Explore brain-inspired machine intelligence for connecting dots on graphs through holographic blueprint of oscillatory synchronization

Interference patterns from spontaneous functional fluctuations

In Young’s double slit experiment conducted in 1801¹⁷, the waves from two slits interfere constructively or destructively at the screen, as shown in Fig. 2a. In computer vision, holography is a stereo-imaging technique that generates a hologram by superimposing a reference beam on the wavefront of interest¹⁸. The resulting hologram is an interference pattern that can be recorded on a physical medium. Following the concept of wave-to-wave interference, we computationally “record” the CFC of time-evolving interference patterns that are formed by superimposing the neural activities across oscillation frequencies, as shown in Fig. 2b. Specifically, we first construct subject-specific graph wavelets from the structural connectome data¹⁹, which serve as the basis functions of geometric scattering transform (GST)²⁰ for each brain region. Applying GST to the fMRI (functional magnetic resonance imaging) data yields a set of “neural oscillators”, each exhibiting fluctuations governed by its own natural frequency. In each brain region, we further capture the presence of CFC patterns in a frequency-to-frequency correlation matrix based on Pearson’s correlation of spontaneous fluctuations between two neural oscillators, as shown in Fig. 2b.

We first show the population average of global CFC patterns (i.e., averaging throughout brain regions and across individuals) for healthy vs. Alzheimer’s disease (AD), healthy vs. Parkinson’s disease (PD), and healthy vs. Frontotemporal dementia (FTD) in Fig. 2c–e, respectively. There is a clear sign that the population-wise CFC average exhibits remarkable off-diagonal striping patterns, which resemble the constructive and destructive interference phenomena in Young’s double slit shown in Fig. 2a. In light of this, we hypothesize that the consistency of CFC degree along these striping patterns may help explain how brain function gives rise to diverse cognitive and behavioral outcomes. Since the striping patterns are visually distinguishable between healthy and disease cohorts, we further speculate that the coherence of maintaining the striping patterns might be an indicator of how brain function is altered as the disease progresses. Following this clue, we compute the degree of sign consistency for each off-diagonal line by calculating the percentage of values exhibiting the predominant sign across instances. In Fig. 2c–e, we plot the distribution of sign consistency for the first three off-diagonal lines for each disease cohort. Values above the 0.5 line indicate a predominant distribution of positive CFCs, while values below 0.5 reflect a predominance of negative CFCs. It is apparent that the sign consistency degree manifests strong differences between healthy and disease connectomes at the significance level p < 0.05. More evidence of CFC patterns in task-fMRI is shown in Supplementary Fig. 1.

Remarks. The presence of interference patterns in CFC matrices implies that discovering the laws governing neural oscillatory fluctuations in the human brain is a necessary precursor to understanding perceptual and behavioral processes. Such understanding could ultimately inspire the development of biologically grounded learning mechanisms in AI.

Identify governing equation of brain rhythms

Mounting evidence shows that neural oscillations play a vital role in synchronizing different brain regions, and the Kuramoto model²¹ has been widely used to study neural synchronization dynamics in both computational neuroscience and empirical brain data²². Suppose the system consists of N oscillators with their pairwise coupling strength denoted by K_ij(i, j = 1, . . . , N). By allowing each oscillator to vibrate at its own natural frequency ω_i, the dynamics of the coupled phase oscillator in the Kuramoto model is formulated as:

where the evolution of the phase \({\theta }_{i}\in {\mathbb{R}}\) associated with the i^th oscillator is synchronized with other coupled oscillators. Initially, each oscillator evolves independently at its intrinsic frequency ω_i. Over time, the coupling term [ ⋅ ] enables the oscillators to adjust their phases relative to one another, leading to collective synchronization. This process results in oscillators locking their phases and forming coherent groups.

