Title: BrainFlow: A Holistic Pathway of Dynamic Neural System on Manifold
Venue: NeurIPS 2025
Accepted: yes
Keywords: brain network analysis; flow matching; neuroimaging analysis
Abstract: A fundamental challenge in cognitive neuroscience is understanding how cognition emerges from the interplay between structural connectivity (SC) and dynamic functional connectivity (FC) in the brain. 
Network neuroscience has emerged as a powerful framework to understand brain function through a holistic perspective on structure-function relationships. In this context, current machine learning approaches typically seek to establish direct mappings between structural connectivity (SC) and functional connectivity (FC) associated with specific cognitive states.
However, these state-independent methods often yield inconsistent results due to overlapping brain networks across cognitive states. 
To address this limitation, we conceptualize to uncover the dendritic coupling mechanism between one static SC and multiple FCs by solving a flow problem that bridges the distribution of SC to a mixed distribution of FCs, conditioned on various cognitive states, along a Riemannian manifold of symmetric positive-definite (SPD) manifold.
We further prove the equivalence between flow matching on the SPD manifold and on the computationally efficient Cholesky manifold. 
Since a spare of functional connections is shared across cognitive states, we introduce the notion of consensus control to promote the shared kinetic structures between multiple FC-to-SC pathways via synchronized coordination, yielding a biologically meaningful underpinning on SC-FC coupling mechanism.
Together, we present BrainFlow, a reversible generative model that achieves state-of-the-art performance on not only synthetic data but also large-scale neuroimaging datasets from UK Biobank and Human Connectome Project.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nBrainFlow proposes a reversible, continuous-time generative model on the SPD manifold, learning an ODE-driven flow to transform structural connectivity into state-specific functional connectivity and back within a single framework. It leverages an isometric embedding into the Cholesky manifold for computational efficiency and adds a Dirichlet-energy consensus term to ensure multi-state SC–FC consistency.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths \n\n* Optimizing on the Cholesky manifold is a clever way to guarantee that all geometric and probabilistic properties are preserved with significantly lower computational overhead.\n\n* The consensus control nicely embeds multi‐state biological consistency directly into the dynamics.\n\n* The paper has a nice flow with solid theoretical grounding.\n\nWeaknesses\n\n* To fit structural connectivity into the SPD manifold, a “small” self-loop is added; however, the authors do not clarify how the choice of self-loop magnitude affects downstream flows or biological interpretability.\n\n* The Dirichlet-energy consensus term is tacked on during reverse ODE integration (eq. (14)) rather than baked into training. It would be interesting for the authors to verify whether this creates a mismatch between the learned vector field and its application at test time. \n\n* The comparisons focus on GAN- and flow-based methods. More recent diffusion/score-based generative models [1,2]  might offer competitive performance and would strengthen the empirical evaluation if included.\n\n[1] Li, Yunchen, et al. \"Spd-ddpm: Denoising diffusion probabilistic models in the symmetric positive definite space.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 38. No. 12. 2024.\n[2] De Bortoli, Valentin, et al. \"Riemannian score-based generative modelling.\" Advances in neural information processing systems 35 (2022): 2406-2422.\n\nQUESTIONS:\nAdding to the concerns raised on the weaknesses section, I have the following questions\n\n* Would jointly learning σ (consensus weight) improve performance?\n\n* How sensitive are results to the choice of consensus weight σ? Could the authors provide an ablation study varying σ and its impact on reconstruction error?\n\n* Could the authors comment on if/how model interpretability has informed any neuroscientific insights?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces BrainFlow, a generative model for learning a reversible mapping between structural (SC) and functional connectivity (FC) in the brain. It uses flow matching on the Riemannian manifold of symmetric positive-definite (SPD) matrices and optimizes on the Cholesky manifold for efficiency. A consensus control mechanism improves reverse flows by aligning FC states from the same subject. Experiments are conducted on synthetic data and three large neuroimaging datasets: UK Biobank, HCP-Aging, and HCP-Young Adult.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n * BrainFlow addresses the problem of SC-FC coupling across tasks using a geometrically principled and computationally efficient approach, that truly models inter-subject and inter-task variance in the domains of structure and function simultaneously.\n* The model is bidirectional and incorporates a creative consensus control mechanism.