BrainFlow: A Holistic Pathway of Dynamic Neural System on Manifold
Authors: Zhixuan Zhou
Tingting Dan
Guorong Wu
Venue: NeurIPS 2025 poster ()
Keywords: brain network analysis; flow matching; neuroimaging analysis
Primary area: neuroscience_and_cognitive_science
TL;DR: We propose an efficient flow matching framework on SPD manifold with a novel control mechanism for predicting functional connectivity and vice versa.

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.

BrainFlows A Holistic Pathway of Dynamic Neural System on Manifold
Zhixuan Zhou
Tingting Dan
Guorong Wu"
Departments of Computer Science and Psychiatry
University of North Carolina Chapel Hill
Chapel Hill, NC 27599
(Tingting Dan grww)@med....e..o unc. unc.udu. zwdu;zzzixuuneeee enc.edu
Abstract
A fundamental challenge in cognitive neuroscience understanding how cognition emerges from the interplay between structural connectivity (SC) and functional connectivity (FC). Current machine learning approaches typically seek to establish direct mappings from SC to FC associated with specific cognitive states. However, these methods often treat SC and FC as distinct endpoints, failing to capture the coupling relationship throughout the progressive transformation between them. To address this limitation, we propose BrainFlow, a reversible generative model designed to parametrize flows between the distribution of SC and the mixed distribu- tion of FCs from different cognitive tasks. Our method explicitly models the SC-FC coupling by training on interpolated states along the symmetric positive-definite (SPD) manifold. We further prove the equivalence between flow matching on the SPD manifold and on the computationally efficient Cholesky manifold, enhancing numerical stability. To mitigate cumulative errors during reverse-flow simulation, we introduce consensus control mechanism that utilizes complementary infor- mation across multiple FC-to-SC pathways, yielding biologically meaningful reconstruction the underlying structural scaffold. Together, BrainFlow achieves state-of-the-att performance on both synthetic data and large-scale neuroimaging datasets from the UK Biobank and Human Connectome Project.
Introduction
A central goal in cognitive neuroscience is understanding how cognition arises from the complex in- terplay between (static) brain structure and (dynamic) functional fluctuations. Structural connectivity (SC), representing the anatomical pathways that physically connect distinct brain regions, provides the structural scaffold that supports the spontaneous synchronization of functional fluctuations across multiple brain regions, shaping the network topology of functional connectivity (FC). This interplay underpins the brain's 's ability perform complex cognitive, sensory, and motor functions, highlighting the necessity of understanding how anatomical structure constrains and shapes brain function, and vice versa
Machine learning techniques have been widely employed to characterize the uni-directional relation- ship between SC and FC. One streamlined approach simply predict one type of connectome from another in a supervised learning scenario 6].
Despite promising prediction accuracy, these computational approaches have several significant issues, from problem formulation to model explainability. (1) Over-simplified problem formulation. Most current methods only focus the one-way mapping function, i.e., either predicting FC from
Conresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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FC manifold
FC
sc
Controlled Reverse Flow
Cogaltive tasks
A Controversial prediction of uarestricted
Ferward flow
Unrestricted Reverse Flow
conditioned on
cognitive states
SC munifold
We model the SC-FC relationship in human brain as a flow problem, characterized by collection of one-to- ment forward flows from single (static) SC to multiple (evolving) FCs, along with corresponding many-to-one reverse flows from state-dependent FCs back to the underlying SC. Recognizing the neuroscience insight that significant portion of neural circuits are shared across different cognitive tasks, we introduce the concept of consensus control to harmonize the reverse flows between cognitive states.
Figure 1:
SC or vice versa. However, mounting evidence from neuroscience studies underscores the interplay between SC and FC. this context, the real biological challenge establishing a new understanding of SC-FC coupling mechanism in human brain using data-driven approaches. (2) Overlook the intrinsic data geometry. is popular to learn feature representation for SC and FC vector space by flattening the region-to-region connectivity matrix into vector. Since the matrix form of connectome data does not live a Euclidean space 77 manifold unique data geometry (such as the symmetric and positive-defiite properties) is needed to guarantee the mathematics and neuroscience insight in operating the connectome data.
algebra tailored for the
To address these limitations, we propose BrainFlow, reversible generative model designed to parametrize flows between the distribution of SC and the mixed distribution of FCs on the Riemannian manifold (Fig. D Conceptually, a flow in this context is not a direct physical process but learned transformation pathway, akin to continuous interpolation, that connects the geometric space of structural connectomes to that of functional connectomes. Specifically, we consider that each SC or FC a particle sampled from its respective distribution on the manifold. this context, relationship between SC and FC can be formulated with a flow governed by time-dependent ODE,
Following the spirit of trajectory inference g 9 we parameterize pathways that transport SC distri- butions to statc-specific FC distributions. This enables generating FC in different states for unseen SC data and, crucially, reconstructing SC from FC without additional training.
We further propose three innovative components to find mapping flows between SC and FCs. (1) We present flow model that only predicts FCs different states from SC but also reverses the process to recover SC from FC. This bidirectional capability is essential for understanding the interplay between the brain's structural and functional organization, enabling insights into how anatomical constraints shape functional dynamics and how functional variations reflect structural substrates. (2) We parametrize the flow on the Cholesky manifold [10], an isometric embedding of the symmetric positive-definite manifold (Fig. 2 This preserves the geometric and probabilistic properties of the original SPD space while avoiding the computational overhead of traditional affine- invariant Riemannian metrics, ensuring efficient and stable operations for digh-dimensionl matrices. By employing the log- Cholesky metric, BrainFlow ensures efficient and stable operations while preserving the intrinsic structure of SC and FC data. (3) We introduce consensus control mechanism that leverages the shared structural foundation underlying multiple functional states (demonstrated in Fig. rright). By treating SC as stable anchor and introducing consensus-driven adjustments into the dynamics, BruinFlow ensures that predictions across different FC states (e.g., resting-stan state and task-specific FCs) are both consistent and biologically reasonable.

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SPD manifold
SC
(5t.g) 9)
Cholesky manifold
Structural Connectivity
Cholsesky map:X LL
(E g)
(SC)
Functional Connectivity
FC
(FC)
Figure 2: Both SC and FC are regarded as data elements drawn from their respective distribution on the SPD manifold. Rather than estimating the flow directly on the SPD manifold, we perform optimization on the Cholesky manifold, which computationally efficient while being equivalent to the SPD manifold.
We compare methods baselines and large-scale neuroimaging datasets from UK Biobank and Human Connectome Project. In addition to competitive or state-of-the performance on different metries, our method shows faster training speed and more stable inference.
2 Preliminaries
2.1 Riemannian, SPD, and Cholesky Manifold
Riemannian manifold. Riemannian manifold (M.g) is a smooth manifold M equipped with a Riemannian metric that defines an inner product on each tangent space TpM. The geodesic ytt between two points on Riemannian manifold represents the locally cength-miimmizinn curve, analogous to straight lines in Euclidean space. Two fundamental operations Riemannian manifolds are the exponential map exp(-) and inverse, the logarithmic map log(-). The exponential map (x) TpM + .M maps tangent vector e to the point reached by following the geodesic starting at p in direction for unit time, while log,()) : M TpM maps a point g to the tangent vector e such that (v) q.
SPD manifold. Symmetric positive definite (SPD) matrices form a SPD space that naturally represents functional brain connectivity 7 IT]. When equipped with Riemannian metrics like commonly
used Log-Euclidean metric 12 affine-invarint metric B S turns into a Riemannian manifold. However, these classical metrics involve matrix exponential and logarithm operations that are computationally expensive and potentially numerically unstable for large- scale ill-conditioned matrices due to eigendecomposition.
Cholesky manifold. Recently, Lin ((T)) proposed the Log-Cholesky metric, which mainly relies on Cholesky decomposition = I.I." where X E S and is a lower triangular matrix whose diagonal elements are positive. Lin ((IO) first introduced the Riemannian manifold lower triangular matrices, which are the Cholesky decomposition factors of SPD matrices, and proves there exists an isometry between the SPD manifold and lower triangular manifold. Here we briefly introduce the Cholesky space. We firstly denote L, as a space whose elements are lower-triangular matrices with positive diagonal. The notation [A] denotes the strict lower triangular part of matrix A, specifically, [A]i, = Ar, if à >> j and zeros otherwise. ((A) represents diagonal whose diagonal elements are those of A. We then introduce some important properties of Cholesky manifold (C G). where is Riemannian metric. First, the tangent space Tr, at L is L., the space of lower triangular matrices. Second, the inner product between P, Q E Ct i.e., the Riemannian metric ÿ, is defined
3