Given the presence of interference patterns in CFC matrices (shown in Fig. 2), we propose a physics-informed deep neural network to parameterize the Kuramoto model, which allows us to understand the neurobiological mechanism of how coupled oscillatory synchronization emerges underlying cognitive tasks. We conceptualize that oscillatory activity across interconnected brain regions together forms a dynamic structure shaped by neural synchronization mechanisms. To that end, we design a neural network architecture that characterizes the whole-brain oscillatory synchronization governed by the Kuramoto model of coupled phase oscillators. Furthermore, we replace the conventional attention module with an optimal control component, inspired by the notion of attending memory in cognitive neuroscience, to focus on task-relevant information while suppressing irrelevant stimuli during dynamic cognitive activities. Drawing inspiration from neuroscience and grounded in the principles of holography, we name our method HoloBrain. In a nutshell, the observed blood oxygenation level dependent (BOLD) signals are first processed using GST, and the resulting output, denoted by X⁽⁰⁾, serves as the input to HoloBrain. The output of HoloBrain includes (1) synchronized neural oscillations X^(L) and (2) the learned control pattern Y^(L) after L layers. By combining the power of machine learning and an unprecedented amount of public neuroimaging data, our HoloBrain method not only yields promising performance in predicting cognitive tasks but also offers an interpretable perspective of functional brain dynamics.

Prediction of cognitive tasks

We evaluate the model performance on the following public datasets. (i) The Lifespan Human Connectome Project Aging (HCP-A) dataset²³. The HCP-A dataset provides a comprehensive view of the aging process and is instrumental in task recognition research. It consists of data from 717 subjects, including both fMRI scans (4864 time series) and diffusion-weighted imaging (DWI) scans (717 in total). The dataset includes four tasks associated with memory: VISMOTOR, CARIT, FACENAME, and REST state. Each fMRI scan comprises 300 time points. In our experiments, these tasks are treated as distinct categories in a four-class classification problem. (ii) The Human Connectome Project – Young Adults (HCP-YA) database²⁴. Each scan contains data from seven cognitive tasks related to memory, including Motor, Relational, Social, Working Memory, Language, Emotion, and Gambling. Each task-based fMRI scan contains 175 time points in our experiments. In addition, the Working Memory task (abbreviated as HCP-WM) involves 2-back and 0-back task conditions with body, place, face, and tool stimuli, interspersed with fixation periods. A resting-state period follows two sequential cognitive task periods in an alternating fashion, resulting in fMRI scans with a total of 405 time points.

For both HCP-A and HCP-YA datasets, the brain is partitioned into 116 regions using the AAL atlas²⁵. The structural connectivity (SC) matrix is a 116 × 116 matrix, where each element quantifies the number of fibers linking two brain regions. The fiber counts are normalized by the total fiber count of each subject. Functional connectivity (FC) is computed as the Pearson correlation between BOLD signals of different regions, representing the temporal synchronization of neuronal activity in the brain. The preprocessing pipelines are fully described and accessible through publicly available tools (fMRIPrep: https://fmriprep.org/en/stable/, QSIPrep: https://qsiprep.readthedocs.io/en/latest/). To evaluate scalability, we employ the Brainnetome Atlas²⁶ to partition the whole brain into 246 regions for the HCP-WM data. In our experiments, the HCP-YA tasks are treated as distinct categories in a seven-class classification problem, while the HCP-WM tasks form an eight-class classification problem. For these experiments, we report 5-fold cross-validation results.

Prediction accuracy. We compare against a diverse set of state-of-the-art graph-based methods, including vanilla GCN¹¹, graph attention network (GAT)²⁷, graph isomorphism network (GIN)²⁸, GCNII²⁹, GraphSAGE³⁰, graph transformer with spectral attention network (SAN)³¹, graph-coupled oscillator networks (GraphCON)³², and a Kuramoto model-based approach (KuramotoGNN)³³. Fig. 5a and Supplementary Table 3 list the quantitative comparison results for three datasets on nine methods. HoloBrain achieves the best performance on all datasets over the existing hand-designed GNN models.

Model interpretability. These results provide compelling evidence that our physics-based oscillation model effectively synchronizes different brain regions, with the resulting synchronization patterns corresponding to distinct brain cognitive states. Especially in HCP-WM, all tasks from the same subject share identical structural connectivity. This design penalizes topology-dependent baselines but favors models that read out temporal coordination. HoloBrain explicitly models oscillatory synchronization and therefore separates the eight states more effectively than static GNNs. To examine the relationship between neural synchronization and cognitive tasks, we examine the phase-space representations of various brain regions, as shown in Fig. 3a. Our analysis reveals that different brain states exhibit unique synchronization patterns. For instance, during the VISMOTOR task, synchronization is predominantly observed in the visual and sensorimotor regions, reflecting the functional demands of the task. In contrast, during the REST state, the synchronization pattern is more pronounced within the default mode network, aligning with its role in self-referential and intrinsic cognitive processes³⁴. In addition, we further investigate task-specific synchronization across all subjects. As neural synchronization progresses (Fig. 3b), tasks gradually synchronize across groups, resembling the phenomenon of runners on a track who initially move at their own speeds but gradually form synchronized clusters through mutual interaction. More visualization analyses are shown in Supplementary Fig. 3.

