W1

• We apologize for the confusion. We will rephrase the technical terms.
• The measurement we use to construct structural connectivity is the fiber count.

W2

• For functional connectivity preprocessing and construction, we use xcp-d, a post-processing pipeline for fMRI data [1].
• As for the structural connectivity, we utilize QSIPrep [2] for processing and reconstruction.
• XCP-D computes functional connectivity matrices using Pearson correlation coefficients between parcellated time series. The pipeline employs the 36P nuisance regression strategy, which includes six motion parameters along with their derivatives and quadratic terms, plus white matter, cerebrospinal fluid, and global signal regressors. The correlation calculations themselves do not include any explicit regularization beyond the standard nuisance regression approach.
• QSIPrep's default reconstruction workflows use iFOD2 (2nd-order Integration over Fiber Orientation Distributions) as the tractography algorithm, and SIFT2 (Spherical-deconvolution Informed Filtering of Tractograms 2) is enabled to provide biologically meaningful connection weights.
• We apologize for the confusion on image demographic information. We clarify that the usable SC-FC pair is 3,592, which is mainly constrained by the number of DWI. The remaining fMRI that don't have corresponding DWI are deprecated.
• We will include all information into the supplement material of the final version.

W3

• We would like to gently note that Table 2 has included metrics for the classifier’s performance on real FCs from other datasets, alongside the details mentioned in the text for UK Biobank.
• To avoid any confusion, we will revise the table caption in the revised manuscript to explicitly highlight this inclusion, ensuring the information is more immediately apparent.
• And other models in Table 2 were conditioned on task with flexible class guidance. However, as the original MGCN-GAN just focused on resting FC instead of task-specific prediction. We didn't include it in the task recognition experiment.

Q1

• Each pair is created according to subject constraint. During training, each pair is sample from the joint distribution, i.e. $(X_0, X_1) \sim q(X_0,X_1)$ instead of $(X_0,X_1)\sim p(X_0)p(X_1)$. Each sampled pair are considered independent.

Q2

• We argue that consensus control may not be suitable to regularize the training.
  • First, the reversible nature of our flow means that Dirichlet energy regularization applied to the reverse flow also constrains the forward flow, thereby reducing the diversity of predicted task FCs.
  • Second, there is a mismatch between training and inference objectives. While consensus control leverages complementary information across different task FCs to reduce simulation error during inference, BrainFlow training is simulation-free and does not account for this simulation error. Therefore, using the consensus control to regularize the training may not be a good choice.

Q3

• We will change to “involve human subject”. All authors have the HIPPA and CITI training certificates and qualified for processing (de-identified) neuroimaging data.

Limitations

We admit that the "larger dataset" claim for UK Biobank performance is speculative. As the performance gain may also be attributed to the simpler task states (only two different states). Different scanning protocols, populations, and task types across datasets may also be the factors that influence the performance. We will clarify these claims more clearly.

Readability issues

• We will add more intuitive explanation with less mathematical language in Section 2.2
• We will add a subsection for notation introduction to improve the readability.
• We will correct the formatting issuses.
• We will place the related work before experiments and then include all the baseline methods in the Related Work section.

[1] Mehta, K., Salo, T., Madison, T., Adebimpe, A., Bassett, D. S., Bertolero, M., Cieslak, M., Covitz, S., Houghton, A., Keller, A. S., Luo, A., Miranda-Dominguez, O., Nelson, S. M., Shafiei, G., Shanmugan, S., Shinohara, R. T., Sydnor, V. J., Feczko, E., Fair, D. A., & Satterthwaite, T. D. (2023). XCP-D: A Robust Pipeline for the post-processing of fMRI data. bioRxiv : the preprint server for biology, 2023.11.20.567926. https://doi.org/10.1101/2023.11.20.567926

[2] Cieslak, M., Cook, P.A., He, X. et al. QSIPrep: an integrative platform for preprocessing and reconstructing diffusion MRI data. Nat Methods 18, 775–778 (2021). https://doi.org/10.1038/s41592-021-01185-5

W1:

• We will give a theoretical analysis on computational efficiency, as follows, in the final version.
• First, the flow-based methods can be summarized with the following steps:
  • Sampling from the source and target distributions.
  • Calculate the interpolated position and velocity between the source and target samples
  • Regress the predicted velocity to ground truth velocity.
  • During inference, integrate the time-dependent velocity.
• We show the additional computation of Riemannian conditional flow matching on SPD manifold (RCFM-SPD) and BrainFlow on these steps compared to baseline vanilla Flow Matching.
  • Before sampling stage, additional preprocessing (Cholesky decomposition) is required for BrainFlow, whose time complexity is $\mathcal{O}(Nd^3), N$ is the size of training set.
  • During interpolation stage, the $\exp(\cdot),\log(\cdot)$ of RCFM-SPD both requires $\mathcal{O}(Bd^3)$, where $B$ is the batch size, while these operations on Cholesky manifold only requires $\mathcal{O}(d)$
  • During loss calculation inner product $<\cdot,\cdot>$ on SPD manifold requires $\mathcal{O}(Bd^3)$ while Cholesky manifold has the same complexity as vanilla Flow Matching ($L_2$ norms)
  • During inference, RCFM-SPD requires eigenvalue clipping to keep the element SPD, requiring $\mathcal{O}(Bd^3)$ each integration step. Hence, it requires additional $\mathcal{O}(LBd^3)$ where $L$ is total step. | Stage | RCFM-SPD | BrainFlow | Flow Matching | |--|--|--|--| | Sampling | Standard | + $\mathcal{O}(Nd³)$ Cholesky preprocessing | Baseline | | Interpolation | +$\mathcal{O}(Bd³)$ $\exp(·), \log(·)$ (SPD)| $+\mathcal{O}(Bd)$ $\exp(·), \log(·)$ (Cholesky) | Baseline | | Loss | +$\mathcal{O}(Bd³)$ (SPD inner product) | Standard (Cholesky inner product) | Baseline | | Inference | +$\mathcal{O}(LBd³)$eigenvalue clipping | Standard | Baseline |

• N: training set size, B: batch size, d: matrix dimension, L: integration steps
• "+": additional computational cost compared to baseline
• The computational cost of Cholesky preprocessing is incurred only once, whereas the costs associated with interpolation, loss calculation, and inference need to be repeated at each step.

W2:

• $\sigma$ in Eq. 14 and that in $a_{ij}$ are different., We will correct it by changing the coefficient in eq.14. to \beta.

W3:

• Although HCP-A includes both resting-state and task FC data, the dataset is predominantly composed of resting-state FC (four runs per subject) compared to task FC (one run per subject). In the experiment in Section 4.2, we specifically focus on task-evoked connectivity patterns, which provide more controlled experimental conditions. Therefore, we chose to utilize the HCP-YA dataset, which offers a more comprehensive collection of task-based FC data, to validate our condition-related experimental hypotheses.

W4, Q3:

We process the task fMRI by fmriprep and xcp-d, which do not include run time correction. Hence, we follow the reviewer’s idea to do experiments to evaluate the performance per task. We can see that each method maintains relatively stable performance across all tasks despite varying run times. What’s more, the relative ranking of methods remains consistent regardless of task, suggesting that run time differences do not substantially alter the fundamental performance characteristics of each approach.

RMSE

GCE

LCE

W5:

• Thank you for this valuable suggestion. We will rephrase it with “we propose BrainFlow, a reversible generative model designed to parametrize flows between the distribution of SC and the mixed distribution of FCs in different tasks”

W6-13 ( typos, figures, notations):

• We will correct these typos and correctly define the notations before being used. We will claim that $q(z)$ is the distribution of the conditioner $z$. We will use $\sigma^2$ in Gaussian kernels

Q1:

• The value of $\epsilon$ we used is fixed to be the largest value in the training set and is not subject-specific. This fixed value enables recovering the prediction of the structural connectivity to the original range.

Q2:

• We first want to correct that the trajectories go from blue circle to green circle instead of from green to blue.
• Then, As for the transport trajectories, they first go to the center is because both the original velocity $-v_\theta$ and the consensus velocity $v_{con}$ have components that point to center, making the trajectories go to center.
• Next, the consensus between the point starting from blue circle and that starting from green circle works like dragging the center point back to the target position.

Dear Reviewer,

Thank you for your question, we'd like to clarify why we keep the positive edges of FC.

