NeuroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Authors: Jun-En Ding
Dongsheng Luo
Chenwei Wu
Feng Liu
Venue: ICML 2025 poster (2025)
Keywords: fMRI
Mental Disorders
Graph Neural Networks
Neuroscience
Primary area: applications->neuroscience_cognitive_science

Abstract: Mental disorders are among the most widespread diseases globally. Analyzing functional brain networks through functional magnetic resonance imaging (fMRI) is crucial for understanding mental disorder behaviors. Although existing fMRI-based graph neural networks (GNNs) have demonstrated significant potential in brain network feature extraction, they often fail to characterize complex relationships between brain regions and demographic information in mental disorders. To overcome these limitations, we propose a learnable NeuroTree framework that integrates a $k$-hop AGE-GCN with neural ordinary differential equations (ODEs) and contrastive masked functional connectivity (CMFC) to enhance similarities and dissimilarities of brain region distance. Furthermore, NeuroTree effectively decodes fMRI network features into tree structures, which improves the capture of high-order brain regional pathway features and enables the identification of hierarchical neural behavioral patterns essential for understanding disease-related brain subnetworks. Our empirical evaluations demonstrate that NeuroTree achieves state-of-the-art performance across two distinct mental disorder datasets. It provides valuable insights into age-related deterioration patterns, elucidating their underlying neural mechanisms.  The code and datasets are available at https://github.com/Ding1119/NeuroTree.

NEUROTREE: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Jun-En Ding'
Dongsheng Luo²
Chenwei Wu
Feng Liu'
Abstract
1 Framework
D CMFC Optimization
Mental disorders are among the most widespread diseases globally. Analyzing functional brain networks through functional magnetic resonance imaging (IMRI) is crucial for understanding men- tal disorder behaviors. Although existing fMRI- based graph neural networks (GNNs) have demon- strated significant potential in brain network fea- ture extraction, they often fail to characterize com- plex relationships between brain regions and de- mographic information in mental disorders. To overcome these limitations, we propose learn- able Neuro'rree framework that integrates k-hop AGE-GCN with neural ordinary differential equa- tions (ODEs) and contrastive masked functional connectivity (CMFC) to enhance similarities and dissimilarities of brain region distance. Further- more, Neuro'Tres Tree effectively decodes fMRI net- work features into tree structures, which improves the capture of high-order brain regional pathway features and enables the identification of hierarchi- cal neural behavioral patterns essential for under- standing disease-related brain subnetworks. Our empirical evaluations demonstrate that NeuroTreee achieves state-of-the-art performance across two distinct mental disorder datasets. It provides valu- able insights into age-related deterioration pat- terns, elucidating their underlying neural mech- anisms. The code and datasets are available at https://gttub..cmm/ comg/ 1ree.roTrre..
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1. Introduction
In recent years, neuropsychiatric disorders and addiction have emerged as substantial public health challenges, with their impact on brain function becoming critical rescarch focus in neuroscience (Sahakian et al., 2015; Hollander et al., 2020; Borumandnia et al., 2022). Substance disorder (SUD) is widely studied in the field of mental disor- ders, enabling the investigation of structural brain alterations among SUD users across various factors (e.g., age or gen- der) (Niklason et al., 2022; Ding et al., 2024b). The develop- ment of brain imaging modalities, such functional magnetic resonance imaging (fMRI), which measures blood oxygen level-dependent (BOLD) signals, and electroencephalogram (EEG), which measures the electrical potentials of the brain (Jiao al., 2024; Yang et al., 2024; Yang & Liu, 2024) has provided a means to analyze brain network activities in mental disorders (Zhang et al., 2016; Li et al., 2022a; Nikla- son et al., 2022: Pan et al., 2025). Moreover, research on the clustering of connectivity in brain regions using fMRI has demonstrated that the brain does not operate statically but instead exhibits dynamic, multi-level hierarchical or- ganization (Meunier et al., 2009; Betzel al., 2023). In particular, hierarchical network analysis has been shown to effectively distinguish these abnormal patterns and identify brain regions associated with specific pathological mecha- nisms psychiatric disorders (Boisvert al., 2024; Segal et al., 2023; Wen al., 2023b).
'Department Systems Engineering, Stevens Institute of Technology, New Jersey, USA 'Department of Computing and Information Sciences, Florida International University. USA DDepartment of Electrical Engineering and Computer Science, University of Michigan, USA. Correspondence to: Feng Liu <fiin2@@stevens.edu>>
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Currently, traditional brain network analysis has success-