Remarks. The collective synchronization emerged from HoloBrain, akin to patterns observed in natural systems such as fireflies flashing in unison or birds flying in formation, highlights the emergent coordination in neural dynamics. Notably, distinct synchronization patterns emerge between tasks, reinforcing the structured nature of this process.

Emergence of interference patterns from HoloBrain. Since HoloBrain is inspired by the interference of cross-frequency coupling, we hypothesize that HoloBrain recognizes neural activities by synchronizing oscillations across frequencies and across brain regions, which eventually give rise to interference patterns observed in Fig. 2.

Emergence of cross-frequency interference patterns associated with cognitive tasks. In HoloBrain, we predict cognitive tasks by modeling the synchronization of neural oscillatory waves throughout the brain, with each wave oscillating at its own natural frequency. In this context, we compute the H × H(H = 10) CFC matrix on each brain region, where each element in the CFC matrix characterizes the interference between two synchronized oscillatory waves. First, we examine whole-brain CFC patterns associated with underlying cognitive tasks by averaging the 10 × 10 CFC matrices across brain regions and individuals. As shown in Fig. 4a, it is evident that our HoloBrain reproduces the constructive and destructive patterns along the off-diagonal in the CFC matrices, suggesting the emergence of interference patterns during the learning process. Through visual inspection, the population-wise CFC matrices exhibit substantial differences across cognitive tasks. Following the quantitative analysis in Fig. 2, we calculate the sign consistency degree for the first, second, and third off-diagonal lines of each subject’s whole-brain CFC matrix, where the distribution of sign consistency degree of each cognitive task is shown in the bar-plot in Fig. 4a. Significant task-to-task differences at p < 0.05 are indicated by ‘*’.

Emerging knowledge of critical brain regions from the perspective of neural synchronization. Furthermore, we investigate constructive/destructive interference patterns in each regional CFC matrix. In Fig. 4b, we display five regions with the highest sign consistency degrees along the first, second, and third off-diagonal lines in older (HCPA: 58.0 ± 18.5 years) and young adults (HCPYA: 29.1 ± 3.6 years). In older adults, the left precentral gyrus (#1), left and right supplementary motor area (#19-20), right calcarine (#44), and left superior parietal gyrus (#59) manifest the highest local synchrony, which form a coherent motor-attention-visual compensatory network. Numerous aging studies^35,36 have shown that older adults exhibiting increased β/γ coupling in primary motor (M1/SMA) circuits have a high reserve to preserve motor function. In addition, the observed full synchrony in the primary visual cortex (calcarine) supports prior reports suggesting that increased local coupling in the aging visual system serves as a compensatory strategy for maintaining perception³⁷. Similarly, the left superior parietal lobule, a core node of the dorsal attention network, often displays elevated short-range connectivity to offset declines in spatial attention. Taken together, our findings imply that neural oscillatory synchronization provides a window for understanding critical dynamics of brain function that characterize healthy brain aging. In young adults, the five regions showing the strongest local synchrony are: #1-left precentral gyrus, #11-left inferior frontal gyrus (opercular part), #19-left supplementary motor area, #60-right superior parietal gyrus, and #61-left inferior parietal gyrus. These brain regions form a tight sensorimotor-attention loop in which primary and supplementary motor regions are anchored to adjacent parietal cortices. This “motor-attentional” network consistently exhibits the highest amplitude-coupling and local coherence at rest in healthy young adults, reflecting its role in maintaining fine motor control and spatial attention even in the absence of overt movement. This finding has been replicated across multiple HCP analyses³⁸, confirming that the sensorimotor network dominates intrinsic neural synchrony in youth.

