Let Brain Rhythm Shape Machine Intelligence for Connecting Dots on Graphs
Authors: Jiaqi Ding
Tingting Dan
Zhixuan Zhou
Guorong Wu
Venue: NeurIPS 2025 poster ()
Keywords: Neuroimage
Brain dynamics
Kuramoto model
Graph Learning
Primary area: neuroscience_and_cognitive_science

Abstract: In both neuroscience and artificial intelligence (AI), it is well-established that neural “coupling” gives rise to dynamically distributed systems. These systems exhibit self-organized spatiotemporal patterns of synchronized neural oscillations, enabling the representation of abstract concepts. By capitalizing on the unprecedented amount of human neuroimaging data, we propose that advancing the theoretical understanding of rhythmic coordination in neural circuits can offer powerful design principles for the next generation of machine learning models with improved efficiency and robustness. To this end, we introduce a physics-informed deep learning framework for \underline{B}rain \underline{R}hythm \underline{I}dentification by \underline{K}uramoto and \underline{C}ontrol (coined \textit{BRICK}) to characterize the synchronization of neural oscillations that shapes the dynamics of evolving cognitive states.  Recognizing that brain networks are structurally connected yet behaviorally dynamic, we further conceptualize rhythmic neural activity as an artificial dynamical system of coupled oscillators, offering a shared mechanistic bridge to brain-inspired machine intelligence. By treating each node as an oscillator interacting with its neighbors, this approach moves beyond the conventional paradigm of graph heat diffusion and establishes a new regime of representation compression through oscillatory synchronization. Empirical evaluations demonstrate that this synchronization-driven mechanism not only mitigates over-smoothing in deep GNNs but also enhances the model’s capacity for reasoning and solving complex graph-based problems.

Let Brain Rhythm Shape Machine Intelligence for Connecting Dots on Graphs
Jiaqi Ding
Tingting D Zhixuan Zhou'
Guorong Wu
'Department of Computer Science Department of Psychiatry
The University of North Carolina at Chapel Hill
{jiaqid, zzhixuan]@cs. unc. edu [Tingting_Dan, grwu)&med unc. unc
"The whole universe a complex system, including the human brain." " Immanuel Kant
Abstract
In both neuroscience and artificial intelligence (AI), is well-established that neural "coupling" gives rise dynamically distributed systems. These systems exhibit self- organized spatiotemporal patterns of synchronized neural oscillations, enabling the representation of abstract concepts. By capitalizing on the unprecedented amount of human neuroimaging data, we propose that advancing the theoretical understanding of rhythmic coordination in neural circuits can offer powerful design principles for the next generation of machine learning models with improved deep
efficiency and robustness. To this end, we introduce a physics-informe learning framework for Brain Rhythm Identification by Kuramoto and Control (coined BRICK) to characterize the synchronization of neural oscillations that shapes the dynamics of evolving cognitive states. Recognizing that brain networks are structurally connected yet behaviorally dynamic, we further conceptualize rhythmic neural activity offering a shared mechanistic bridge to brain-inspired machine intelligence. By treating each node as oscillator interacting with its neighbors, this approach moves beyond the conventional paradigm of graph heat diffusion and establishes new regime of representation compression through oscillatory synchronization. Empirical evaluations demonstrate that this synchronization diriiaation-diveen mechanism not only mitigates over-smoothing in deep GNNs but also enhances the model's capacity for reasoning and solving complex graph-based based problems.
artificial dynamical system of coupled oscillators,
Introduction
The evolution of artificial intelligence (AI) has long been intertwined with efforts to decipher the principles of human cognition. Rather than existing as separate disciplines, Al and neuroscience share conceptual foundations in how information is represented and transformed. In the human brain, massive ensembles neurons form densely interconnected circuits that coordinate activity across multiple spatial and temporal scales (4. Many artificial graph systems, ranging from transportation networks online communities, show comparable organizational motifs, including efficient long- range connectivity and modular substructures (57).
Both domains can be viewed through the lens collective dynamics. Neural assemblies give rise to oscillatory rhythms through intricate coupling between regions 200 while graph neural
Equull contribution.
Corresponning author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

networks (GNNs) update node states by iteratively exchanging information, leading to structured representations that emerge across layers [58].
Machine learning shows remarkable proficiency in extracting statistical patterns from vast datasets, yet it remains fundamentally limited in its capacity for flexible reasoning and context-aware integration. Biological neural systems, in contrast, constantly fuse inputs from different sensory streams in real time [24]. Everyday experiences, for example, hearing approaching footsteps before seeing someone, highlight how sensory predictions interact across modalities to construct unified percepts.
Neural oscillations serve as core mechanism that enables coordinated information exchange across different sensory modalities [33] 49]. At the microscopic scale, accumulating evidence indicates that neurons communicate through lateral connections [31, and their rhythmic patterns of excitability, commonly referred to as brain rhythms, emerge from the intrinsic dynamics of local circuits and the biophysical properties of ionic channels 8 Neighboring neurons often synchronize their activity, forming competitive clusters that interpret sensory inputs [25] 45 This process, known as "competitive learning"
[2] compresses information during layer-wise propagation and enhance abstraction. Such synchronization also drives functional specialization among higher-order cortical populations, aligning their activity (analogous fireflies synchronizing their flashes) to generate compact and meaningful neural representations 77]
While modern GNNs such graph convolution network (GCN) 34 and Graph Transformer [63 excel at structural representation learning, they lack the adaptive dynamical properties that charac- terize biological neural systems. On the flip side, cognition in humans and animals is governed by large-scale, dynamic oscillations that support perception, memory and decision-making. This parallel between biological and artificial systems motivates coupled neural oscillators, which presents new opportunities to reshape GNN architectures.
rethink graph learning through the lens of
Following this spirit, we aim to establish a novel learn- ing mechanism for graph- structured data grounded in neural oscillatory synchro- nization, as illustrated in Fig. [1. First, we introduce BRICK, a physics-informed deep learning framework for brain rhythm identification. This model unifies the syn- chronization dynamics of the Kuramoto model (36] with the concept of attending mem- ory [32). allowing us to cap-
Neuroscience Iuspiration:
Machine Learning Innovation
Bruin Rhythms & State Decoding
Graph Neural Oscillatory Syochronization Conventional: Graph hear. diffusion
Brain #bythan
Control
state-related
Oversmosthed state
syme patterus
Proposed: Synch clistering
Sensor
BIG-NDS
feedback
stimuli
synced
Control- driven
Neural sctivity
Brain states
The link between the human brain and artificial intelligence inspires the development of novel learning mechanism of graph-structured data based neural oscillatory synchronization.
Figure 1:
ture rhythmic neural oscillations that underlie cognitive state transitions. By bridging control theory with oscillatory dynamics, BRICK provides a principled foundation for extracting interpretable task- related synchronization patterns from complex neural signals. Second, we generalize this oscillatory synchronization principle to graph learning and propose BIG-NOS, which a biologically inspired graph neural oscillatory synchronization framework. Here, each graph node functions as coupled oscillator whose state evolves dynamically through network interactions. In contrast to conventional graph message passing and convolution paradigms constrained by heat diffusion [II. which often suffer from over- smoothing and information dilution [37 17). BIG-NOS preserves oscillatory co- herence across graph topology. By steering feature propagation via controlled synchronization, the model naturally captures emergent interference patterns in the spatial domain, unveiling meaningful modular structures (e.g., red-blue clustering in Fig. T.. This biologically grounded formulation draws inspiration from how rhythmic synchronization supports information integration in the human brain, enriching the theoretical connection between brain dynamics and graph learning and offering a pow- erful foundation for next-generation machine intelligence, capable of interpretable reasoning, robust pattern integration, and scalable performance real-world graph datasets. In summary, we propose a neural oscillatory synchronization- based graph learning mechanism, uniting a physics-informed model (BRICK) and a biologically grounded GNN (BIG-NOS) with decent real-world performance.
2

---

2 Related Works
Graph neural networks for brain network modeling. GNNs have become foundational tools for analyzing structured data, spanning areas such as chemistry, recommendation systems, biological networks, and social graphs. Initial developments focused on spectral formulations like GCN [3
and GAT [56], which were later extended by more expressive variants including GIN [59] and GraphSAGE (27. Beyond standard tasks of node and graph classification, GNNs have been adapted
for spatio-tempora modeling [60], link prediction, and graph generation (38), with scalability
improved through methods such as clustering and sparse attention [15] 351
Applications to brain networks have followed a similar trajectory, using both structural and functional connectomes to represent the complex topology of neural systems. Early work such as Brain- NeICNN [33] demonstrated the feasibility of disease classification from brain graphs, while more recent studies leverage hierarchical pooling [22] and attention mechanisms [29] to uncover multi-scale
functional organization and enhance interpretability. Temporal extensions [00 have further enabled
dynamic modeling of fMRI and EEG signals, offering insight into evolving neural states over time.
System identification approaches. A key objective in computational neuroscience has been to uncover mechanistic principles governing how the brain regulates perception, action, and cogni- tion 66_ Motivated by parallels between cognitive control and classical control theory, prior work has represented neural dynamics as linear or bilinear control systems 266 40
enabling controllability
analyses through energy-based formulations [1]. More recently, research has shifted toward inte- grating recurrent neural architectures with physics-inspired continuous- dynamics [14] 28 [6].
The Kuramoto model, in particular, has been a central tool for linking structural connectivity to emergent functional activity 9 10), showing that spontaneous neural fluctuations can be explained as coordinated interactions between local oscillators and large-scale network architecture. Our proposed BRICK builds on this line of work by parameterizing these governing dynamics through deep learning framework.
Kuramoto dynamics in machine learning. A closely related effort is the recent development of artificial Kuramoto oscillatory neurons for unsupervised vision representation 4 which leverage synchronization achieve compact and structured feature embeddings. In contrast, our approach emphasizes the bidirectional exchange between neuroscience and machine learning through dynamical systems principles [8, and places additional focus on probing the intrinsic potential of the model. Specifically, we aim to (1) use machine learning models to reveal the mechanisms by which fluctuating brain activity maps to cognitive states, and (2) translate biologically grounded synchronization mechanisms into graph learning to enable large-scale network reasoning without explicit supervision.
3 Methods
In this work, we model both human brain networks and graph- suup-sstructured data as dynamical systems
whose collective behavior is governed by the Kuramoto model [18]. Consider a network of N interacting oscillators with pairwise coupling strengths K.ş (i.j I...., N). Each oscillator oscillates at its intrinsic frequency wi, and the temporal evolution of its phase 0, can be expressed as
d0; = W. + Σ Kij sin (0, - (0).
N
dt
where 0: E R denotes the instantaneous phase of the i-th oscillator. When coupling is absent, the oscillators evolve independently following their own natural frequencies. As interactions accumulate over time, the sinusoidal coupling term in Eq. drives gradual phase alignment across units, giving rise to collective synchronization. This interaction enables groups oscillators to entrain to common rhythms, ultimately forming stable phase-locked clusters (as shown in Fig. bottom-right).
3.1 Deep Model for Brain Rhythm Identification
growing body of research has highlighted the central role of neural oscillations coordinating activity across distributed brain regions g 55
To mechanistically characterize these large-scale synchronization phenomena, the Kuramoto model has become a widely used framework in both thcoretical neuroscience and neuroimaging research [9].
3

---

Problem setup. We formalize the brain network as a weighted graph G = (V.W), where V represents N brain regions (nodes) and W [w,]],== denotes the weighted adjacency matrix. Each weight wi; encodes the coupling intensity between node and Vj. derived from neuroimaging measurement [3) Let the observed BOLD signal be X = x;(() E RR, l..... T, where (t) the signal measured from brain region U at time t. We then construct a physics-inforee informed deep learning model to map neural activity patterns to cognitive outcomes.
Vectorized neural oscillators. To move beyond scalar-valued signals (1) and better characterize complex neural dynamics, we adopt the geometric scattering transform (GST) [23]. GST leverages harmonic wavelets derived from the graph Laplacian construct representations that encode both multi-scale structure and frequency content. We begin by defining the lazy random walk matrix [42]:
G = (AN + W D ?), where IN denotes the N X N identity matrix and D is the diagonal degree matrix associated with W. From this operator, sequence of graph wavelets { A== is generated recursively, enabling a compact and structured embedding of neural oscillatory activity across multiple
=-1
scales. Vo =- IN -G, V/=- == G GG2 1 S h < H. Let xr() E R be the BOLD snapshot at time t. For each scale h, the GST output a" (t) E RN is computed by: (1) applying the wavelet transform (L', =t)). (2) taking element- wise absolute value, and (3) applying the low-pass filter
- G2" yielding (1) = 4 (V,,, x (())..
Vectorized Kuramoto model for multi- mueti-frre neural synchronization. Following GST decom- position, each brain region v yields set of rresovecy-resolved BOLD signals (t) [+) (1)
capturing temporal fluctuations across multiple scales. The evolution of these oscillator phases can be compactly described by vector-valued differential equation:
dē: = + [T dz. Σ (
where f E Rd denotes the oscillator state of region v; on a unit hypersphere, wi its intrinsic frequency, and r global coupling gain parameter. The mapping ø constrains the aggregated phase input to the tangent space of d., enforcing smooth manifold-constrained dynamics.
Rather than simply fitting temporal fluctuations, the aim is to represent neural synchronization in way that reflects how the brain achieves coordinated oscillatory activity with minimal energetic cost. Accordingly, we seek an oscillator-bass -ased representation and a control strategy that are both mathematically stable and biologically meaningful.
Kuramoto model with attending memory. The standard Kuramoto model (Eq. relies on global dynamics, which are not well suited for capturing short-lived functional fluctuations tied to specific cognitive operations. In contrast, the human brain can flexibly emphasize task-relevant inputs while down-weights wwinhwegg irrelevant signals, effectively deciding "what to remember" and "what to ignore." This selective filtering resonates with the concept of attending memory (32) and motivates us to augment the Kuramoto model with a biologically inspired, memory -gnided control mechanism. We incorporate a global optimal control term:
dē:
wi + [T (1: +
(3)
dt
where yi is a feedback control signal derived from attending memory, reflecting observed neural activity. Each y, represents behavior-specific, population-level memory traces that dynamically modulate the influence of oscillator i in given cognitive context. This control formulation allows the system to continuously adapt synchronization dynamics as neural states evolve.
Intuition behind the Kuramoto model with attending memory. Assuming uniform contribution from each brain region 2/ and a symmetric coupling matrix W, the extended Kuramoto dynamics in Eq. 3 can be viewed as a gradient flow that evolves along the steepest descent direction of the following energy functional
3 43
E
Σ i.j
a. Wis - Σ TT
(4)
di The term E acts Lyapunov function that guarantees the system's 's stability, dE(a(()) < 0. Its
first component captures pairwise coupling, promoting synchronization among oscillators è; and i..
'For example, Wy can represent the normalized fiber count in structural connectivity (SC) or the strength of temporal co co-fluctuation in functional connectivity (FC).