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NeuroTreee Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
fully represented regions of interest (ROIs) as nodes with functional connectivity (FC) as edges (Kulkarni al., 2023; Xu et al., 2024; 2016). Specifically, research on graph neu- ral networks (GNNs) has shown promising results, demon- strating strong predictive capabilities for neuropsychiatric disorders by learning patterns across brain networks (Tong et al., 2023; Kazi et al., 2019).
classification performance but also effectively inter- prets how addiction and schizophrenia disorders cause changes in FC related to brain age.
2. Related Work
Neural ODEs for Dynamic BOLD fMRI Signal Model- ing. The BOLD fMRI signal is an indirect and delayed reflection of neural activity. Traditional approaches like DCM use ordinary differential equations (ODEs) to infer connectivity, but they are computationally intensive and rely on prior assumptions (Cao et al., 2019). Recent work has demonstrated the effectiveness of ODEs in modeling continuous-time neural dynamics (Han et al., 2024; Tang et al., 2024). The evolution of brain states can be described by the equation de the brain state at time t, and f()) is a neural network that learns the rules governing state transitions. This frame- work provides natural way to model the continuous nature of brain dynamics while maintaining biological plausibil- ity (Havlicek et al., 2015).
Differential equations can capture the dynamic processes underlying changes in brain activity over time. CortexODE further advances these approaches by combining ordinary differential equations (ODEs) with deep learning methods to model MRI images as point trajectories, thereby enabling the reconstruction of complex cortical surfaces (Ma et al., 2022). Moreover, STE-ODE (Tang et al., 2024) combines the advantages of ODEs with continuous-ting fMRI and structural MRI to model dynamic brain networks while leveraging GNN architectures for embedding learning. Cur- rently, the existing methods have several limitations. First, although GNNs demonstrate excellent node feature learn- ing capabilities in graph structures, they are generally con- regarding
dX(()) f(X (t), f), where X (t) represents
strained by limited
satrreaablity-pprtiuuuule the explicability of auxiliary factors underlying mental dis- orders. Second, graph-based networks for human brain FC restricted to feature learning from dynamic graph se- quences, overlooking the brain's 's hierarchical clustering of network activity. Consequently, there is a lack of effective methodologies for predicting functional brain alterations across different age stages in psychiatric disorders.
Path-Aware Learning in GNNs. GCKN computes node distances using kernel functions along vertex-connected paths (Chen et al., 2020). Additionally, PathNNS enhance the expressive power of GNNs by aggregating information from multiple paths between nodes to improve performance across multiple graph classification and regression tasks and capturing complex graph structures (Michel et al., 2023). The HEmoN model (Huang et al., 2025) has provided pre- liminary insights into human emotions through tree path analysis; however, it lacks aggregation of higher-order brain pathways in complex brain region interactions, as well as consideration of different demographic factors between in- dividuals. In this study, we propose a weighted higher-order brain tree pathway framework that enhances pathway repre- sentation and effectively explains the predictions of changes in FC networks associated with mental disorders across different age groups.
To address these limitations, we leverage the properties of ODEs for fMRI modeling similar to STE-ODE and extend continuous-time dynamics methods. In this study, we pro- pose NEUROTREE, framework designed to characterize brain network patterns through hierarchical analysis and rep- resent the fMRI network as tree-structured pathways. This approach decodes brain network features and facilitates vi- sualization of disorder propagation through brain pathways. Additionally, we develop k-hop ODE-based AGE-GCN to capture complex interactions between adjacent and distant brain regions and to measure the convergence of various mental disorder subtypes. We incorporate demographic features like age into model learning and message passing to enhance interpretability. Moreover, we implement con- trastive masking optimization to identify key FC patterns. Our main contributions are as follows:
3. The Proposed Architecture
The proposed NEUROTREE framework, illustrated in Fig. 1, constructs dynamic FC graphs from subjects' time-series data. It encodes dynamic graph fMRI features using k- hop ODE-Based AGE-GCN incorporate demographics for graph embedding learning for downstream tasks (c.g., graph classification, brain age prediction). Subsequently, a CMFC optimization is performed adjust the FC strength between similar and dissimilar brain regions distance. Then, we build the hierarchical brain tree by reranking predicted brain regions of interest (ROIs) scores and high-order tree path weights, followed by tree pruning to create anatomically meaningful hierarchical brain network structures. The re- sulting brain tree structures are then employed to interpret
We integrate demographic information into Neural ODEs for modeling both static and dynamic brain net- works via k-hop graph convolution.
We investigate two types of brain disorder datasets to decode fMRI signals, constructing disease-specific brain trees while employing trainable framework to identify hierarchical functional subnetwork regions.
Our research not only achieves ofte-offthe-art graph
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NeeroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
disease cohorts, where functional networks primarily affect- ing mental disorders manifest different tree levels.
x= ((((),
(2)
where u(t) denotes the external experimental stimuli, We consider the influence of an external parameter, 0, on FC across the entire temporal time series (t), which is repre- sented by adjacency matrix Ad. The following ODE models this relationship:
4. Preliminaries
Problem Formulation and Notations. In this section, we formulate the classification of mental disorders using both static and sequential segmented instance brain networks constructed from fMRI as supervised graph classification task for each patient. Let temporal brain activity measure- ments from N patients, X; E RUXT, represent the complete BOLD IMRI signals with regions of interest (ROIs) and a total length of T time series. We denote (t) E RR* is segmented time points with dimension T for the ith subject. For each subject, we construct a static or dynamic FC graph E(V,E), where V represents the of vertices (i.e., brain regions) and & denotes the set of edges (i.e., functional con- nections). The FC between regions is typically computed using Pearson correlation methods. Based on these tempo- ral correlation patterns, we derive both a static adjacency matrix A E R from the complete time sequence and time-varying dynamic FC matrices (t) R The static matrices capture averaged temporal dependencies, while the dynamic matrices preserve the temporal evolution of neural interactions (Zhang et al., 2016; Li et al., 2022a).
d.((1) dt ----1 (x Cu((),
+
where the matrix C' E Rxx encodes the reception of exter- nal stimuli u(t) E RR" and parameters 17, p E R. To simplify the analysis, we assume that Cu(() = 0, indicating the absence of additional external stimuli due to resting-state IMRI task. By discretizing Eq. 3, we can derive the effec- tive connectivity matrix A (t). Its discrete representation given by:
7
1 d( 1
(1
X(1) - p p.0) X(t + 1) -X(t)
(4)
1)
dt
At=1 4
pp.0
(5)
X(t)
where 4p = and are scale factors ranging from 0 to 1. To address the limitations of first-order feature convo- lution filters in traditional GCNs, we propose an extended spatial feature filter that operates over a k-hop connectivity operator. This approach enables the spatial k-order brain network representation to be learned via generalized graph convolution, as defined in definition 4.1.
4.1. Graph Convolutional Networks
Traditional GCNs operate through neighborhood aggrega- tion, whereby each node's representation is updated by ag- gregating features from its adjacent nodes. The classical GCN layer is formulated as
Definition 4.1 (k-hop graph convolution). The spatial k- order brain network representation can be learned through a generalized graph convolution as follows:
H aD
ADD -D- (1 (((D))((-)).
where A is the adjacency matrix, D is the degree matrix, H represents the node features at layer I, and W con- tains learnable parameters. However, this formulation is limited to first-order neighborhoods and may not effec- tively capture the complex interactions among brain re- gion (Veličković et al., 2017).
H H0+D o
Σ T (t)
(6)
where o denotes the ReLU function, and (x( = ò- $ A, (t) represents the normalized k-hop connec- tivity adjacency operator. The adjacency operator can be defined as:
4.2. K-hop ODE-Based Graph Embedding
The ODE effectively captures the long-term dependencies of dynamic changes in effective connectivity between different ROIs Maanhez-RRment et al., 2019). Inspired by existing studies (Cao et al., 2019; Wen et al., 2024; Tang al., 2024) that utilize ODEs to model BOLD signals or biomarkers in Alzheimer's is disease (AD) research, extend the ODE modeling framework by incorporating scalar parameter, E R+ (i.e., age), representing the chronological age of subjects. This enhancement aims to improve the prediction and interpretability of mental disorders in fMRI analysis.
Ax(1) = PO -
(7)
k-hop connectivity
where I' is a learnable parameter, the k-hop connectivity operator retains directional information, evidenced by the fact that (x(t) * (ÁA (t)) when À + 1. This result is derived using the binomial theorem, as detailed in Appendix A. Intuitively, Eq. 7 considers that A" provides connectivity information for the entire static fMRI sequence, while A can regulate the bidirectional combination of dynamic fMRI in the brain through the parameter X.
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NeeroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Theorem 4.2. Suppose the Pe-norm of the k-hop connec-
Here, xt) = (B-) denotes the age-modulated
tivity adjacency operator [Ax(t)]lz can be derived from Eq. (7). Then, as the k-hop approaches infinity (k- + 0o ),
feature matrix with learnable parameter 3 (0, 1) for age modulation strength at layer 1, and Wh represents the learn- able weights associated with the k-th hop filter weight at layer l.
the [[Ax(t)l2 is bounded by following inequality:
lim (1)x())l2 S ||T |1 max(AA 1- X) 8-000
(8)
For t 2 1, the temporal age-modulated graph embedding update is expressed as:
The detailed proof of Eq. 8 is provided in Appendix B. To ensure numerical stability when computing powers of (t), we establish the bounded condition: [t)x2)]22 I 1, which holds for all values of k and 4. In proposition 4.3, we demonstrate that & (t) exhibits smooth temporal evolution of the graph structure while maintaining well- defined upper bound.
Z(1+ + 1) = Z(1) + AGE-GCN& (G' (G+
K-1
= Z(t) + o Øx(t + 1)x²² (t + ())
(12)
where AG 0) denotes the goemooule
Proposition 4.3 (Convergence and Uniqueness). Let (x(t) be the normalized adjacency operator that satisfies the Lip- schitz condition (Coddington& & Levinson, 1955) (11) - b (t2)))2 I Ellt--tyy tzllz for some constant L 2 O. Then, for Lipschitz continuous activation function a and bounded weights ((O, the k-hop ODE-GCN admits a unique solu- tion.
k-hop graph convolution applied to the graph G (t 1) at the final layer L.
4.3. Optimization of Objective Function
To enhance the distinction between brain regions, we de- veloped node-wise contrastive masked functional connec- tivity (CMFC) loss function that simultaneously minimizes similarities and maximizes dissimilarities. This loss effec- tively amplifies the distances between mntra-FCsttenngths. For each node i, we calculate the FC strength (t)
The proof from Picard's existence theorem can be found in Appendix C.
Theorem 4.4 (Discretization of Age-Aware Continuous-
Time Graph Convolution (Tang et al., 2024)). Consider a series of brain networks (G(t; o) parameterized by an age variable 0. Let (t) denote the embedding (or feature representation) of the STE- STE-ODE network at time t. The continuous temporal evolution of the brain network can be expressed through the following age- age-modulated graph convolution representation:
M ΣM (t)( at timestamp t, where [V] represents the total number of nodes in the network. We then map this FC strength to the latent space using a multilayer percep- tron (t) = MLP(C, (t)). Subsequently, we generate posi- tive and negative masks for connectivity pains according to M Ej (1) T (L) こे , (1) A (C) 전 1 (1)). where i (C(() is the mean value of FC and I(-) is the indicator func- tion. The positive and negative sample sets are then defined
j=l
Z(t + At) = Z(t)+ S. ++
as A+ (t) = {(i,j) (t) = 1}. Finally, the CMFC loss
F(Z(T),T,0) dr. (9)
is expressed as:
where F(-) is the graph embedding function that captures the spatiotemporal and age-related information within the network.
exp(Si,())
P
Σ (i.))A
log
ΣKEV (t)p(St(t)) )
IA+1
Specifically, the temporal evolution function defined as:
Lueg
AA-1
1
Σ
log 1 EKED exp (Sth (4) +
exp (s;, (t))
(K)))A
(13)
F(Z(),1,0) Σ H (1,0))
(10)
[O][]]])))()]] computes the cosine similar- ity of latent vectors of FC for nodes and j at time f, and 10 is added to prevent division by zero To opti- mize the model, we minimize the combined CMFC loss LCMFC = Spss + Lueg and combine it with the binary loss L. for the overall loss function L = LCMPC + Ls-
where Siy(t) = (h,(
where (t, 8) representing the age modulated layer-wise transformation at time t. Using a first-order approximation, the temporal dynamics described in Eq. 6 and theorem 4.4 can be discretized into the following form:
Z(t At) ~ Ztt) + At (t))
Node Score Predictor
L
K-1
= z(t) + A Σ
Σ
(x (1)