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Jr. (P,Q)) - {>1
+ Σ
Pas Prs
(I)
j=1
Lin (110)) further induced the log-Cholesky metric on S. Specifically, given X E S and
W.V V T the Jog-Choleksy metric is defined as:
gx gx = ÅL (L(L-"WL-T) LLL -(-
(2)
where the Cholesky factor such that X LL and ()) L.J (D.)/2 2 for input square matrix.
Next, the geodesic between two points K, L. E 4t on the Cholesky manifold defined as:
(t) P + t[| []] + D())exx((((oog) (K) (ogD(L))),
where the strict lower triangular part can be regarded as the straight line in Euclidean space and the diagonal part is designed to meet the constraint of Cholesky space. More basic properties about Cholesky manifold and SPD manifold can be found in Appendix.
2.2 Flow Matching on Riemannian Manifold
Riemannian Flow Matching, Given a flow thy that generates a probability path Pt connecting two distributions Po and Pl, we can know the vector field 4s by de thx (x) = ue (x))) Then we can fit
parametrized vector field of to by optimizing the loss CRFM = E, (x U| (x²) ) - ue (x)))) l1
uniform (0, 1). This method is named (Riemannian) Flow Matching (RFM) [15
16
17 8 However,
uy intractable in most cases. Conditional flow matching therefore is then proposed to use of to approximate conditional vector field iy (zall) defined each sample which also generates a conditional probability path pr(xx|=). Then we can recover the unconditional vector field and
probability path by (xt) = SM ux ((2g22))=(e Pe(ae (x)(() dvol, and Pc(xt) [M (=(22a)))((2ddvl, [
se(os))
The Riemannian flow matching objective then becomes
CROFM E,, Bx.pp
un (xx
It has been proved that the loss CREM and CROFM have same gradients 8 conditional velocity field as target to fit u/.
w.E.t. 6. We can use the
2.3 Brain Connectivity
The human brain's 's connectivity can be characterized through two distinct mathematical representa- tions defined on set of n brain regions V = { .... }.
Structural connectivity (SC) describes the anatomical architecture through matrix A R where each element A 2 0 represents the connection strength between regions U; and By. Structural connectivity is typically modeled an undirected weighted/binary graph.
In contrast, functional connectivity (FC) represented by A E Raxn quantifying temporal rela- tionships, where A/.5 is the correlation between Blood Oxygen Level Dependent (BOLD) signals of regions Y; and dy- SPD matrices naturally arise in the analysis of functional brain connectivity patterns through its derivation from correlation or covariance measures. Though SC is not guaranteed to be SPD, it is common satisfy SPD property by adding a (small) self-loop connection to the symmetric SC matrix.
3 Related Works
3.1 Prediction between Functional Network and Structural Network
There have been many works aimed at predicting functional connectivities from structural connec- tivities or vice versa. Traditional methods 2 18 typically used the statistical properties (e.g. connection strength, distances, shortest paths, eigenvalues of Laplacian matrices) of one type of connectivities to predict the other connectivities. However, these methods mainly focused on small- scale even single subject, ignoring potential common properties among the population. Some

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machine learning methods are also used for predicting the connectivity. MGCN-GAN[ uses CycleGAN-base based network predict structural connectivities with functional connectivities while DSBGM proposed a signed graph encoder to map functional connectivities structural connec- tivities. These methods considered only the one-way mapping and focused only on resting-state functional connectivities.
3.2 Transformation between two distributions
Transformation between distributions is important problem in natural sciences 20 21
Re-
cently, many simulation-free distribution interpolation techniques were proposed 8
15 17
22
23
These methods trained time-dependent ODE evolve initial distribution the target distribution Most them worked with the distributions on Fuclidean space. RCFM [8] proposed conditional flow matching on Riemannian manifold. Based on RCFM, we further explore flow matching among isometries, which enables applying flow matching on a simpler manifolds and then maps it back to the complex isometry, making the framework more efficient.
4 Flow Matching for Bridging Structural and Functional Connectivity
4.1 Problem formulation
In this work, aim to predict resting-state state and task-specific functional connectivity (FC) matrices given structural connectivity (SC) matrices from the same individual. In addition, we also aim to predict SC matrices based on given FC matrices. Specifically, we denote the distribution of structural connectivity matrices A as the source distribution Po. and the distribution of functional connectivity matrices A the target distribution P1. Elements sampled from these distributions are represented as Xo po and X ~ Pt. where Xo corresponds to an SC matrix A and X corresponds to an FC matrix A?.
Our objective is to train a vector field Uy such that, given an SC matrix X3 for subject i, the corresponding resting task-specific FC matrix can be predicted via integration:
X S. (Xi.
x
where y² is the label of the target FC (e.g., resting-state, motor, working memory). On the other hand, the method can also predict SC given a labeled FC matrix.
4.2 Flow matching on Cholesky Manifold
Motivation. Our approach aims to construct conditional vector field ux(XiZz) Sx that transports Xy ~ Po to X1 Pi and use neural network of to fit (X,
Subject-speciffe prior conditioner. Following (17), define the prior conditioner Z as joint distribution of source and target:
(X,xXi) ~ x(Z) = ((oo,Xi) Due to the individual-specific nature of connectivities, however, we do employ either independent cou- pling (Xo, X1) == po (X (Xi) ) or optimal transport coupling qf (x,,, X1) = ग (Xo, X1). Instead, we define a specialized joint distribution (Z) that ensures each sample Z corresponds to an individ- ual. In this formulation, Xo and X1 represent SC and FC matrices derived from the same subject, preserving biologically coherent and individualized relationships between structural and functional domains.
Construct flow with geodesic. then construct flow ti to connect Xo and X1. Analogous the straight line in Euclidean, it is natural to use geodesic connecting X, and
X, Xx (t) = expx, (flogx, (X1))
(6)
where the exponential and logarithmic maps are computed under a Riemannian metric on the symmetric positive definite (SPD) space S...
Flow matching with log-Cholesky metries. A natural choice of Riemannian metric on St is the affine-invariant Riemannian metric (AIRM), used in [ While AIRM is effective in low- dimensional cases, its computational complexity poses challenges high-dimensional SPD mani- folds due to the reliance matrix cigendecomposition, which becomes numerically unstable when
5

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eigenvalues approach zero. Additionally, the computations of fractional matrix powers and matrix inverses contribute high computational costs. (See Appendix|B22 BB2 details.) These factors make AIRM difficult to apply to higher-dimensional SPD manifold. To address these challenges, we adopt the log-Cholle Cholesky metric in eq. 2 whose main overhead of geodesic and inner product computation only matrix inverse. Then the objective becomes:
(0) (
Xxllxx.
where = X, (x, and ue (X, 2) is the velocity the geodesic.
Flow matching on Cholesky manifold. Recall the definition log-Cholesky metric (eq. 2), we find that many operations on (S directly rely on those in (C, 9). Therefore, seek to optimize the flow on Cholesky manifold directly (Fig. 2). Before that, we first need to investigate the feasibility of this approach.
Lemma 4.1 1OJ The Cholesky map L S + C by LLLL = L and its inverse y are isometries between (St .9) and (Ct,
Based on the conservation of probability measure 22 and Lemma 4.1
we have:
Proposition 4.2 Given two manifolds (S and (C+.9) a probability density Ps defined on (S (9) together with the isometry L induces another probability pt on (C via the relationship:
Ps (x PL VX E St L = (X)
(8)
The proof of this proposition is given in Appendix.
E.1
This proposition enables to transform optimization problem from the SPD manifold to the Cholesky manifold while preserving probability measures. Furthermore, we can establish that (To [1) (No,, X1) (Z), leading modified objective L
(RC (L., (L. - L.ll
(9)
where the velocity L, has close-form solution derived in Appendix.
B
Theorem 4.3 Given two points Xx E S and their corresponding Cholesky decomposition L. let Xt be the geodesic on S connecting
factors Lo, L1 f where Xu LoLY and X1 LLLT Xu and X1, and L, be the corresponding geodesic on C, connecting Lp and L,. Then at the optimal solution:
10 (X.. - x. gx, = oL -
(10)
gL.
where L T) + (L,, y) L and üg (Lt) is the optimal velocity field on L, These results demonstrate that minimizing the objective on L is equivalent to minimizing the objective on St.
The proof of this theorem is given in Appendix. guarantee to optimize on Р directly.
E.2
With these properties, we have the theoretic
This formulation offers several benefits. First, direct optimization on the Cholesky manifold eliminates the need for frequently mapping between (S g) and (Ct, The more important point is that, this approach avoids the numerical instabilities associated with eigendecomposition in AIRM, particularly for matrices with near-zero eigenvalues.
Algorithm
T0 summarizes the training procedure of BrainFlow.
4.3 Reversing Flows with Consensus Control
Before detailing the mechanism, it is crucial to understand its neuroscientific motivation. The consensus control module designed enforce the fundamental biological principle of shared structural scaffold. The brain's 's relatively stable anatomical wiring (SC) serves as the foundation for a multitude dynamic functional states (FCs). Therefore, when reversing the flow from different FCs of the same individual, these pathways should converge towards single, consistent SC. Our consensus mechanism operationalizes this principle guide the reverse-flow predictions and ensure they are biologically plausible.