Remarks. Leveraging the principles of the Kuramoto model, our HoloBrain provides a powerful explanatory framework that captures the majority of variability in the relationship between functional fluctuations and cognitive tasks. The inference patterns underlying cross-frequency coupling have been replicated by synchronizing neural oscillations in our physics-informed neural network, which allows us to investigate criticality through the lens of local synchrony.

Clinical value in disease diagnosis. In this experiment, we evaluate the potential of HoloBrain in the early diagnosis of neurodegenerative diseases, using the following three public neuroimaging data cohorts for AD, PD, and FTD. Specifically, we involve (i) Alzheimer’s Disease Neuroimaging Initiative (ADNI): This dataset includes resting-state fMRI data from 135 subjects, comprising individuals diagnosed with Alzheimer’s disease (AD) and cognitively normal (CN) controls. It is designed to track brain changes associated with AD progression. (ii) Parkinson’s Progression Markers Initiative (PPMI): A multi-center study that collects neuroimaging data from 173 subjects, including individuals with Parkinson’s disease (PD), scans without evidence of dopaminergic deficit (SWEDD), prodromal PD, and CN. (iii) Neuroimaging Initiative for Frontotemporal Lobar Degeneration (NIFD): This dataset focuses on frontotemporal dementia (FTD) and includes resting-state fMRI data from 1010 subjects. Participants are categorized into CN, logopenic variant of primary progressive aphasia (LVPPA), behavioral variant frontotemporal dementia (BV), progressive non-fluent aphasia (PNFA), and semantic variant (SV) groups.

Consistent with the preprocessing pipeline applied to the HCP dataset and the comparison methods, we obtain the regional mean BOLD signals and 116 × 116 SC matrices based on the AAL atlas²⁵. The graph embeddings of these methods are BOLD signals, and the adjacency matrices are SCs. For the disease diagnosis task, the ADNI dataset is formulated as a binary classification problem (AD vs. CN), the PPMI dataset as a four-class classification problem (PD, SWEDD, Prodromal, and CN), and the NIFD dataset as a five-class classification problem (CN, LVPPA, BV, PNFA, and SV). We evaluate model performance using accuracy (Acc), precision (Pre), and F1-score (F1), reporting results based on 5-fold cross-validation.

Diagnosis accuracy is shown in Fig. 5a and Supplementary Table 3, our HoloBrain method significantly outperforms the counterpart approaches in identifying AD, PD, and FTD subjects, where the improvement is significant (*p < 0.05, paired t-test) on PPMI and NIFD datasets, demonstrating the effectiveness of physics-informed deep model in disease diagnosis. Compared to other GNN models, the governing equation in our HoloBrain offers a systems-level perspective on functional fluctuations, helping to elucidate the mechanisms underlying dysfunctional syndromes in neurodegenerative diseases³⁹, as detailed below.

Neural synchronization metrics as biomarkers for neurodegenerative disease. We hypothesize that the reduced degree of neural synchronization might be a putative indicator of the neurodegeneration process in aging brains. To validate our hypothesis, we analyze the phase synchronization between different brain regions by computing the Kuramoto Order Parameter (KOP), thus the population-wise KOP for i^th region is defined as:\({R}_{i}=\left\vert {e}^{\sqrt{-1}(arg({{{{\bf{x}}}}}_{i}^{(L)}))}\right\vert\), where \(arg({{{{\bf{x}}}}}_{i}^{(L)})\) is the synchronized oscillation (see Supplementary Information for details) in phase space (output of L^th layer of HoloBrain) of i^th brain region. The population average of whole-brain KOP is also defined as: \({R}_{whole}=\frac{1}{N}\left\vert {\sum }_{i}^{N}{R}_{i}\right\vert\), where N denotes the number of brain regions. In this context, we examine the group difference of synchronization level for ADNI, PPMI, and NIFD datasets (as shown in Fig. 5b) to investigate whether and how the neural synchronization mechanism is altered in neurodegenerative diseases.