Different mental disorders exhibit distinct intrinsic FC pat- terns between various brain regions (Gallo et al., 2023; Fu et al., 2021). By analyzing these connectivity differences
1=1
K=0
(1))
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NeeroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
in specific brain areas, we can predict regional scores to identify similar abnormal functional networks (Li et al., 2020). We leverage FC strength latent vectors h2 combined with age-aware k-hhpp graph embeddings parameterized by O {A, 0, w() } to generate predictive scores for individ- ual brain regions. The score for the ith node is expressed as follows:
(+) Original Graph
High Order Aggregation
(c) Final Stracture
Onder Higv rash
Lon bOOK
Figure 2. Aggregation for hierarchical neighborhood paths. Fig. (a) illustrates the original graph structure with weighted connec- tivity before pruning, providing subnetworks that represent the differences between brain regions. In Fig. (b), the zero-order path is depicted; here, aggregation initiates the highest-scoring node, which integrates its immediate neighhorhood by combining edges Cab and end. i.e., two direct paths connecting the -echeig- node to its neighbors. Fig, (c) shows the aggregation of higher- order paths, where isolated nodes are connected along the shortest weighted paths, thereby capturing more complex connectivity pat- terns beyond immediate neighbors.
S, - (1) [2,(0) ( EE (1.2...... MD
jEV
(14)
where ((-) denotes a non-linear transformation function and Z()) E M represents the d-dimensional node embeddings learned by AGE-GCN with the parameter set +
Definition 4.5 (MST algorithm). Given a weighted undi- rected graph G = (V, E, w"), where w" (EE) assigns a weight the edge Eij E E between vertices and j. we define the pruned weighted graph T(G") ET; w"). where ET C E, |&T|= << |E] represents the set of edges retained in the minimum spanning tree (MST) after prun- ing. Our goal is to find the minimum spanning tree while satisfying:
P3 (v),0)) E TT(G), where each edge ev,y, E E(P) repre- sent the path of neighbor connectivity. The composite path weight expressed as:
s
W(P) a
Σ 0( PEP
Σ
E
F(O) Bily
min T
Σ w (eig).
(15)
E (B.P))EE(P)
E3JEET
Node Score Contribution
Hlgh-Order FC Contribation
where T' is spanning tree with edges &T C E and diameter upper bound diam(T) S (MI-110712)) (Spira & Pan, 1975). In practice, we can apply Kruskal's 's algorithm (Kruskal, 1956) to find the minimum spanning tree.
(16)
2
Here, Fere, E R represents the s-th order FC strength between nodes i and by along the tree path. To solve Eq. 16, we identify the path that minimizes ((P) by adjusting the parameter E [0. 1] to balance the contributions of the scoring function S(v; +) and the connectivity strengths along the weighted path. Additional experimental results can be found in Fig. 7. According to theorem D.1 in the Appendix, the optimal path can be determined when a* exist in the interval [aL, au] (Xue, 2000).
5. Hierarchical Brain Tree Construction
To characterize the hierarchical levels of FC in brain net- works associated with mental illness, we construct tree structure from a weighted undirected FC graph G" based on predicted node scores and weighted edges (i.e., FC strength), which contributes to identifying the most significant func- tional pathways. Note that Eq. 7 represents a model trained on directional graphs to investigate complex asymmetric connection patterns for graph embedding. The constructed tree of edge weights w"(e,)) is derived from the model's learnable FC strengths measured for each ROI connectome. Following Definition& 4.5, the pruning process ensures that the set E retains only the significant edges that satisfy the optimization criterion.
5.2. High-Order Tree Path Aggregation
In high-order tree path traversal, our objective to aggregate information of direct and indirect about the connectivity of tree paths within the pruned graph T(G). In practice, we aggregate connectivity information from neighboring edges at each node, forming a weighted tree path. We define the second term in Eq. 16, which represents the high-order FC contribution for an s-order connectivity along the p-th path, as follows:
5.1. Optimal Weighted Tree Path
Given the pruned graph T(G), we examine paths that inte- grate node importance and high-order FC across multiple tree depths. Let P = {ro, vi,..., } represent a path of length k in T(G), where each node v E P is associated with a scalar score S(e; θ) E R, computed from Eq. 14. The edge set of P is denoted by E(P) {(0, 2/)] Us By E
F(a) F(U)
Σ P.(p)
(17)
DEP,(
where [[) = F(wz)) + F(u;) represents the FC of a 0- th order path, and (u 05) denotes the set of all possi- ble s-order paths between nodes and j in T(G). For
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Nree:Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Algorithm Hierarchical Tree Trunk Extraction with
6. Experiments
Weighted Tree Path Learning
In this section, we first evaluate the proposed NEUROTREE performance in brain disease classification (section 6.2) and its application predicting chronological age (section 8), In section 6.4, we analyze the convergence of the spectral norm for k-hop connectivity. Finally, we examine how hi- erarchical brain trees interpret FC patterns between brain regions and their corresponding seven subnetworks in sec- tion 7. Detailed experimental settings for NEUROTREE are provided in Appendix
1:
Input: Node scores (S(e; θ)) eev, pruned graph
T() with nodes V and edges εT, maximum levels Lmax. scaling factor a.
2: Output: Hierarchical trunks T = (Ti, .2...., Timas
3: Initialize T < 0.
4: Let (T) be the pruned graph using MST.
5: for (
1-11t to Lmxx do
6: Identify connected components {Ki, K2.......... Km)
in G.
7: for component K3 do
8: Find Ustart arg max,ex, S(v).
6.1. Datasets
9: Compute shortest paths (UUsarrt u) for all Kj
with weighted path:
We validated two publicly available IMRI datasets -one fo- cusing on cannabis use disorder and the other on schizophre- nia. @ Cannabis (Kulkarni et al., 2023): The cannabis dataset comprises fMRI data from two distinct sources. The data were preprocessed from 3-Tesla fMRI acquisitions. and the mean time series for each subject was computed across 90 ROIs using the Stanford atlas parcellation (Shirer et al., 2012). 2 COBRE (Calhoun et al., 2012): The Cen- ter for Biomedical Research Excellence (COBRE) dataset includes resting-state IMRI data collected from healthy con- trols and individuals diagnosed with schizophrenia. All MRI data were parcellated into 118 ROI regions using the Harvard-Oxfen atlas. We summarized the statistics and demographics of both datasets in Appendix Table 5.
W(P) = a E ((; +)+((-01))
Σ F(a)
((E)))) 0,
PEP
10:
Let Vend maximize W(P), and define P
P( (Vssaatt Vend).
11:
Append p* Tí.
12:
end for
13:
Remove edges in Ti from G and update the graph.
14:
Combine paths: T CTUUT
15:
end for
16:
Return T.
each s-order path index p (i, k2,.... J), where all intermediate nodes are distinct (kim # lin Vm + n), its contribution is calculated as Pe(p) = Evep F."). For a path of order s, its extended connectivity at order + 1-th through the neighboring node & can be com- x(o+1)- 0. = EKEA((,,,)))
6.2. Qualitative Results
This section is dedicated to evaluating the performance of NEUROTREE against state-of-the-art (SOTA) GCN ar- chitectures including 1) Pearson GCN 2) k-NN GCN 3) GAT (Veličković et al., 2017) 4) BrainGNN (Li et al., 2021) 5) BrainUSL (Zhang et al., 2023) and BrainGSL (Wen et al., 2023a) for graph classification tasks. Additionally, we in- clude the tree path-based model PathNN (Michel et al., 2023). The comparative analysis was conducted using 5- fold cross- volidaaalo two distinct datasets, benchmark- ing NEUROTREF against both baseline models and SOTA approaches. The comprehensive experimental results are presented in Table 1.
(Flile + Fogey) where (1,, 0y) represents the set of common neighbors between nodes and j. and F(?)), denotes the s-th order path connec- tivity from node i node k. This formulation allows us to capture direct connections and higher-order structural rela- tionships within the brain network, as illustrated in Fig. 2.
puted as
Fine-Grained Tree Trunk
We aim to construct a hierarchical tree structure com- prising trunk and branches distributed across different levels in TG). Our method groups nodes with simi- lar scores into the same hierarchical level integrates the most relevant paths into the (-th trunk Ti and excludes unrelated paths. For the i-th level connected component G = (Vi, Et). the numbers of nodes and edges decrease monotonically as the level increases- that is, |V|- < [V-1] and | < E Following Section 5.1, we identify the shortest path set P(ustart, Vend) = {Vatart, U2, Vena) that incorporates the weighted tree path and the optimal trunk set T = [Ti, T2...., Tx), as presented in algorithm
Compare with Brain GCN Models. We quantitatively assessed brain network construction using various metrics, employing Pearson correlation and KNN as connectivity measures. As shown in Table the GAT model effec- tively integrates attention mechanisms into fMRI node fea- tures, achieving mean AUC scores of 0.72 and 0.67. In contrast, the conventional GCN is limited to learning solely from topological structures. Furthermore, NEUROTREE en- hances ACC performance in IMRI-based models, improving BrainGNN (ACC t 23.72%) BrainUSL (ACC 12.30%), and BrainGSL (ACC + 12.30%) on the cannabis dataset. On the COBRE dataset, BrainGNN, BrainUSL, and BrainGSL
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NeuroTree:: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Table 1. Evaluating graph classification performance with five-fold cross-validation. We computed the most competitive baseline for each
method. We compared the second- second-best methods denoted by blue color and calculated the improvement rate, denoted as "Improv. (9))"
Cannabis
COBRE
Model
AUC
Acc.
Prec.
Rec.
AUC
Acc.
Prec.
Rec.
Pearson GCN
0.67.0.06
0.55=0.07
1.591
0.5510.08
0,5410.11
0,55
0..1
0.5510.10
k-NN GCN
0.64
0.02
0.63.10.3
0.63...03
0.6610.07
0.62
0.6310.08
0.631..08
GAT (Veličković et al., 2017)
0.72=0.05
0.67=0.04
0.70==0.0
0.67=00.04
0.67++..8
0.60+0.11
0.57+0.221
0.60=a.11
BrainGiNN (li et al., 2021)
0.67=0.13
0.59=0.16
=.
0.59
0.55
0.50+0.02
+0.1
0.50
BrainUSL (Zhang et al., 2023)
0.63110.11
0.65
0.62.10.13
0.63
10.11
0.5410.04
0.41 18.18
0.57
BrainGSL (Wen et al., 2023a)
0.59
0.65
0.67.1.0.17
0.60.2.02
0.551
0,5140.00
0.451...1
0.51
MixHop A-b--E--Haipp GPC-GCN (Li et al., 20226)
al., 2019)
0.73.00.5
0.69
0.70.10.04
0.69
1..0s
0.61
0.6210.07
0.611
0.53..0.05
0.00
0.30/10.08
0.60.1.08
0.5010.00
0.47.0.00
0.221.0.4
0.47
PathNN (Michel et al., 2023)
0.70=
0.67=0004
0.722+0.12
0.83=
0.49++0.01
+0.05
0.32++0.27 27
0. 0.444
Ours (w/o 8)
0.49.10.01
0.00
0.37.10.08
0.60
0.50
0.4710.04
0.2210.01
0.4710.04
Ours (w/o LCMPC) )
0.74=0.08
0.05
0.73:t0.04
0.73==0.05
0.
0.63++0.10
0.61+
0.63+0.10
NEUROTREE
0.80
0.73=0004
0.73=0.04
0.74
0.71 =0.10
0.65#t0.08
0.66+t0.88
0.65+a0.08
Improv. (%)
8.11%
1.37%
2.89%
3.17%
3.12%
3.17%
Convergence Analysis of By(t)
Convergence for Different A (COBRE)
Convergence for Different Á (Cannabis)
COBRE
0.0060
A=02
Spectral Norm
Cannabis
Spectral Norm
0.0055
0.0050
0.4
0.007
0.006
0.666
Am0.4
0.004
0.0045
A 0.9
Spectral
0.005
Am 0.9
0.003
0.0040
0.004
0.0035
0.002
0.0030
0.003
0.0025
0.002
0.001
0.0020
10
10
30 k-hop converge
20
40
50
10
20
30
40
50
k-hop converge
k-hop converge
Figure 3. Convergence analysis of (t) over k-hop The spectral norm of (t) reveals differential convergence rates across varying k-orders among distinct mental disorders, notably demonstrating that cannabis exhibits a steeper convergence gradient compared to COBRE as A increases.
further improve ACC by 30.00%, 20.37%, and 27.45%, respectively.
6.3. Ablation Study
The Impact of Objective Function and Demographics. In our experiments, we evaluate the efficacy of CMFC loss integration within our proposed NEUROTREE. The quantita- tive analysis presented in Table I indicates modest improve- ments in the cannabis dataset: however, its implementation in the COBRE dataset achieved an AUC of 0.71, with con- sistent performance gains exceeding 3% across metrics, par- ticularly in the characterization of mental disorder features. This enhancement stems from combining the loss function with CMFC loss optimization framework that simultane- ously suppresses connectivity between dissimilar regions while amplifying patterns among similar neurological areas. Furthermore, our ablation experiments demonstrate that age serves critical modulator in AGE-GCN by adapting parameter 0, substantially influencing the classification of changes in brain connectivity.
Compare with Path-Based Models. As shown in Table I, NEUROTREF outperforms existing high-connectivity view field-aware graph GCN models, such as the high-order con- catenation layer-based MixHop, achieving over 6% average performance improvement in AUC across all metrics on the cannabis dataset. Furthermore, compared to MixHop on the COBRE dataset, NEUROTREE not only achieved 2% increase in AUC, but also demonstrated mean perfor- mance improvement of 6.3% to 6.5% in ACC, Precision, and Recall. Notably, although GPC-GCN can aggregate multi- level connectivity information by considering the concept of paths, it does not significantly improve classification perfor- mance on critical connectivity paths. Similarly, the PathNN framework aggregates path information starting from nodes to generate node representations but shows only limited im- provements. In contrast, our NEUROTREE achieves a 14% improvement in AUC on the cannabis dataset, while PathNN underperforms compared to MixHop on the COBRE dataset. The experimental results indicate that NEUROTRER TREE is su- perior in capturing temporal FC patterns and predicting outcomes in addiction and psychiatric disorders.
6.4. Convergence Analysis
The intuition for analyzing the rate of spectral norm decay with increasing k-hop allows us to examine how rapidly information from distant nodes attenuates across the dy- namic brain network structure. More detalil explanation