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The trained velocity field optimized from eq given SC. On the flip side, given functional connectivity, we can get a SC by simulating the reversed vector field - (-))
e enables predicting the FCs by simulating (-) with
Proposition 4.4 Given the Cholesky factor of a functional connectivity L1 of class y and vector field ig by simulating the following ODE from 0 to 1.
d T, - = v), Lo = L1, E [0
(11)
dt
Then can get a structural connectivity T4 corresponding to the FC from same subject if (f) u ( IT). Here Ln means the result of forward flow (FC), and T12 means the result of reverse flow (SC).
However, two factors may cause the simulation to diverge. First, the trained oy might not accurately approximate the velocity along the geodesic path. Second, during integration, cumulative errors can cause the trajectory to deviate progressively from the ideal path. Our solution to address these issues described below.
Enforces consensus by Dirichlet energy. We propose a training-free ffree consensus control module based on the idea of consensus in the network system[25]. It aims to minimizing of the Dirichlet energy functional, which enforces consistency among different FC states of the same subject. A fundamental property of the human brain is that its functional processes structure. This implies that FCs across different states share common information rooted in SC. Moreover, SC is relatively stable compared to the varying FCs, making shared anchor for all FC states of a subject.
supported by its underlying anatomical
Under this assumption, we define functional for consistency, E ([ 1 Dirichlet energy of the fully connected network formed by m FC states {X] X km
1= representing the
J==1
1
E E(LT
ai
L
-
(12)
j7
where each FC factor L is regarded as a node of the network and the edge weight between them is defined as 0ij = exp (-1) || - r. I²). the Gaussian kernel that weights the similarity between the FC states. The Dirichlet energy measures the smoothness of the connectivity states over the fully connected network graph defined by {. Intuitively, minimizing E ensures that all FC states become more consistent with each other by reducing their pairwise differences, weighted by their similarity. The negative gradient E w.r.. X gives the consensus velocity:
con
V L.
E
> ai (L L jti
(13)
which constructs another flow that as a force pulling L to toward its neighbours in the network.
To integrate this consensus mechanism with the original dynamics of the trained velocity field og. modify the ODE governing the evolution of L as:
d L. di = + ov
(14)
where a controls the balance between the original dynamics and the consensus term.
After that, we can calculate the Fréchet mean of these m final SC, specifically,
ma
L
L'th exp(m [L)
(15)
This formulation ensures that the evolution of the FC states is guided by both their individual geodesic paths and a global consensus mechanism rooted in the minimization of Dirichlet energy. We summarize the inference procedure in Algorithm 10
7

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Experiments
In this section, we conduct extensive experiments on synthetic datasets and real brain structural- functional network datasets for validating the effectiveness of BrainFlow. The detailed implementa- tion and experiment settings are shown in Appendix and the data description Human Connectome Project-Aging (HCP-A) [26], HCP- Appendix D.4
HCP-Young Young Adult (HCP-YA) [27]. UK Biobank [28] are listed in
View
Side
View
Side View
8
Ground Truth
Source Distribution
8
Geverated
Transforingg
Transforming
Figure-"8"
Figure 3: Results of textural distribution by our BrainFlow. The top is the target distribution on SPD manifold. The bottom row shows the distribution transformed from the source distribution. Although the flow operates on the Cholesky manifold, SPD manifold, demonstrating the effectiveness of our model.
accurately recover the results on the
5.1 Experiment on Synthetic Dataset
Fig. B and Fig. 5 demonstrate the effectiveness of optimization the Cholesky manifold and consensus control, respectively, where we demonstrate the promising results of finding a manifold flow from pre-detined source distribution to the target distribution. Table 7 iillusstrate the metrics of BrainFlow on synthetic dataset. Due to space constraints, further details and analysis are provided in Appendix
5.2 Functional Connectivity Prediction
Experimental setting. We predict functional connectivity from structural connectivity on all three datasets (UK Biobank, HCP-A, A, HCP-YA). Baselines include the one-way connectivity predic- tion method MGCN-GAN (6] as well the flow-based method Conditional Flow Matching (CFM) Riemannian Flow Matching on SPD (RCFM) R, Sariance-Presert Flow Matching (VPFM) 29 |Action Matching (2M) Schrödinger Bridge model ([SF) and our BrainFlow. To comprehensively evaluate the prediction performance, we employ multiple complementary met- rics. The Moot-Mean-SSuse Error (RMSE) quantifies the edge-wing wise prediction accuracy. To assess the preservation network topology, we compute the relative error local clustering coefficients (denoted LCE, calculated by (Ceeal- ), which reflects the accuracy of predicted local network organization around individual nodes, and global clustering coefficient error (denoted as GCE, calcu-
Creat
[C 1Carad- -Craal to evaluate the overall network transitivity prediction. (See Appendix D.2 for
lated by details)
Greal
Results. Table and Fig. 6 present the results of FC prediction. While MGCN-GAN consistently achieves the lowest RMSE across all three datasets, BrainFlow excels in preserving network topo- logical properties, as indicated by the lowest GCE and LCE values across all datasets. The SPD variant of RCFM fails to converge across all datasets, indicated by the divergence notation. This issue arises because, during inference, the intermediate SPD matrix becomes ill-conditioned, with some cigenvalues approaching zero. Moreover, the training time per epoch is significantly longer, taking approximately 100 seconds under the experimental setting described in Appendix D whereas BrainFlow requires only seconds per epoch.
Discussion. This performance trade-off between edge-wise aceuracy (RMSE) and topological preservation (GCE, LCE) suggests that different models prioritize distinct aspects of the prediction task. While MGCN-GAN excels minimizing direct edge-wise errors, BruinFlow more effectively preserves the network's 's hierarchical organization at both local and global scales. The visualization

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in Fig. 6 highlighhss that our method better preserves the contrast between different connectivity strengths. maintaining clearer separation between strongly connected regions (yellow) and weakly connected areas (blue)-a crucial factor for accurately representing brain network organization.
Table
Performance comparison across different datasets. Bold is the best.
Model
UK Biobunk
ICPAA
HCP YA
(S)
GCE(S)
RMSE
LCE(S5)
GCE
RMSE
GCE (9))
0.209-1-.003
19.04-144
15.74-14
19.0-338
CFM
0.206:10.004
9.11:044
6.1::0.5
0,259:100003
11.41:033
8,3:20.5
0.200
13.3:0033
9.4.10.88
RCPM (SPD)
diverge
diverge
diverge
diverge
diverge
diverge
diverge
diverge
diverge
AM
0.962-+0.001
47.04-0.2
46.6--0022
0.962+-00001
48.64-0.22
48,24-0.2
0171+-0.022
49.74-0.2
49.1--0.2
VPTM
0.274:1:0001
10441:03
0:615:1::211
17.2::13.3
19.1:2:2.2
8.8:1118
[SP) ad
0.202+000006
10.0+-:16
6.64:116
0,267
9.1--003
5.9+0088
0.355--00001
9.9-0.4
7.4+:100
BrulsFlow
0.200:10.001
891.0.3
5.5
0.291 4:0002
8.1::022
0.5
0.336-1:0001
8.7 0.2
58::0..5
Table 2: Generated Samples Classification Performance Across Datasets (N/A means the metrics fall below meaningful threshold for application).
UK Biobank (Binary)
HCP-A (4 tasks)
HCP-YA (7 tasks)
Model
Fl (%)
ACC (%)
Fl (%)
ACC (%)
F1 (%)
ACC (%)
Real
99.12 N/A
99.10
95.51
93.48
92.31
92.06
AM
N/A
N/A
N/A
N/A
N/A
VPFM
98.44:1:..04
98.28:t1.22
44.60:134.79
42.22-4:34.48
16.74
60.16
[SF]
08.65:10.26
98.67-10.26
77.05
82.99-1:.44
51.36:+3,9
53.87-13.80
CFM
0..59==0.30
98.59=00.28
77.03=2.12
85.48:11.85
57.89:3.55
59.87=3.14
BrainFlow
98.85:10.17
98.93-1..19
77.17=2.34
83,40+2.55
61.14::.677
64.23:14.49
Table 3: Reverse Flow Performance Evaluation Across Different Datasets.
Revens Flow
UK Biobank
HCP-A
HCP-YA
RMSE
LCE (%)
(S)
RMSE
LCE
(C)
RMSE
(CE
GCE (%)
AM
1.17::0.00
0.24-0.02
0.214:033
1.50=0.00
1.0234:1330
26.0011.00
1.46:0000
25.51+1.70
13
VPFM
0.27:2:0001
0.18:0.01
0.18:0:001
0.55=00.9
10.71:::2:4
11.72::1665
0.3120000
0.55:20.04
0.57:20.04
[8P] M
0.23:::000
0.20:0001
0.21:0:00
0.48:10.06
0.49=1005
0.30:2000
0.512002
0.5120022
0.25=0.00
CFM
1.14::0000
24.6110002
26 15:20000
1.05=0.00
20.23:20004
21 46:22.004
0:07:10000
21 21.28:20.01
:000
BrainFlow
0.00:10000
0,00:10.00
0.08:10.00
0.36=0.00
0.59:10.03
0,59
0.24:1..00
0,47:00.02
0.46:10.02
Table 4: Ablation study on noise level a across different datasets. Bold indicates the best performance for each metric.
Weight Parameter
HCP-
JICP-YA
UK Biobank
MSE
(CEE
(C)
(C)
GCE(5)
LCE
0.10::0.00
0.71:20.00
0.69:10.00
0.14:1..00
0.55:1..00
0.59:10.00
0.07:10.00
0.39
0.30
0.4
0.46:20.01
0.95:20.00
0.93:20.01
0.52:1:000
0.90:1:0000
0.88:1:0.00
0.89::0000
1.35:20.00
1.30=0.00
L.52:0.03
3,17:50.00
3,18-10.02
1.32--0.02
252-4-001
3,02--002
189--00:02
10,93-4-0022
11.16+00011
2.37::0.01
20.25:::.05
2.12:10.01
15.74:10.03
16.40:10.02
2.52:1:000
27.88:1:0002
28.16:1:002
5.3 Brain Task Recognition Using The Predicted FC
Experimental description. In this experiment, we evaluate the conditional prediction ability of BrainFlow. However, the differences in connectivity patterns across cognitive tasks are often highly variable across subjects, making it challenging to establish consistent signatures for specific task states. Additionally, no standardized metric exists to evaluate the quality of speciicc connectivity predictions. In this context, employ proxy evaluation metrics such accuracy and Fl-score within a downstream aas-claasifiiation framework. Specifically, we first train a classifier on the real FC data. We then apply this pre-trained classifier to the generated FC to assess whether they retain sufficient task-specific information for accurate task discrimination. Additionally, we generate samples for each task in the HCP-YA dataset and then average them within each task to obtain representative sample each task. We then compare them with the real average representation. Appendix D.3 shows related result.
Results. Table illustrates the effectiveness of BrainFlow in preserving task-specific information across datasets with varying task complexity. For the UK Biobank dataset, which involves binary