First, we conduct a t-test to compare the whole-brain KOP degree between CN individuals and those with diseases. As illustrated in the shaded panels in Fig. 5b, all disease groups (AD, PD, and FTD) consistently show a significantly lower whole-brain KOP degree compared to the CN group (p < 0.05). Second, we scale up the group comparison by expressing the disease outcome via the regional KOP degree. At a significance level p < 10⁻⁵, we have identified group differences in 11 brain regions between CN and AD. For CN vs. PD, 5 cerebellar regions and 6 other brain regions showed significant differences. Similarly, the comparison between CN and FTD revealed significant differences in 4 cerebellar regions and 7 additional brain regions. We display these disease-specific brain regions in Fig. 5b, along with the distributions of regional KOP degree R_i in CN and disease groups. Specifically, the majority of brain regions show a reduction in KOP degree (effect size β < 0 in the linear regression model) in the disease group (marked in green), indicating that disease-related pathology may disrupt neural synchronization. This pattern is quite consistent in PD and FTD. However, four brain regions—the right precuneus, right superior parietal gyrus, right fusiform gyrus, and right parahippocampal area—exhibit an opposite trend (marked in red), all associated with Alzheimer’s disease, where pathological changes appear to enhance synchronization levels. One possible reason is that these regions are located on the spreading pathway of tau aggregates (one of the key pathological hallmarks) at the early stage of AD. As a result, increased synchronization in these areas may facilitate the spread of AD pathology across the brain, thereby contributing to disease progression.

Uncover disease-specific interference patterns. To replicate the hypothesis of wave-to-wave interference (Fig. 2b), we compute the CFC matrix for each brain region between synchronized oscillations determined by the Kuramoto model in HoloBrain (i.e., the final-layer node representations). By averaging the CFC matrix across regions and individuals, we show the average CFC matrices in Fig. 6a for AD, PD, and FTD. Compared to the average CFC matrices in Fig. 2, our HoloBrain reproduces quite similar interference patterns along the off-diagonal lines, while revealing more pronounced group differences. This finding implies that the neurobiological process underlying oscillatory synchronization might be disrupted during the progression of neurodegenerative diseases.

Furthermore, we investigate the hypothesis that the contrast between constructive and destructive interference patterns is linked to neurodegenerative diseases. To do so, we first select five brain regions with the strongest local synchrony, i.e., the total degree of sign consistency along the first, second, and third off-diagonal lines is higher than in other regions, for AD, PD, and FTD subjects. We conduct this step on ADNI, PPMI, and NIFD datasets separately. Note, clinical label information is not used in region selection. As shown in Fig. 6b, the left hippocampus, bilateral superior frontal gyrus, and bilateral cuneus exhibit the highest levels of synchrony in older adults with AD, forming a hippocampo-occipital constellation. This pattern recapitulates medial-temporal and occipital short-range hyperconnectivity, previously associated with early tau deposition, memory decline, and visual-attentional symptoms in AD⁴⁰. In PD subjects, the top five regions with the highest level of local synchrony are located in primary/secondary visual cortex—right/left cuneus and lingual gyri plus the left superior occipital gyrus (Fig. 6b). The resulting “visual belt” mirrors the ‘β/γ’-band hypersynchrony and tremor-related cerebello-thalamo-visual loop repeatedly reported in PPMI cohorts⁴¹. In Fig. 6b, five regions in the FTD population are strictly peri-Sylvian and left-lateralized, comprising the inferior frontal operculum, middle frontal gyrus, angular gyrus, inferior parietal lobule, and the supplementary motor area. These nodes form the executive-speech circuit that shows early structural loss and local FC increases in non-fluent primary progressive aphasia and behavioral variant FTD^42,43.

After that, we perform a group comparison to examine whether the total degree of sign consistency is associated with the clinical label using a linear regression model. At a significance level of p < 0.05, the synchrony degrees at the left hippocampus and left/right cuneus are strongly associated with AD, where cognitively normal subjects manifest higher local synchrony degrees than AD subjects. All of the top five highly synchronous brain regions demonstrate significant differences between CN and PD. Likewise, all regions—except the inferior frontal operculum—show significant differences between CN and FTD. Notably, the group difference patterns observed in PD and FTD are consistent with those in AD, with both conditions exhibiting reduced local synchrony.

Remarks. The results presented in Figs. 5, 6 provide evidence that the neurobiological mechanisms underlying oscillatory synchronization may be disrupted throughout the progression of neurodegenerative diseases.