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Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
(a-1)
(b-1) The Brain Tree Structure of Schizophrenia Level
Level Level 2
Level
Level
Level
Level
Level
Level
(a-2)
(b-2)
Defosit
Subcortical
Visual
Defaut
Natworaa
Dersal Attention
Somatomotor
Semnatomstor
Frontopariecal
Subcortical
Number of connectivity
Figure The visualization of brain tree illustrates psychiatric disorders structured into three hierarchical trunk levels. Panels (a-1) (b-1) mark the most significant nodes along the tree path. The 1-1 pathways represent regional connectivity, corresponding the level three brain maps on the right. Panels (a-2) and (b-2) depict the number connections using color gradients across the hicrarchical levels.
be found in Appendix I. To further examine the vergence speed of k-hop FC in various brain disorders, we analyze the spectral norm under the A: = 1 setting for the convergence of (t) ) in Fig. 3 (left). Although the COBRE dataset initially exhibits slower convergence, particularly with the boundary defined around order 15, the overall con- vergence in the cannabis dataset is faster than in COBRE across different k-hop values. Furthermore, when analyzing the convergence of & (t) under varying À values, shown in Fig. 3 (middle) and (right), the À values for cannabis are more concentrated and stable. In contrast, COBRE requires larger values to achieve convergence. The empirical find- ings demonstrate that examining the spectral norm of 2 (t) across various values effectively reveals the relationship between high-order patterns and decay characteristics of FC in different mental disorders.
vary across different hierarchical levels of brain subnet- works, NEUROTREE predicts each ROI score and performs reranking, illustrated in Fig. 1. Higher levels represent trunk pathways of high-scoring nodes, strongly connected edges, and their associated brain ROIs. We mapped the atlas parcellations of both datasets (i.e., 90 ROIs and ROIs) to Yeo's seven-network parcellation (Yeo al., 201 1). which includes the Visual Network (VN), Somatomotor Network (SMN), Dorsal Attention Network (DAN), Ventral Attention Network (VAN), Frontoparietal Network (FPN), and De- fault Mode Network (DMN). Additionally, we incorporated the Subcortical System (SUB) in our analysis to examine the subcortical structures. Atlas regions not belonging to these networks were classified as 'Others." More details can be found Appendix L.2.
Next, discuss the addiction brain tree illustrated in Fig. 4 (a-1) 1) and (a-2), which features three trunk levels. At the primary trunk level (red pathway), the addiction brain tree predominantly exhibits connectivity among DMN (red) and VN (purple) nodes; similarly, in Fig. 4 (b-1) and (b-2), the schizophrenia brain tree at the main trunk level primarily involves SUB (pink) nodes and partial VN nodes. At the second hierarchical level (pink pathways), the addiction co- hort exhibits predominant connections within DAN, DMN, and VN nodes. In contrast, the schizophrenia group at this level demonstrates connectivity primarily among VN nodes and some FPN (light blue) nodes. At the third hierarchical level (yellow pathways). the terminal branches in the addic- tion group show minimal connections with the SMN nodes (green) and DAN nodes, whereas the schizophrenia cohort displays several connections among DMN nodes.
7. Brain Tree Analysis
This section investigates brain disorders by analyzing hier- archical brain networks using decoded fMRI-derived con- nectomes. NEUROTREE deconstructs the fMRI brain net- work into tree-structured component graph consisting of main trunk and radial branches. We identify key regions associated with brain disorders within each trunk by analyz- ing optimal weighted high-order tree paths. Our analysis highlights the most relevant functional network regions de- termining critical pathways within the brain connectome tree.
Exploring Hierarchical Regional Patterns in Brain Disor- ders. To better understand how embedded brain features
8

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Nree:Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Table 2. NEUROTREE prediction for different age groups.
(a) Addiction
(b) Schizophrenia
Datasets
Age Group
MSE
Pearson (r)
hearthy tilepteree bny
Healthy reloowoh 1h0 lagreer poopeook Divesu
18-25 (n=
1.67
0.27+0.02
50
Caosse (ilnewddar)) Chooss ffai
Cannabis
25-40 (n=
27.32+3.15
0.18+0.01
Predtced Brain AgS
4.5
Prod crdd Brain Age
-0
40-54 (n=41)
25.12:11.4
0.1810.00
18-25 (n=35)
4.48=0.42
RD
COBRE
25-40 (n=56)
37.64+2.00 02
0.10=0.01
40-66 (n=61) 32.6413.43
0.49
30
20
were primarily concentrated in networks governing cogni- tive functions, including the DMN, the visual task VN and the decision-making FPN. This network-specific disruption pattern aligns with previous findings (Kleinhans et al., 2020; Kulkarni et al., 2023; Ding et al., 2024b).
20 25
30
35
40
45
50
55
20
30
40
50
60
Chronological Age
Chronologicc call Age
Figure 5. The scatterplot shows the gaps hetween IMRI-predicted brain age and chronological age for healthy control and mental disorder groups.
Prior studies have established disrupted connectivity pat- terns in schizophrenia, particularly within the DMN (Ishida et al., 2024; Yang et al., 2022). We identified intermittent ab- normal coupling between subcortical regions, especially the connections with the prefrontal cortex and cerehellum (Ding et al., 2024a; Hancock et al., 2023). Furthermore, areas of the visual cortex were correlated with impaired motor coor- dination and sensorimotor integration (Verma et al., 2023; Zhao et al., 2022), consistent with the high FC regions illus- trated in Fig. 4. Additionally, our analysis revealed distinct age-related differences in brain FC between addiction and schizophrenia (Table 2). We further verify finding in Fig. 5, which illustrates the relationship between predicted brain age and chronological age. The predicted brain age for both mental disorders is lower than the chronological age, particularly in the higher age group. Both conditions appear to affect age-related brain characteristics, leading to an abnormal relationship between brain age and chronologi- cal age. Notably, addiction demonstrates more significant impact on brain development and aging than schizophrenia, which is consistent with our observations in section 8.
8. Brain Age Estimation
Comparing estimated IMRI brain age against actual chrono- logical age in individuals with mental disorders can provide valuable insights into the severity and progression of these conditions (Stankevičiūté et al., 2020). 'he physiological actual age of the human brain is unknown and cannot be di- rectly measured; however, we can predict brain age through regression tasks on IMRI data using NeuroTree learned fMRI graph embeddings. We hypothesize that there exist certain gaps between the actual chronological age of healthy individuals and the brain age of those with mental disorders. For example, when the gap between model-predd prdeel-reddcted brain age and actual chronological age is positive, it indicates accelerated brain aging.
To validate model prediction in brain age, we trained Neu- roTree on a healthy control cohort and predicted non-healthy subjects to better measure brain age gaps. In Table 2, we demonstrate NeuroTree's predictive performance across dif- ferent age groups in two datasets. Notably, the younger age group (18-25 years) exhibited markedly lower Mean Squared Error (MSE) values of 9.51 in the Cannabis dataset and 4.48 in the COBRE dataset compared to other age groups, indicating greater accuracy in brain age prediction for younger individuals. Compared to younger age groups, NeuroTree's 's prediction MSE across both mental disorder datasets provides valuable insight, showing that brain age estimation for older individuals may be affected by greater individual variability in brain changes associated with men- tal disorders after middle age, while younger groups likely exhibit more consistent patterns of FC during neurodevelop- ment.
10. Conclusion
This study proposed NEUROTREF, novel hierarchical brain tree framework that integrates h-hop AGE-GCN with neural ODEs and contrastive mask FC loss to decode functional brain pathways in psychiatric disorders. NEU- ROTREE effectively incorporates demographic to improve brain disease classification and enhances interpretability in predicting brain age while revealing distinct hierarchi- cal connectivity patterns within the DMN, FPN, and VN across different mental disorders. Additionally, these find- ings demonstrate that different subtypes of mental disor- ders exhibit varying trajectories of FC changes across age periods. These insights contribute to the neuroscientific understanding of functional alterations induced by mental disorders, potentially informing therapeutic interventions and pharmacological treatments.
9. Discussion
In section 7, our analysis of the tree-structured topology revealed distinct patterns of brain subnetwork alterations associated with addiction and schizophrenia. These changes
9