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task classification, the model achieves an impressive 98.93% accuracy with the generated functional connectivity, closely matching the real data performance of 99. 12%. This result demonstrates that BrainFlon effectively captures the critical connectivity patterns necessary for distinguishing between the two cognitive states binary task discrimination. As task complexity increases, prediction performance declines but remains superior to the other methods in most scenarios.
5.4 Reverse Structural Connectivity Prediction
Experimental description. In this experiment, we evaluate the performance of consensus control by reversing the trained BrainFlow, using different FC inputs (We have multiple fMRI scans for each subject) from the same subject. We employ the same evaluation metrics in Sec. 5.2] including RMSE, LCE, and GCE.
Results. Table TableBhiggttn highlighss the effectiivness of our proposed consensus control mechanism across three datasets. Notably, most approaches achieve consistently low error rates, largely due to the inherent stability of SC patterns across subjects, in contrast to the more dynamic nature of FC. This fundamental characteristic of brain architecture ensures that the core graph topology remains well-preserved during reconstruction, reflected by the consistently low clustering coefficient errors observed across all datasets. For a visual comparison, please refer to Fig. 7 in Appendix D.5
Discussion. The most notable improvements are observed in the UK Biobank dataset, where the consensus control mechanism significantly reduces both global and local clustering coefficient errors while maintaining a low RMSE This indicates that the consensus approach effectively integrates shared information across multiple FC states to reconstruct more accurate structural patterns.
5.5 Parameter Sensitivity Analysis
Experimental description. We conduct an experiment to test the sensitivity of the consensus control parameter We evaluate BrainFlow's reverse flow performance across five different values: 0.0, 0.2, 0.4, 0.6, 0.8 on all three datasets (UK Biobank, HCP-A, HC)-YA). For each configuration, we use the same evaluation metries in Sec. 5.2 including MSE, LCE, and GCE. The 0.0 setting represents vanilla reverse flow without consensus control, serving as our baseline for comparison.
Results. Table presents the results across different values. The optimal performance is consistently achieved at o = 0.2 across all three datasets, with substantially lower error metrics compared to 0.0 (vanilla reverse flow).
Discussion. We can see that when a is small (~ 0.2), the model has significant performance gain compared to vanilla reverse flow. However, as a increases, the consensus control comes to dominate the reverse flow, causing the trajectories become biased. This degradation occurs because excessive consensus control overly constrains the reverse trajectories, causing them to converge prematurely and lose fidelity to individual structural patterns. We then argue that a may not 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 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-frr free and does not account for this simulation error. Therefore, optimizing a during training is unsuitable for the intended inference procedure.
6 Conclusion
In this work, we propose a flow matching approach on the Cholesky manifold to bridge SC and FC, improving numerical stability and efficiency. A consensus control mechanism enforces consistency across functional states, mitigating errors from vector field approximations and numerical integration. Our results demonstrate effective structural connectivity inference, with applications in neuroscience, brain disorder diagnosis, and connectivity-basse dynamic functional connectivity modeling and integrating domain-specifie constraints. It could be extended model both macro-scale lifespan connectivity changes and micro-scale, within-scan fluctuations by lcarning flows between consecutive time windows.
interventions. Future work includes extending to
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Acknowledgement
This work was supported by the National Institutes of Health (AG091653, AG068399, AG084375) and the Foundation of Hope. The views and conclusions contained in this document are those of the authors and should not be interpreted representing the official policies, either expressed or implied, of the NIH.
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A Additional Basic of Log-Cholesky Metric
We follow the notations [0]. Denote the Cholesky decomposition of P S as P= LL the Cholesky map is defined as (P) = and its inverse map (L) = P. Table. shows properties
of Riemannian manifolds ( 9) and (9)
Table 5: Basic properties Riemannian manifolds.
Busic properties of Riemannian manifolis (C and (8%
(4
($ 9)
lungent space
tangent space
Xicmannian metrle
Kiemantian metric
(x, Y) Eis X1 + Ei X N3
g V) (z1,s((0)):21,,
goodesic emanating from L with direction X
gendesic emanating from P with direction W
x1,xx (4) = [L] qX] + (X))
TPP. (WW) (1tp://0000000000000000000000000
Rlemannian exponential
Riemannian exposemtial map
Expg [G] + [X] + (XX)))))))
Exp 17
Riemannina logaritimic map
Klemannian logarithmic map
Logg, [K] - [6] D(F) (Tx( (T() D(K)}
1 = (D D (1ng (Q))
goodesic distance between and K
goodesic distance betwezn P and
d 2. (L, K) (1LL)) - [K]] + |o D(L) log 1/
(P. Q) de, (P)) (Q)))
B Extensions of methods
B.1 Solution for closed-form velocity.
Given the flow/geodesic L,, the constant velocity L, in eq 9999 calculated by di L, = i. [ meaning that the conditional velocity field u, (L, can be obtained by calculating the geodesic and its derivative w. time. While this could be calculated using autograd during the forward pass, derive a more efficient closed-form expression based on the geodesic definition (eq. 3) on (
L, = // L, (t) = [o] L1 + (L))exx(tMM
(16)
M logIl (Lo) (L,)
B.2 Computation Cost Analysis
We provide a thcoretical analysis comparing the computational complexity of BrainFlow against RCFM-SPD and vanilla Flow Matching. Flow-based methods can be summarized with the following pipeline steps:
Sampling from the source and target distributions
Calculate the interpolated position and velocity
Regress the predicted velocity to ground truth velocity
During inference, integrate the time-dependent velocity
We analyze the additional computation of Riemannian conditional flow matching on SPD manifold (RCFM-SPD) and BrainFlow on these steps compared to baseline vanilla Flow Matching.
Sampling Stage. Before sampling stage, additional preprocessing (Cholesky decomposition) is required for BrainFlow, whose time complexity O(Nd²), where the size of training set and the matrix dimension. Importantly, 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.
Interpolation Stage. During interpolation stage, the exp(-), log(-) operations of RCFM-SPD both require O(BdS), where B is the batch size, while these operations Cholesky manifold only
21