Q1.1

1. Definition conflict

The standard clustering-coefficient formula $C_i=\frac{ \sum_{j, k} A_{i j} A_{j k} A_{k i}}{k_i(k_{i-1})}$ assumes non-negative weights. Negative values can make the numerator negative and the metric no longer interpretable as a “fraction of closed triplets.” For consistency with graph-theory practice [1,2], we set $A_{ij}^- =0$ when computing LCE and GCE. In addition, clustering coefficient measures the tendency of nodes to form cohesive groups. Positive FC represents cooperative relationships between brain regions, which directly corresponds to the clustering concept. Negative FC represents competitive/anti-correlated relationships, which contradicts the notion of forming cohesive clusters

2. No impact on BrainFlow optimisation

BrainFlow itself receives the signed FC; its transport solver is unchanged. Only the post-hoc metric uses the positive-weight mask, so discarding negatives here cannot bias training or flow estimation.

In addition, we also have tried to use absolute values to calculate the clustering coefficients. However, taking absolute value turns the FC into a fully connected graph, the local topology becomes uniform across subjects. According to the definition of $C = \frac{\text{number of closed triplets}}{\text{number of all triplets (open and closed)}}$, GCE will equal to $0$ constantly.

[1] Fagiolo, G. (2007). Clustering in complex directed networks. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 76(2), 026107.

[2] Rubinov, M., & Sporns, O. (2010). Complex network measures of brain connectivity: uses and interpretations. Neuroimage, 52(3), 1059-1069.

Q1.2

• You are correct: by definition $\mathrm{GCE}$ is a weighted average of $\mathrm{LCE}$, where the weight is related to the neighbors of nodes. Hence they are not independent. Our goal was to give both local and global perspectives, not to introduce an independent metric. In the revision, we will state this relationship explicitly in Sec. 4.2 and keep only one of the two values in the main table, with the other moved to the supplement for completeness.

Q2

• Thanks for the reviewers' suggestions. We have included more experiments beside neuroimages. We evalutated BrainFlow on both low-dimension ($2\times 2$) and high-dimensional ($100 \times 100$) SPD data. We tested if the baselines and BrainFlow can keep the SPD property for targets and the Wasserstein distance between generated and real target distribution.

We hope this addresses the reviewer’s concern and will incorporate all the results and discussion in the final version.

We sincerely appreciate your time and efforts of acknowledging our response.

W1

• We choose the value of \epsilon we used is fixed to be the largest value in the training set such that we can recover the prediction of the structural connectivity to the original range. Hence this operation will not affect the biological interpretability and is just for training stability.

W2, Q1, Q2

• This “mismatch” wais intentional and aligns with our design objectives. As noted in the main text, vanilla reverse flow introduces integration errors during prediction. Consensus control specifically addresses this by using complementary information across task FCs to reduce such errors at test time. Therefore, the apparent mismatch between training (simulation-free) and inference (with consensus control) is intentional—it allows us to correct for the limitations of the learned vector field during application.

• We also performed sensitivity analysis on the consensus control weights $\sigma$ to validate this strategy. We can see that when $sigma$ is small (~0.2), the model has significant performance gain compared to vanilla reverse flow. However, as $\sigma$ increases, the consensus control comes to dominate the reverse flow, causing the trajectories to become biased.

• We then argue that $\sigma$ may not be suitable to optimize during training.

  • First, the reversible nature of our flow means that Dirichlet energy regularization applied to the reverse flow also constrains the forward flow, thereby reducing the diversity of predicted task FCs.
  • Second, there is a mismatch between training and inference objectives. While consensus control leverages complementary information across different task FCs to reduce simulation error during inference, BrainFlow training is simulation-free and does not account for this simulation error. Therefore, optimizing σ during training is unsuitable for the intended inference procedure.

• HCP-A | $\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | | :--- | :---: | :---: | :---: | :---: | :---: | | MSE | 0.65±0.00 | 0.13±0.00 | 0.44±0.01 | 1.52±0.03 | 2.37±0.01 | | LCE(%) | 0.74±0.00 | 0.71±0.00 | 0.95±0.00 | 3.17±0.03 | 20.28±0.05 | | GCE(%) | 0.73±0.00 | 0.69±0.00 | 0.93±0.01 | 3.18±0.02 | 20.50±0.03 |

• HCP-YA | $\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | | :--- | :---: | :---: | :---: | :---: | :---: | | MSE | 0.42±0.00 | 0.14±0.00 | 0.52±0.00 | 1.32±0.02 | 2.12±0.01 | | LCE(%) | 0.62±0.00 | 0.55±0.00 | 0.93±0.00 | 2.92±0.01 | 15.74±0.03 | | GCE(%) | 0.59±0.00 | 0.59±0.00 | 0.88±0.00 | 3.02±0.02 | 16.40±0.02 |