---

Graph embedding
Kuramoto Solver
R
(b)
Initialization
Xne
X
W
y(
y p
BOL signal
networks
Costroller
Brain rhythhm identification
Node
Edge
classification
Figure 2: Overview of the proposed framework. (a) BRICK (solid eyan box) models neural oscillatory synchronization for brain rhythm identification. (b) BIG-NOS (dashed orange box) extends the same Kuramoto- based dynamics to graph learning.
proportional the interaction strength Wij. The second component represents task-driven feedback control, aligning each oscillator's is dynamics with its control signal ys. which encodes population-level attention and seraavioo-specffee modulation. Together, these two objectives describe biologically motivated trade-off between network coherence and cognitive specialization.
From a control-theoretic standpoint, the feedback variables Y {uili = I,..,, N can be interpreted as spatial control fields, providing physically grounded attention mechanism distinct from graph attention [56]. These feedback patterns quantify the controllability of each brain region [39], bridging the link between oscillatory synchronization and adaptive regulation of cognitive states. Importantly, the energy formulation also offers gradient-flow interpretation. ensuring that the system mono- tonically decreases E' along its trajectory, thereby maintaining smooth and stable evolution on the hyperspherical manifold.
This unified view reveals why the proposed model alleviates the over-smoothing problem observed in deep GNNs: oscillatory synchronization fosters global coherence while preserving discrimina- tive, high-frequency information. Consequently, this mechanism provides both a neurobiologically meaningful and computationally efficient pathway toward large-scale graph reasoning. forming the theoretical basis of the BIG-NOS model introduced in Sec. 3.2
Physics-informet deep model for brain rhythm identification. Building upon the above formu- lation, we introduce BRICK, a thysics-informer framework designed to predict cognitive states through neural synchronization dynamics. The overall architecture of the model is illustrated in the solid cyan box in Fig. P2.. Specifically, the intrinsic frequency of each oscillator is determined by set of learnable vectors I * ( whose magnitudes control the rotation speed, while the control signal y: is generated through a neural mapping I (f;), with a and j denoting network parameters. The synchronization process evolves in three key steps: (1) Coupling influence aggregation. For each
oscillator ê,, we compute the aggregate input from its neighbors as 2; y: Ey= wiy) â where = [s is a symmetric, trainable reweighting matrix that adaptively modulates the coupling strengths W. This step captures how local interactions shape the global phase landscape. (2) Tangent-space projection. Each z; is then projected onto the tangent space of the unit hyper- sphere at & through dØ. (2:) = 24 - (zt, &. This ensures that the updated direction respects the geometric constraint of the manifold and maintains stability during the synchronization process [12). (3) Phase-state update. Finally, d; is updated via forward Euler integration of Eq. while the controller y4 transformed using mapping function fo to remain on the hypersphere and capture ihase-invariant patterns. These three steps together implement a geometry-awant synchronization mechanism in which oscillator states evolve under both coupling dynamics and task-driven control. The procedure is summarized in Algorithm
The downstream objective combines the task-driven loss L, defined as cross-entropy between pre- dicted and ground-truth cognitive task labels, with the synchronization energy E. For unseen subjects, the trained BRICK synchronizes large oscillator ensemble X using learned parameters and μ., producing both the predicted cognitive state and its associated control pattern Y.
3.2 BIG-NOS: Graph Learning via Neural Oscillation
New learning mechanism of GNN. Conceptually, BIG-NOS extends BRICK from continuous brain dynamics to arbitrary graph data (as shown in the dashed orange box of Fig. (2)). By treating graph nodes analogs of brain regions and the adjacency matrix proxy for inter-regional coupling
5

---

Algorithm 1: Iterative Solver for BRICK Dynamics (Eq. 3)
Input: BOLD signal X ( coupling matrix W
Output: Final oscillator states ( and controller (
Initialization; Obtain initial oscillator states (0) by applying GST to X set
y (x)
Parameterize natural frequencies Ω (X(())
for [ = to L do
for q = 1 to Q do
// Step 1: Aggregate coupling influence
7 7-y0) + ( W) W)
// Step 2: Project to tangent space of the hypersphere
& (Z) 7 - (Z, (
Step 3: Integrate oscillator dynamics
AX + + xx (Z);
((9+1) i + 3 AX
11 Step 4: Renormalize stay manifold X(++1) X ||
end
// Step 5: Update control signal on hypersphere ((+1) t fx(A(at));
end
strength, seamlessly adapt the physics-informed architecture of BRICK general graphs. Both models share the same governing equation rooted in Kuramoto synchronization but differ in their data domains and learning objectives.
Let X |2,],"1 denote the initial graph embeddings. In conventional GNNs, feature propagation follows a graph heat diffusion process, formalized as 2X V (VX), where V and V denote the graph gradient and divergence operators [11]. Prior studies have shown that excessive message passing under this diffusion mechanism leads to over-smoothing, where node representations become indistinguishable (37. Inspired by BRICK, we mitigate over-smoothing by evolving graph features within a latent oscillatory phase space rather than diffusing them over the graph domain. The Kuramoto model's intrinsic oscillatory dynamics, driven by coupling interactions and intrinsic frequencies, naturally prevent convergence to static equilibrium (as in diffusion-based GNNs (51) and instead produce partial synchronization (I].
Unlike conventional attention mechanisms that re-weight neighbor contributions during aggrega- tion [56], our model introduces control patterns Y task-dependent feedback signals that guide synchronization dynamics (Eq. 3 These patterns provide a new perspective on graph controllability through the lens of complex systems theory [3]].
Taken together, conventional GNNs are typically governed by a diffusion equation, in which node features evolve according to a heat-diffusion process. This formulation leads to passive information averaging across neighboring nodes, an approach that effectively captures local smoothness but often suffers from over-smoothing and limited expressiveness in deeper architectures. In contrast, our brain brain-rhythmn rettt-hhttmmmmmim thythm-inspired BIG-NOS is formulated as a neural oscillation system, where each node behaves coupled oscillator whose state evolves according to both its intrinsic frequency and the phase interactions with connected nodes. Rather than diffusive averaging, information propagation in BIG-NOS emerges from oscillatory synchronization, allowing it to represent complex, nonlinear dependencies across the network. Furthermore, unlike conventional attention mechanisms that rely on fixed weighting matrices, BIG-NOS incorporates task-tde feedback control mechanism that dynamically modulates coupling strength, enabling flexible coordination and improved generalization across diverse graph structures.
Network architecture. We discretize the BRICK formulation into a GNN framework, denoted as BIG-NOS, to enable general graph learning. In this architecture, each node is represented as

---

oscillator, and its interactions are governed by the underlying graph topology. The model takes initial graph embeddings X as input, and an encoder (c.g.. a GCN layer) is used to initialize the control term. As illustrated in Fig. 2p. feature evolution in BIG-NOS follows Kuramoto-based dynamics, which are explicitly described in Algorithm
Experiments
In this section, present extensive experiments validate the effectiveness and interpretability of our proposed models across a variety of brain-related and general graph learning tasks.
We conduct a comprehensive evaluation of our proposed models across multiple tasks. We evaluate BRICK for brain rhythm identification and explore its effectiveness unsupervised brain parcel- lation, demonstrating its potential for uncovering latent neural patterns in neural dynamics. We further evaluate BIG-NOS on standard benchmarks including node and graph classification. To analyze its robustness to "over-smoothingg, we examine performance across increasing network depths. We compared BIG-NOS with diverse set of graph-based baselines: GCN [34,, GAT (56), GIN [59), GCNII [13). GraphSAGE (27]. SAN [35), GRAND (I]]. GTN (63]. GraphCON (88 and KuramotoGNN (KGNN) [46). To validate scalability, we also evaluate BIG-NOS on the large-scale ogbn-arxiv dataset. For this task, compare not only against the above baselines but also with
top-performing models from the official OGB leaderboard
4.1 Data Description
To evaluate the effectiveness of BRICK and BIG-NOS, we use the publicly available neuroimaging and graph datasets, respectively.
Human brain datasets
1. HCP-Aging (HCP-A) 15). This dataset includes 717 subjects with 4,846 fMRI scans (300 time points each), covering four tasks (VISMOTOR, CARIT, FACENAME, Resting State).
2. HCCP-Yuugg Adults (IICP-YA) (54). This includes seven cognitive tasks: Motor, Relational, Social, Working Memory, Language, Emotion, and Gambling (175 time points each). The Working Memory task (HCP-W) WM) involves alternating 2-back and O-back conditions with body, place, face, and tool stimuli with total of 405 time points. Preprocessing follows [9].
In both HCP-A and HCP- YA datasets, the brain is parcellated into 116 regions via the AAL atlas [52 Structural connectivity (SC) matrices with the size of 116 x 116 encode fiber counts between regions (normalized per subject), while functional connectivity (FC) derived from Pearson correlations of BOLD signals across brain regions. To test scalability, granularity to 246 regions for HCP-WN. M. We perform 5-fold cross- valsdavali for both the 4-class (HCP-A), 7-class (HCP-YA) and 8-class (HCP-WM) classification tasks.
use the Brainnetome atlas 22 to increase
Graph-based datasets
1. Node classification. We apply BIG-NOS to homophilic and heterophilic graphs sorted by homophily ratio [64]: Texas (h=0.11), Wisconsin (0.21). Actor (0.22), Squirrel (0.22), Chameleon (0.23),
Cornell (0.3), Citeseer (0.74), Pubmed (0.80), Cora (0.81). We also evaluate on the large-scale
ogbn-arxiv dataset from the Open Graph Benchmark (OGB)
30 Details are in Table (Appendix).
Graph classification. We evaluate on ENZYMES and PROTEINS from TUDataset 44 detailed in Table. 5] For homophilic graph data (Cora, Citeseer, Pubmed) in node classification tasks, adopt the semi-supervised 20-per-class training split from (34) and average over random seeds. For heterophilic graph data and TUDataset, we follow [47 and (21], using 10-fold cross-validation and reporting test-set averages. All datasets are evaluated using Accuracy (Acc), weighted-Precisionn (Pre) and F1-Score (F1).
4.2 Performance on Human Brain Data
4.2.1 Brain rhythm identification
Results. We evaluate BRICK on the task of decoding dynamic brain states through neural synchro- nization. Fig. 3 reports the performance across three human brain datasets and seven baseline
7

---

HCP-A
HCP-YA
HCP-WM
0.45
Accuptry
Preciven
000
0.90
P 0001
Px
0.001
000
5t0re
P
*pcs 0.001
0.80
Accuray
Precision
500
0.73
04
Accuraty
Precision
0.70
0.0
0.2
Time
9
8.0
acan
GraphSAGE
6AH
BRICK
GCN
GIN
GAT
acwi
GraphEAGE
SAN
BRICK
GCN
GEM0
GraPhSACE
SAN
BRICK
Figure 3: (a) Performance (%) on human brain data. indicates statistically significant improvement (p
0.001). (h) Inference time (ms/subject) on three human brain datasets.
a HCPA
Visual Attention
Limbic
HCPA
Sensorlmotor
Ventral Attention
Frontoparietal
Resting State
X
X
X
Real Label
CARIT
BRICK
FACENAME
Spectral clustering
VEMOOOR
(
(
Figure 4: (a) Distribution of regional features across different brain states (tasks) the HCP-A dataset. Each brain region's 's feature trajectory over time is projected onto the unit phase space (magnitude = 1), revealing consistent synchronization of the same regions across tasks. (b) Phase-space feature aggregation dataset, where features from different categories each time point mapped onto the unit phase space illustrate their clustering patterns. (c) Unsupervised clustering comparison between BRICK classical spectral clustering, where BRICK yields more coherent and neurobiologically meaningful functional communities.
HCP-A
models. BRICK consistently achieves the best performance (p < 0.001), surpassing all existing hand-designed GNN models.
Diseussion.
These results demonstrate that our physics-based oscillation model effectively synchro- nizes brain regions, producing patterns closely aligned with specific cognitive states. To further explore this relationship, we visualize the phase-space representations of brain regions in Fig. 4a. BRICK captures task-specific synchronization across functionally defined brain network in HCP-A. For instance, Visual network regions (red) cluster clearly during the VISMOTOR task in the deeper coordination. Similarly, tighter phase clus-
representation (L), indicating strong tering emerges in the Sensorimotor and Dorsal Attention networks under tasks such as CART and FACENAME. These results suggest that our model respects functional boundaries in biologically meaningful manner.
We also investigate task-specific synchronization across all subjects. As shown in Fig.4p. individuals performing the same task (e.g., "VISMOTOR" or "FACENAME" in HCP-A dataset) form well- separated synchronization dynamics consistent across subjects and datasets, supporting the notion that neural phase alignment underlines cognitive state representation.
deeper layers (e.g.. x()).. This indicates BRICK captures latent
4.2.2 Potential in unsupervised brain parcellation
To evaluate our model's capability in unsupervised settings, we examine functional parcellation on the HCP-A dataset, which partitions 116 brain regions (AAL atlas) into eight functional subnetworks