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Stage requires (Bd), as the strict lower triangular part follows element-wise addition/subbraationn and the diagonal part uses diagonal matrix exponential and logarithm.
Loss Calculation Stage. During loss calculation, the inner product (..)) on SPD manifold requires ( while Cholesky manifold has the complexity vanilla Flow Matching (T₂2 norms).
Inference Stage. During inference, RCFM-SPD requires eigenvalue clipping to keep the element SPD, requiring O each integration step. Hence, it requires additional OLLBd) where L is total step, while BrainFlow naturally maintains the Cholesky structure without additional operations.
Table 6 summarizes the computational complexity across different stages. The key insight is that while BrainFlow incurs one-time preprocessing cost, RCFM- SPD repeatedly pays O (Bd at every iteration and (ILBa)) during inference, making BrainFlow substantially more efficient in practice.
Table 6: Computational complexity comparison. N: training set size, B: batch size, d: matrix dimension, L: integration steps. "+" denotes additional computational cost compared to baseline.
RCFM-SPD
BrainFlow
Flow Matching
Sampling
Standard
+ O(Nd") Cholesky preprocessing
Baseline
Interpolation
+B(B))) exp(-). log(-) (SPD)
+ +O(Bd) exp(-), log(-) (Cholesky)
Baseline
Loss
+O(Bd* (SPD inner product)
Standard (Cholesky inner product)
Bascline
Inference
+O(LBd) eigenvalue clipping
Standard
Baseline
For typical brain connectivity matrices with d = 116 regions and L = 100 integration steps, the repeated eigendecompositions in RCFM-SPD accumulate to significantly higher computational costs than BrainFlow's 's one-time preprocessing. Hmpirically, reported in Sec. 5.2] BrainFlow achieves a 12.5x speedup (8 seconds vs. 100 seconds per epoch) while maintaining numerical stability throughout training and inference.
B.3 Algorithm
Here we briefly introduce the algorithm pipeline in Alg. 10
Algorithm 1 Training of BrainFlow
Algorithm 2 Predict FC/SC
Require:
Joint distribution q(Xo. X1)
Require: SC X, with desired task state y or
Initialize parameters of ii
m FCs from same subject (X.') ) trained v
while not converged do sample time f uniform(0, 1) sample training pair (Xo,Xi) q with task label y
if SC (forward flow) then
Lo = (((o)
L = ode ode_solve(Lo, (. (,,)), (0,
Lo. L1 = (Xx). (X,)
X; L1Lب
L. expr.., log L.. (L1)
else if FC (reverse flow) then
calculate L, from eq.
16
L (X('), L...m
()) ر: (L,, v)
Le gu.
I ode ode_solve( T. -
0
o t= (0)
end while
calculate L1' using eq with [L 'e M
= 4,
end if
return Predicted FC X, or SC X
22

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presents three C Experiment on Synthetic Dataset
C.1 Generation of Artificial Textures on SPD Manifold
We trained a simple MLP-baser Flow Matching (FM) model on Cholesky manifold. Two synthetic datasets whose elements are SPD matrices on S are used visualize the trained dynamics. They are the siwssroll and figure-eight. We can represent the space of x 2 SPD matrices S as points above a circular cone in 3D space by mapping each SPD matrix to a point in R The cone itself represents singular matrices with zero determinant. Every point above this cone corresponds exactly one SPD matrix for visualization.
Result. We apply Cholesky decomposition both the prior and target dataset and transformed them back to SPD manifold. The results of data generation are shown in Fig. [ Fig. 3 orthogonal views (Front, Top, and Side) of both Figure- "8" pattern and Swissroll configuration. For each pattern, we compare the original data distribution (shown in green) with the samples generated by the FM on Cholesky manifold (shown in purple). All generated points strictly remain within the SPD manifold, as evidenced by their positioning above the determinant-zero cone in all views. The visual consistency between the generated and original samples across both patterns, coupled with the strict adherence to the SPD manifold constraints, demonstrates that learning flow on the Cholesky manifold practical.
C.2 Quantity Performance on Synthetic Datasets
Dataset Besides the low-dimensional artificial textures data on SPD manifold, we also generate two 100 x 100 SPD matrix distributions for further analysis. Specifically, we first generate two groups of random matrices with different mean values and standard variance. Then we can get SPD matrices by X LL el where is the random matrix and I is the identity matrix. We evaluate the Wasserstein distance between the generated and true distributions. The ratio of generated samples that lie on SPD manifold is also considered.
Result Table 7 shows that our method can keep the SPD property well and get lower Wasserstein distance.
Table 7: Model Performance across Synthetic Datasets. (W2) means the quadratic Wasserstein distance
Swiss-roll
Fig-8
High-dim SPD
Model
SPD ratio (%)
Wz(x10-1)
SPD ratio (%)
Wa(x10
SPD ratio (%)
W2(x10
AM
100.00
9.03
100.00
1.42
0.00
9.13
VPFM
99.96
6.28
99.99
1.93
0.00
7.98
M
99.98
6.05
99.69
1.81
71.98
7.98
CFM
98.84
5.98
99.95
1.83
23.78
7.98
RCFM
100.00
4.02
100.00
1.43
100.00
8.00
BrainFlow
100.00
4.07
100.00
1.43
100.00
7.97
C.3
Controlled Reverse Flow Synthetic Datasets
In this experiment we validate the reverse ODE flow with the proposed consensus control. For simplicity, synthetic datasets and FM model Euclidean space to illustrate the effect of consensus control.
Dataset. The synthetic dataset consists of points in 2D Euclidean space constructed as follows.
The source distribution contains points uniformly distributed on circle with radius 0.5, with small random perturbations (a 0.05) added to create natural distribution (Green Circle in Fig. 5) Each
23

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Average Predicted FC
Emotion
Gambling
Language
MOTOR
Relational
Social
WM
Average True FC
Figure The results of average predicted task FC and average true task FC in HCP-YA dataset
point in the source distribution is assigned unique identifier (ID). These points are then transformed into the target distribution through two distinct mappings corresponding to different classes. Points mapped to first class are expanded
circle with radius 1.0 and rotated clockwise by (Orange circle in Fig. 55) while points mapped to the second class are contracted to another larger circle with radius 1.5 and rotated counter-clockwise by 6 (Blue circle in 5). Both transformations include small Gaussian noise (a 0.05) to simulate real-world variations. Finally, points in target distribution inherit their IDs from their source points, meaning that points from different classes can share same ID if they originated from the same source point. This design allows us to clearly demonstrate how consensus control can help points with the same ID converge to their true source, even when the conditional optimal transport coupling suggests different correspondences.
Start point
End point
Ground truth Trajectory
Trajectory without Control
Trajectory with Control
Figure 5: Examples of consensus control. The left panel illustrates reverse flow without control, deviating significantly from the ground truth. The right panel demonstrates reverse flow with control, effectively steering it closer to the desired outcome.
Result. The FM model trained with conditional OT path and we sample points independently from source distribution and target distributions. Fig. (lett) shows the reversed trained flow from the target distribution to the source distribution and the right one shows the reversed flow with consensus control. Two groups of sample points and their trajectories shown in the Fig. with different colours (purple and orange).
In the case without control (left), the trajectories of points sharing the same ID (highlighted by blue dashed circles) fail to converge to their true source positions. The flow merely follows the learned conditional optimal transport paths, leading to divergent endpoints even for points that originated

---

Table 8: Task-specific functional connectivity prediction performance HCP-YA dataset. Best results in bold.
Task
Metric
AM
VPFM
[SF]
CFM
BrainFlow
RMSE
0.980:10.002
0.339-10.003
0.367:10.002
0.314:10.001
0.3291100002
Emotion
LCE (%)
0.503++0.002
0.08
0.093+00005
0.138+00002
0.085=00001
GCB
0,499:00.003
0.080100000
0.064:10.010
0.095=00.004
0.059=00.003
RMSE
0.974+00.002
0.313+00002
0.343+0.003
0.288+0.002
0.303=0.0002
Gambling
LCE (%)
0.481=#0.00%
0.100
0.104
0.125=00.00
0.097:10.002
GCB
0.473+00003
0.060#:.022
0.076+0.017
0.072+00.002
0.071=00003
RMSE
0.965++0.001
0.287+00002
0.303++0.003
0.268+00.001
0,282++0.002
Language
LCE (%)
0.526++0.002
0.090+000006
0.097+0.007
0.119=00.002
0.084=0.001
GCE (%)
0.521
0.057:10001
0.077:10.016
0.083+0.002
0.059=0.001
RMSE
0.969=00.003
0.302
0.325++0.002
0.280=0..002
0.295=000002
Motor
LCE (%)
0.526++0.005
0.089+00008
0.085+0.004
0.117+00.001
0.081=00.001
GCE
0.504
0.08310.016
0.053+0.011
0.083+000011
0.057=00002
RMSE
0.976+0.002
0.317+00003
0.302++0.003
0.293+000001
0.307=000003
Relational
LCE (%)
0.453++0005
0.0011
0.112+00011
0. 0.12000.002
0.101=:0002
GCE (%)
0.442+00005
0.071
0.084+0.026
0.096+000022
0.070=000003
RMSE
0.971 +:0002
0.003
0.330+L0.003
0.201
0.295+10.002
Social
LCE (%)
0.491+0.000
0.0959
0.102
0. 0.133+0.001
0.090=10001
GCE
0.482:10.003
0.0631:.021
0.084
0.064:000002
0.066:10.003
RMSE
+0.600
0.269
0.298
0.249=00.002
0.262::0.002
WM
LCE (%)
0.502-40.006
0.095
0..012
0. 0.125000002
0.090:-0.003
GCE
0.493:0.007
0.089::.012
0.076#00018
0.075=00.02
0.059=0.003
from the same source. In contrast, when consensus control is applied (right), the trajectories of points with shared IDs successfully converge to their corresponding ground truth source positions, as indicated by their alignment with the square markers representing ground truth locations. This convergence is achieved despite the points starting from different target distributions (orange and blue circles) and having different initial flow directions. The controlled trajectories exhibit smooth adaptation from their initial paths toward consensus points, demonstrating the effectiveness of the control mechanism in enforcing consistent reverse mapping.
D Experiment Settings
D.1 Model Implementation
We base our implementation on 30 The flow-based approach uses an 8-layer Transformer without positional encoding maintain permutation equivariance of connectivity matrices. We use
25