• UKB | $\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | | :--- | :---: | :---: | :---: | :---: | :---: | | MSE | 0.09±0.00 | 0.07±0.00 | 0.69±0.00 | 1.89±0.002 | 2.52±0.01 | | LCE | 0.52±0.00 | 0.39±0.00 | 1.33±0.01 | 10.93±0.02 | 27.88±0.02 | | GCE | 0.53±0.00 | 0.39±0.00 | 1.30±0.01 | 11.16±0.01 | 28.16±0.02 |

W3

• We appreciate the reviewer's suggestion to include diffusion/score-based methods. However, there are fundamental differences in problem formulation that make direct comparison challenging.
• Diffusion models are designed for generation from noise distributions to target data distributions, whereas our task requires deterministic transport between two specific, biologically-coupled distributions. We need to map from a given structural connectivity (SC) to its corresponding functional connectivity (FC) from the same individual, preserving subject-specific relationships rather than generating samples from scratch. Thus, diffusion models are not included in the comparison.
• Besides, we include Schrodinger Bridge based method, which is a variant of diffusion model that can transfer between distributions into our experiments. We believe our current comparison provides an appropriate baseline given the transport nature of our problem.

Q3

• Our model's design is directly informed by neuroscientific principles to ensure its interpretability. For instance, the consensus control feature is based on the biological reality that different cognitive functions rely on shared neural circuits anchored by the stable underlying structural connectivity (SC). This ensures our model's predictions are consistent and biologically plausible.
• The model also captures how functional networks dynamically reconfigure for different cognitive demands, successfully generating meaningful, task-specific connectivity patterns that align with established neuroscience literature.

I thank the authors for their thorough responses. Their explanations have mostly addressed my concerns and as a result I am increasing my score.

W1, Q1

• We agree with the reviewer that the visualization from the same structural connectivity to different task functional connectivity is missing. We will update the corresponding figure in the final manuscript as external links are not allowed.

W2, Q2

• We appreciate this very insightful comment.
  • At the edge level, we completely agree that the relationship between structural and functional connectivity is far from a simple one-to-one mapping, due to the biological mechanisms mentioned by this reviewer. To address this challenge, we consider each SC and FC matrix as a data instance on the manifold. In this context, the collection of SC and FC matrices form the distribution on the manifold. Our method is trained to learns a mapping from the entire structural connectome distribution to the entire functional connectome distribution.
  • Recall that during training, it leverages paired SC-FC data from the same subject to construct a joint distribution, enabling the flow to capture statistical properties of both structural and functional connectivity including those arising from indirect pathways that manifest in FC but are not explicitly encoded in SC.
  • To better address this important point in our manuscript, we will add a paragraph to the Limitations section (Appendix F). This new text will explicitly acknowledge the challenges of the SC-FC mismatch and tractography limitations. We will clarify that BrainFlow's strength lies in its ability to learn the statistical regularities within the available imaging data and thereby contributing to understanding this well-known mismatch.

W3

• We appreciate this constructive comment. To improve clarity, we will integrate the concise interpretation of BrainFlow. For example, a "flow" in our model is not a physical process but a learned transformation pathway like interpolation between SC and FC. Consensus control enforces the biological principle of a shared structural scaffold. The brain's stable anatomy (SC) supports many dynamic functional states (FCs). We will integrate these explanations into the Introduction and Methods sections to improve the manuscript's accessibility.

W4

• We agree with this reviewer’s opinion. We therefore add a robust graph neural network baseline, GNN$^+$ [1], which showed excellent performance on most graph-level tasks with simple architecture. We set the objective as the MSE loss. Other details will be displayed on the final version. The result shows that GNN$^+$ performs consistently across three datasets, with superior RMSE and relatively high GCE/LCE. However, another observation from the visualization (not allowed during rebuttal) is that most of the predictions of GNN$^+$ (i.e. the non-flow method) show very similar patterns. We argue that this is because the MSE loss drives predictions toward the center of the training distribution (similar to regression to the mean), minimizing overall training loss but sacrificing individual-specific patterns. This results in good aggregate metrics but poor preservation of subject-level variability, as evidenced in the visualization.
• ||HCP-A RMSE | HCP-A GCE | HCP-A LCE | HCP-YA RMSE | HCP-YA GCE | HCP-YA LCE | UKB RMSE | UKB GCE | UKB LCE | | :--- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | | GNN$^+$ | 0.211±0.000 | 13.4±0.0 | 10.4±0.2 | 0.242±0.001 | 13.7±0.1 | 10.3±0.0 | 0.216±0.001 | 13.8±0.0 | 9.9±0.0 |

W5

• We apologize for the confusion. In our experiment, we have applied a very strict validation plan. Specifically, the pre-trained classifier was first trained on 5-fold cross-validation to select hyperparameters of the model. We then use the whole training set to train this classifier. Finally we classify the generated samples.