---

heterophilic graphs
homophilic graphs
Accuracy &
Wisconsin
Chameleon
Cornell
Pubmed
ENZYMES PROTEINS
Figure 5: Accuracy (%) node and graph classification tasks.
indicates significant improvement (p < 0.001).
(default mode, frontoparietal, limbic, ventral/dorsal attention, sensorimotor, visual, and cerebellar networks). We train an unsupervised variant of BRICK that identifies functional communities by minimizing pairwise distances between synchronized oscillators 2; and Tjj using a Rayleigh quotient objective [50]. As shown in Fig. E., compared to classical spectral clustering, BRICK more effectively synchronizes related brain regions into meaningful clusters, demonstrating strong potential for personalized brain parcellation.
4.2.3 Ablation study and model efficiency
We conduct an ablation study on the GST to assess the impact of different frequency scales. Specifi- cally, we vary the wavelet order (from 0 2) and level (1 or the HCP-YA dataset (fixed-group split), which define a multi-branch wavelet tree: the level determines the tree's depth, while the order sets the branching factor. As shown in Table increasing either the level order generally improves performance by incorporating additional frequency components. However, this comes at the cost of increased computational complexity. For example, using level=2 and order= [0, 1] expands the feature dimension from [N] to [N, 7]. due to the increased number of wavelet filters.
Table 1: Ablation results of GST with different levels and orders on HCP-YA YA.
Level
Order
Ace (%)
Pre (%)
F1 (%)
[0]
-0.41
-0.45
-0.42
[0, I]
-0.14
-0.05
-0.16
[0, 21
85,20
85,53
85.13
[0]
-0.23
-0.24
-0.24
[0, 1]
+0,21
+0.25
+0.22
We provide detailed comparison of inference time per subject across three human brain datasets in Fig. 3b. Overall, all baseline GNNs demonstrate low computational cost, with inference typically below 2 ms per subject. We further PDE-based GRAND and KuramotoGNN are notably slower. In contrast, BRICK achieves comparable efficiency to lightweight baselines such as GCN and GraphSAGE. Even the larger HCP-WM dataset (246 regions), BRICK maintains modest runtime of 1.56 ms per subject (as shown in Fig. Bb-the rightmost data point), confirming its scalability and computational practicality for large-scale brain analysis.
4.3 Performance on Graph Data
4.3.1 Node classification
Result. Fig. 5 (left and middle) presents the results of eleven models across nine node-classification benchmarks. Overall, BIG-NOS achieves the best or second-best accuracy on the majority of datasets, showing strong generalization across both homophilic and hetcrophilic graphs. For homophilic graphs (Citeseer, Pubmed, Cora). BIG-NOS maintains competitive performance on par with advanced architectures like GCNII and GRAND. For heterophilic graphs, it achieves the highest performance on Actor, Squirrel, and Chameleon, where conventional message-passing GNNs typically fail due to inconsistent neighborhood features. In contrast, BIG-NOS continues to improve through phase-basee coupling, which allows nodes to synchronize dynamically rather than relying solely feature similarity. This oscillatory mechanism enables coherent message propagation even when node attributes are diverse, making BIG-NOS especially suitable for heterophilic settings.
Scalability. To assess the scalability of BIG-NOS, we evaluated its performance on the large-scale agbn-arxiv dataset. As reported in Table 2 our model achieved accuracy of 0.86, outperforming

---

Table 2: Performance ogbn-arxiv test set. Bold and underline indicate the best and second best results.
Dataset
Metric
GCN
GIN
GAT
GCNII
GraphSAGE
SAN BiGTex SinTTGGHTAPE-ROGAAT BIG-NOS
Acc
0.70
0.71
0.70
0.71
0.71
0.69
0.89
0.78
0.86
ogbn-arxiv
Pre
0.70
0.70
0.69
0.71
0.70
0.68
0.86
F1
0.69
0.69
0.68
0.70
0.69
0.68
0.85
BiGTex and SMnEEE---AEE-Reeeeee are the two models on the OGB Le at the time of wriling. Pre and F1 are not reported.
all standard baselines and ranking second on the leaderboard, just behind BiGTex (0.89). These results underscore the strong generalization ability of our oscillatory synchronization mechanism on large graphs.
These results show that our mes- sage aggregation strategy effectively promotes synchronization among nodes of the same class while preserving the structural dependencies de- fined by the coupling strength. In contrast con- ventional GNNs, which tend to drive neighboring nodes toward indistinguishable embeddings, our model maintains representational diversity and prevents excessive feature homogenization. As result, it inherently resists over-smoothing the network depth increases. This property is empirically verified Fig. where we observe consistently high classification performance even with up to 128 layers, demonstrating the robustness of our synchronization framework against the depth-ine induced degradation common in traditional GNNs.
Discussion.
Performance
Accuracy
I-score
0.75
Precision
0.7
0.65
16
32
64
128
Number of layers
Figure 6: Performance of BIG-NOS as the network depth increases Cora (red) and Citeseer (blue).
4.3.2 Graph classification
Result. Fig. (right) reports graph classification accuracy the ENZYMES and PROTEINS datasets
from TUDataset 44 BIG-NOS achieves the best accuracy on ENZY MES (60.0014.2s) and performs competitively PROTEINS (75.0212.6) despite using minimal batch size (batch size=1). These results indicate that our model maintains strong generalization from node-level to graph-level tasks without architectural modification.
Discussion. Graph classification is conceptually analogous to brain rhythm identification, where the goal is to predict a global state emerging from local interactions. The success of BIG-NOS in this setting further validates the ability of our synchronization mechanism to generate global patterns that effectively distinguish different graphs, which reinforces its applicability beyond node-level tasks.
Table summarizes the average accuracy across homophilic, heterophilic, and all datasets. The balanced performance is particularly encouraging: whereas many existing models tend to specialize single regime (for example, GCN performs well primarily on homophilic graphs), our model demonstrates consistent generalization across both homophilic and heterophilic graphs. This robust- ness suggests that the underlying neural oscillation mechanism inherently generalizable and largely independent of specific graph structures.
Table 3: The average accuracy homophilic, heterophilic, and all datasets.
GCN
GIN
GAT
GCNII
GraphSAGE
SAN
GRAND
GTN
GraphCON
KGNN
BIG-NOS
Hete. Avg
42.19
43.99
62.01
58,19
57.18
50.55
54.99
56.28
51.23
69.18
Homo. Avg
76.68
66.81
75.77
79.20
75.46
72.99
78.32
74.77
78.61
71.96
76.76
Total Avg
53.69
48.58
54.58
67.74
63.95
62.45
59.80
61.58
63.73
58.14
71.70
Conclusion
In this work, we introduce BRICK, new deep learning framework inspired by the biological mechanisms underlying fluctuating brain activity and cognitive state formation. Building on this foundation, we extended the concepts of neural oscillatory synchronization and attentional memory to graph domain and derive a novel graph-neura architecture, BIG-NOS. Extensive experiments demonstrate that BIG-NOS effectively alleviates over-smoothing problem conventional GNNs, and delivers competitive performance across diverse GNN benchmarks. These results highlight the model's promise for tackling challenging graph-learning problems while offering biologically grounded view of information propagation.
10

---

Acknowledgement
This work was supported by the National Institutes of Health [AG091653, AG068399, AG084375) and the Foundation of Hope. The views and conclusions contained in this document are those of the authors and should not interpreted as representing the official policies, either expressed or implied, of the NIH.
References
[1] Juan Acebrón, Luis L Bonilla, Conrad Pérez Vicente, Félix Ritort, and Renato Spigler. The kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics, 77(1):137-18, 2005.
[2] Shun-Ichi Amari and Michael A Arbib. Competition and cooperation in neural nets. Systems Neuroscience, pages 119-163, 1977.
[3] Toshio Aoyagi. Network of neural oscillators for retrieving phase information. Physical Review Letters, 74(20):4075, 1995.
[4] Danielle S Bassett and Olaf Spors. Network neuroscience. Nature Neuroscience, 20(3):353- 364, 2017.
[5] Susan Y Bookheimer, David H Salat, Melissa Terpstra, Beau M Ances, Deanna M Barch, Randy I. Buckner, Gregory Burgess, Sandra W Curtiss, Mirella Diaz-Santos, Jennifer Stine Elam, al. The lifespan human connectome project in aging: overview. Neuroimage, 055:335-44,, 2019.
[6] Matthew M Botvinick and Jonathan D Cohen. The computational and neural basis of cognitive control: charted territory and new frontiers. Cognitive Science, 38(6)):249-1222 2014.
[7] Adrian Burgess. Towards unified understanding of event-related changes in the eeg: The firefly model of synchronization through cross cros-ffeeyency phase modulation. PLOS ONE, 7()):445630, 09 2012.
[8] György Buzsáki. Rhythms of the Brain. Oxford University Press, 10 2006.
[9] Joana Cabral, Etienne Hugues, Olaf Sporns, and Gustavo Deco. Role of local network oscilla- tions in resting-state functional connectivity. Neuroimage,
2011.
[10] Katerina Capouskova, Morten L Kringelbach, and Gustavo Deco. Modes of cognition: Evidence from metastable brain dynamics. Neuroimage, 260:1 19489, 2022.
[11] Ben Chamberlain, James Rowbottom, Maria I Gorinova, Michael Bronstein, Stefan Webb, and Emanuele Rossi. Grand: Graph neural diffusion. In International Conference Machine Learning, pages 1407-1418. PMLR, 2021.
[12] Sarthak Chandra, Michelle Girvan, and Edward Ott. Continuous versus discontinuous transitions in the d-dimensional generalized kuramoto model: Odd d is different. Physical Review. X, 9(1), 2019.
[13] Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In International Conference Machine Learning, pages 1725-1735. PMLR, 2020.
[14] Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 2018.
[15] Wei-Lin Chiang, Xuanqing Liu, Si Si, Yang Li, Samy Bengio, and Cho-Jui Hsieh. Cluster-gen: An efficient algorithm for training deep and large graph convolutional networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery Data Mining, pages 257-266, 2019.

---

[16] Tingting Dan, Hongmin Cai, Zhuobin Huang, Paul Laurienti, Won Hwa Kim, and Guorong Wu. Neuro-rdm: an explainable neural network landscape of paction-diffusion model for cognitive task recognition. In International Conference on Medical Image Computing and Computer-Asiisted Intervention, pages 365-374. Springer, 2022.
[17] Tingting Dan, Jiaqi Ding, Ziquan Wei, Shahar Kovalsky, Minjeong Kim, Won Hwa Kim, and Guorong Wu. Re-think and re-design graph neural networks in spaces of continuous graph diffusion functionals. Advances in Neural Information Processing Systems, 36:59375-59387, 2023.
[18] Tingting Dan, Jiaqi Ding, and Guorong Wu. Explore brain-inspired machine intelligence for connecting dots graphs through holographic blueprint of oscillatory synchronization. Nature Communications, 16(1 16(1):9425, 2025.
[19] Tingting Dan, Ziquan Wei, Won Hwa Kim, and Guorong Wu. Exploring the enigma of neural dynamics through a
mixer landscape for riemannian manifold. International Conference on Machine Learning (ICMI), 2024.
[20] Tahnée Engelen, Marco Solcà, and Catherine Tallon-Baudy Interoceptive rhythms in the brain. Nature Neuroscience,
2023.
[21] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification. arXiv preprint arXiv:1912.0989) 2019.
[22] Lingzhong Fan, Hai Li, Junjie Zhuo, Yu Zhang. Jiaojian Wang. Liangfu Chen, Zhengyi Yang, Congying Chu, Sangma Xie, Angela R Laird, al. The human brainnetome atlas: new brain atlas based on connectional architecture. Cerebral Cortex, 26(8):3508 -3260, 2016.
[23] Feng Gao, Guy Wolf, and Matthew Him. Geometric scattering for graph data analysis. In International Conference on Machine Learning, pages
PMLR, 2019.
[24] Asif A. Ghazanfar and David J. Lewkowicz. The emergence of multisensory systems through perceptual narrowing. Trends in Cognitive Sciences, 13(11))470-478, 2009.
[25] Charles M Gray, Peter König, Andreas K Engel, and Wolf Singer. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature, 338(6213):334-3333 1989.
[26] Shi Gu, Fabio Pasqualetti, Matthew Cieslak, Qawi K Telesford, Alfred B Yu, Ari E Kahn, John D Medaglia, Jean M Vettel, Michael B Miller, Scott T Grafton, et al. Controllability of structural brain networks. Nature Communications, 6(1):8414, 2015.
[27] William L. Hamilton, Rex Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Proceedings of the 31 International Conference on Neural Information Processing Systems, NIPS' 17, page 1025-1035. 2017.
(28) Ramin Hasani, Mathias Lechner, Alexander Amini, Daniela Rus, and Radu Grosu. Liquid time-constant networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 7657-7666, 2021.
[29] SM Hadi Hosseini, Fumiko Hoeft, and Shelli R Kesler. Gat: graph- theorrcchltttlll analysis toolbox for analyzing between-group differences in large-scale structural and functional brain networks. PloS One, 7(7):c40709, 2012.
[30] Weihua Hu, Matthias Fey, Marinka Zitikk Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. Advances Neural Information Processing Systems, 33:22118 18-22133, 2020.
[31] David H Hubel and Torsten N Wiesel. Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. The Journal of Physiology, 160(1):106, 1962.
[32] Jamie B. Hutchinson and Nicholas B. Bark-Browne. Memory-guided attention: control from multiple memory systems. Trends in Cognitive Sciences, 16(12):576-57, 2012.
12