\n* Results are evaluated on large-scale datasets with a variety of metrics and relevant baselines.\n* Figures are informative and suggest strong qualitative preservation of structure.\n* The paper seems honest in its reporting.\n\n# Weaknesses \n## Misleading brain connectivity terminology\nTerms like \"dendritic coupling mechanism\" are misleading: MRI does not provide dendritic-level information. SC is described as \"number of nerve fibers,\" which oversimplifies tractography. The terminology should be more rigorous, and authors should outline what dMRI derivatives they are using (tractography counts?).\n\n\n## Missing Details on Connectivity and Dataset Composition\nFC metric construction (e.g., correlation type, GSR, regularization) is unspecified. SC pipeline details (e.g., tractography method, filtering) are missing. The number of usable SC-FC pairs in HCP-YA is unclear; only ~300 subjects have DWI.\n\n\n## Evaluation Framing Issues\nTable 2 lacks the classifier's performance on real FCs, making interpretation difficult (although this is indicated in the text for UK biobank). It’s unclear if other models in Table 2 were conditioned on task; MGCN-GAN is missing.\n\n# Conclusion \nOverall assessment: BrainFlow combines an elegant theoretical framework with competent empirical work and a welcome level of transparency. Despite weaknesses in domain framing, missing methodological detail, and some presentation issues, the paper advances the modeling of SC-FC relationships in a meaningful way. I recommend acceptance, provided revisions improve clarity and align biological claims with the data.\n\nQUESTIONS:\nThe model uses subject-paired SC-FC training pairs, but are those pairs treated independently during training? \n\nIs the consensus only used for reverse inference, or could it play a regularizing role in training as well? \n\nThe ethics checklist claims the study does not involve human subjects. This is incorrect, as secondary analysis of human neuroimaging data qualifies as human subjects research. Depending on local policies, secondary analyses may or may not be exempt from ethics approval, but this is still regulated human research. UK biobank access requirements in particular are quite strict and accessing these data now requires mandatory ethics training. Please clarify this aspect of the submission.\n\nLIMITATIONS:\n## Speculative Claims About Scaling\nThe performance gain on UK Biobank is attributed to dataset size without a scaling experiment. The opportunity to explore scaling across task complexity in HCP datasets is missed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces BrainFlow, a reversible flow-based generative model that bridges structural connectivity (SC) and functional connectivity (FC) distributions via conditional flow matching on the symmetric positive definite (SPD) manifold. The method leverages the Cholesky manifold as a computationally efficient surrogate for the SPD manifold, ensuring geometric fidelity and stability. A key novelty is the consensus control mechanism, which harmonizes the reverse flow across multiple FC states to produce structurally coherent outputs. The approach is validated on synthetic data and large-scale neuroimaging datasets (HCP-YA, HCP-A, UK Biobank), demonstrating strong performance on both SC-to-FC and FC-to-SC mappings, with a particular advantage in preserving topological properties of brain networks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nOriginality: Introduces a novel bidirectional model between SC and FC using flow matching on the Cholesky manifold, a less-explored but mathematically rigorous space. The integration of consensus control is a biologically meaningful enhancement.\n\nTechnical Quality: The paper rigorously defines the mathematical framework, includes theoretical guarantees (e.g., isometry between SPD and Cholesky manifolds), and provides closed-form velocity expressions and optimization procedures.\n\nSignificance: Addresses a central problem in cognitive neuroscience: modeling the SC–FC relationship. The reversible formulation allows both predictive modeling and insight into underlying mechanisms.\n\nEmpirical Validation: Experiments on real datasets with various evaluation metrics (RMSE, clustering error, classification accuracy) convincingly support the model’s claims. The method achieves the best performance in preserving topological properties.\n\nWeaknesses:\n\nLack of Visualization of Learned Connectivities: Despite the focus on modeling mappings between SC and FC, the paper provides limited visualization of the actual predicted connectomes. Figures illustrating specific examples of how a single SC gives rise to distinct FC patterns across cognitive states would help demonstrate the biological plausibility and interpretability of the model's outputs.\n\nSC–FC Mismatch in Biology: The assumption that FC can be reconstructed from SC (and vice versa) may be overly strong. In practice, many FC relationships—especially those reflecting long-range integration or indirect modulation—are not directly observable in structural scans due to tractography limitations. For instance, interhemispheric and long-range connections are frequently underestimated. The paper could benefit from a more explicit discussion of these biological mismatches and whether the model compensates for them.