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NeeroTree:: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Impact Statement
Calhoun, V. D., Sui, J., Kiehl, K., Turner, J., Allen, E., and Pearlson, G. Exploring the psychosis functional connectome: aberrant intrinsic networks in schizophrenia and bipolar disorder. Frontiers in psychiatry, 2:75, 2012.
In this study, we examine the increasing societal impact of mental illness and addiction, highlighting the critical im- portance of understanding the underlying mechanisms of brain disorders. Traditional GNNs designed for fMRI data have proven effective in capturing network-based features; however. many of these approaches lack interpretability and fail to incorporate external demographic information within a deep learning framework. To address these limitations, we introduce DFR,,,REF, a novel framework that establishes robust foundation for advancing neuroscience research, identifies influential brain network regions associated with disorders, and elucidates potential therapeutic targets for future drug development and intervention strategies in spe- cific brain regions. In future work, our NeurTree framework can be extended to related research fields such as EEG and social networks.
Cao, X., Sandstede, B., and Luo, X. A functional data method for causal dynamic network modeling of task- related finri. Frontiers in neuroscience, 13:127, 2019.
Chen, D., Jacob, L.,, and Mairal, J. Convolutional kernel networks for graph-structured data. In International Con- ference on Machine Learning, pp. 1576-1586. PMLR, 2020.
Coddington, E. A. and Levinson, N. Theory of ordinary differential equations. McGraw-Hill New York, 1955.
Cole, J. H. and Franke, K. Predicting age using neuroimag- ing: innovative brain ageing biomarkers. Trends in neu- rosciences, 40(12):68 1699,, 2017.
Reproducibility
Ding, C., Li, A., Xie, S., Tian, X., Li, K., Fan, L., Yan, H., Chen, J., Chen, Y., Wang, H., et al. Mapping brain synergy dysfunction in schizophrenia: Understanding individual differences and underlying molecular mecha- nisms. Advanced Science, pp. 2400929, 2024a.
All of our experiments are reproducible, with results pre- sented in Table 1 and Table 2. We will upload the prepro- cessed data to Google Drive and release our code on GitHub upon acceptance of the paper. Please note that, based on our experience, there may be some variations in the results shown in Table and Table due to differing GPU config- urations. For accurate results, please refer to released checkpoints.
Ding, J.-E., Yang, S., Zilverstand, A., Kulkami, K. R., Gu, X., and Liu, F. Spatial craving patterns in marijuana users: Insights from fmri brain connectivity analysis with high- order graph attention neural networks. IEEE Journal of Biomedical and Health Informatics. 2024b.
References
Fu, Z., Iraji, A., Sui, J., and Calhoun, V. D. Whole-brain functional network connectivity abnormalities in affective and non-affective early phase psychosis. Frontiers in Neuroscience, 15:682110, 2021.
Abu-El-Haija, S., Perozzi, B., Kapoor, A., Alipourfard, N., Lerman, K., Harutyunyan, H., Ver Steeg, G., and Gal- styan, A. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In in- lernational conference on machine learning, pp. 21-29. PMLR, 2019.
Gallo, S., El-Gazzar, A., Zhutovsky, P., Thomas, R. M., Javaheripour, N., Li, M., Bartova, L., Bathula, D., Dannlowski, U., Davey, C., et Functional connec- tivity signatures of major depressive disorder: machine learning analysis of two multicenter neuroimaging stud- ies. Molecular Psychiatry, 28(7):3013-3022 2023.
Betzel, R. F., Cutts, S. A., Tanner, J., Greenwell, S. A., Varley, T., Faskowitz, J., and Sporns, O. Hierarchical organization of spontaneous co-fluctuations in densely sampled individuals using fmri. Network Neuroscience, 7(3):926-99) 949, 2023.
Han, K., Yang. Y., Huang, Z., Kan, X., Guo, Y., Yang. Y., He, L., Zhan, L., Sun, Y., Wang, W., et al. Brainode: Dynamic brain signal analysis via graph-aided neural ordinary dif- ferential equations. In 2024 IEEE EMBS International Conference on Biomedical and Health Informatics (BHI), pp. 1-8. IEEE, 2024.
Boisvert, M., Dugré, J. R., and Potvin, S. Patterns of abnor- mal activations in severe mental disorders a transdiagnos- tic data-drives meta- meta-analyy analysis of bask- fmri studies. Psychological Medicine, (111))33312-62222 2024.
Borumandnia, N., Majd, H. A., Doosti, H., and Olazadeh, K. The trend analysis of neurological disorders as ma- jor causes of death and disability according to human development, 1990-2019. Environmental Science and Pollution Research,
Hancock, F., Rosas, F. E., McCutcheon, R. A., Cabral, J., Dipasquale, O., and Turkheimer, F. E. Metastability as candidate neuromechanistic biomarker of schizophrenia pathology. Plos one, (28)),00222777,22222
2022.
10

---

NeuroTree:: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Havlicek, M., Roebroeck, A., Friston, K., Gardumi, A., Ivanov, D., and Uludag, K. Physiologically informed dynamic causal modeling of fmrr data. Neuroimage, 122: 355-372, 2015.
Li, G.-z., Liu, P.-h., Zhang, A.-x,, Andari, E., and Zhang, K.-R. A resting state fmri study of major depressive disorder with and without anxiety. Psychiatry research, 315:114697,, 2022a.
Hollander, J. A., Cory-Slechta, D. A., Jacka, N.N., Szabo, S. T., Guilarte, T. R., Bilbo, S. D., Mattingly, C. J., Moy, S. S., Haroon, E., Hornig, M., et al. Beyond the look- ing glass: recent advances understanding the impact of environmental exposures on neuropsychiatric disease. Neuropsychopharmmccoogy 45(7):1086-1000 2020.
Li, X., Zhou, Y., Dvornek, N. C.,, Zhang, M., Zhuang, J., Ventola, P., and Duncan, J. S. Pooling regularized graph neural network for fmmi biomarker analysis. In Medical Image Computing and Computer Assisted Intervention- MICCAI 2020: 23rd International Conference, Lima, Peru, October 4-8, 2020, Proceedings, Part VII 23, pp. 625-635. Springer, 2020.
Huang, Z., Du, C., Li, C., Fu, K., and He, H. Identifying the hierarchical emotional areas in the human brain through information fusion. Information Fusion, 113:102613, 2025.
Li, X., Zhou, Y., Dvornkk N., Zhang, M., Gao, S., Zhuang, J., Scheinost, D., Staib, L. H., Ventola, P., and Duncan, J. Braingnn: Interpretable brain graph neural network for fmri analysis. Medical Image Analysis, 74: 102233, 2021.
Ishdda, T., Yamada, S., Yasuda, K., Uenishi, S., Tamaki, A., Tabata, M., Ikeda, N., Takahashi, S., and Kimoto, S. Aber- rant brain dynamics of large-scale functional networks across schizophrenia and mood disorder. Neurolmage: Clinical, 41: 41:103474, 2024.
Li, Y., Zhang, X., Nie, J., Zhang, G., Fang, R., Xu, X., Wu, Z., Hu, D., Wang, L., Zhang, H., et al. Brain connectivity based graph convolutional networks and its application to infant age prediction. IEEE transactions on medical imaging, (110):7644-7766 2022b.
Jiao, M., Xian, X., Wang, B., Zhang, Y., Yang, S., Chen, S., Sun, H., and Liu, F. Xdl-esi: Electrophysiological sources imaging via explainable deep learning framework with validation on simultaneous eeg and ieeg. Neuroimage, 299: 120802, 2024.
Lombardi, A., Diacono, D., Amoroso, N., Monaco, A., Tavares, M. R., Bellotti, R., and Tangaro, S. Explain- able deep learning for personalized age prediction with brain morphology. Frontiers in nearoscience, 15:674055, 2021.
Kazi, A., Shekarforoush, S., Arvind Krishna, S., Burwinkel, H., Vivar, G., Kortüm, K., Ahmadi, S.-A., Albarqouni, S., and Navab, N. Inceptiongen: receptive field aware graph convolutional network for disease prediction. In Informa- tion Processing in Medical Imaging: 26th International Conference, 1PMI 2019, Hong Kong, China, June 2-7, 2019, Proceedings 26, pp. 73-85. Springer, 2019.
Ma, Q., Li. L.,, Robinson, E. C., Kainz, B., Rueckert, D., and Alansary, A. Cortexode: Learning cortical surface reconstruction by neural odes. IEEE Transactions on Medical Imaging, 42(2):430 44000044334400 2022.
Ma, X., Yang, W. F. Z., Zheng, W., Li, Z., Tang, J., Yuan, L., Ouyang, L., Wang, Y., Li, C., Jin, K., et al. Neuronal dysfunction in individuals at early stage of schizophrenia, a resting-state fmri study. Psychiatry Research, 322: 115123, 2023.
Kleinhans, N. M., Sweigert, J., Blake, M., Douglass, B., Doane, B., Reitz, F., and Larimer, M. Fmri activation to cannabis odor cues is altered in individuals at risk for cannabis use disorder. Brain and Behavior, 10(10): c01764, 2020.
Meunier, D., Lambiotte, R., Fornito, A., Ersche, K., and Bullmore, E. T. Hierarchical modularity in human brain functional networks. Frontiers in neuroinformatics, 3: 571, 2009.
Kruskal, B. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical society, 7(1):48-50 1956.
Kulkarni, K. R., Schafer, M., Berner, L. A., Fiore, V. G., Heflin, M., Hutchison, K., Calhoun, V., Filbey, F., Pandey, G., Schiller, D., et al. An interpretable and predictive connectivity-based neural signature for chronic cannabis use. Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, 8(3):320-333 202.
Michel, G., Nikolentzos, G., Lutzeyer, J. F., and Vazirgian- nis, M. Path neural networks: Expressive and accurate graph neural networks. In International Conference on Machine Learning, pp. 24737-24755. PMLR, 2023.
Niklason, G. R.,, Rawls, E., Ma, S., Kummerfeld, E., Maxwell, A. M., Brucar, I. R., Drossel, G., and Zilver- stand, A. Explainable machine learning analysis reveals sex and gender differences in the phenotypic and neuro- biological markers of cannabis use disorder. Scientific reports, 12(1):15624, 2022.
Langley, P. Crafting papers on machine learning. In Langley, P. (ed.), Proceedings of the 17th International Conference on Machine Learning (ICML 2000), pp. 1207-1216, Stan- ford, CA, 2000. Morgan Kaufmann.