---

[33] Jeremy Kawahara, Colin J Brown, Steven P Miller, Brian G Booth, Vann Chau, Ruth E Grunau, Jill G Zwicker, and Ghassan Hamarneh. Brainnetenn: Convolutional neural networks for brain networks; towards predicting neurodevelopment. Newroimage, 146: 146:1038-149) 2017.
[34] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv: 1009.02009 2016.
[35] Devin Kreuzer, Dominique Beaini, Will Hamilton, Vincent Létoureau, and Prudencio Tossou. Rethinking graph transformers with spectral attention. Advances in Neural Information Pro- cessing Systems, 34:21618 21629, 2021.
[36] Yoshiki Kuramoto. Chemical Turbulence, pages 111-140. Springer Berlin Heidelberg, Berlin, Heidelberg, 1984.
[37] Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional net- works for semi- eemi-supervised learning. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Con- ference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence, MAAI'18/AAM 18/EAAI 18. AAAI Press, 2018
[38] Renjie Liao, Yujia Li, Yang Song, Shenlong Wang. William L. Hamilton, David Duvenaud, Raquel Urtasun, and Richard Zemel. Efficient graph generation with graph recurrent attention networks. Curran Associates Inc., Red Hook, NY, USA, 2019.
[39] Yang- Yu Liu, Jean-Jacques Slotine, and Albert-Láázzl Barabási. Controllability of complex networks. Nature, 167713,, 2011.
[40] Amanda L McGowan, Linden Parkes, Xiaosong He, Ovidia Stanoi, Yoona Kang, Silicia Lomax, Mia Jovanova, Peter J Mucha, Kevin N Ochsner, Emily B Falk, et al. Controllability of structural brain networks and the waxing and waning of negative affect in daily life. Biological Psychiatry Global Open Science, 2(4))432 439, 2022.
[41] John Dominic Medaglia, Fabio Pasqualetti, Roy H Hamilton, Sharon L Thompson Scchil,, and Danielle S Bassett. Brain and cognitive reserve: Translation via network control theory. Neuroscience & Biobehavioral Reviews, 75:53-64, 2017.
[42] Yimeng Min, Frederik Wenkel, and Guy Wolf. Scattering gen: Overcoming oversmoothness in graph convolutional networks. Advances in Neural Information Processing Systems, 33: 33:14498- 14508, 2020.
[43] Takeru Miyato, Sindy Löwe, Andreas Geiger, and Max Welling. Artificial kuramoto oscillatory neurons. arXiv preprint arXiv:2410. 13821, 2024.
[44] Christopher Morris, Nils M Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neumann. Tudataset: A collection of benchmark datasets for learning with graphs. arXiv preprint arXiv:2007.08661 2020.
[45] Vemon B Mountcastle. The columnar organization of the neocortex. Brain: a journal of neurology, 120(4):701-722, 1997.
[46] Tuan Nguyen, Hirotada Honda, Takashi Sano, Vinh Nguyen, Shugo Nakamura, and Tan Minh Nguyen. From coupled oscillators to graph neural networks: Reducing over-smoot via kuramoto model-based approach. In International Conference on Artificial Intelligence and Statistics, pages 2710-2716 PMLR, 2024.
[47] Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. Geom-gen: Geometric graph convolutional networks. arXiv preprint arXiv:2002.05: 052887 2020.
[48] T Konstantin Rusch, Ben Chamberlain, James Rowbottom, Siddhartha Mishra, and Michael Bronstein. Graph-coupled oscillator networks. In International Conference on Machine Learning, pages 18888 18988811899..188 PMLR, 2022.
[49] Daniel Senkowski, Till R Schneider, John J Foxe, and Andreas K Engel. Crossmodal binding through neural coherence: implications for multisensory processing. Trends in Neurosciences, 3 3108):401440, 2008.
13

---

[50] Uri Shaham, Kelly Stanton, Henri Li, Boaz Nadler, Ronen Basri, and Yuval Kluger. Spectralnet: Spectral clustering using deep neural networks. In Proc. ICLR 2018, 2018.
[51] Steven Strogatz. From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 143(1-4): 1231-, 2000.
[52] Nathalie Tzourio-Mazoyeer Brigitte Landeau, Dimitri Papathanassiou, Fabrice Crivello, Octave Etard, Nicolas Delcroix, Bernard Mazoyer, and Mare Joliot. Automated anatomical labeling of activations in spm using macroscopic anatomical parcellation of the mni mri single-subject brain. Neuroimage, 15()):273-289 2002.
[53] Nienke Van Atteveldt, Micah M Murray, Gregor Thut, and Charles E Schroeder. Multisensory integration: flexible use of general operations. Neuron, 81(6): 1240-1124-2 2014.
[54] David C Van Essen, Stephen M Smith, Deanna M Barch, Timothy Behrens, Essa Yacoub, and Kamil Ugurbil. The wu-minn human connectome project: An overview. Neuroimage, 80:62-79, 2013.
[55] Francisco Varela, Sean-Philippe Lachaux, Eugenio Rodriguez, and Jacques Martinerie. The brainweb: phase synchronization and large-scale integration. Nature Reviews Neuroscience, ((4):229-399) 2001.
[56] Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. arXiv preprint 1710:/770..0000 2017.
[57] Dunean J Watts and Steven H Strogatz. Collective dynamics of 'small-words netwol'.s.. Nature, 393(6684):440 442, 1998.
[58) Louis-Pascal Xhonneux, Michaël Defferrard, and Pierre Vandergheynst. Continuous graph neural networks. In Proceedings of the 37th International Conference on Machine Learning, pages 10432-10441, PMLR, 2020.
[59] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanic Jegelka. How powerful are graph neural networks? arXiv preprint 1eXiv:/800.0000 2018.
[60] Sijie Yan, Yuanjun Xiong, and Dahua Lin. Spatial temporal graph convolutional networks for skeleton-based action recognition. In Proceedings of the "hirtt-Second AAAI Confer- ence on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence. AAAI 18/AAAI' 18/A'// 18/EAAI 18. AAAI Press, 2018.
[61] Junhan Yang. Zheng Liu, Shitao Xiao, Chaozhuo Li, Defu Lian, Sanjay Agrawal, Amit Singh, Guangzhong Sun, and Xing Xie. Graphformers: Gnn-nested transformers for representation learning on textual graph. Advances in Neural Information Processing Systems, 34:28798- 28810, 2021.
[62] Rex Ying, Jiaxuan You, Christopher Morris, Xiang Ren, William L. Hamilton, and Jure Leskovec. Hicrarchical graph representation learning with differentiable pooling. In Pro- ceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS'18, 18, page 4805 4815. Curran Associates Inc., 2018.
[63] Seongjun Yun, Minbyul Jeong, Rachyun Kim, Jaewoo Kang, and Hyunwoo J. Kim. Graph transformer networks. In Advances in Neural Information Processing Systems (NeurIPS), volume 32, 2019.
[64] Jiong Zhu, Yujun Yan, Lingxiao Zhao Mark Heimann, Leman Akoglu, and Danai Koutra. Beyond homophily in graph neural networks: current limitations and effective designs. In Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS '20. Curran Associates Inc., 2020.
14

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Please refer to Abstract and the last paragraph of Introduction
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results
be expected to generalize other settings.
is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Please refer to Appendix A.4
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low images are taken in low lighting. Or speech-to-text system might be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
15

---

Justification:
The paper does not include theoretical results.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification:
The code is released at Anonymous GitHuub
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm. (b) the contribution is primarily a new model architecture, the paper should describe the architccture clearly and fully.
(c) If the contribution is new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. the case of closed-source models, may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
16

---

Answer: [Yes]
Justification:
The code is released at Anonymous GitHub
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.cm
iy for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
/ cppppi.ccppulli//uuuiu wwwtii/gguie///eeeeceoocc CoodeSubeiiieeee ssi wsionnolli onnolicy)))) for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Please refer to Appendix A.2
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: We use paired-ttest.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, at least for the experiments that support main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
should be clear whether the error bar is the standard deviation the standard error of the mean.
17

---

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of enrors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: Please refer Appendix A.2
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g.. preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: wwt.rrissccppubblii httprppp......b.....hh..ccccaeea
Answer:
[Yes]
Justification:
We have reviewed the NeurIPS Code of Ethies.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is a special consid- eration due to laws or regulations their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: Please refer Appendix A.4
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to
18

---

generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative socictal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: The paper poses such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: Please refer to Appendix A.2 for the use data.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, wwppppswpptseeee wwworeewwthhhe....c. com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's is creators.
13. New assets
Question: Are new assets introduced in the paper well documented and the documentation provided alongside the assets?
19

---

Answer: [Yes]
Justification: We release a new codebase implementing the proposed models, with documen- tation and instructions provided in the Anonymous GitHub
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According the NeurIPS Code of Bthies, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for rescarch with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsoureing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMSs if it is an important, original, or non-standard component of the methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality the research, declaration is required.
Answer: [NA]
20

---

The core method development in this research does not involve LLMs as any important, original, or non-standard components.
Justification:
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non- standad components.
Please refer to our LLM policy hattps://neurips... 200p://ee////oofeeeeee////////// for what should or should not be described.
21

---

A Technical Appendices and Supplementary Material
A.1 Data Description
Table 4: Dataset description for node classification.
Texas
Wisconsin
Actor
Squirrel
Chameleon
Cornell
Citeseer
Pubmed
Cora
ogbn-arxiv
Hom, ratio
0.11
0.21
0.22
0.22
0.23
0.3
0.57
0.74
0.81
Nodes
185
251
7,600
5,201
2,277
183
3,327
19.717
2.708
169,343
# Edges
295
466
26,752
198,493
31,421
280
4,676
44,327
5,278
1,166,243
Classes
5
40
Table 5: Dataset description for graph classification.
ENZYMES
PROTEINS
Aug # Nodes
32.63
39.36
Avg # Edges
62.14
39,06
Classes
# Graphs
600
1,113
A.2
Hyperparameters
As human data example, for all methods, the hidden dimension is set to 256. The network depth and batch size are set to 2 and 64 for the comparison methods, while for our BRICK, they are set to 16 and 256, respectively.
A.3
Connections and Distinctions from Prior Approaches
A.3.1 Relevance to neuroscience domains
In principle, the learning behavior of BRICK is shaped by the same governing equations (i.e., Kuramoto model) that describe the neural oscillatory dynamics responsible for generating cognition and behavior. We further introduce structural- structuralfuunctional coupling, implemented through the geometric scattering transform, to emulate the cros-ffreyuecy coupling observed in cognitive neuroscience. In addition, we incorporate adaptive control term to enhance task-relevant modulation. Drawing inspiration from neural oscillatory synchronization, our BRICK model not only improves predictive accuracy but also enhances interpretability, thereby offering novel insights into cognitive processes.
A.3.2 Advances over conventional methods brain rhythm identification
While traditional models, including CNNs, RNNs, Transformers, and GNNs, have demonstrated strong empirical performance brain rhythm identification, they largely remain data-driven and agnostic to the underlying neurobiological mechanisms, they often treat the brain as a generic data source without sufficient domain knowledge. In contrast, BRICK explicitly incorporates neural synchronization dynamics, well-established principle in cognitive neuroscience, into the learning architecture. This alignment with neuroscience allows us to interpret learned patterns (e.g., coupling strengths or synchronization clusters) in biologically meaningful terms (as shown in Figure 3), such as inter-regional communication or pathological desynchronization. Therefore, compared to conventional approaches, BRICK is not just deep model for feature representation learning, it shows the potential to establish a biologically inspired reasoning system. By embedding core principles of brain dynamics into learnable, differentiable architecture, our proposed deep model enables achieving more interpretable, temporally- awaar, and robust brain state identification.
A.3.3 Differences from prior dynamical models
While both BRICK and Cabral et al. 9 adopt the Kuramoto model explore brain dynamics, their roles and applications diverge significantly. In their work, the Kuramoto model is used as forward simulator reproduce biologically observed functional connectivity (FC) patterns from empirical structural connectivity (SC), with the goal of explaining emergent brain phenomena. In contrast, our BRICK repurposes the Kuramoto model as computational module within a machine learning framework. Rather than passively simulating dynamics, BRICK actively learns to perform representation learning, reasoning, and predictive tasks on graphs using dynamic synchronization mechanisms.
22

---

For methodological paradigm, while Capouskova et al. [10] follow data-driven analytical paradigm, using autoencoders and clustering to uncover latent cognitive states from empirical fMRI data, they encode BOLD phase coherence data using modern machine learning tools without Kuramoto model. Their focus lies in neuroscientific pattern discovery and interpretation. duces physics-informed learning framework that embeds the principles of neural synchronization, specifically via the Kuramoto model, directly into the learning process. The outcomes of these two approaches are also distinct: Capouskova et al. [10] generate empirical insights about cognitive brain states, whereas BRICK yields new machine learning models that not only excel in brain-related tasks but also generalize broader graph-based Al problems.
contrast, BRICK intro-
Compared with 14), our BRICK framework offers two key advantages. First, neuroscientific grounding: starting from fMRI-derived brain rhythms, we augment the Kuramoto core with a task- driven feedhack controller that mimics cognitive control, thereby preserving biological interpretability. Second, domain-specific impact: on real HCP fMRI data BRICK achieves the best task decoding and unsupervised parcellation, while on graph benchmarks delivers competitive accuracy and remains resistant to oversmoothing even at 128 layers.
Taken together, these properties make BRICK a biologically inspired but practically efficient module that bridges brain dynamics and general-purpose Al.
A.4 Discussion and Limitation
Although our approach demonstrates promising accuracy brain connectomes and node-level benchmarks, its computational efficiency remains limited due to the iterative solver required to integrate BRICK dynamics. Unlike feed-forward GNNs, model relies repeated geometric projections, oscillator updates, and manifold renormalization, all of which introduce substantial per-batch computational overhead. This bottleneck becomes particularly pronounced datasets such as TUDataset.
Despite these efficiency constraints, method still achieves competitive graph classification performance, underscoring the robustness of our dynamical-systene perspective beyond ncuroscience applications. In future work, plan develop more efficient numerical solvers and adaptive graph batching strategies to significantly improve scalability. We also aim to apply our framework to broader multimodal and clinical datasets further assess its generalizability and translational value.
More broadly, this line of research highlights the potential of brain-inspired synchronization mecha- nisms for building more interpretable and principled Al systems.
More broadly, this line of research highlights the promise of brain-inspired synchronization mech- anisms for building more interpretable and efficient AI systems, while also drawing attention to important ethical considcrations surrounding the use of neural data and the socictal impact of au- tonomous decision pocision-making technologies.
23