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a label embedding to handle conditional predictions, while time information follows the encoding scheme from 30]. For inference, we employ an Euler solver with 0.01 step size.
D.2 Clustering Coefficient Calculation
We provide detailed explanations for the clustering coefficient metrics used throughout this paper.
Graph Notation. An undirected graph G EV formally consists of of vertices V and set of edges E between them. An edge ezj connects vertex 8; with vertex Uj. For functional connectivity matrices, we only consider positive edge weights.
Local Clustering Coefficient (LCC). The local clustering coefficient undirected graphs quantifies the degree to which neighbors of given node are connected to each other, defined as:
C = 2/e5x Uj. UN E Ni, Cjkk E
(17)
ki - 1)
where N₂ denotes the neighborhood of node i, and
Ej
Aky is the degree node
Expressed in terms of the adjacency matrix A, this becomes:
C
1 (k. 1)
Σ j.k
AijAjAAsi
(18)
which counts the number of closed triangles (triplets) involving node i.
Global Clustering Coefficient (GCC), The global clustering coefficient measures the overall ten- dency of nodes to form clusters across the entire network:
number of closed triplets
(19)
number all triplets (open and closed)
In matrix form:
== Eiji Ax
(20)
&
where when the denominator is zero (i.e., no connected triplets exist).
Error Metrics. We compute the relative error for both local and global clustering coefficients as:
Cprred Creat
LCE
(21)
Greal
C pred
Cleal
GCE
(22)
Clcal
where C denotes local clustering coefficient and C" denotes global clustering coefficient. For isolated nodes where Creat = 0, we exclude these nodes from the LCE calculation as they represent degenerate cases and occur very rarely in brain connectivity networks.
D.3 Training Details
The datasets are split into train, validation, test set with ratio 7:1:2. All the models are trained with 1000 epoch using AdamW [32) with Cosine Annealing learning rate schedule (33). We do grid search on the following hyperparameters: (1) learning rate: {0.001, 0.0005, 0.0001} } (2) Layer of Model: (4. 6,8, 12} and (3) Batch size: f256, 512. 1024). We do model evaluation every 50 epochs. All the experiments are repeated 5 times with different random seed for data split and model initialization. The structural connectivity data is preprocessed by first A log(A* 1), where the log(.) is element-wise logarithm. This step can constraint the scale of SC to reasonable range. Then we add a large enough e/ to the SC, transforming it to SPD matrix. We run all the experiments using single
NVIDIA A6000 GPU.
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D.4 Data Description
Data Processing For functional connectivity preprocessing and construction, we use xep-d [34] post-ppoooee post-processing pipeline for fMRI data. As for the structural connectivity, we utilize QSIPrep 331 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 iFOD2 (2nd-order
nuisance regression approach. QSIPrep's default reconstruction workflows 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.
HCP-A Human Connectome Project-Aging (HCP-A) dataset includes data from 717 subjects, encompassing both fMRI (4,846 time series) and Diffusion Weighted Imaging (DWI) (717) scans 26]. This rich collection facilitates in-depth analyses of both functional and structural connectivity. The HCP-A dataset includes data from four brain tasks associated with memory: VISMOTOR, CARIT, FACENAME, and resting state. In the following experiments, these tasks are treated as distinet categories. In the following experiments, we treat the data as a four-class classification problem.
HCP-YA Human Connectome Project- Young Adults (HCP-YA) database 27 involves 11,608 IMRI and 328 DWI. Each IMRI scan includes seven cognitive tasks associated with memory, including Motor, Relational, Social, Working memory, Language, Emotion, and Gambling. In the following experiments, treat the data as seven-class classification problem.
UK Biobank a large-scale neuroimaging dataset that includes MRI data 28 Specifically, contains preprocessed fMRI (n=14,619) and DWI (n=5,731 data, following the pipeline described in (36. The dataset includes recordings from a brain task designed to engage both cognitive and sensorimotor functions [37]. In the following experiments, we treat the data as two-class classification problem.
For all these three datasets, we partition each SC and FC into I 16 regions using AAL atlas 38 Thus, SC is 116 x 116 matrix where each element quantified by the number of nerve fibers linking two brain regions.
D.5 Further Results
SC-to-FC. The visualization Fig. reveals that our method better preserves the contrast between different connectivity strengths, maintaining more distinct separation between strongly connected regions (yellow) and weakly connected areas (blue), which is crucial for accurate representation of brain network organization. This qualitative observation reinforces quantitative findings that BrainFlow excels in preserving the hierarchical organization of brain networks, even though it may not minimize edge-wise wise errors to the same degree MGCN-GAN.
FC-to-SC. In Fig. [ we first calculate the absolute value of the difference between the prediction and ground truth. Then we binarize them with defined threshold. The large difference (>>hreshold) set to dark and small difference set to light. We can see that the controlled reverse flow (bottom row) shows better prediction result with significantly less difference.
Task Functional Connectivity Generation We explain our findings on HCP-YA datasets here. We first record all the predicted functional connectivities from test sets. Then, we also record the ground-truth functional connectivities of the whole dataset. Next, we group them by different tasks and averages within the groups on the predicted (at the top of the Fig. { and ground-truth FC (at the bottom of the Fig. 4.. Fig. reveals several biologically meaningful patterns that go beyond mere prediction accuracy: (1) Network Organization Preservation. We observe block-like structures and prominent diagonal elements that reflect strong intra-hemispheric and inter-regional interactions. (2) Task-Specific Network Re-organization. The difference across tasks in the predicted FC matrices reveals that the model is sensitive to the task-induced reconfiguration of functional connectivity, consistent with the literature on dynamic functional networks. For example, in the MOTOR task,
27

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Sample
Sample
Ground Truth
MGCN-GAN
CFM
Ours
-0.5
0.0
0.5
1.0
Figure 6: Then predicted results different methods. Two randomly drawn samples are displayed on two rows.
Large difference (>threchold)
Without Control
Small differance (SIbreshold)
With Control
Figure 7: The results of reverse prediction. The large difference (>thhreholld) is set to dark and small difference set to light. Each column represents a result predicted from one specific state FC from same subject. The top column is the result without control and the bottom row is the result using consensus control.
28

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can see the strong activation in the top-left part of the matrix (The motor networks are indexed with about 1-20), which aligns with the findings in work by Tzourio- Tzourio-Mazoyer Mazoyer N et.al. [3
Evaluation on each Task of HCP-YA.
Experimental description. We conduct task-specific functional connectivity prediction for each of the seven cognitive tasks in the HCP- YA dataset. Using the same experimental setup as described Sec. 5.2 we train all baseline methods (AM, VPFM, [SF] CFM) and BrainFlow to predict task-specific FCs from the same structural connectivity. For each task, we evaluate the prediction quality using three metrics: RMSE for edge- edge-wise accuracy. LCE for local network topology preservation, and GCE for global network structure preservation. All experiments are repeated times with different random seeds, and report mean = standard deviation for each metric-task combination.
Results. Table presents the comprehensive task-specific prediction performance across all seven cognitive tasks. BrainFlow consistently achieves the best or near-best performance on topological metrics (LCE and GCE) across all seven tasks. Specifically, BrainFlow obtains the lowest LCE in all seven tasks, with values ranging from 8. % (Motor) to 10.1% (Relational), demonstrating superior local network topology preservation. The standard deviations across all metrics remain small (typically <0.3%), indicating stable and reproducible performance across different random initializations. Among the remaining baselines, CFM demonstrates competitive RMSE performance but falls short in preserving network topology compared to BrainFlow.
Discussion. The task-speeific analysis reveals three key insights. First, each method maintains stable performance across all seven cognitive tasks, with small standard deviations indicating consistent behavior regardless of the functional state. Second, the relative ranking of methods remains consistent across tasks: BrainFlow consistently achieves the best topological preservation (LCE/GCE), while CFM obtains the lowest RMSE This stability suggests that model performance is primarily driven by architecture and training objectives rather than task-specific properties. Third, the narrow performance range across tasks (e.g., BrainFlow's 's LCE varies only from 8.1% 1% to 10.1%) 1%) demonstrates that our method has learned unified SC-FC relationship that generalizes robustly across diverse cognitive states, supporting the hypothesis that a shared structural scaffold underlies multiple functional configurations.
E
Proof
E.1
Proof of Proposition 4.2
Proof. We first introduce conservation of probability measure on the Riemannian manifold. Assume a probability density Pu : MM - RR defined on a Riemannian manifold Mg and transformation T : Mu - M1 that maps the Riemannian manifold Mo to M1. Then the density Po together with transformation T induce probability density P1 Mi + R++ via the relation:
S. Ma
-f,, pi (11))vool,
po(zo)
According to 24 one can derive:
(1(21) = po (x:) Je
(23)
where det J (x,
means the determinant of the Jacobian of g-1, Here we apply this theorem to Riemannian manifold (S g) and (Ct, with Cholesky decomposition L, an isometry between them. We can directly induce that:
Ject .0)
= -f PL()ddvol,
Pe Px(Xvvvoxx
(24)
(Ct..)
Because L is an isometry between (St g) and (C;t, ,)), then L preserves the metric and, conse- quently, the volume measure. This directly implies:
dvolx dvolz (under the isometry IP)
Thus, the Jacobian determinant adjustment [det.J (I)) is everywhere (notice that the inverse of an isometry is also an isometry). Therefore,
Ps (X) (LL) = L, S
29