---

NeuroTreee: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Pan, W., Ling, G., and Liu, F. mgnn-bw: Multi-scale graph neural network based biased random walk path aggre- gation for asd diagnosis. IEEE Transactions Neural Systems and Rehabilitation Engineering, 2025.
Veličković, P., Cucurull, G., Casanova, A., Romero, A.,
Lio, P., and Bengio, Y. Graph attention networks. arXiv preprint arXiv: Seti::1710.10000 2017.
Verma, S., Goel, T., Tanveer, M., Ding, W., Sharma, R., and Murugan, R. Machine learning techniques for the schizophrenia diagnosis: a comprehensive review and fu- ture research directions. Journal of Ambient Intelligence and Humanized Computing, 1 14(5):4795 4807, 2023.
Passiatore, R., Antonucci, L. A., DeRamus, T. P., Fazio, L., Stolfa, G., Sportelli, I.., Kikidis, G. C.,, Blasi, G., Chen, Q., Dukart, J., et al. Changes in patterns of age- related network connectivity are associated with risk for schizophrenia. Proceedings of the National Academy of Sciences, 120632):c222153312220 2023.
Wen, G., Cao, P., Liu, L., Yang. J., Zhang, X., Wang, F., and Zaiane, O. R. Graph self-supervised learning with application to brain networks analysis. IEEE Journal of Biomedical and Health Informatics, 2023a.
Sahakian, B. J., Bruhl, A. B., Cook. J.. Killikelly, C.,, Savulich, G., Piercy, T., Hafizi, S., Perez, J., Fernandez- Egea, E., Suckling, J., et al. The impact of neuroscience on society: cogmitive enhancement in neuropsychiatric disorders and in healthy people. Philosophical Transac- tions of the Royal Society B: Biological Sciences, 370 (667))201402144 2015.
Wen, S., Yang. S., Ju, X., Liao, T., and Liu, F. Prediction of cannabis addictive patients with graph neural networks. International Conference on Brain Informatics, pp. 297-307 Springer, 2023b.
Wen, Z.. Ghafouri, A., and Biros, G. Biophysics- based data assimilation of longitudinal tau and amyloid-3 pet scans. In International Conference on Medical Image Computing and Compuler-Assisted Intervention, pp. 14- 24. Springer, 2024.
Sanchez-Romero, R., Ramsey, J. D.. Zhang. K., Glymour, M. R., Huang, B., and Glymour, C. Estimating feed- forward and feedback effective connections from fmri time series: Assessments of statistical methods. Network Neuroscience, 3(2):274-306, 2019.
Wojtalik, J. A. and Barch, D. M. An fmri study of the influ- ence of a history of substance abuse on working memory- related brain activation in schizophrenia. Frontiers in Psychiatry, 5:1, 2014.
Segal, A., Parkes, L., Aquino, K., Kia, S. M., Wolfers, T., Franke, B., Hoogman, M., Beckmann, C. F., Westlye, L. T., Andreassen, O. A., et al. Regional, circuit and net- work heterogeneity of brain abnormalities in psychiatric disorders. Nature Neuroscience, 2609):1633-1629 2023.
Xu, H., Li, J., Huang, H., Yin, B., and Li, D.-D. Abnormal developmental of structural covariance networks in young adults with heavy cannabis use: a 3-year follow-up study, Translational Psychiatry, 14(1):45, 2024.
Shirer, W. R., Ryali, S., Rykhlevskaia, E., Menon, V., and Greicius, M. D. Decoding subject-driven cognitive states with brain connectivity patterns. Cerebral cortex, 22(1):158-165 2012.
Xu, T., Cullen, K. R., Mueller, B., Schreiner, M. W., Lim, K. O., Schulz, S. C., and Parhi, K. K. Network analysis of functional brain connectivity in borderline personality disorder using resting- state fmri. Neurolmage: Clinical, 1 11:302-315, 2016
Spira, P. M. and Pan, A. On finding and updating spanning trees and shortest paths. SIAM Journal on Computing, 4 (3):375-380, 1975.
Xue, G. Primal-dual algorithms for computing weight- constrained shortest paths and weight-constrrineed mini- mum spanning trees. In Conference Proceedings of the 2000 IEEE International Performance, Computing, and Communications Conference (Cat. No. 00CH37086), pp. 271-277. IEEE, 2000.
Stankevičiūté, K., Azevedo, T., Campbell, A., Bethlehem, R. and Lio, P. Population graph gnns for brain age prediction. biorxiv, pp. 2020-0220022 122000662200 2020.
Tang, H., Liu, G., Dai, S., Ye, K., Zhao, K., Wang, W., Yang, C., He, I.., Leow, A., Thompson, P., et al. Interpretable spatio-tempor embedding for brain structural-effective network with ordinary differential equation. In Inter- national Conference on Medical Image Computing and Cosputer-Assisted Intervention, pp. 227-237. Springer, 2024.
Yang, H., Zhang, H., Meng, C., Wohlschläger, A., Brandl, F., Di, X., Wang, S., Tian, L., and Biswal, B. Frequency- specific coactivation patterns in resting-state and their alterations in schizophrenia: An fmni study. Human Brain Mapping, 43(12):3792-3800 2022.
Tong, W., Li, Y.-X., Zhao, X.-Y., Chen, Q.- Gao, Y.-B., Li, P., and Wu, E. Q. fmri-based brain disease diagnosis: graph network approach. IEEE Transactions Medical Robotics and Bionics, 5(2):312-32,, 2023.
Yang, S. and Liu, F. Electrophy brain network estimation with simultaneous scalp eeg and intracranial eeg: Inference algorithm and applications. bioRxiv, pp. 2024 04, 2024.
12

---

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Yang, S., Jiao, M., Xiang, J., Fotedar, N., Sun, H., and Liu, F. Rejuvenating classical brain electrophysiology source localization methods with spatial graph fourier filters for source extents estimation. Brain Informatics. 11(1):8, 2024.
Yeo, B. T., Krienen, M., Sepulcre, J., Sabuncu, M. R., Lashkari, D., Hollinshead, M., Roffman, J. L., Smoller, J. W., Zöllei, L.. Polimeni, J. R., et al. The organization of the human cerebral cortex estimated by intrinsic func- tional connectivity. Journal of neurophysiology, 2011.
Zhang, J., Cheng, W. Liu, Z., Zhang, K.,, Lei, X., Yao, Y., Becker, B., Liu, Y., Kendrick, K. M., Lu, G., et al. Neu- ral, electrophysiologica and anatomical basis of brain- network variability and its characteristic changes in men- tal disorders. Brain, 139(8):2307-232 2016.
Zhang, P., Wen, G., Cao, P., Yang, J., Zhang, J., Zhang, X., Zhu, X., Zaiane, O. R., and Wang, F. Brainusl: U nsupervised graph s tructure earning for functional brain network analysis. International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 205-214. Springer, 2023.
Zhao, L., Bo, Q., Zhang, Z., Chen, Z., Wang. Y., Zhang, D., Li, T., Yang. N., Zhou, Y., and Wang, Altered dynamic functional connectivity early psychosis between the salience network and visual network. Neuroscience, 491: 165-175,1 2022.
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NreroTree:: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
A. Appendix: Proof Formula 7 for K-hop Connectivity
Theorem A.1 (K-hop Connectivity). Given the k-hop connectivity adjacency operator:
(x( = r A
(t)+ (1 (1-))A4(t)))
(18)
the k-hop connectivity can be expressed as:
k +1- ( - (A
(19)
Proof. For k 1, the formula holds trivially:
[AA()) (1 -M0
Assume the formula holds for n
(1- A1 " -
For = n + 1, expand:
[1=4(t)+11 (0) (t + +(1++111-1)11 (1A"0))"AA4"000+0000000000"00110 (1- (1-))(40))]
Substituting the inductive hypothesis and applying Pascal's 's identity:
+ ( i 1 + ((*)
We obtain:
n+1
(1 (1-1000001)))
n+
Σ
x'11- (*+))x(1
n +++1
(N0) (1))))
i
Thus, by induction, the formula holds for all k. And we have following corollary in A.2
Corollary A.2 (Asymmetric Property of K-hop Operator). The k-hop operator preserves directional information:
As(t) + (A,(t)), VA 2
(20)
Proof. By the definition of the k-hop connectivity operator:
(x = 7 A" (AA + (1- ())(()))***
(21)
Applying the binomial expansion from A.l:
Ax(1)= T 1
(t)
k
x²(1 X(1-)))1(((()))((()))))))111
(22)
Taking the transpose of (t)::
Σ (+) X0
1=0
(23)
k
1b
X1 - -))))((((((()))()()))=+))+
=T 4 O
i
0
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Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
When A + 3, the weights X' and (1 A)k- asymmetric, meaning:
- ~ X)
(24)
Therefore, Ax(t) # (A)( unless all terms in the expansion are zero.
B. Appendix: Proof of Theorem 4.2
We prove the [1)2e has an upper bound h-hop approaches infinity:
lim ||Ax(0]l2 < ||]|2 ||A"||2 max 2)
(25)
Proof. Using matrix norm properties:
||A e Bll2 S ||A||2 ||B1|2
(26)
Applying the triangle inequality when À E [0, 1], we obtain:
||d + - Bl & max(A, 2)(111122 + (B
(27)
Applying the (2-norm to (t):)):
(t)l2 || A"o + (1-))(00+00)00)0)))
(28)
And we have:
(1)22) S || ||4"|22 (1(A4((1))))))))))))))))))))))() (1 (1=)))((0)7TT1l
(29)
Using Eq. 27, we further bound:
(t) + (1 (1)) max(A, 1 + (1(
(30)
Since (1(4())))1 = (t)l()) we obtain:
+1- (L-)(((())]] 3 2max(A, 1 -))]]
(31)
Taking the power of & on both sides of Eq. 31, we have (1)l2 < [[A()) and the following inequality holds:
(t) - max(A,1 2)
(32)
Substituting into equation 29, get:
(11l < [[T||2 ||4 max(A, x)*
(33)
Since max(A, X) < for E 10, the series converges geometrically. This result indicates that [1) converges to zero - 20, provided the spectral radius of A, denoted by X. satisfies max(A, A) < Consequently. the rate of decay of (t)ll is governed by max(A, 1 2)4, which demonstrates the dependence the spectral properties of A and ensures convergence for sufficiently small X.
15

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Nree:Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
C. Appendix: Proof for Proposition 4.3
To prove proposition 4.3, we express the neural ODE system as:
(H) (t)) dt
(34)
where
K-1 (t) -a Σ t (1)
(35)
Proof. Assume is differentiable function and continuous in t. We show that F satisfies Lipschitz continuity in H and continuity Let (1)) and H! be two output states:
|FH b) F(H22) 1)ll
K
K-1 (1)
Σ
t (t) (T)
Σ
(36)
since [[ (1) - -F [)]l2 2 is bounded by the Lipschitz constant of a, denoted by Lig, we obtain:
K
|| (( s Tho
Σ
Φ (1)(H - H w
(37)
Using matrix norm properties and [[) S and /
()) are bounding the norms. The bound equation can be written as:
K-1
[[F .) - " ,1)l2 S Lo
Σ
l
[2
H 112
(38)
k=
Let M = max; |[W 122 Then:
[F FF,, ,11l2 S LLKMM -H1511 112
(39)
Thus, F Lipschitz continuous in H with Lipschitz constant LF Lg KMM. For continuity in t, the Lipschitz condition of (():
(+x(t)) - (x((2)l22 - L| - t2ll2
(40)
Thus, Eq. 40 ensures that is continuous According to Picard's existence theorem, the k-hop ODE-GCN has a unique solution.
D. Appendix: Finding the Shortest Paths in Weighted Brain Tree
Theorem D.1 (Optimal Path Extremes). Let W(S) = Evep S(v; θ) represent the node score contribution and W(C) EeyeEr, Ci.j represent the connectivity strength contribution in the pruned graph T(G). For an arbitrary integer threshold X², the optimal path pa satisfies the following interval constraint W(S) X W(C).
In practice, the minimum interval can be determined using the bisection method within [aLL, az. using bisection method such that W(S) < Waa (P*) a* S War (P*) < W(C) (Xue, 2000).
16