Reviews:
[{"review_id": "EdepfYv1Ox", "reviewer": "Reviewer_NLZ3", "summary": "Inspired by neural oscillation mechanism, the authors propose novel deep learning methods that utilize neural oscillation, BRICK and BIG-NOS. BRICK was applied for brain fMRI datasets. Specifically, BRICK achieved the best performance in brain analysis tasks, compared to simple GNN baselines. BIG-NOS was applied for graph benchmark datasets, and the authors report its superior node and graph classification accuracies compared to simple GNN baselines. The authors argue that the proposed approaches represent a new graph learning mechanism with SOTA performance.", "questions": "See weaknesses", "limitations": "Yes, but the authors only discussed marginal limitations.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The **key strengths** include:\n\n- [S1 novelty]. The proposed approaches seem somewhat novel.\n- [S2 visualization]. The visualizations are among the most beautiful ones I have seen in AI conferences.\n\nHowever, I note two **critical limitations** to the present work.\n\n- [W1 poor experimental setting]. **Many of the reported baseline performances in Table 3 are substantially lower than those reported in eariler studies**. For instance, GCNII performance on Cora is reported as 79.92, whereas the other works generally report up to 86 in the same train/val/test split. In fact, I have run many experiments with the baselines before, and the reported performances are definitely not optimal. I assume this is due to poor choice of hyperparameters. According to Appendix A.2, the authors did not tune some of the key hyperparameters for all baseline methods, and some other key hyperparameters, such as learning rate, are not reported. With such poor experimental setting, I find it hard to trust the experimental outcomes.\n- [W2 lack of method justification]. I am not convinced of the justification to join neural oscillation with graph learning. Due to the issue in W1, empirical justification is hardly achieved. No formal theories were developed to justify them. The motivation illustrated in the introduction section (e.g., node in a graph as a couple oscillator) is only speculative and have not been analyzed or supporte by the prior works. Thus, I am not convinced why the proposed method is necessary or important.\n\nBesides, I also find some **notable limitations**.\n\n- [W3 missing details]. Some of important details are missing. For instance, formal description of the proposed method BIG-NOS seems to be missing. Is the Governing Equation in Table 1 the exact functional form of BIG-NOS? If so, please clarify. Also, the Figure 3 results should be buttressed with numeric outcomes. With only the plots, it is hard to objectively discern how they support the authors’ claims.\n- [W4 missing prior works]. There have been some previous works that connect neural oscillation with graph neural networks [1,2]. However, the authors cited none of them. The authors should clarify how the proposed method compare to those earlier works.\n    - [1] Graph-Coupled Oscillator Networks, ICML 2022\n    - [2] From Coupled Oscillators to Graph Neural Networks: Reducing Over-smoothing via a Kuramoto Model-based Approach, ICML 2024\n- [W5 unsupported claims]. The authors claim for SOTA performance. However, all the baselines are outdated, with the most recent one published in 2021. Even if the issue raised in W1 is somewhat justified, there is no evidence to demonstrate that the proposed method is not SOTA.", "was_revised": true, "final_justification": "I raised concerns about the paper's unfair experimental setting for graph learning benchmarks. During rebuttal, the authors reported results with improved experimental settings and revised their argument accordingly. Additionally, the authors reported SOTA performance of the proposed method in more brain application benchmarks. Considering all those factors, I consider a weak accept to be a proper final score.", "rebuttal": "We sincerely thank the reviewer for acknowledging the novelty and the value of our work. Below, we provide detailed responses to all concerns.\n### *[W1 poor exp setting]，[W5 unsupported claims]*\n**[Exp setting and model selection]**\n\nWe acknowledge that some of the reported accuracy for baseline models are lower than those in prior studies that specifically optimized each model/dataset. However, our intention was to ensure a **fair and controlled comparison** across all models under **uniform exp settings**, particularly important consideration in neuroscience, where the focus lies in uncovering mechanistic and functional insights.\n\nFor all models, we fix major hyperparams as hidden dim 256;  #layers 4 (2 for GCN/GAT); #epochs 1500; lr 5e-4 – 1e-3; weight decay 5e-4. We will report all the details in final draft. \n\nWhile these settings may not be optimal for every model, e.g., GCNII perform best with 32–64 layers on Cora,which is not applicable for GCN or GIN due to over-smoothing or instability. Our choice thus ensures comparability under identical architectural constraints. We would also like to emphasize that the accuracy of GCN on Cora in our setting (81.66%) **is highly consistent** with prior reports (e.g., 81.5% in GRAND), which validates our implementation.\n\nIn addition, since **our main focus is on neuroscience-related applications aligns with the Neuroscience and cognitive science track**, we prioritize baseline models that have been validated in computational neuroscience, such as GCN, GAT, GIN etc. [1]. Nevertheless, we have extended our comparisons by including GTN (2019), GRAND (2021), GraphCON (2022) and KuramotoGNN (KGNN) (2024). The results show that our models achieve significant improvements in brain datasets and retain strong performance in general graph benchmarks.\n\n**Brain states identification:**\n|Acc(%)|HCP-A|HCP-YA|HCP-WM|\n|-|--|-|-|\n|GRAND|86.18±1.79|47.36±0.67|33.27±2.63|\n|GTN|82.56±0.87|53.42±1.21|30.68±0.36|\n|GraphCON|87.87±2.29|66.29±1.04|44.12±2.1|\n|KGNN|85.53±1.68|45.68±1.12|35.79±1.54|\n|BRICK|95.55±0.77|84.20±1.60|89.22±1.71|\n\n**Node classification:**\n|Acc(%)|Texas|Wisconsin|Actor|Squirrel|Chameleon|Cornell|Citeseer|Pubmed|Cora|\n|-|-|-|-|-|-|-|-|-|-|\n|GRAND|67.42±7.81|73.09±4.83|33.34±1.16|34.45±1.84|38.43±3.33|56.55±9.15|72.1±1.07|79.12±0.3|83.73±0.58|\n|GTN|74.52±10.2|68.89±11.9|35.42±1.5|36.45±1.72|55.84±4.08|58.81±15.09|68.8±1.69|76.90 ± 0.77|78.60±1.72|\n|GraphCON|82.43±4.72|83.72±4.48|35.13±1.38|26.9±2.17|33.75±3.77|74.59±2.48|62.9±5.95|71.21±7.36|62.73±5.79|\n|KGNN|71.75±5.57|69.43±6.18|31.5±0.82|35.02±1.67|38.5±2.78|61.16±9.16|72.06±1.53|67.24±0.36|76.57±1.4|\n|BIG-NOS|81.35±3.51|82.16±3.56|35.8±0.6|68.06±1.65|74.5±1.13|73.24±4.75|70.63±0.36|77.8±0.23|81.86±0.25|\n\n**Graph classification:**\n|Acc(%)|ENZYMES|PROTEINS|\n|-|-|-|\n|GRAND|28±6.67| 71.97±4.26|\n|GTN|20.15±2.27| 75.48±4.27|\n|GraphCON|44.83±3.53| 69.82±6.12|\n|KGNN|29.26±2.84| 63.76±5.17|\n|BIG-NOS|60±4.28| 75.02±2.61|\n\n### *[W2 lack of method justification]*\n\nWe now clarify the **motivation**, **justification**, and **necessity** of our method.\nOur primary focus is ***brain rhythm identification*** from **neural oscillations**, which are fundamental in coordinating communication and cognitive functions [2]. Moreover, Kuramoto model has long been used to study coupled neural oscillators [3]. **Building on these foundation**, *BRICK* explicitly models oscillatory dynamics of brain function by Kuramoto dynamics, which not only achieves strong performance in brain state decoding (Tab.2), but also yields interpretable, biologically meaningful synchronization patterns (Fig.3). We believe this justifies both the **necessity and novelty** of our approach in the perspective of computational neuroscience.\n\nEncouraged by the promising results on brain data, we sought to genelize neural oscillation principles to **general graph domains** in AI/ML, which share the same notion of coupled dynamical systems as in brain networks. We hypothesize that **learning distinctive graph representations** is analogous to **coordinating functional roles (graph-level) across brain regions (node-level)**. \n\nWe also provide the following methodology justifications. If there are **specific aspects of the theoretical formulation** that the reviewer would like further clarification on, we would be more than happy to provide detailed explanations as needed.\n\n**Proof of Lyapunov conditions.**  \nLet $\\hat{\\mathbf{x}} = [\\hat{x}\\_1, ..., \\hat{x}\\_N] \\in \\mathbb{R}^{N \\times d}$ and recall dynamical system Eq.(4) and energy $E$ Eq.(5). Because $W = [w\\_{ij}]$ is symmetric (undirected),\n\n$\n\\nabla_{\\hat{\\mathbf{x}}} E = - (W \\hat{\\mathbf{x}} + \\mathbf{y}), \\quad \\mathbf{y} = [y_1 \\ldots y_N]\n$\n\nDefine $u_i := y_i + \\sum_j w\\_{ij} \\hat{x}\\_j$. Then Eq.(4) may be rewritten as:\n\n$\n\\dot{\\hat{x}}_i = \\omega_i + \\gamma \\phi\\_i(u_i) = \\omega\\_i - \\gamma \\phi\\_i(-u\\_i) = \\omega\\_i - \\gamma \\phi\\_i([\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i)\n$\n\nThen the time derivative of $E$ is defined as:\n\n$\n\\dot{E} = \\left\\langle \\nabla\\_{\\hat{\\mathbf{x}}} E, \\dot{\\hat{\\mathbf{x}}} \\right\\rangle = \\sum\\_i \\underbrace{ \\langle [\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i, \\omega_i \\rangle}\\_{\\text{(i)}} - \\gamma \\underbrace{\\langle [\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i, \\phi\\_i([\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i) \\rangle }\\_{\\text{(ii)}}\n$\n\nThe $\\omega_i$ induces a pure phase rotation; it is orthogonal to the gradient direction, so\n$\n \\langle[\\nabla_{\\hat{\\mathbf{x}}}E]_i, \\omega_i\\rangle=0\n$\n\nAnd $\\phi\\_i(\\cdot)$ is\n**(i)** odd: \\$\\phi\\_i(-z) = -\\phi\\_i(z)\\$, and\n**(ii)** monotone increasing:\n$\n \\langle \\phi_i(z_1)-\\phi_i(z_2), z_1-z_2\\rangle\\geq 0\n$. \nWith \\$z = \\[\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i\\$, we have \\$\\langle z, \\phi\\_i(z) \\rangle \\geq 0\\$, this therefore yields (ii) $\\leq 0\\$. Combining (i) and (ii),\n\n$\n\\dot{E}=0-\\gamma\\sum_i\\langle z, \\phi_i(z)\\rangle\\leq0, \\quad z=[\\nabla_{\\hat{\\mathbf{x}}} E]_i\n$\n\nIf \\$W \\succeq 0\\$ and both \\$w\\_{ij}\\$ and \\$y\\_i\\$ are finite, then Eq.(5) is at most quadratic with a finite lower bound:\n\n$\nE(\\hat{\\mathbf{x}})\\geq-\\lambda_{\\max}(W) \\|\\hat{\\mathbf{x}}\\|^2-\\|\\mathbf{y}\\| \\|\\hat{\\mathbf{x}}\\| \\geq c_1 \\|\\hat{\\mathbf{x}}\\|^2-c_2\n$, \nfor suitable constants $c_1 > 0$, $c_2 \\geq 0$.\n\nEq.(5) and the bound above provide the classical Lyapunov conditions:\n$\nE(\\hat{\\mathbf{x}}) \\geq c_1 \\|\\hat{\\mathbf{x}}\\|^2 - c_2, \\quad \\dot{E}(\\hat{\\mathbf{x}}) \\leq 0\n$\n\nHence **\\$E\\$ is a Lyapunov function** for the Eq.(4): it is lower-bounded and monotonically non-increasing along every trajectory, guaranteeing stability.\n\n### *[W3 missing details]*\n\n**[Details of _BRICK_ and _BIG-NOS_]**\n\nThe governing equation presents in Tab.1 for BIG-NOS is shared with BRICK. This alignment is intentional and central to our design, as it enables a seamless adaptation of BRICK from human brain to general graph learning tasks. We'll give more details in final version.\n\n**[Interpretation of Fig.3]**\n\nWe now elaborate how the discovered patterns in Fig. 3 support neurobiological plausibility:\n* **Alignment with functional subnetworks (Fig.3a):**  \n*BRICK* captures task-specific synchronization across functional brain network in HCP-A (e.g., Visual network regions (red) cluster clearly during the VISMOTOR task in the deeper feature $X^{(L)}$, indicating strong within-network coordination). It suggest that *BRICK* respects functional boundaries.\n* **Inter-subject consistency and task-relevant encoding (Fig.3b):**  \nSubjects performing the same task (e.g., “SOCIAL” in HCP-YA) form well-separated synchronization in deeper $X^{(L)}$. This indicates BRICK captures consistent latent dynamics across subjects, supporting that neural phase alignment underlines cognitive state representation.\n* **Unsupervised functional differentiation (Fig.3c):** \n*BRICK* can produce meaningful cortical parcellations without supervision. Compared to spectral clustering, *BRICK* yields phase-based partitions that are more spatially aligned with known functional networks [4]. \n\nTo provide stronger quantitative evidence, we have already included numeric results in Tab.2 and Appendix A.4. However, we will also include the following numeric figures to support Fig.3:\n\n- Within-network vs. between-network phase variance for Fig.3a to quantify the degree of synchronization.\n- Purity and NMI to measure how well the unsupervised parcels in Fig.3c align with the existing functional network.\n\n### *[W4 missing prior works]*\nWe have carefully reviewed both papers and will include them in the paper with a discussion of differences.\n\n1. **Compare with GraphCON**: GraphCON is based on **second-order** ODEs representing *damped* nonlinear oscillators. Its formulation is **inspired by mechanical oscillators** to simulate oscillatory dynamics. *BRICK* is explicitly **informed by neural oscillation**, which is **first-order and phase-based**, aiming to model functional coordination in brain dynamics and translate those principles into learnable graph architectures. \n\n2. **Compare with KuramotoGNN**: Their work demonstrates the use of synchronization dynamics for improving message passing in GNN, thus mitigating oversmoothing. However, *BRICK* incorporates **vector-valued oscillators** and **adaptive control mechanisms**. we ground our model in **neuroscientific applications** and demonstrate its performance and interpretability on **neural data**, beyond generic graph benchmarks.\n\n[1] Dan, T. et al. (2024). Exploring the enigma of neural dynamics through a scattering-transform mixer landscape for riemannian manifold.  \n[2] Fries, P. (2005). A mechanism for cognitive dynamics: Neuronal communication through neuronal coherence.  \n[3] Cabral, J.et al. (2011). Role of local network oscillations in resting-state functional connectivity.  \n[4] Yeo, B.T. et al. (2011). The organization of the human cerebral cortex estimated by intrinsic functional connectivity. \n\nWe are glad to provide further clarification or discuss in any format that would be helpful."}, {"review_id": "9F0E5Kmdzf", "reviewer": "Reviewer_SVVA", "summary": "The paper presents BRICK, a novel deep learning-based framework designed to decode cognitive states with strong biological inspiration. Unlike traditional artificial neural networks (such as the perceptron or multilayer perceptron), which drew only loose inspiration from neuroscience and abstracted away much of the brain’s complexity, BRICK integrates neuroscientific principles, such as fluctuating brain activity, neural oscillatory synchronization, and attentional memory to more closely emulate neurobiological processes.\t \n\nSpecifically, BRICK extends these neuroscience principles into the graph domain, employing a physics-informed model for brain rhythm identification. This model utilizes a Kuramoto model-inspired control mechanism to capture memory-guided neural synchronization. This approach enables the development of a neurobiologically inspired GNN, called BIG-NOS, which specifically addresses main limitations like over-smoothing while offering scalability across node and graph classification tasks.\n\nExtensive evaluation on publicly available neuroimaging and graph datasets demonstrate that BRICK effectively synchronizes brain regions, yielding synchronization patterns closely aligned with specific cognitive states. BIG-NOS achieves state-of-the-art (SOTA) performance on a range of GNN benchmarks.", "questions": "- What are the pros of BRICK model w.r.t. to related work [9,10] that used Kuramoto model to describe the coupling between brain structure and function? What are the advantages of BRICK compared to the work in [44]?\n\n- Can you clarify why equation 4 can be interpreted as gradient flow? I guess that in line 185 the authors want to provide intuition behind eq. 5 not 4.\n\n- What kind on nerural nertwork architecture  did the author use to o parameterize the natural frequency Ω (lines 201-202)?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths \n- Originality.\nThe main strength of this paper is to introduce a new design paradigm for developing GNNs through the lens of coupled neural oscillators. The authors highlight a link between the neuroscience and machine learning, where the last one is used learn the parameters the Kuramoto model, where a new term is included to incorporate the concept of the attending memory.\n\n- Empirical validation. \nThe claim related to BRICK ant to BIG-NOS are well-supported by the experimental results, that show:\n   - BRICK ability to synchronize brain region. Especially even the experiments in the absence of ground truth, show the BRICK’s capability of identifying clusters of functional subnetworks.\n  - BIG-NOSE achieve SOTA performance on node and graph classification showing robustness to over-smoothing and scalability issues.\n\n\nWeaknesses:\n\nClarity. The main weakness is related to the clarity of the paper. Specifically, the two following points limits the readability of the paper.\n- Related work.