W6, Q5

• We completely agree with this reviewer. In this work, we put the spotlight on the method development of solving the foundational "one-to-many" mapping problem, i.e., bridging a single static structural connectome (SC) to multiple, distinct static functional connectomes (FCs) associated with different cognitive states. However, the "flow" concept at the core of our model is highly general and well-suited for modeling various types of neural dynamics.
• In macro-scale, BrainFlow framework could be used to model the slow evolution of brain connectivity over months or years (lifespan).
• In micro-scale, the framework can be adapted to model rapid, within-scan fluctuations in functional connectivity. In this context, the model would learn a flow that maps an FC matrix from one time window to the next.

Q3

• We conduct an experiment to test the sensitivity of the consensus control parameter σ. We can see that when $\sigma$ is small (~0.2), the model has significant performance gain compared to vanilla reverse flow. However, as $\sigma$ increases, the consensus control comes to dominate the reverse flow, causing the trajectories to become biased.

• We then argue that $\sigma$ may not be suitable to optimize during training.

  • First, the reversible nature of our flow means that Dirichlet energy regularization applied to the reverse flow also constrains the forward flow, thereby reducing the diversity of predicted task FCs.
  • Second, there is a mismatch between training and inference objectives. While consensus control leverages complementary information across different task FCs to reduce simulation error during inference, BrainFlow training is simulation-free and does not account for this simulation error. Therefore, optimizing $\sigma$ during training is unsuitable for the intended inference procedure.

• HCP-A | $\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | | :--- | :---: | :---: | :---: | :---: | :---: | | MSE | 0.65±0.00 | 0.13±0.00 | 0.44±0.01 | 1.52±0.03 | 2.37±0.01 | | LCE(%) | 0.74±0.00 | 0.71±0.00 | 0.95±0.00 | 3.17±0.03 | 20.28±0.05 | | GCE(%) | 0.73±0.00 | 0.69±0.00 | 0.93±0.01 | 3.18±0.02 | 20.50±0.03 |

• HCP-YA | $\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | | :--- | :---: | :---: | :---: | :---: | :---: | | MSE | 0.42±0.00 | 0.14±0.00 | 0.52±0.00 | 1.32±0.02 | 2.12±0.01 | | LCE(%) | 0.62±0.00 | 0.55±0.00 | 0.93±0.00 | 2.92±0.01 | 15.74±0.03 | | GCE(%) | 0.59±0.00 | 0.59±0.00 | 0.88±0.00 | 3.02±0.02 | 16.40±0.02 |

• UKB | $\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | | :--- | :---: | :---: | :---: | :---: | :---: | | MSE | 0.09±0.00 | 0.07±0.00 | 0.69±0.00 | 1.89±0.002 | 2.52±0.01 | | LCE | 0.52±0.00 | 0.39±0.00 | 1.33±0.01 | 10.93±0.02 | 27.88±0.02 | | GCE | 0.53±0.00 | 0.39±0.00 | 1.30±0.01 | 11.16±0.01 | 28.16±0.02 |

Q4

• BrainFlow can not directly generalize to unseen cognitive states. However, based on the trained model, one can easily extend to a new task state by inserting a new task state embedding and optimizing with new data.
• Similarly, interpolation between known task-specific FCs can not be directly achieved. We can do that by training a new model.
• It would be interesting to embed the connectivity (both SC and FC) into a unified latent space to achieve flexible transformation.

Limitations

• We clarify that the model may fail when functional connectomes (FCs) from the same subject and task exhibit significantly distinct patterns, rendering it difficult for the model to capture the underlying dependencies of SC-FC pairs.

[1] Luo Y, Shi L, Wu X M. Can Classic GNNs Be Strong Baselines for Graph-level Tasks? Simple Architectures Meet Excellence[C] //Forty-second International Conference on Machine Learning