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Nree:Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
E. Prediction of FC Alterations in Brain Age Across Different Age Groups
This section presents our application of NEUROTREE for predicting mental disorders and examining their impact on brain FC changes across various age cohorts. We utilized AGE-GCN embeddings to reconstruct FC values for different age groups. Subsequently, we analyzed FC values for various age groups, as illustrated in Fig. 6. The results presented in Fig. (a) and (b) reveal that significant within-group differences were observed exclusively among cannabis users in the 18-25 age group. We observe significant changes in FC values associated with cannabis use disorder and schizophrenia across various age groups, indicating age-dependent variations in FC patterns among individuals with these psychiatric disorders.
< 0.001
0.056
HC Cennabis
14
p 0.001
0:007
FC Value
10
0.017
p. 0.643
p < 0.001
p 0.081
p 0.001
18-25
25-40 Age Groups
40-55
(a) Cannabis
00.000011
0.003
p < 0.001
HC Schizophrenia
3.5
3.0
p = 0.160
FC Value
2.5
p 0,027
p 0.006
2.0
1.5
p 0.595
1.0
p 0.244
0.5
18-25
25-40 Age Groups
40-66
(b) COBRE
Figure 6. Age-related changes predicted FC values across different groups. (a) cannabis users versus healthy controls (HC) and (b) the COBRE dataset. Note that asterisks indicate statistical significance ("p < 0.05, "'p < 0.01,
schizophrenia patients versus HC using *""p< < 0.001). Error bars represent standard error of the mean.
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Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
F. High-Order Tree Path Influence
further examining the characterization of IMRI the tree path, we investigated the contribution FC strength through Eq. 16 across two distinct subtypes of fMRI representations, as illustrated in Fig. Our observations revealed that cannabis-related pathways exhibited relatively higher average FC weights compared the COBRE dataset. Notably, both the addiction and schizophrenia groups demonstrated significant decreases following increase in a, while the HC cohort maintained relatively stable patterns. These findings provide compelling evidence that higher-order brain tree paths effectively capture the differential contributions of FC strength arising from alterations related to addiction or schizophrenia.
Cannabis
COBRE
(a)
Average FC Strength Weight
(b)
Average FC Strength Weight
0.1485
0.1180
0.1480
0,1175
0.1475
0.1170
0.1470
0.1165
Schizophrania
0.1465
0.1160
0.1460
HC
0.1155
0.1455
Cannabis
0.1150
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 7. Average FC value weights versus in high-order paths, comparing (a) HC cannabis and (b) HIC versus schizophrenia patients. The results indicate reduced connectivity patterns with increasing a in both clinical groups compared to the control group.
G. More Related Work
Schizophrenia (SCZ) and substance abuse (SA) both profoundly impact brain structure and function, with a particularly high comorbidity rate observed among younger populations. Research indicates that SCZ patients frequently exhibit functional abnormalities critical brain regions, including the DMN, central executive network (CEN), and visual pathways (Ma al., 2023). However, findings remain inconsistent regarding whether comorbid SA history SCZ patients exacerbates these neural impairments. (Wojtalik & Barch, 2014) conducted IMRI scans during working memory tasks and discovered significantly enhanced activation in frontal-pariett parietal-thalamii regions among control subjects with SA history. Interestingly, SC7 patients showed no significant activation differences regardless of SA history, suggesting that pre-existing neural deficits in SCZ may obscure the additional effects of SA (Wojtalik & Barch, 2014). In addition, (Passiatore et al., 2023) further demonstrated that FC alterations associated with SCZ risk emerge during adolescence, particularly in cerebellum- occipitoparictal and
sensorimotor regions, strongly correlating with polygenic risk scores (PRS).
Beyond network abnormalities, SCZ and SA patients commonly exhibit accelerated brain aging, where their brain-predices age significantly exceeds their chronological age, indicating heightened functional deterioration of the brain-a phenomenon with critical implications for understanding cognitive decline and disease risk (Cole & Franke, 2017; Lombardi et al., 2021). With the emergence of GNN applications to fMRI data, researchers can now model and analyze brain regions nodes and functional connections edges in graph structures. The Pooling Regularized GNN (PR-GNN) proposed by (Li et al., 2020). Applying this methodology to SCZ with comorbid SA may simultaneously facilitate classification and biomarker identification, while incorporating demographic variables such as age, gender, and SA history enables individualized modeling. In summary, SCZ and SA have age-related and highly intertwined effects on brain functional networks. GNN methodologies provide technical tools with both explanatory and predictive capabilities, offering promise for deepening our understanding of the neural mechanisms underlying psychiatric disorders and their potential clinical applications.
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NeuroTree:: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
H. Experiments Detail
Experiments Sefting
Baseline Models. The classical methods for constructing brain networks include Pearson correlation and k-nearest neighbor approaches combined with GCN, such as: as:These methods are as follows::) Pearson GCN: This model constructs an fMRI graph using Pearson correlation coefficients represent relationships between ROIs. 2) k-NN GCN: This model builds brain networks by connecting k-nearest neighbors based similar imput features within brain regions in the graph. 3) GAT (Veličković et al., 2017): This model extends the GCN framework by incorporating attention mechanism that enhances the aggregation and learning of features from neighboring nodes. 4) BrainGNN (Li et al., 2021): BrainGNN is an interpretable GNN framework designed for analyzing fMRI data. This framework effectively identifies and characterizes significant brain regions and community patterns associated with specific neurological conditions. 5) BrainUSL (Zhang al., 2023): BrainUSL learns the graph structure directly from BOLD signals while incorporating sparsity constraints and contrastive learning capture meaningful connections between brain regions. Finally, we compare NEUROTREE with other methods that incorporate higher-order node information, specifically MixHop (Abu-El-Haijp al., 2019) and graph path-based models such as GPC-GCN (Li et al., 2022b) and PathNN (Michel et al., 2023).
Evaluation Metries In our IMRI graph classification experiments (Section 6.2), evaluate NEUROTREE using Area Under the Curve (AUC), Accuracy (Acc.), Precision (Prec.), and Recall (Rec.). In brain age prediction (Section 8), we evaluate NEUROTREE predicted age and chronological age using MSE metric.
Parameters Setting 0.001.
In this study, two datasets were trained with batch size of 16 for 100 epochs using a learning rate of
Table 3. Training Parameters.
Notation
Meaning
Value
Scale parameter
0.5
T
Number of time segment
k-hop connectivity balance between dynamic FC matrices (1)
(0,1)
Learnable age- modulated parameter
(0.1)
Learnbbe graph weight parameter
Learnable weight matrix in Ith layer
Additional Ablation Study In this subsection, we conduct ablation experiments on the performance of brain networks classification with respect to two parameters À and I' on two mental disorders dataset. The experimental results demonstrate that for Cannabis, the model achieves optimal accuracy (0.8) and AUC value (0.87) at À = 0.2, with performance decreasing as A increases to 0.4, followed by recovery at X = 0.6. In contrast, COBRE exhibits a different parameter sensitivity pattern, with accuracy showing a decreasing- wwwtttiinnnnaeeee accuracy (0.8) at À = 0.8. Regarding the parameter, Cannabis shows an overall decline in accuracy as I increases. On the other hand, COBRE reaches its optimal accuracy (0.8) and AUC value (0.79) at r 0.6, but experiences sharp performance drop at I 0.8. These findings suggest that Cannabis-related brain network structures more effectively captured lower À values, whereas roBRt-related brain functional abnormalities more efficiently identified higher and moderate I' values.
trend as À increases from 0.2 to 0.8, ultimately reaching peak
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Nree:Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
Accuracy vs
Accuracy vs
0.800
0.800
Cannabis COBRE
0.775
0.775
0.750
0.750
Accuracy
0.725
Accuracy
0.725
0.700
0.700
0.675
0.675
0.650
0.650
0.625
Cannabis
0.625
COBRE
0.600
0.600
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0.3
0.4
0.5
0.6
0.7
AUC À
AUC VS [
Cannabis COBRE
0.85
0.8
0.80
AUC
0.75
AUC
0.7
0.6
0.65
Cannabis
0.60
COBRE
0.2
0.3
0.4
0.5 A
0.6
0.7
0.8
0.2
0.3
0.4
0.5
Figure 8. Evaluation of brain network classification performance based model parameters À and
Environment Setting our experiments, we follow the environment setting:
Table 4. Experiment Environment Details
Component
Details
GPU
NVIDIA GeForce RTX 3070
Python Version
3.7.6
Numpy Version
1.21.6
Pandas Version
1.3.5
Torch Version
1.13.1
Operating System
Linux
Processor
x86 64
Architecture
64-bit
Logical CPU Cores
32
Physical CPU Cores
24
Total Memory (GB)
62.44
Biological Interpretation of k-hop Convergence
A faster decay in the spectral norm implies that higher-order information becomes less influential, indicating more localized brain interaction pattern, while slower decay suggests that longer-range dependencies are preserved in the FC network.
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Nree:Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
As the k-value increases, more distant neural connections are considered, which may reflect the integration process of long-distance FC in the brain. Thus, the spectral norm convergence analysis provides quantitative insights into the scale and extent of connectivity alterations, complementing the interpretability of our tree-structured model.
Spectral Norm Convergence
Spectral Norm
Faster decay leads to more localized brain interaction patterns
Slower decay preserves longer-range dependencies
K-value
Figure 9. The visualization of different decay meanings in spectral norm convergence. The left side of the curve (blue section) demonstrates faster decay, which leads to more localized brain interaction patterns, visualized compact, hierarchical network structure with blue nodes. The right side (orange section) shows slower decay, which preserves longer-range dependencies, illustrated by a more distributed network with orange nodes connected across greater distances.
J. Age- Age-Rel Brain Network Changes in Addiction and Schizophrenia
In this section, we identified age-related FC subnetwork changes associated with mental disorders, as shown in Fig. 10. his study reveals distinct patterns of brain network connectivity across different age groups in individuals with addiction and schizophrenia through FC analysis. In addiction patients, age emerges critical variable significantly influencing brain network organization. Young addiction patients (18-25 years) demonstrate strong connectivity between the DMN, VN, and FPN, which may reflect the collective influence of these critical networks during the early stages of addiction development. As age increases, middle-aged addiction patients (25-40 years) show connectivity extending to the DAN, indicating enhanced attentional control mechanisms during persistent addiction. Notably, older addiction patients (40-55 years) predominantly exhibit Somatomotor network connectivity, possibly representing specific effects of long-term addiction on motor control systems. In contrast, schizophrenia patients display differennt age-related network change trajectories. Young schizophrenia patients (18-25 years) primarily characterized by strong connections between Somatomotor and VN, which may relate to abnormal sensory-moto integration in early schizophrenia. Middle-aged schizophrenia patients (25-40 years) demonstrate stronger FPN activity, suggesting the importance of higher-order cognitive control networks in disease progression. Older schizophrenia patients (40-66 years) are mainly characterized by enhanced connectivity between DAN and VN, potentially reflecting changes attentional processing and visual integration mechanisms in late-stage disease. From our research findings, we demonstrate significant heterogeneity across different age groups various mental disorders, particularly in the progression patterns of age-dependent network changes.
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Tree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
(a) Addiction
(b) Schizophrenia
Age:18-25
Age:25-40
Age:40-55
Age:18-25
Age:25-40
Aga:40-66
Sucond
Kaedi
Voe
Frontoparictal
Default Visual & Frontoparietal
Somatomotor
Somatomotor & Visual
Frontoparietal
Dorsal Attention & Visual
Visual
Somatomotor
Dorsal Attention
Ventral Attention
Frontoparietal
Default Mode
Figure 10. Predicted FC changes in different age groups and affiliated functional subnetworks.
K. Effects of Age and CMFC Loss Brain Tree Structure
According to Fig. 11, without parameter age and CMFC loss, the tree branches appear disorganized and fragmented, with paths lacking anatomical continuity and interpretability. However, with parameter that dynamically regulates FC based on individual age, it presents more coherent branches and clearer hierarchical structure.
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NreroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
(A)
w/o Age Parameter
With Age Parameter
(B)
w/o CMFS Loss
With CMFS Loss
Levdl Level
Level
Figure 11. Constructed brain Tree based on w/o and w/o CMFC loss. We present different modeling approaches for brain connectivity networks and their robust in brain tree construct. (A) The left panel illustrates brain connectivity network models without age parameters, while the right panel demonstrates models incorporating age parameters. (B) The left panel displays brain connectivity network models without CMFS loss function, whereas the right panel shows models utilizing CMFC loss function.
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Nree:Tree:: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
L. Dataset
presents the summary statistic of two public fMRI dataset below:
Table S. Summary statisties of demographics in Cannabis and COBRE
Dataset
Sample size
Age
(Mean SD)
Cannabis
HC (n=128)
30.06 + 10,47
Cannabis (n=195)
27.23 + 7.73
COBRE
HC (n=72)
38.31 + 12.21
Schizophrenia (n=70)
36.04 + 13.02
L.1. IMRI preprocessing in cannabis dataset
fMRI Data Processing Pipeline (Kulkarni et al., 2023) The fMRI data underwent a comprehensive preprocessing pipeline. Raw data was acquired using MRI scanners (Siemens Trio or Philips) with w2e-weighted and high-resolution TI-weighted structural images. The preprocessing, implemented through FMRIPREP, included intensity non-uniformity correction, skull stripping, and brain surface reconstruction using FreeSurfer. Spatial normalization to the ICBM 152 template performed, followed brain tissue segmentation into cerebrospinal fluid (CSF), white matter, and gray matter. The functional data underwent slice-timing correction, motion correction, and co-registration to corresponding Tlw images using boundary-based registration. Physiological noise was addressed through CompCor regression. Additional denoising steps included removing participants with framewise displacement > 3mm in over 5% of scan length and applying ArtRepair for remaining nigh-motion volumes. Combined task/nuisance regression was performed using motion parameters and aCompCor as covariates. The processed whole-brain time series were then parcellated using the Stanford functional ROIs atlas, resulting in 90 regions of interest. Quality control analyses confirmed that motion metries did not significantly differ between groups (t-test p=0.86) and showed no correlation with classification accuracy (p=0.475).
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NreroTree: Hierarchical Functional Brain Pathway Decoding for Mental Health Disorders
L.2. Different Atlases According to Seven Affiliated Functional Subnetworks
Table Comparison of Brain Networks between Stanford (90 ROIs) and Harvard-Oxford (118 ROIs) Atlases
Network
Stanford (90 ROIs)
Harvard-Oxford (118 ROIs)
Default
bilateral mPFC, L/R lateral angular gyrus, supe- nior frontal gyrus, bilateral posterior/middle/midd posterior cingulate, bilateral anterior thalamus, L/R parahippocampal gyrus, L/R inferior/mid- temporal cortex, L medial angular gyrus, crus cerebellum (lang), L inferior parietal cortex, L/R middle frontal gyrus (vent_DMi) R inferior cere- bellum (vent_DMN)
Left/Right Frontal Pole, Superior Frontal Gyrus, Middle Temporal Gyrus (anterior/posterior)) Angu-
Mode
lar Gyrus, Precuneous Cortex
Visual
L/R mid occipital cortex, L/R precuneus (post sal), bilateral medial posterior/medial pre- cuneus, R angular gyrus precuneus, L/R ventral precuneus, L/R fusiform gyrus, L/R mic-occipital cortex
Left/Right Middle/Inferior Temporal Gyrus (tem- porooccipital), Lateral Occipital Cortex (supe- rior/inferior), Cuneal Cortex, Lingual Gyrus, Temporal Occipital Fusiform Cortex, Occipital Fusiform Gyrus, Occipital Pole
Dorsal Attention
L/R superior frontal gyrus, L/R superior/inferior parietal cortex, L/R precentral/fronto-oprrra region, L/R inferior temporal cortex, L/R cerebel- lum (visuospatial), R crus cerebellum
Left/Right Middle Frontal Gyrus, Superior Parietal
Lobule, Supramarginal Gyrus (anterior/posterion)
Frontoparictal
L/R middle frontal/dIPFC, L inferior frontal (tri- angular), L/R inferior parietal/angular gyrus, L inferior temporal gyrus, R/L crus cerebellum (LECN/RECN), L posterior thalamus, R middle orbito-frontal cortex, R superior medial frontal gyrus, caudate
Left/Right Inferior Frontal Gyrus (pars triangu- laris/opercularis, Frontal Operculum Cortex
Somatomotor
L/R superior temporal/auditory, thalamus, L/R pre/post-central gyri, bilateral supplementary tor area, L/R ventral posterior nucleus of thalamus, cerebellar vermis
Left/Right Precentral Gyrus, Superior Temporal Gyrus (anterior/posterior), Postcentral Gyrus, Jux- tapositional Lobule Cortex, Central Opercular Cor- tex
Subcortical
R striatum/thalamus (2), L/R inferior frontal gyrus, pons/dropout region
Left/Right Thalamus, Caudate, Putamen, Pallidum, Stein, Hippocampus, Amygdala, Accum- bens Left/Right Insular Cortex, Temporal Pole, Frontal Medial/Orbital Cortex, Subcallosal Cortex, Paracingulate Gyrus, Cingulate Gyrus (ante- rior/posterior), Parahippocampal Gyrus, Temporal Fusiform Cortex, Parietal Operculum Cortex, Planum regions, Cerebral White Matter/Cortex, Lateral Ventricle
Others
prefrontal cortex, L/R anterior insula, bilateral ACC, L/R crus cerebellum (ant_sal), L/R mid- dle frontal gyrus, L/R supramarginal cortex, L/R middle thalamus, L/R anterior cerebellum, L/R superior temporal gyrus, bilateral calcarine cortex, L.N
25