\n     - It is not clear what are the differences between the authors' modified Kumamoto model in BRICK and the ones proposed in [9,10]. \n    -  It is not clear what are the differences between the Kumamoto model in BRICK and the work [44]. In that regard, the authors contribution is to put the spotlight on bridging the reciprocal relationship between neuroscience and AI through the lens of governing equations in the dynamical system.\n\n- Methods. The readability will improve to incorporate more information about the Big-NOS network architecture.\n\n[9] Joana Cabral, Etienne Hugues, Olaf Sporns, and Gustavo Deco. Role of local network oscillaions in resting-state functional c\tonnectivity. Neuroimage, 57(1):130–139, 2011.\n\n[10] Katerina Capouskova, Morten L Kringelbach, and Gustavo Deco. Modes of cognition: Evidence\nfrom metastable brain dynamics. NeuroImage, 260:119489, 2022.\n\n[44] Takeru Miyato, Sindy Löwe, Andreas Geiger, and Max Welling. Artificial kuramoto oscillatory neurons. arXiv preprint arXiv:2410.13821, 2024.", "was_revised": true, "final_justification": "Dear authors, thank you for the rebuttal. After carefully reviewing all the comments and your response, I am confirming my original rating.", "rebuttal": "We sincerely thank the reviewer for recognizing the originality of our work and the comprehensiveness of our experimental evaluation. We also appreciate the thoughtful feedback regarding **related work** and **methodology details**. Below, we carefully address each concern and provide clarifications or revisions where needed.   \n\n### *1. It is not clear what are the differences between the authors' modified Kuramoto model in BRICK and the ones proposed in [9,10]. What are the advantages of BRICK compared to the work in [44]?*\n\n**[Differences between our model and [9], [10], [44]]**    \n**BRICK and [9]:**  \nWhile both *BRICK* and Cabral et al. [9] adopt the Kuramoto model to explore brain dynamics, their **roles and applications** diverge significantly. In their work, the Kuramoto model is used as a forward simulator to reproduce biologically observed functional connectivity (FC) patterns from empirical structural connectivity (SC), with the goal of explaining emergent brain phenomena. In contrast, our *BRICK* repurposes the Kuramoto model as a computational module within a machine learning framework. Rather than passively simulating dynamics, *BRICK* actively learns to perform representation learning, reasoning, and predictive tasks on graphs using dynamic synchronization mechanisms.\n\n**BRICK and [10]:**  \nFor **methodological paradigm**, while *Capouskova* et al. [10] follow a data-driven analytical paradigm, using autoencoders and clustering to uncover latent cognitive states from empirical fMRI data, they encode BOLD phase coherence data using modern machine learning tools **without Kuramoto model**. Their focus lies in neuroscientific pattern discovery and interpretation. In contrast, *BRICK* introduces a physics-informed learning framework that embeds the principles of neural synchronization, specifically via the Kuramoto model, directly into the learning process.\n\nThe **outcomes** of these two approaches are also distinct: *Capouskova* et al. generate empirical insights about cognitive brain states, whereas *BRICK* yields new machine learning models that not only excel in brain-related tasks but also generalize to broader graph-based AI problems.\n\n**BRICK and [44]:**  \nCompared with [44], our *BRICK* framework offers two key advantages.\n* **First, neuroscientific grounding**: starting from fMRI-derived brain rhythms, we augment the Kuramoto core with a task-driven feedback controller $y_i$ that mimics cognitive control, thereby preserving biological interpretability.\n\n* **Second, domain-specific impact**: on real HCP fMRI data *BRICK* achieves the best task decoding and unsupervised parcellation, while on graph benchmarks it delivers competitive accuracy and remains resistant to oversmoothing even at 128 layers.  \n\nTaken together, these properties make *BRICK* a biologically inspired but practically efficient module that bridges brain dynamics and general-purpose AI.\n\n### _2. Methods. The readability will improve to incorporate more information about the Big-NOS network architecture._  \n\n **[Details of BIG-NOS]**  \nConceptually, *BIG-NOS* can be viewed as an extension of *BRICK* to graph data. By interpreting graph nodes as analogs of brain regions and the adjacency matrix as a proxy for coupling strength among regions, we seamlessly adapt the physics-informed architecture of *BRICK* to arbitrary graphs. Both models share the same governing equations rooted in Kuramoto synchronization, but differ in their data domains and learning objectives.\n\nIn *BRICK*, the inputs consist of BOLD signals over brain regions along with structural or functional connectivity; in *BIG-NOS*, the inputs are general node features and graph topology. Likewise, the outputs diverge in form: *BRICK* predicts cognitive or disease-related brain states (analogous to graph-level classification), while *BIG-NOS* predicts node/graph labels in standard graph learning settings.\n\n**We are committed to include the formal network detail of *BIG-NOS* into Section 3.2 as part of the overall architecture description.** \n\n### _3. Can you clarify why equation 4 can be interpreted as gradient flow?_  \n\n **[Gradient flow interpretation of Eq.(4)]**\n\nGradient flow refers to a class of dynamical systems that evolve along the steepest descent direction of an energy functional $E$. The evolution equation takes the form\n$\n\\dot{\\hat{x}}_i = -\\frac{\\partial E}{\\partial \\hat{x}_i}\n$\n\nwhich guarantees that the energy $\\dot{E}$ decreases over time, i.e.,\n$\n\\dot{E} = \\frac{dE}{dt} \\leq 0\n$\n\nleading the system toward a local minimum of $E$. The energy function defined in Eq.(5)\n$\nE = -\\sum_{i,j} w_{ij} \\hat{x}_i^\\top \\hat{x}_j - \\sum_i y_i^\\top \\hat{x}_i\n$\nencodes two key objectives:  \n\n* The first term promotes pairwise alignment (**synchronization**) among oscillators $\\hat{x}\\_i$ and $\\hat{x}\\_j$ with strong coupling $w\\_{ij}$.\n\n* The second term aligns each oscillator with its task-specific control pattern $y_i$ (**controlled synchronization**).\n\nTaking the gradient of $E$ with respect to $\\hat{x}_i$ yields \n$\n\\frac{\\partial E}{\\partial \\hat{x}\\_i} = -\\sum_j w\\_{ij} \\hat{x}\\_j - y\\_i\n$,\n\nso that the negative gradient becomes\n$\n-\\frac{\\partial E}{\\partial \\hat{x}\\_i} = \\sum_j w\\_{ij} \\hat{x}\\_j + y_i\n$\n\nSubstituting this into Eq.(4) we obtain:\n$\n\\dot{\\hat{x}}_i = \\omega_i + \\gamma \\phi_i \\left( -\\frac{\\partial E}{\\partial \\hat{x}_i} \\right)\n$\n\nThus, Eq.(4) is considered a gradient flow because its dynamics are equivalent to evolving along the negative gradient direction of an energy functional $E$. Here, the operator $\\phi$ does not destroy the gradient flow structure, but instead imposes structural constraints on the descent direction (e.g., ensuring motion remains on the unit sphere).\n\n### _4. What kind on nerural nertwork architecture did the author use to parameterize the natural frequency Ω (lines 201-202)?_ \n\n**[Parameterization of natural frequency Ω]**  \nThe parametrization of natural frequency Ω does not rely on a deep neural network. Instead, Ω is parameterized using **a set of learnable vectors** whose norms determine the rotation speed (i.e., natural frequency) of oscillators. In the forward pass, these frequency values are used to rotate the 2D oscillator states (like turning a point in a circle), simulating how each node evolves over time due to its intrinsic dynamics.\n\n[9] Joana Cabral, Etienne Hugues, Olaf Sporns, and Gustavo Deco. Role of local network oscillations in resting-state functional connectivity. Neuroimage, 57(1):130–139, 2011.373  \n[10] Katerina Capouskova, Morten L Kringelbach, and Gustavo Deco. Modes of cognition: Evidence from metastable brain dynamics. NeuroImage, 260:119489, 2022.375  \n[44] Takeru Miyato, Sindy Löwe, Andreas Geiger, and Max Welling. Artificial kuramoto oscillatory neurons. arXiv preprint arXiv:2410.13821, 2024  \n\n**Should there remain any ambiguities or if the reviewer has further questions, we would be more than happy to engage in continued discussion and resolve any remaining issues.**"}, {"review_id": "YPi6bHFecg", "reviewer": "Reviewer_zxED", "summary": "This framework presents a physics and neuroscience-informed deep learning framework. It integrates with a synchronization mechanism of neural oscillations inside a graph representation learning framework. Motivated by addressing the oversmoothing issue of conventional GNNs, it proposed to utilize the brain rhythms in the artificial dynamical system. This method integrates a Kuramoto model with attending memory for modeling oscillatory synchronization in brain regions. It has been applied to solve two tasks on brain rhythm identification and conventional tasks on graph data, demonstrating superior performance compared to existing baselines.", "questions": "1. What might be challenges for the optimization solver for the proposed graph learning procedure, would this guarantee to converge, and how it is sensitive to initialization and hyperparameters?\n2. Would this be more computational expensive compared to conventional graph learning methods, and how it will be scale to larger graph?", "limitations": "1. Include more discussions with neuroscience literatures and mechanisms would be more impactful.\n2. Include more discussion and evaluations on the computational cost, and scalability.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. This work presents a novel solution to address the oversmoothing issue of conventional GNNs, it incorporated with a physics-informed structure with Kuramoto model with attending memory inside a graph representation framework. This system has novel designs in governing equation, learning mechanism, and attention mechanism compared to conventional GNN models.\n2. The model demonstrates superior performance on two tasks (brain rhythm identification and graph tasks) compared to existing baselines. \n3. This model with good interpretability, and demonstrate interesting visualizations of synchronization patterns in Fig 3.\n4. Multiple ablations studies are conducted.\n\n**Weaknesses**\n1. It would be more impactful to link the synchronization mechanisms to neuroscience domains, and explains why it might be more superior compared to conventional approachs in brain rhythm identification task, and how the discovery in Fig 3 might be relevant to existing studies.\n2. Missing comparisons of the computational cost of the proposed methods compared to existing methods.", "was_revised": true, "final_justification": "Thanks for the authors' detailed responses. I am especially impressed by the scientific insights and motivations clarified by the authors. I also appreciates the mechanistic understanding on neuronal synchronization that this work could bring. The responses on baselines comparisons, initialization, convergence also mostly addressed my concerns. I would like to increase my score to accept (5). Thanks for your efforts.", "rebuttal": "We sincerely thank the reviewer for recognizing the novelty and comprehensive nature of our work. We also greatly appreciate the reviewer’s interest and thoughtful feedback regarding the **biological interpretability**, **computational efficiency** and **scalability**, as well as the **optimization dynamics** underlying our framework. Below, we address each of the reviewer’s concerns in detail.\n\n### _1. It would be more impactful to link the synchronization mechanisms to neuroscience domains, and explains why it might be more superior compared to conventional approachs in brain rhythm identification task, and how the discovery in Fig 3 might be relevant to existing studies._\n\n**[Link to neuroscience domains]**  \nIn principle, the learning behavior of _BRICK_ is shaped by the same governing equations (i.e., Kuramoto model) that describe the neural oscillatory dynamics responsible for generating cognition and behavior. We further introduce structural-functional coupling, implemented through the geometric scattering transform (Line 159), to emulate the cross-frequency coupling observed in cognitive neuroscience. In addition, we incorporate an adaptive control term to enhance task-relevant modulation. Drawing inspiration from neural oscillatory synchronization, our BRICK model not only improves predictive accuracy but also enhances interpretability, thereby offering novel insights into cognitive processes (Section 3.1).\n\n**[Advances compared to conventional approaches in brain rhythm identification task]**  \nWhile traditional models, including CNNs, RNNs, Transformers, and GNNs, have demonstrated strong empirical performance in brain rhythm identification, they largely remain data-driven and agnostic to the underlying neurobiological mechanisms, they often treat the brain as a generic data source without sufficient domain knowledge. In contrast, _BRICK_ explicitly incorporates neural synchronization dynamics, a well-established principle in cognitive neuroscience, into the learning architecture. This alignment with neuroscience allows us to interpret learned patterns (e.g., coupling strengths or synchronization clusters) in biologically meaningful terms (as shown in Figure 3), such as inter-regional communication or pathological desynchronization. Therefore, compared to conventional approaches, _BRICK_ is not just a deep model for feature representation learning, it shows the potential to establish a biologically inspired reasoning system. By embedding core principles of brain dynamics into a learnable, differentiable architecture, our proposed deep model enables achieving more interpretable, temporally-aware, and robust brain state identification.   \n\n**[Relevance of Figure 3 to existing studies]**  \nBelow is a detailed explanation of how the discovered patterns in Figure 3 align with known neurobiological structures and functions:  \n* **Alignment with functional subnetworks (Figure 3a):**  \n*BRICK* captures task-specific synchronization across functionally defined brain network in HCP-A. For instance, Visual network regions (red) cluster clearly during the VISMOTOR task in the deeper representation $X^{(L)}$, indicating strong within-network coordination. Similarly, tighter phase clustering emerges in the Sensorimotor and Dorsal Attention networks under tasks such as CART and FACENAME. These results suggest that our model respects functional boundaries in a biologically meaningful manner.\n* **Intersubject consistency and task-relevant encoding (Figure 3b):**  \nThis figure shows that individuals performing the same task (e.g., “SOCIAL” or “EMOTION” in HCP-YA) form well-separated synchronization in deeper layers (e.g., $X^{(L)}$). This indicates BRICK captures latent dynamics consistent across subjects and datasets, supporting the notion that neural phase alignment underlines cognitive state representation.\n* **Unsupervised Functional Differentiation (Figure 3c):** \nHere, we assess whether BRICK can produce meaningful cortical parcellations without supervision. Compared to traditional spectral clustering, BRICK yields phase-based partitions that are more spatially localized and better aligned with known functional networks [1]. This supports the claim that BRICK captures biologically coherent dynamics.  \n\nBy mapping learned feature vectors onto a phase manifold, we can interpret inter-regional phase alignment as a proxy for functional coordination. In this way, _BRICK_ offers a mechanistic perspective grounded in neural oscillation theory, which allows us to move beyond black-box representations toward interpretable latent dynamics.  \n\n### _2. Missing comparisons of the computational cost of the proposed methods compared to existing methods. Would this be more computational expensive compared to conventional graph learning methods, and how it will be scale to larger graph?_  \n\n**[Comparisons of the computational cost]**  \nWe have provided a detailed comparison of the inference time per subject for all models across three brain datasets in **Appendix A.2 Table 7**. In addition, we have now included a summary table that reports the average inference time of each model on all brain datasets. As shown in the results, our proposed model **BRICK** has slightly higher inference time compared to lightweight baselines such as **GCN**, **GIN**, and **GraphSAGE**. However, it remains more efficient than several other models, while also delivering significantly better performance on brain datasets.\n\n| Model                           | GCN | GIN  | GAT  | GCNII | GraphSAGE | SAN  | BRICK |\n| ------------------------------- | --- | ---- | ---- | ----- | --------- | ---- | ----- |\n| Avg inference time (ms/subject) | 0.9 | 0.74 | 1.31 | 1.09  | 0.72      | 1.62 | 1.25  |\n\n**[How to scale to larger graph]**  \nTo ensure the scalability of our method to large-scale graphs, we implement all matrix operations using sparse representations. This design allows our model to scale comparably to lightweight models such as GCN, and enables full-batch training on large graphs like ogbn-arxiv without exceeding memory limits.   \n\nMoreover, for even larger graphs, e.g., fine-grained applications such as vertex-level cortical modeling in neuroscience, our framework can readily incorporate subgraph sampling strategies [2] to enable smooth training.