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E.2 Proof of Theorem 4.3
To enhance the readability of the proof, we omit the input of V.(() and Vz(.). According to Section
3.2 of [10], the inverse map of the Cholesky decomposition :: C - S by (LL) = I.I." an isometry between (L+. and (St .)). Specifically,
IL P. Q) (z)) for all L E L+ and P.Q E T.C+. Here the differential DLy: : TIL+ - TLL- St is given by
(L = LP + PL
Next, again because y is an isometry and Xo - (LL),, X1 - L (LI), according to Proposition
5.6. of [39], the map the geodesic 5 (Ix) between Lo and L is the geodesic between Xo and X1, i.c., (Le) = Xx,
Now have
Tr. (P.Q) - gx.
Vi 1 ØL.
According to the definition of a norm, we have
= (Ve Ln, V. - i.). Since I is
an isometry between (C+, and (S .)). the metrics satisfy:
JL. (Vi L., Vi L.) gx. (DLL, L.), (DL, L.)
It can be verified that Dr, l linear map. Then the optimal Vi2 implies that
(Di.))i L.) (Dr, DV, 9LL
V₁,- X,
Hence, when we find an optimal Vr, we can get V by V = L.V. + V. LT and
gr. (V. 14, V. i.) = gx. (V. Xz, V x.)
Therefore, by the isometry property of sf and Proposition 4.2 the optimization problems in (C and (S g) are equivalent.
F Limitations
Here we discuss the potential limitations of BrainFlow. The first point is that though we prove the equivalence of the optimal solution between SPD manifold and Cholesky manifold, the convergence speed is not rigorously discussed. We only evaluate them empirically.
Second, the model shows better performance on larger dataset like UK Biobank compared with the other two datasets, revealing that the method requires large number of data for good performance, which is not easy for neuroimaging data.
Third, acknowledge the well-known challenge that the relationship between structural and functional connectivity is not simple one-to-one mapping at the edge level. Complex biological mechanisms, such as polysynaptic pathways and neuromodulatory effects, mean that strong functional links can exist between regions with no direct anatomical connection. Furthermore, the inherent limitations of diffusion MRI and tractography algorithms can lead to both false positives and false negatives in SC estimation. Our model does not explicitly simulate these underlying biological mechanisms. Instead, BrainFlow addresses this challenge by learning mapping between the entire distribution of structural connectomes and the distribution of functional connectomes. By leveraging paired SC-FC data from the same individuals to construct joint distribution, the model captures the high-level statistical regularities that emerge from these complex, indirect interactions. Therefore, BrainFlow's 's strength lies in its ability to learn and represent the statistical SC-FC coupling within the available imaging data, thereby contributing to understanding of this well known mismatch.
30

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G
Impact Statement
This paper advances the field of Machine Learning by introducing BrainFlow, generative model that captures the coupling between structural and functional brain connectivity. By leveraging flow-based optimization on the SPD manifold, our approach enhances the interpretability and robustness of learning structured and dynamic data. While our work has potential implications for neuroscience and AI, there are no specific societal consequences that must be highlighted here.
31


