftward and rightward choices. The $p$ -values indicate two-sample two-sided Kolmogorov–Smirnov test ( $p>0.1$ ).

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