\n\n### _3. What might be challenges for the optimization solver for the proposed graph learning procedure, would this guarantee to converge, and how it is sensitive to initialization and hyperparameters?_   \n\n**[Initialization and hyperparameters ablations]**   \n**Sensitivity to initialization.**\nTo assess the sensitivity of our model to random initialization, we conducted experiments on the Cora, Citeseer, and Pubmed datasets using **5** different random seeds. As reported in Table 2, our model achieves highly consistent results, with the maximum standard deviation across these runs being only **0.36%**. This indicates that our method is stable and not significantly affected by initialization variance.  \n\n**Sensitivity to hyperparameters.** We conducted **several** ablation studies on the most critical hyperparameters and reported them in the appendix.\n\n* In **Appendix A.2**, we ablate two parameters specific to the **BRICK** architecture: *L* (outer iterations for control update) and *Q* (inner time steps for oscillator dynamics). Increasing both parameters increases inference time, while having only a marginal effect on accuracy. This suggests the model is not overly sensitive to these parameters within a reasonable range.\n\n* In **Appendix A.3**, we further analyze the sensitivity of geometric scattering transform (GST)-related hyperparameters: *level* and *order*. Our results show that increasing either improves performance by capturing more frequency components, but at the cost of slightly increased computation.\n\n**[Optimization challenge and converge guarantee]**    \nWe appreciate the reviewer’s insightful question regarding the optimization behavior of our proposed method. From a theoretical perspective, the dynamics of our model are governed by a *gradient flow* is:\n$\n\\frac{d\\hat{x}_i}{dt}=\\omega_i+\\gamma\\cdot \\phi_i (y_i+\\sum\\_{j=1}^{N} w\\_{ij}\\hat{x}_j)\n$, \n\nwhich corresponds to a gradient descent flow on the following energy landscape:\n$\nE = - \\sum_{i,j} w_{ij} \\hat{x}_i^\\top \\hat{x}_j - \\sum_i y_i^\\top \\hat{x}_i\n$.\nThis energy function $E$ acts as a natural Lyapunov function, satisfying the condition: $\\frac{dE}{dt} \\leq 0$,\nthus ensuring that the system always evolves towards a (local) steady state. Therefore, the entire system remains a valid gradient flow on a constrained manifold, which theoretically guarantees convergence.\n\nIn practice, we also observe stable optimization behavior during training. To support this claim, we will include the **training loss trajectories** of our model on representative datasets in **Appendix**.\n\n[1] Yeo, B. T. T. et al (2011). The organization of the human cerebral cortex estimated by intrinsic functional connectivity. Journal of Neurophysiology, 106(3), 1125–1165.   \n[2] Zeng, H. et al (2019). Graphsaint: Graph sampling based inductive learning method. arXiv preprint arXiv:1907.04931.\n\nThanks again for your feedback, if anything is still unclear or the reviewer would like additional clarification, we would gladly continue the discussion and address any further concerns."}, {"review_id": "npuRdOUvN2", "reviewer": "Reviewer_Rarc", "summary": "This paper proposes a new learning framework called BRICK. It develops more effective machine learning algorithm design principles by simulating brain rhythm mechanisms, thereby connecting neuroscience and artificial intelligence. The goal of BRICK is to identify brain rhythms, and to develop an oscillator representation and control mechanism closely integrated with fundamental neuroscience principles. The representations output by this model highly reflect neuroscientific principles (i.e., brain rhythms). Furthermore, building upon this, the paper extends this concept to the graph domain, introducing a new graph neural network architecture called BIG-NOS. This model improves existing graph learning models and possesses the ability to avoid the over-smoothing issue.", "questions": "See weaknesses. Overall, I believe this paper is a good work. I’m happy to raise my score if the authors address my concerns well.", "limitations": "See weaknesses.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. This paper presents novel and significant work. It introduces a new perspective to the AI field, particularly in graph neural networks, by proposing a biologically inspired mechanism of brain rhythms for machine learning. \n2. This paper experimentally demonstrates the model's ability to identify brain rhythms. For example, BRICK can decode dynamic brain states based on neural synchronisation patterns and achieves significantly superior performance compared to existing GNN models on human brain datasets such as HCP-A, HCP-YA, and HCP-WM.\n3. This paper effectively avoids the over-smoothing issue in GNNs. The proposed BIG-NOS explicitly avoids over-smoothing by evolving graph features within a latent oscillatory phase space, rather than through traditional heat diffusion methods.\n\nWeaknesses:\n1. The paper does not compare with popular models like graph diffusion and graph transformer, thus lacking a comprehensive comparison with the latest state-of-the-art graph learning models [1-3].\n2. The interpretability (biological plausibility) analysis is insufficient. In brain rhythm identification, the authors demonstrate visualisation results of brain region synchronisation, but do not provide further analysis on whether the output representations are consistent with biological characteristics of the brain. For instance, Line 305 states that the proposed mode uncovers functional communities that are not\nonly align with known subnetworks but also capture novel patterns in brain organisation, but a more detailed biological interpretation is lacking.\n3. The paper lacks theoretical proof for its core mechanism (e.g., how oscillatory synchronisation mitigates over-smoothing). Although an energy function $E$ is proposed, it does not provide complete evidence to support its theoretical results. Although $E$ is claimed to be a \"natural\" Lyapunov function, no detailed mathematical derivations or proofs are given.\n\n[1] Yang, H., Wang, B., & Jia, J. (2024). Gnncert: Deterministic certification of graph neural networks against adversarial perturbations. In The Twelfth International Conference on Learning Representations.\n\n[2] Chamberlain, B., Rowbottom, J., Gorinova, M. I., Bronstein, M., Webb, S., & Rossi, E. (2021, July). Grand: Graph neural diffusion. In International conference on machine learning (pp. 1407-1418). PMLR.\n\n[3] Yun, S., Jeong, M., Kim, R., Kang, J., & Kim, H. J. (2019). Graph transformer networks. Advances in neural information processing systems, 32.", "was_revised": true, "final_justification": "The authors have addressed my concerns, so I'd like to raise my score to 5.", "rebuttal": "We sincerely thank the reviewer for recognizing the novelty and importance of our work. We also greatly appreciate the constructive feedback regarding **empirical performance**, **neuroscientific interpretability**, and **theoretical grounding** of the proposed model. Below, we provide detailed responses to each of the reviewer’s concerns. \n\n### *1. Lacking a comprehensive comparison with the latest state-of-the-art graph learning models [1-3].*\n\n**[Comparison with additional graph learning methods]**  \nWe have added two baseline models as reviewer suggested: GRAND, GTN. In addition, we also included KuramotoGNN (KGNN)[1] and GraphCON [2], which are both recent graph models for oscillatory dynamics. While we initially considered GNNCert, it focuses on adversarial robustness and its reported clean accuracy is based on existing backbones GCN, GAT, and GIN, which are already part of our benchmark. Nonetheless, we would include GNNCert to the related work to acknowledge its relevance in the broader graph learning. The updated results are presented in Table below, the results show that our models still achieve significant improvements in brain datasets and retain strong performance in general graph benchmarks.\n\n**Brain states identification:**\n|Acc(%)|HCP-A|HCP-YA|HCP-WM|\n|-|--|-|-|\n|GRAND|86.18±1.79|47.36±0.67|33.27±2.63|\n|GTN|82.56±0.87|53.42±1.21|30.68±0.36|\n|GraphCON|87.87±2.29|66.29±1.04|44.12±2.1|\n|KGNN|85.53±1.68|45.68±1.12|35.79±1.54|\n|BRICK|95.55±0.77|84.20±1.60|89.22±1.71|\n\n**Node classification:**\n|Acc(%)|Texas|Wisconsin|Actor|Squirrel|Chameleon|Cornell|Citeseer|Pubmed|Cora|\n|-|-|-|-|-|-|-|-|-|-|\n|GRAND|67.42±7.81|73.09±4.83|33.34±1.16|34.45±1.84|38.43±3.33|56.55±9.15|72.1±1.07|79.12±0.3|83.73±0.58|\n|GTN|74.52±10.2|68.89±11.9|35.42±1.5|36.45±1.72|55.84±4.08|58.81±15.09|68.8±1.69|76.90 ± 0.77|78.60±1.72|\n|GraphCON|82.43±4.72|83.72±4.48|35.13±1.38|26.9±2.17|33.75±3.77|74.59±2.48|62.9±5.95|71.21±7.36|62.73±5.79|\n|KGNN|71.75±5.57|69.43±6.18|31.5±0.82|35.02±1.67|38.5±2.78|61.16±9.16|72.06±1.53|67.24±0.36|76.57±1.4|\n|BIG-NOS|81.35±3.51|82.16±3.56|35.8±0.6|68.06±1.65|74.5±1.13|73.24±4.75|70.63±0.36|77.8±0.23|81.86±0.25|\n\n**Graph classification:**\n|Acc(%)|ENZYMES|PROTEINS|\n|-|-|-|\n|GRAND|28±6.67| 71.97±4.26|\n|GTN|20.15±2.27| 75.48±4.27|\n|GraphCON|44.83±3.53| 69.82±6.12|\n|KGNN|29.26±2.84| 63.76±5.17|\n|BIG-NOS|60±4.28| 75.02±2.61|\n\n### *2. The interpretability (biological plausibility) analysis is insufficient.*\n\n**[More detailed interpretability (biological plausibility) analysis]**\nWe now elaborate how the discovered patterns in Figure 3 support neurobiological plausibility:\n* **Alignment with functional subnetworks (Figure 3a):**  \n*BRICK* captures task-specific synchronization across functionally defined brain network in HCP-A. For instance, Visual network regions (red) cluster clearly during the VISMOTOR task in the deeper representation $X^{(L)}$, indicating strong within-network coordination. These results suggest that our model respects functional boundaries in a biologically meaningful manner.\n* **Intersubject consistency and task-relevant encoding (Figure 3b):**  \nThis figure shows that individuals performing the same task (e.g., “SOCIAL” in HCP-YA) form well-separated synchronization in deeper layers $X^{(L)}$. This indicates BRICK captures latent dynamics consistent across subjects, supporting the notion that neural phase alignment underlines cognitive state representation.\n* **Unsupervised Functional Differentiation (Figure 3c):** \nHere, we assess whether BRICK can produce meaningful cortical parcellations without supervision. Compared to spectral clustering, BRICK yields phase-based partitions that are more spatially localized and better aligned with known functional networks [3]. This supports the claim that BRICK captures biologically coherent dynamics.\n\nIn this way, BRICK offers a mechanistic perspective grounded in neural oscillation theory, which allows us to move beyond black-box representations toward interpretable latent dynamics.\n\n### *3. The paper lacks theoretical proof for its core mechanism (e.g., how oscillatory synchronisation mitigates over-smoothing). Although E is claimed to be a \"natural\" Lyapunov function, no detailed mathematical derivations or proofs are given.*\n\n**[Why *E* is a *natural* Lyapunov function?]**\n1. **Physical intuition** \n\nThe dynamical system (Eq.(4))\n$\n\\frac{d\\hat{x}\\_i}{dt} = \\omega_i + \\gamma \\phi\\_i( y_i + \\sum\\_{j=1}^N w\\_{ij} \\hat{x}\\_j)\n$\nis a network of weakly-coupled oscillators driven by an external signal $y_i$. This suggests the following energy function (Eq.(5)):\n\n$\nE(\\hat{x}) = -\\sum\\_{i,j} \\hat{x}\\_i^\\top w\\_{ij} \\hat{x}\\_j - \\sum\\_i y_i^\\top \\hat{x}\\_i\n$,\n\nwhere the first sum represents the **coupling term** and the second sum is the **driving term**. Because the update $\\frac{d\\hat{x}_i}{dt}$ explicitly attempts to align $\\hat{x}_i$ with its neighbors $(w\\_{ij} \\hat{x}\\_j)$ and with $y_i$, the quantity measuring *how well aligned* they are, namely \\$E\\$, is the physically most natural Lyapunov candidate.\n\n 2. **Lyapunov property** [4]\n\nLet $\\hat{\\mathbf{x}} = [\\hat{x}\\_1, ..., \\hat{x}\\_N] \\in \\mathbb{R}^{N \\times d}$ and recall dynamical system Eq.(4) and energy $E$ Eq.(5). Because $W = [w\\_{ij}]$ is symmetric (undirected),\n\n$\n\\nabla_{\\hat{\\mathbf{x}}} E = - (W \\hat{\\mathbf{x}} + \\mathbf{y}), \\quad \\mathbf{y} = [y_1 \\ldots y_N]\n$\n\nDefine $u_i := y_i + \\sum_j w\\_{ij} \\hat{x}\\_j$. Then Eq.(4) may be rewritten as:\n\n$\n\\dot{\\hat{x}}_i = \\omega_i + \\gamma \\phi\\_i(u_i) = \\omega\\_i - \\gamma \\phi\\_i(-u\\_i) = \\omega\\_i - \\gamma \\phi\\_i([\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i)\n$\n\nThen the time derivative of $E$ is defined as:\n\n$\n\\dot{E} = \\left\\langle \\nabla\\_{\\hat{\\mathbf{x}}} E, \\dot{\\hat{\\mathbf{x}}} \\right\\rangle = \\sum\\_i \\underbrace{ \\langle [\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i, \\omega_i \\rangle}\\_{\\text{(i)}} - \\gamma \\underbrace{\\langle [\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i, \\phi\\_i([\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i) \\rangle }\\_{\\text{(ii)}}\n$\n\nThe $\\omega_i$ induces a pure phase rotation; it is orthogonal to the gradient direction, so\n$\n \\langle[\\nabla_{\\hat{\\mathbf{x}}}E]_i, \\omega_i\\rangle=0\n$\n\nAnd $\\phi\\_i(\\cdot)$ is\n  **(i)** odd: \\$\\phi\\_i(-z) = -\\phi\\_i(z)\\$, and\n  **(ii)** monotone increasing:\n$\n \\langle \\phi_i(z_1) - \\phi_i(z_2), z_1 - z_2 \\rangle \\geq 0\n$. \nWith \\$z = \\[\\nabla\\_{\\hat{\\mathbf{x}}} E]\\_i\\$, we have \\$\\langle z, \\phi\\_i(z) \\rangle \\geq 0\\$, this therefore yields (ii) $\\leq 0\\$. Combining (i) and (ii),\n\n$\n\\dot{E}=0-\\gamma\\sum_i\\langle z, \\phi_i(z)\\rangle\\leq0, \\quad z=[\\nabla_{\\hat{\\mathbf{x}}} E]_i\n$\n\nIf \\$W \\succeq 0\\$ and both \\$w\\_{ij}\\$ and \\$y\\_i\\$ are finite, then Eq.(5) is at most quadratic with a finite lower bound:\n\n$\nE(\\hat{\\mathbf{x}})\\geq-\\lambda_{\\max}(W) \\|\\hat{\\mathbf{x}}\\|^2-\\|\\mathbf{y}\\| \\|\\hat{\\mathbf{x}}\\| \\geq c_1 \\|\\hat{\\mathbf{x}}\\|^2-c_2\n$, \nfor suitable constants $c_1 > 0$, $c_2 \\geq 0$.\n\nEq.(5) and the bound above provide the classical Lyapunov conditions:\n$\nE(\\hat{\\mathbf{x}}) \\geq c_1 \\|\\hat{\\mathbf{x}}\\|^2 - c_2, \\quad \\dot{E}(\\hat{\\mathbf{x}}) \\leq 0\n$\n\nHence **\\$E\\$ is a Lyapunov function** for the Eq.(4): it is lower-bounded and monotonically non-increasing along every trajectory, guaranteeing stability.\n\n**[Heuristic Analysis: Why *BRICK* alleviates over-smoothing]**\n\nWe analyze why the *BRICK* can theoretically alleviate the oversmoothing by examining its simplified dynamics, steady-state solution and spectral response.\n\n**Simplified BRICK Dynamics.**\nTo derive an interpretable steady-state solution, we consider a simplified version of BRICK dynamics. We assume a constant natural frequency $\\omega_i$ across all nodes and linearize the nonlinear projection $\\phi$:\n\n$\n\\frac{d\\hat{\\mathbf{x}}}{dt} = -\\hat{\\mathbf{x}}+W\\hat{\\mathbf{x}}+\\mathbf{y}\n$\n\nThis corresponds to a linear consensus-like system, where the $-\\hat{\\mathbf{x}}$ term mimics a dissipative force, ensuring stability.\n\n**Equilibrium Solution.**\nAt steady state $( \\frac{d\\hat{\\mathbf{x}}}{dt} = 0)$, we obtain:\n\n$\n(I - W)\\hat{\\mathbf{x}}^* = \\mathbf{y} \\quad \\Rightarrow \\quad \\hat{\\mathbf{x}}^* = (I - W)^{-1} \\mathbf{y}\n$\n\nThe inverse exists provided the spectral radius of $W$ is strictly smaller than 1 (i.e., $\\lambda_k \\neq 1, \\forall k$), a condition that is typically satisfied for normalized adjacency or attention matrices used in practice.\n\n**Spectral Interpretation.**\nLet $W$ be symmetric and decomposed as $W = U \\Lambda U^\\top$, where $U$ is the orthonormal eigenvector matrix and\n$\\Lambda = \\text{diag}(\\lambda_1, ..., \\lambda_n)$. Projecting into the spectral domain:\n\n$\n\\hat{\\mathbf{x}}^* = U(I - \\Lambda)^{-1}U^\\top \\mathbf{y}\n$\n\nLet $\\tilde{\\mathbf{y}} = U^\\top \\mathbf{y}$ and $\\tilde{\\hat{\\mathbf{x}}}^* = U^\\top \\hat{\\mathbf{x}}^*$, then:\n\n$\n\\tilde{\\hat{x}}^*_k = \\frac{1}{1 - \\lambda_k} \\tilde{y}_k\n$\n\nThus, each spectral mode is scaled by a transfer function:\n$\nh(\\lambda_k) = \\frac{1}{1 - \\lambda_k}\n$\n\n**Comparison with Diffusion-based GNNs.**\nIn standard diffusion-type GNNs (e.g., GCN), applying $L$ layers is equivalent to using a transfer function\n$e^{-L\\lambda_k}$ in spectral space. This leads to exponential suppression of high-frequency signals (large $\\lambda_k$), causing oversmoothing. In contrast, *BRICK* uses a transfer function that decays much slower:\n$h(\\lambda_k) = \\frac{1}{1 - \\lambda_k}$, which corresponds to $1/\\lambda$-level suppression, allowing high-frequency, discriminative signals to persist.\n\n[1] Rusch, T. K. et al. (2022). Graph-coupled oscillator networks.  \n[2] Nguyen, T. et al. (2024). From coupled oscillators to graph neural networks: Reducing over-smoothing via a kuramoto model-based approach.  \n[3] Yeo, B. T. T. et al. (2011). The organization of the human cerebral cortex estimated by intrinsic functional connectivity.  \n[4] Takeru M. et al. Artificial kuramoto oscillatory neurons.\n\n**Thanks again for your feedback, if the reviewer has additional questions, we would be glad to provide further clarifications.**"}]