Brain rhythm-inspired learning mechanism for graphs

Suppose \({{{\bf{X}}}}={[{{{{\bf{x}}}}}_{i}]}_{i=1}^{N}\) is the initial graph embeddings, where the graph consists of N nodes. Current GNNs follow the principle of graph heat diffusion to aggregate features on top of the graph topology by \(\frac{\partial {{{\bf{X}}}}}{\partial t}+\Delta {{{\bf{X}}}}=0\), where Δ denotes the graph Laplacian operator¹⁵. Since the fundamental assumption in current GNNs is the graph smoothness \({\sum }_{i,j}{{{{\bf{x}}}}}_{i}^{\top }\Delta {{{{\bf{x}}}}}_{j}\), excessive information exchange often leads to over-smoothing issues⁴⁴. Inspired by HoloBrain for modeling functional dynamics, we present a solution to address the over-smoothing issues by (1) considering each graph node as an oscillator and (2) compressing the graph feature representations towards partial/full synchronization in the latent oscillatory phase space. In this regard, the oscillatory nature of the brain-inspired system prevents the GNN model from converging to a static and trivial state where all graph embeddings become identical by the end of graph heat diffusion. In a nutshell, we translate the design principle of HoloBrain to the cliché of GNN, yielding a GNN model coined HoloGraph.

Integrate the Kuramoto model into HoloGraph. We treat each graph node as an oscillator, with interactions determined by the graph topology. HoloGraph takes the graph embedding after the GST transform as the initial condition for oscillatory synchronization, denoted by x(0), operating at different harmonic frequencies. In this context, the phase dynamics of these oscillators are governed by a vector-valued equation¹⁰:

where ϕ represents a projection of the vector K_ijx_j onto the tangent space at point x_i on the unit sphere and ρ is a scalar hyperparameter.

Integrate optimal control into HoloGraph. The global nature of dynamics in the Kuramoto model poses challenges to capturing transient patterns of functional fluctuations that are closely associated with cognitive tasks. To address this issue, we introduce the notion of optimal control from cognitive neuroscience⁴⁵ into our model by:

where y_i is a learnable memory controller of the past activity of oscillator x_i. Intuitively, each y_i captures behavior-specific episodic memories and characterizes the mechanistic role of each oscillator engaged in a given cognitive task. By tracking the progression of cognitive states, we can continuously assess feedback from neural oscillations and adjust the synchronization in the Kuramoto system through episodic control.

Taken together, feature learning in HoloGraph is cast as a dynamic process governed by the Kuramoto model. As shown in Fig. 7a, the backbone consists of two layers: the Kuramoto layer and the optimal control layer, which together update the feature representations X^(l+1) and control patterns Y^(l+1). Kuramoto layer is used to compress the feature \({{{{\bf{x}}}}}_{i}^{(l)}\) on each oscillator by \({{{{\bf{x}}}}}_{i}^{(l+1)}=norm({{{{\bf{x}}}}}_{i}^{(l)}+\epsilon \cdot \Delta {{{{\bf{x}}}}}_{i}^{(l)})\), where \(\Delta {{{{\bf{x}}}}}_{i}^{(l)}={\omega }_{i}^{(l)}+\rho \cdot \phi ({{{{\bf{y}}}}}_{i}^{(l)}+{\sum }_{j\ne i}{w}_{ij}^{(l)}{{{{\bf{x}}}}}_{j}^{(l)})\) and norm(⋅) denotes the normalization operation that ensures the oscillator stays on the sphere. In parallel with the governing equation in HoloBrain, our HoloGraph compresses feature representations by aligning the oscillators with similar properties or labels via the Kuramoto model, thus preventing the over-smoothing problem in modern GNNs. Control layer reads out patterns encoded in the oscillators by a mapping function Y^(l+1)= f_φ(X^(l+1)) with the parameter φ.