\n\nClarity for Neuroscience Audience: While the model is technically sound, the biological interpretation of flows and consensus control could be elaborated further for neuroscientists less familiar with manifold-based generative modeling.\n\nBaseline Limitations: Some recent deep learning baselines (e.g., graph neural networks tuned for dynamic FC prediction) are not compared. The study primarily benchmarks against flow-based or conventional methods.\n\nTask Classification Evaluation: The use of pre-trained classifiers for evaluating generated FC could benefit from cross-validation or additional controls to ensure robustness across tasks and datasets.\n\nGeneralization to Dynamic FC: Although briefly mentioned as future work, the current approach is limited to static FCs. Its extension to time-varying FC would be important for broader applicability.\n\nQUESTIONS:\nQuestions\nVisualization of Output: Can the authors provide explicit examples of SC–to–FC mappings across multiple tasks? It would be helpful to see how one SC gives rise to several FCs and whether those match known task-specific patterns.\n\nBiological Gaps in SC–FC Modeling: How does BrainFlow handle functional relationships not directly observable in SC data (e.g., polysynaptic or modulatory pathways)? Has the model been evaluated on such cases where the SC–FC coupling is weak?\n\nConsensus Control Sensitivity: How sensitive is performance to the consensus control parameter σ? Could this be optimized during training rather than set manually?\n\nTask State Generalization: Does the model generalize to unseen cognitive states or does it require retraining for each state? Can it interpolate between known task-specific FCs?\n\nDynamic FC Extension: Can BrainFlow be extended to model time-varying FC trajectories? This would seem a natural next step given the flow-based foundation.\n\nScore Increase Criteria: Inclusion of clear visualizations of estimated SC–FC mappings, and a discussion or empirical treatment of cases where SC underrepresents functional pathways, would significantly improve interpretability and trust in biological validity.\n\nLIMITATIONS:\nNo major omissions. However, the paper could better address the potential disconnect between SC and FC due to neuroimaging limitations, and clarify when the model’s assumptions may not hold.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a novel way to connect subject-specific functional connectivity (FC) and structural connectivity (SC) in MRI data. The authors model SC and FCs as elements of the SPD manifold, overcoming the common Euclidean assumption. The authors provide a computationally efficient method to learn the flow from SC to FCs by proving equivalence of SPD and Cholesky manifolds. Using Dirichlet energy to constrain the backwards flow from multiple FCs to SC, the authors are effectively able to learn the correct subject-specific SC.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:  \n\n1. The proposed method – BrainFlow – is well-motivated and appears to be mathematically well-grounded  \n\n2. Proving the equivalence between optimizing on the Cholesky manifold and SPD manifold opens up avenues for other researchers to use the computationally efficient version  \n\n3. BrainFlow performs well on graph-based metrics (quantified with LCE and GCE) in addition to RMSE, suggesting that it is able to capture intended patterns  \n\nWeaknesses:  \n\n1. The evidence for computational efficiency is sparingly mentioned  \n\n2. Eq. 14: Is the \\sigma the same as the one used in a_{ij}? If so, why?  \n\n3. Section D.3 – HCP-YA: Why was resting-state excluded (unlike in HCP-A)?  \n\n4. Section D.3 – HCP-YA: It is well-known that FC estimation is affected by run time. Since different tasks have different run times, did the authors perform any correction? If not, do the run times affect the quality of predicted FCs? Prediction quality (RMSE, LCE, GCE) per task would be a valuable add to the manuscript.  \n\n5. Line 46: “evolving” may not be the right term since temporal changes in FCs are not modelled  \n\n6. Figure 2: Cholsesky -> Cholesky  \n\n7. Line 83: Mention S_n^+ before use  \n\n8. Line 96: Missing element notation  \n\n9. Equation 4: Mention q(z) before use  \n\n10. Figure 4: Using different colors for circle and dots (orange) will improve clarity  \n\n11. Line 221: typical convention is to use \\sigma^2 in Gaussian kernels  \n\n12. Line 225: gradient is w.r.t L_1^{y_i}\n\n13. Title should read \"Manifold\" (it now says \"Mainfold\")\n\nQUESTIONS:\nLine 520: What is the \\epsilon value used? Was it subject-specific?  \n\nFigure 4 right panel: shows a very interesting way the model has chosen to transport from the green circle to the blue – i.e., move to the center first and then to the blue circle. Could the authors comment on this or provide a reasoning why this could be happening?  \n\nA suggestion on Section D.3 – HCP-YA: Conditioning on the subtask (instead of the task) could be useful in informing finer details in the SC maps\n\nLIMITATIONS:\nN/a", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}