Author rebuttal:

Meta-review: The paper integrates a synchronization mechanism of neural oscillations with a graph representation learning framework. Motivated by addressing the oversmoothing issue of conventional GNNs, it proposed to utilize the brain rhythms in the artificial dynamical system. This method integrates a Kuramoto model with attending memory for modeling oscillatory synchronization in brain regions. The authors argue that the proposed approaches represent a new graph learning mechanism with SOTA performance.

Strengths identified by the reviewers include:

1. The approach seems novel. However as noted by a reviewer there is prior work integrating neural oscillation with graph neural networks which was not cited in the original manuscript.

2. Experiments on human brain datasets were appreciated by most reviewers. One reviewer however expressed some concerns which were addressed during rebuttal.

3. Good visualizations


Weaknesses identified include:

1. Important related prior work on neural oscillators not cited. The authors indicate in their rebuttal that they will cite missing work

2. A reviewer expressed concerns about the fairness and rigor of experimental setting, especially with respect to hyperparameters chosen. 

3. More information needs to be reported on the computational cost of the proposed methods compared to existing methods

The authors did a good job addressing the reviewer concerns, which led reviewers to update their score. In particular the authors did a good job providing a better theoretical justification of their work, and a better justification of the biological plausibility, which is my understanding will be included in the final paper.

Overall the paper received 3 Accept ratings and 1 Borderline accept. Its score was significantly higher than the minimum threshold needed for acceptance so I recommend acceptance. However one of the reviewers did raise some valid concerns. The authors indicated in their rebuttal that they would address those concerns in their revised manuscript.
Decision: Accept (poster)