Evaluation on graph node classification. We apply our HoloGraph to both heterophilic and homophilic datasets (sorted by homophily ratio ℏ⁴⁶): Texas (ℏ = 0.11), Wisconsin (ℏ = 0.21), Actor (ℏ = 0.22), Squirrel (ℏ = 0.22), Chameleon (ℏ = 0.23), Cornell (ℏ = 0.3), Citeseer (ℏ = 0.74), Pubmed (ℏ = 0.8) and Cora (ℏ = 0.81), where ℏ indicate the fraction of edges that connect nodes with the same label. We also evaluate on the large-scale OGBN-Arxiv dataset from the Open Graph Benchmark (OGB)⁴⁷. The detailed data description is shown in Supplementary Table 1. Fig. 7b, c and Supplementary Table 4 present the comparison results across nine datasets using nine different methods. HoloGraph achieves competitive performance on heterophilic (Fig. 7b, Table Supplementary Table 4) and decent performance on homophilic graphs (Fig. 7c, Table Supplementary Table 4), outperforming existing hand-designed GNN models. Notably, in heterophilic graph data (such as Texas, Wisconsin, Actor, Squirrel, Chameleon, and Cornell), node connections don’t depend on similarity. Particularly, traditional GNNs, which rely on the assumption that neighboring nodes have similar features (message passing mechanism), struggle with heterophilic graphs where this assumption doesn’t hold. This leads to poor performance. In contrast, our HoloGraph captures complex interactions through oscillator synchronization. By modeling coupling and phase synchronization, it allows information to be transmitted based on dynamic relationships, not just similarity. This makes it well-suited for heterophilic graphs, as it can propagate information through “asynchronous synchronization” even when node features are far apart in the initial condition. In short, HoloGraph effectively handles heterogeneity and improves performance on such graphs. To evaluate model scalability, we further tested HoloGraph on the large-scale ogbn-arxiv dataset. As shown in Fig. 7d and Supplementary Table 5, our model outperforms all standard baselines and achieves the best performance, with an accuracy of 0.86. These results highlight the generalization and discriminative power of our oscillatory synchronization mechanism on large graphs. Discussion. These results provide strong evidence that our message aggregation mechanism effectively synchronizes nodes of the same class while adhering to the constraints of the adjacency matrix (as shown in Fig. 7f). This approach is fundamentally distinct from traditional message-passing methods in GNNs, as it does not enforce identical feature representations for nodes within the same class. Consequently, our method mitigates the risk of over-smoothing as the network depth increases. To validate this claim, we evaluated classification performance across multiple network layers, as shown in Fig. 7g. It is clear that the classification performance remains stable even as the number of network layers increases, reaching up to 128 layers. This indicates that our method effectively mitigates the issue of over-smoothing.

Evaluation on graph classification. We apply HoloGraph to ENZYMES and PROTEINS (TUDataset⁴⁸). The detailed data description is shown in Supplementary Table 2. For homophilic graph data (Cora, Citeseer, Pubmed), we adopt the data-splitting method used in the vanilla GCN¹¹, specifically the Public semi-supervised Split manner, which allocates a fixed 20 nodes per class for training, and we report results averaged over 10 runs with different random seeds. For heterophilic graph data and TUDataset, we follow the data-splitting manner introduced in ref. ⁴⁹ and ref. ⁵⁰, respectively, with 10-fold cross-validation, and we report the averaged metric on all test folds. Each dataset is evaluated by accuracy, precision, and F1-score. Fig. 7e and Supplementary Table 6 present the quantitative comparison of nine methods across two datasets. HoloGraph demonstrates decent performance (batch_size = 1) in graph classification. Discussion. The graph classification task shares similarities with brain rhythm identification, as both aim to classify entire graphs rather than individual nodes. Consequently, the synchronization mechanism in our model is designed to exhibit distinct synchronization patterns for different graphs, validating its effectiveness on graph classification tasks.

Additional implementation details and ablation studies are provided in the Supplementary Information (Supplementary Tables 7–10).

Explore the potential of solving problems without ground truth

In the HCP-A dataset, we use the AAL atlas to divide the brain’s 116 regions into eight functional subnetworks: the default mode, frontoparietal, limbic, ventral attention, dorsal attention, sensorimotor, visual subnetworks, and cerebellum. To investigate the potential for identifying population-level functional communities in the absence of ground truth, we train an unsupervised neural oscillatory synchronization model designed to cluster brain regions into distinct regions. This model minimizes the Euclidean distance between x_i and x_j through the optimization of the Rayleigh quotient⁵¹. According to Eq. (7) in the Method session, our approach effectively identifies population-level functional communities that not only align with established subnetworks but also reveal potential patterns in brain organization. Fig. 5c shows the clustering results on our HoloBrain and classic spectral clustering. It is clear that, our proposed HoloBrain closely synchronized related brain regions through oscillation, generating specific functional communities. This demonstrates the potential of our approach to collaboratively enable personalized brain parcellation.