Reviews:
[{"review_id": "eisePwQRH0", "reviewer": "Reviewer_fciF", "summary": "This paper introduces NEUROTREE, a novel framework for analyzing functional brain networks derived from fMRI data.  The framework integrates k-hop Graph Convolutional Networks (GCNs) with neural Ordinary Differential Equations (ODEs) to enhance the learning of dynamic functional connectivity (FC) features and capture high-order brain regional pathway features in a tree topology.  The authors demonstrate the effectiveness of NEUROTREE in predicting psychiatric disorders and elucidating their underlying neural mechanisms across two distinct mental disorder datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **M1 \\& W2:** **The study's evaluation is limited by using only two datasets, including disorder representation, dataset heterogeneity, and demographic diversity.**\n\nWe thank the reviewer concern for data diversity! \nDue to data privacy concerns and the limited availability of public fMRI data for individuals with mental disorders that also include demographic information. **We believe our proposed NeuroTree method's contributions are beneficial not only for decoding specific disease fMRI datasets but also extend to broader research applications such as EEG in neuroscience.**\n\n> **M2:**  **Summary statistics of demographics in two datasets and sample size.**\nWe thank the reviewer for taking the time to review our work. Due to page limitations, we have placed the demographic statistics details for both datasets in `Appendix H. Dataset`. The total Cannabis sample size is `323` and COBRE is `142`.\n\n> **W1:** **Model computational cost and complexity, and the scalability for future studies.**\n\nWe appreciate for pointing out this issue.\n`1).Computational cost`:\nWe have additionally supplemented experiments conducted for 100 epochs including a comparative table of computational costs with and without parameters ($\\theta$,$\\Gamma$,$\\lambda$) across two datasets. To facilitate comparison of computational costs, we have included a Graph Transformer-based module with higher computational cost for comparison in the table below:\n\n| Dataset (Model)             | Type    | Training Time (sec) | GPU Memory (MB) | Inference Time (sec，avg over 10 runs) |\n|-----------------------------|--------------|----------------|------------------------|---------------------------|\n| Cannabis (NeuroTree)        | With params       | 13.958         | 20.62                  | 0.000870                  |\n|                             | NO params     | 5.396          | 14.25                  | 0.000160                  |\n| Cannabis (Graph Transformer)| With params       | 19.065         | 170.38                 | 0.002445                  |\n|                             | NO params     | 5.232          | 18.53                  | 0.000346                  |\n| COBRE (NeuroTree)           | With params       | 6.157          | 18.12                  | 0.001771                  |\n|                             | NO params     | 2.137          | 9.21                   | 0.000351                  |\n| COBRE (Graph Transformer)   | With params       | 8.948          | 213.08                 |0.002608                    |\n|                             | NO params     | 2.173          | 13.46                  | 0.000356                    |\n\n`2). Code Implementation`:\nIn addition, the model we designed can be easily trained on a personal computer (include GPU, Google Colab Jupyter Notebook), the related environment setting can be found in our paper `Appendix G Table 4`. Compared to Graph Transformer modules with their extensive matrix calculations, NeuroTree significantly  reduces computational demands through strategic parameter configurations. Additionally, NeuroTree enables researchers to observe parameter effects, enhancing model interpretability in alignment with specific research objectives.\n\n> **W3:**  **Low reproducibility: No code/models are provided.**\nThank you very much for the reviewer's discussion on the reproducibility of our research! \n\n`1). Code acquisition`: We will release the complete code (including tree plot) and processed fMRI data on GitHub at the camera-ready stage when the paper is accepted. \n\n`2). Reproducible`:\nWe demonstrate relevant links to the training and testing process in Tensorboard according to the reviewer's (**gVcG**) suggestion for your reference:https://anonymous.4open.science/r/anonymous_ICML/anonymous_loss_comparison.png\n\n> **O1:** **Redundant Sentence Correction.**\nWe appreciate your careful review and corrections! We express our sincere apologies regarding the extra sentence mistake. We have corrected the English grammar expression to maintain logic.  *However, current approaches face two fundamental limitations. First, .. Second,..*\n\n> **O2 \\&  O3:** **Consider (1) clearly distinguishing between matrix and scalar values in equations, and (2) specifying the dimensions of each matrix.**\nWe thank the reviewer for the detailed review of our work! \nWe have added explanations for $\\sigma$ being the sigmoid function in Eq (1) and $\\eta, \\rho \\in  \\mathbb{R}^+$ for the scalar values in Eq (2) of the paper. We clarify that the external input vector $u(t) \\in \\mathbb{R}^{v}$ interacts with the external stimulus encoding matrix $C \\in \\mathbb{R}^{v \\times v}$, while both the static adjacency matrix $A^{s} \\in \\mathbb{R}^{v \\times v}$ and the time-dependent dynamic adjacency matrix $A^{d}(t) \\in \\mathbb{R}^{v \\times v}$ contribute to the overall network connectivity at time $t$, and $D^{-\\frac{1}{2}} A D^{-\\frac{1}{2}} H^{(l-1)} W^{(l-1)} \\in \\mathbb{R}^{v \\times d_{\\text{out}}}$ with output $d$ dimension.", "claims_and_evidence": "Looks good to me.", "methods_and_evaluation_criteria": "The authors only evaluated on two datasets, Cannabis and COBRE. While the use of publicly available datasets is commendable for reproducibility and comparison purposes, the selection is limited in scope with the following concerns: 1) limited disorder representation; 2) dataset heterogeneity; 3) lack of demographic diversity. \n\nAdditionally, the authors are encouraged to explicitly mention the sample sizes of the datasets used. Larger datasets would provide more robust validation and enhance the credibility of the findings.", "theoretical_claims": "The authors make several theoretical claims, which appear to be correct. Theorem 3.2 posits that the l2-norm of the k-hop connectivity adjacency operator is bounded as k-hp approaches infinity. Theorem 3.4 describes the discretization of Age-Aware Continuous-Time Graph Convolution, building upon previous work by Tang et al. (2024). Detailed proofs are provided in the appendix.", "experimental_designs_or_analyses": "The authors may consider providing significance analysis as they claimed that their method significantly outperforms SOTA models.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n1. Novelty and technical soundness: NEUROTREE offers a novel approach by integrating k-hop GCNs with neural ODEs. It is well-explained, with clear descriptions of the k-hop ODE-GCN, Contrastive Masked FC Strength (CMFS) optimization, and hierarchical brain tree construction. The authors also provide theoretical support for their methods, including theorems and proofs in the appendices.\n2. Interpretability: A key strength of NEUROTREE is its interpretability.  The framework facilitates the identification of hierarchical neural behavioral patterns and provides insights into age-related deterioration patterns, enhancing the understanding of underlying neural mechanisms in psychiatric disorders.\n\nWeaknesses:\n1. Complexity and computational cost: The proposed NEUROTREE framework is complex, involving multiple components and parameters.  This complexity may make it challenging for researchers to implement and apply the model. The use of k-hop GCNs and neural ODEs can be computationally expensive, potentially limiting the scalability of the model to larger datasets.\n2. Limited generalization: The study focuses on two specific psychiatric disorders.\n3. Low reproducibility: No code/models are provided.", "other_comments_or_suggestions": "Some writing issues:\n1. Ln 043 - 045 (right): \"Nevertheless, ... However...\" The sentence begins with \"Nevertheless\" and then uses \"However\" shortly after. Both words serve a similar purpose (signaling contrast or limitation), so using both is repetitive and disrupts the flow.\n2. Consider explaining what is sigma in Equ. 1. \n3. Consider (1) clearly distinguishing between matrix and scalar values in equations, and (2) specifying the dimensions of each matrix.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JI6GMvITJL", "reviewer": "Reviewer_XciT", "summary": "This paper proposes NeuroTree as a framework for feature learning from functional connectivity for brain disease characterization. NeuroTree integrates standard graph convolutional network with neural ordinary differential equations.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**We thank the reviewer for valuable suggestions and insightful comments, and we have clarified a few things about accuray and brain age so that our contribution can be better understood.**\n\n> **Q1. About the terminology of 'interpretability' and 'causal' modeling.**\n\n`1.) The three meanings of interpretability in NeuroTree:` \n\n* In this work, we integrate the deep learning model concept from the literature [1,2,3] to enhance brain disorders classification. We define \"interpretable variables\" as observable latent factors in our model, not only providing an fMRI modeling but also showing the level of variations. We have corrected our terminology to \"demographics\" in the revised manuscript.  \n\n* In section 3.1, we integrate age as parameter $\\theta$ into ODE-GCN and  predict disease-relevant regions, visualized through node importance scores (brain regions) in tree explanations, enhancing model interpretability (Fig. 4).\n\n* NeuroTree can completely perform model training to hierarchically decompose the fMRI brain network into a tree structure, with each level showing significant brain regions on the tree path to help explain brain disease, which provides further interpretability. \n \n`2.) Statistical Interpretation:` We will conduct hypothesis testing through alation study, comparing our framework with ODEs and without ODEs to show the statistical significance of using causal network analysis. \n\n`3.) The definition of 'causal' modeling:`  We clarify that our use of the term \"Causal\" originates from explanations in deep learning literature [4,5]. For example, dynamic causal graph learning in [4] enables neural networks to automatically learn \"time-varying causal graphs\" or causal structures from traffic data. In Section 3.1, NeuroTree employs the ODE model form Eq (1) to simulate influences between brain regions, reflecting a continuous-time causal dynamic relationship. Although we have not yet used traditional Granger causality or dynamic causal modeling to directly present a directed graph node method, the time-varying graphs (dynamic FC) derived from ODE provide a framework that approximates causal relationships. We appreciate the reviewer's perspective and we have revised the manuscript to remove the \"causal\" word to make our paper more clear.\n\n[1]Zheng, et al. \"Brainib: `Interpretable` brain network-based psychiatric diagnosis with graph information bottleneck.\" IEEE TNNLS, 2024.\n\n[2]Cui, et al. \"`Interpretable` graph neural networks for connectome-based brain disorder analysis.\" MICCAI, 2022.\n\n[3]Chen, et al. \"Learnable subdivision graph neural network for functional brain network analysis and `interpretable` cognitive disorder diagnosis.\" MICCAI, 2023.\n\n[4]Lin, et al. \"Dynamic causal graph convolutional network for traffic prediction.\" IEEE CASE. 2023.\n\n[5]Wein, et al. \"A graph neural network framework for causal inference in brain networks.\" Scientific reports, 2021.\n\n> **Q2. About model accuracy in brain network classification and comparison of models.**\n\nWe clarify this from two perspectives: \n\nOur model NeuroTree combines achieved the state-of-the-art performance using dynamic fMRI compared to similar models like PathNNs, BrainGNN, etc., given the variance in the individual fMRI data [8,9]. However, we believe this is an important yet essential step to predict the patients with mental disorders, especially since we also achieved an **AUC of 0.71** on the **public COBRE dataset**. In the future, NeuroTree can integrate with other modalities (e.g., DTI, genetic information), allowing us to examine the connection alterations among patients.\n\n[8]Zhao,et al. \"Enhancing major depressive disorder diagnosis with dynamic-static fusion graph neural networks.\" IEEE JBHI, 2024.\n\n[9]Peng, et al. \"Gate: Graph CCA for temporal self-supervised learning for label-efficient fmri analysis.\" IEEE TMI, 2022.\n\n\n> **Q3. About 'Chronological Brain Age Prediction' terminology definition problems and model prediction usefulness.**\n\n`1. Clarification of definition and interpretation:` In our work, chronological age refers to the subject’s actual age, which is used as a regulatory parameter to model age-related changes in FC. Specifically, in our AGE-GCN, we incorporate the age parameter $\\theta$ to learn how FC strength evolves over time, thus enhancing the interpretability of aging patterns in mental disorders (See Eq. (3)–(12)).\n\n`2. The utility of predicting actual age in disease populations: ` We fully agree with the reviewer that the difference between chronological age and brain age is a clinically meaningful biomarker. However, NeoroTree is not solely designed to predict age accurately. Instead, it uses age as a modulation variable to explore how FC patterns vary with age across different clinical groups (e.g., cannabis users, schizophrenia patients).\n\n3. In the revised version, we conduct the prediction of brain ages among healthy controls, while comparing the prediction accuracy from patients with addictive disorders.", "claims_and_evidence": "I find various claims and evidence in this paper problematic.\n\n1. The paper claims to integrate `interpretable variables into both static and dynamic causal graph convolutional modeling'. Here, it's very unclear how the `interpretability' of a specific variable is gauged. Moreover, I do not see any convincing statistical evidence of causal modeling or its evaluation within this paper. \n\n2. The proposed model achieves the maximum mean classification accuracy of 73% at best among all experiments considered. Therefore, it is clearly not an effective predictor of brain disease. I understand that the authors claim improvements over other methods. However, if 73% is the best you can achieve on given datasets as your main result, then there is perhaps a clear mismatch between the considered model and the task at hand. \n\n3. I also find the results within the section 'Chronological Brain Age Prediction' problematic. Firstly, chronological age and brain age can differ in populations with brain diseases (which forms the primary motivation for this line of research). Therefore, the authors should rigorously clarify their interpretation and definition of 'Chronological Brain Age'. Furthermore, the gap between chronological age and brain age is often the biomarker, and predicting chronological age within disease cohorts with high Pearson correlation has no practical utility.", "methods_and_evaluation_criteria": "See Claims And Evidence section.", "theoretical_claims": "I did not check the correctness of theoretical claims.", "experimental_designs_or_analyses": "See Claims And Evidence section.", "relation_to_broader_scientific_literature": "The paper lacks solid conceptual contributions relative to broader scientific literature.", "essential_references_not_discussed": "The paper completely ignores the literature on graph convolutional networks from the lens of graph signal processing, and their applications as age prediction models, fMRI characterization, and interpretable biomarker construction. I am not naming specific studies here but I would recommend that the authors review this line of literature as well.", "other_strengths_and_weaknesses": "See Claims And Evidence section for weaknesses.", "other_comments_or_suggestions": "I don't have any other comments.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DXa8PmyVaU", "reviewer": "Reviewer_gVcG", "summary": "The paper introduces Neurotree, a graph convolutional network that employs ordinary differential equations (ODEs) to model neural dynamics and learns a tree topology using contrastive loss to identify functional connectivity (FC) pathways. The model is evaluated on two datasets, achieving state-of-the-art performance. The primary contribution lies in its ability to characterize differences in functional connectivity between patients and healthy subjects, offering insights into neural circuit disruptions in psychiatric disorders.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **C1. Cannabis addicts have stronger FC compared to schizophrenia patients.**\n\nWe appreciate the reviewer's careful inquiry about this claim! Study [1] found cannabis users show higher baseline functional connectivity in reward circuits than schizophrenia patients. Our `Fig 4. (a-2) and (a-3)` results predict higher FC numbers in cannabis users across Yeo's seven brain networks, with additional FC degree centrality predictions available at the link:https://anonymous.4open.science/r/anonymous_ICML/anonymous_DC.png.\n\nReferences:\n[1]Fischer et al. Journal of Schizophrenia Research, 2014.\n\n> **M3 & Q2.: Other parcellation schemes methods and robust approaches using NeuroTree.**\n\nThank you for your valuable comment!\n\n`1). Other parcellation schemes`: The addiction and COBRE atlases contain 90 and 118 ROIs respectively. As shown in `section 7 and Fig 4`, regardless of which atlas was used, the extracted tree-structured pathways successfully mapped to Yeo's 7 networks, enabling cross-dataset interpretation. Our framework consistently revealed distinct hierarchical structures in psychiatric conditions across both parcellations, demonstrating the stability and generalizability of our tree construction method. In future work, NeuroTree can easily extend to additional parcellations (e.g., AAL, BASC) for different disease research problems.\n\n`1). The robust of NeuroTree`:  As described in `Section 6.1` and shown in Table 1 with SOTA performance, despite differences in parcellation schemes, we can improve stability through training the NeuroTree framework to obtain nodes (regions) prediction and weighted paths. This indicates that the hierarchical brain tree structures constructed by NEUROTREE are robust to the choice of ROI definitions. According to Definition 3.5, Kruskal's algorithm ensures tree decomposition with the shortest paths.\n\n> **E1 & E3 & M2: NeuroTree uses the CMFS loss model performance and the biological significance of k-hop using spectral norm**\n\nWe appreciate the reviewer’s insightful question! `1). Loss ablation`:  According to the experiment w/o CMFS loss in `Table 1`, we found that adding CMFS loss can make the overall (+ CE loss) loss more robust in our tensorboard plot at link:https://anonymous.4open.science/r/anonymous_ICML/anonymous_loss_comparison.png.\n\n`2.) K-hop convergence`: The intuition for analyzing the rate of spectral norm decay with increasing k-hop allows us to examine how rapidly information from distant nodes attenuates across the dynamic brain network structure. A faster decay in spectral norm indicates more localized brain interactions as higher-order information becomes less influential. Conversely, slower decay preserves longer-range dependencies in the FC network. Higher k-values incorporate more distant neural connections, potentially reflecting the brain's long-distance functional integration processes. \n\n> **M2 & Q1 & Q4:How does the inclusion of age as an input during training affect the model's ability to predict age groups in Table 2? Was age also provided as an input during prediction?**\n\nWe appreciate your thoughtful review.\n`1.) Effectiveness of including age variable in the model`: Recent research [2,3] shows incorporating demographic data (especially age) into GNN for fMRI enables more precise learning of subject differences. NeuroTree's ODE design uses age to regulate features during message passing, enhancing graph classification through age-aware GCN. This approach accounts for individual fMRI differences, demonstrating how personal features like age can stabilize dynamic graph convolutional neural network performance.\n\nReferences:\n\n[2]Zhang, Hao, et al. IEEE TMI, 2022\n\n[3]Wang, Xuesong, et al. MICCAI, 2022.\n\n`2.) Effects of age and CMFS loss on tree structure`:\nAccording to our supplementary figure at link https://anonymous.4open.science/r/anonymous_ICML/anonymous.png, without parameter $\\theta$ and CMFC loss, the tree branches appear disorganized and fragmented, with paths lacking anatomical continuity and interpretability. However, with parameter $\\theta$ that dynamically regulates functional connectivity based on individual age, it presents more coherent branches and clearer hierarchical structure.\n\n`3.) Is age included as a prediction in Table 2?`: \nThe advantage of NeuroTree is that it is age-aware GCN to learn the influence of age on dynamic FC patterns. We follow the currently existing literature [2,3] conventional practices, incorporating age information as part of the model input for feature learning. We input age during the training phase, `but to avoid data leakage, we do not include age in the prediction phase.` \n\nWe supplement that the complete results of Table 2 the `NeuroTree can also be used in the training and testing process without including age` at the link:https://anonymous.4open.science/r/anonymous_ICML/anonymous_table.png.\n\n**Thank you for your thoughtful review. We've addressed each question as fully as possible within the word limit.**", "claims_and_evidence": "- The claim that \"addiction may lead to stronger brain connectivity signals compared to patients with psychiatric disorders\" lacks justification. The authors should clarify why this is intuitive and provide supporting evidence. Perhaps I missed this. \n\n- The authors claim to model causal graph structures but construct a weighted undirected graph as mentioned \"we construct a weighted undirected graph\". Why was directed graph not used and how can we identify causal relationships instead? Is this based on the tree hierarchy?", "methods_and_evaluation_criteria": "- The model was evaluated on two datasets, but the training process was not thoroughly described, leaving gaps in reproducibility. The authors said that they will release the code upon acceptance which should address this. \n\n- Neurotree was compared against two baseline models and four state-of-the-art models using classification accuracy as the primary metric. While the results are promising, the choice of evaluation metrics could be expanded to include additional performance measures, such as the loss training and test curves, including when adding age as an input $\\theta$ and CMFS objective. Alternatively, more details on the dataset and how the corresponding performance metrics e.g. AUC, Acc, Prec, Rec makes sense in this context should be discussed. \n\n- The authors used Yeo's 7 parcellations to construct the brain tree. It is unclear whether other parcellation schemes were considered and whether they yield similar hierarchical structures.", "theoretical_claims": "I reviewed the theoretical considerations but did not perform a thorough verification. The proposed bounds and theorems appear valid, though a more detailed examination would be necessary to confirm their correctness.", "experimental_designs_or_analyses": "- An ablation study was conducted on contrastive masked FC strengths and age modulation as inputs. However, the necessity of analyzing the rate of spectral norm decrease with k-hop is unclear. The authors should further elaborate on the insights gained from this analysis. What does spectral norm mean and why are we interested in K-hop convergence?\n\n- Table 1 shows a significant performance improvement when age is included as an input. The authors should explain why age contributes to performance and clarify whether previous methods also used age as an input. Additionally, if age is provided during training, how was it predicted for age groups in Table 2?\n\n- The contrastive masked FC strength (CMFS) loss does not significantly improve performance in Table 1. Is this loss more critical for learning the functional hierarchy in Figure 4? How does the brain tree structure change without θ and CMFS loss? Perhaps additional figures similar to Fig. 4 could be included in the appendix with different objectives. \n\n- Could directed graphs be constructed using the temporal $A^d(t)$ matrix? If so, how would this affect the formation of the brain tree network?", "relation_to_broader_scientific_literature": "Neurotree is highly relevant to computational psychiatry and neuroscience. Its application to other datasets and signal modalities, such as EEG or multi-region electrophysiology, could further validate its utility. Additionally, the framework could be adapted to study other network types, such as social or power networks.", "essential_references_not_discussed": "No essential references appear to be missing. However, the authors could consider expanding on advancements in graph-based neural network architectures for brain connectivity analysis.", "other_strengths_and_weaknesses": "- Originality: While the framework builds on existing ideas, the theoretical improvements and the ability to visualize brain tree structures are notable contributions.\n- Significance: The model's ability to elucidate functional hierarchies in brain networks is appealing and could have broad applications in neuroscience.\n- Clarity: The paper is generally well-written, but some sections, such as the training process and the rationale behind certain analyses, could be clarified.", "other_comments_or_suggestions": "No additional comments or suggestions.", "questions_for_authors": "- How does the brain tree structure change when θ and CMFS loss are omitted during network construction?\n- Have the authors considered using alternative parcellation schemes to verify the robustness of the brain tree hierarchy predictions?\n- Could the authors elaborate on the insights gained from analyzing the rate of spectral norm decrease with k-hop?\n- How does the inclusion of age as an input during training affect the model's ability to predict age groups in Table 2? Was age also provided as an input during prediction?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper received one accept, one weak accept, and one reject. While one reviewer raises several concerns, including the lack of strong justification for focusing on psychiatric disorders, limited experimental validation, and unclear interpretations of "chronological brain age," the paper also introduces technically novel and interpretable methods. The proposed NEUROTREE framework, combining k-hop GCNs with neural ODEs, offers a fresh and theoretically grounded approach to modeling hierarchical brain structures. Its focus on interpretability and age-related neural patterns is particularly valuable. Although computational complexity and limited generalization are valid concerns, and reproducibility could be improved with code release, the Area Chair finds that the core issues are largely addressable. Given the interdisciplinary relevance of the work to neuroscience applications, the authors are encouraged to revise the final version to address the reviewers’ concerns and clarify the conceptual framing and empirical contributions.
Decision: Accept (poster)