Reviews:
[{"review_id": "usQC4zrI62", "reviewer": "Reviewer_nEq7", "summary": "The 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.", "questions": "Line 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", "limitations": "N/a", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:  \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\")", "was_revised": true, "final_justification": "I have read the authors' rebuttal responses and discussed these responses, am happy to upgrade my score.", "rebuttal": "# W1: \n+ We will give a theoretical analysis on computational efficiency, as follows, in the final version.\n+ First, the flow-based methods can be summarized with the following steps:\n  + Sampling from the source and target distributions. \n  + Calculate the interpolated position and velocity between the source and target samples \n  + Regress the predicted velocity to ground truth velocity. \n  + During inference, integrate the time-dependent velocity.\n+ 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.\n  + 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.\n  + 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 $\\mathcal{O}(d)$\n  + 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)\n  + 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.\n| Stage | RCFM-SPD | BrainFlow | Flow Matching |\n|--|--|--|--|\n| **Sampling** | Standard | + $\\mathcal{O}(Nd³)$ Cholesky preprocessing | Baseline |\n| **Interpolation** | +$\\mathcal{O}(Bd³)$ $\\exp(·), \\log(·)$ (**SPD**)| $+\\mathcal{O}(Bd)$ $\\exp(·), \\log(·)$ (**Cholesky**)  | Baseline |\n| **Loss** | +$\\mathcal{O}(Bd³)$ (**SPD inner product**) | Standard (**Cholesky inner product**) | Baseline |\n| **Inference** | +$\\mathcal{O}(LBd³) $eigenvalue clipping | Standard | Baseline |\n- N: training set size, B: batch size, d: matrix dimension, L: integration steps\n- \"+\": additional computational cost compared to baseline\n- 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__.\n# W2:\n+  $\\sigma$ in Eq. 14 and that in $a_{ij}$ are different., We will correct it by changing the coefficient in eq.14. to \\beta.\n# W3:\n+ 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. \n# W4, Q3:\nWe 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. \n\n__RMSE__\n| Method | Emotion | Gambling | Language | Motor | Relational | Social | WM |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| AM | 0.980±0.002 | 0.974±0.002 | 0.965±0.001 | 0.969±0.003 | 0.976±0.002 | 0.971±0.002 | 0.961±0.001 |\n| VPFM | 0.339±0.003 | 0.313±0.002 | 0.287±0.002 | 0.302±0.003 | 0.317±0.003 | 0.303±0.003 | 0.269±0.003 |\n| [SF]$ ^2$M  | 0.367±0.002 | 0.343±0.003 | 0.313±0.003 | 0.325±0.002 | 0.352±0.003 | 0.330±0.003 | 0.298±0.003 |\n| CFM | 0.314±0.001 | 0.288±0.002 | 0.268±0.001 | 0.280±0.002 | 0.293±0.001 | 0.281±0.002 | 0.249±0.002 |\n| BrainFlow | 0.329±0.002 | 0.303±0.002 | 0.282±0.002 | 0.295±0.002 | 0.307±0.003 | 0.295±0.002 | 0.262±0.002 |\n\n__GCE__\n| Method | Emotion | Gambling | Language | Motor | Relational | Social | WM |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| AM | 0.499±0.003 | 0.473±0.003 | 0.521±0.002 | 0.524±0.005 | 0.442±0.005 | 0.482±0.003 | 0.493±0.007 |\n| VPFM | 0.080±0.009 | 0.060±0.022 | 0.057±0.013 | 0.083±0.016 | 0.071±0.022 | 0.063±0.021 | 0.089±0.012 |\n| [SF]$^2$M | 0.064±0.010 | 0.076±0.017 | 0.077±0.016 | 0.053±0.011 | 0.084±0.026 | 0.084±0.013 | 0.076±0.018 |\n| CFM | 0.095±0.004 | 0.072±0.002 | 0.083±0.002 | 0.083±0.001 | 0.096±0.002 | 0.064±0.002 | 0.075±0.002 |\n| BrainFlow | 0.059±0.003 | 0.071±0.003 | 0.059±0.001 | 0.057±0.002 | 0.070±0.003 | 0.066±0.003 | 0.059±0.003 |\n\n __LCE__\n| Method | Emotion | Gambling | Language | Motor | Relational | Social | WM |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| AM | 0.503±0.002 | 0.481±0.003 | 0.526±0.002 | 0.526±0.005 | 0.453±0.005 | 0.491±0.003 | 0.502±0.006 |\n| VPFM | 0.089±0.005 | 0.100±0.010 | 0.090±0.006 | 0.089±0.008 | 0.106±0.010 | 0.095±0.009 | 0.095±0.007 |\n| [SF]$^2$M  | 0.093±0.005 | 0.104±0.008 | 0.097±0.007 | 0.085±0.004 | 0.112±0.011 | 0.102±0.006 | 0.101±0.012 |\n| CFM | 0.138±0.002 | 0.125±0.001 | 0.119±0.002 | 0.117±0.001 | 0.128±0.002 | 0.133±0.001 | 0.125±0.002 |\n| BrainFlow | 0.085±0.001 | 0.097±0.002 | 0.084±0.001 | 0.081±0.001 | 0.101±0.002 | 0.090±0.001 | 0.090±0.003 |\n# W5:\n+ 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__” \n# W6-13 ( typos, figures, notations):\n+ 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\n# Q1:\n+ 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.\n# Q2:\n+ We first want to correct that the trajectories go __from blue circle to green circle__ instead of from green to blue.\n+ 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.\n+ 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."}, {"review_id": "7JEI9ezR8J", "reviewer": "Reviewer_mEsD", "summary": "This 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.", "questions": "Questions\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.", "limitations": "No 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.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "Thanks for the detailed response. Most of my concerns are addressed. Given that the original paper lacks visualizations on connectivities (I understand no figures can be provided during rebuttal, however, I expect to see these as they can be important to justify the model's plausibility), I will maintain my score.", "rebuttal": "# W1, Q1\n+ 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.\n# W2, Q2\n+ We appreciate this very insightful comment.\n  + 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. \n  + 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.\n  + 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.\n\n# W3\n+ 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.\n\n# W4\n+ 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.\n+ ||HCP-A RMSE | HCP-A GCE | HCP-A LCE | HCP-YA RMSE  | HCP-YA GCE | HCP-YA LCE   | UKB RMSE    | UKB GCE     | UKB LCE     |\n| :--- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- | :----------- |\n| 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 |\n# W5\n+ 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. \n# W6, Q5\n+ 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.\n+ In macro-scale, BrainFlow framework could be used to model the slow evolution of brain connectivity over months or years (lifespan).\n+ 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.\n# Q3\n+ 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. \n+ We then argue that $\\sigma$ may not be suitable to optimize during training. \n  + 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.\n  + 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.\n+  __HCP-A__\n| $\\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | \n| :--- | :---: | :---: | :---: | :---: | :---: | \n| MSE | 0.65±0.00 | 0.13±0.00 | 0.44±0.01 | 1.52±0.03 | 2.37±0.01 |\n| LCE(%) | 0.74±0.00 | 0.71±0.00 | 0.95±0.00 | 3.17±0.03 | 20.28±0.05 |\n| GCE(%) | 0.73±0.00 | 0.69±0.00 | 0.93±0.01 | 3.18±0.02 | 20.50±0.03 |\n\n+ __HCP-YA__\n| $\\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | \n| :--- | :---: | :---: | :---: | :---: | :---: | \n| MSE | 0.42±0.00 | 0.14±0.00 | 0.52±0.00 | 1.32±0.02 | 2.12±0.01 |\n| LCE(%) | 0.62±0.00 | 0.55±0.00 | 0.93±0.00 | 2.92±0.01 | 15.74±0.03 |\n| GCE(%) | 0.59±0.00 | 0.59±0.00 | 0.88±0.00 | 3.02±0.02 | 16.40±0.02 |\n\n+ __UKB__\n| $\\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | \n| :--- | :---: | :---: | :---: | :---: | :---: | \n| MSE | 0.09±0.00 | 0.07±0.00 | 0.69±0.00 | 1.89±0.002 | 2.52±0.01 |\n| LCE | 0.52±0.00 | 0.39±0.00 | 1.33±0.01 | 10.93±0.02 | 27.88±0.02 |\n| GCE | 0.53±0.00 | 0.39±0.00 | 1.30±0.01 | 11.16±0.01 | 28.16±0.02 |\n\n# Q4\n+ 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.\n+ Similarly, interpolation between known task-specific FCs can not be directly achieved. We can do that by training a new model.\n+ It would be interesting to embed the connectivity (both SC and FC) into a unified latent space to achieve flexible transformation.\n\n## Limitations\n+ 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.\n---\n[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"}, {"review_id": "hQfahgSpdq", "reviewer": "Reviewer_67by", "summary": "This 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.", "questions": "The 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.", "limitations": "## 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.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "# 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.", "was_revised": true, "final_justification": "After reading the rebuttal and the other reviews, I am comforted in my positive evaluation of this work.\n\nMy initial assessment identified the technical description of neuroimaging experiments as a weak point, which the authors fully addressed in their rebuttal. This led to increase my score on clarity to 4. \n\nI was also unsure about the choice of baselines. This was also clarified by the rebuttal, and I now think there is a strong mixture of reference models and tasks for the models proposed in the paper. \n\nI would have liked to see more rigorous experiments on the data efficiency of the methods, but the authors have expanded the range of applications to include other types of datasets with varied dimensions, so I do not see this point as a real weakness. \n\nOverall, I believe this work to be a substantial contribution to the field, with strong methodological and experimental original developments, and I recommend acceptance.", "rebuttal": "# W1\n+ We apologize for the confusion. We will rephrase the technical terms. \n+ The measurement we use to construct structural connectivity is the fiber count.\n\n# W2 \n+ For functional connectivity preprocessing and construction, we use xcp-d, a post-processing pipeline for fMRI data [1].\n+ As for the structural connectivity, we utilize QSIPrep [2] for processing and reconstruction.\n+ 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.\n+ 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.\n+ 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.\n+ We will include all information into the supplement material of the final version. \n# W3\n+ 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.\n+ 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.\n+ 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.\n# Q1\n+ 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. \n# Q2\n+ We argue that consensus control may not be suitable to regularize the training. \n  + 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.\n  +  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.\n# Q3 \n+ We will change to “involve human subject”. All authors have the HIPPA and CITI training certificates and qualified for processing (de-identified) neuroimaging data.\n# Limitations\nWe 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.\n# Readability issues\n+ We will add more intuitive explanation with less mathematical language in Section 2.2\n+ We will add a subsection for notation introduction to improve the readability.\n+ We will correct the formatting issuses.\n+ We will place the related work before experiments and then include all the baseline methods in the Related Work section.\n---\n[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\n\n[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"}, {"review_id": "JKyD8uNBTD", "reviewer": "Reviewer_5mrG", "summary": "BrainFlow 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.", "questions": "Adding 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?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \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.", "was_revised": true, "final_justification": "The authors have addressed most of the concerns,", "rebuttal": "# W1\n+ 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.\n\n# W2, Q1, Q2\n+ __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. \n+ 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.\n+ We then argue that $\\sigma$ may not be suitable to optimize during training. \n   + 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.\n   + 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.\n\n\n+  __HCP-A__\n| $\\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | \n| :--- | :---: | :---: | :---: | :---: | :---: | \n| MSE | 0.65±0.00 | 0.13±0.00 | 0.44±0.01 | 1.52±0.03 | 2.37±0.01 |\n| LCE(%) | 0.74±0.00 | 0.71±0.00 | 0.95±0.00 | 3.17±0.03 | 20.28±0.05 |\n| GCE(%) | 0.73±0.00 | 0.69±0.00 | 0.93±0.01 | 3.18±0.02 | 20.50±0.03 |\n\n+ __HCP-YA__\n| $\\sigma $ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | \n| :--- | :---: | :---: | :---: | :---: | :---: | \n| MSE | 0.42±0.00 | 0.14±0.00 | 0.52±0.00 | 1.32±0.02 | 2.12±0.01 |\n| LCE(%) | 0.62±0.00 | 0.55±0.00 | 0.93±0.00 | 2.92±0.01 | 15.74±0.03 |\n| GCE(%) | 0.59±0.00 | 0.59±0.00 | 0.88±0.00 | 3.02±0.02 | 16.40±0.02 |\n\n+ __UKB__\n| $\\sigma$ | 0.0| 0.2| 0.4 | 0.6 | 0.8 | \n| :--- | :---: | :---: | :---: | :---: | :---: | \n| MSE | 0.09±0.00 | 0.07±0.00 | 0.69±0.00 | 1.89±0.002 | 2.52±0.01 |\n| LCE | 0.52±0.00 | 0.39±0.00 | 1.33±0.01 | 10.93±0.02 | 27.88±0.02 |\n| GCE | 0.53±0.00 | 0.39±0.00 | 1.30±0.01 | 11.16±0.01 | 28.16±0.02 |\n\n# W3\n+ 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.\n+ 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.\n+ 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.\n\n# Q3\n+ 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.\n+ 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."}]

Author rebuttal:

Meta-review: The authors present a flow matching approach to relate structural and functional connectivity, e.g. in MRI studies, leveraging Riemannian geometry by treating both kinds of connectivity matrices as SPD matrices on the SPD manifold. They also present a theoretical result proving equivalence between flow matching on the SPD manifold and on the Cholesky manifold, which has broader significance. 

The paper was reviewed by four expert reviewers. 
The reviewers generally agree on the novelty and creativity of the BrainFlow approach, the theoretical contribution, and its potential impact beyond neuroscience. The experimental results for multiple neuroscience datasets were generally well-received by the reviewers. There were multiple concerns regarding the quality of the writing, primarily the abstract and accurate use of terminology, and three reviewers expected a comparison with a broader class of baselines.
Following the rebuttal and discussion, during which the authors clarified details regarding implementation and evaluation, added additional experiments, and promised changes to the writing, the reviewers agreed their concerns were addressed by the detailed explanations and recommended acceptance. 

The AC strongly encourages the authors to carefully consider all feedback from the reviewers and implement the promised changes to improve clarity and use of correct terminology, as well as add visualizations requested by the reviewers. The additional baselines and experiments conducted during the rebuttal should also be added to the manuscript to better demonstrate the performance of the approach.
Decision: Accept (poster)