Machine-enhanced reconstruction of functional connectomes unravels discriminative brain sub-systems in health and disease

Abstract

Reconstructing functional brain networks from functional Magnetic Resonance Imaging (fMRI) data typically relies on statistical pruning, where pairwise correlations between brain Regions of Interest (ROIs) are independently thresholded, ignoring potentially relevant collective co-activation patterns within sub-networks. Here we introduce a functional pruning approach that identifies which connections to retain based on their collective importance for distinguishing healthy from affected individuals, rather than on their individual statistical significance. We use geometric deep learning to learn network representations and an Explainable Artificial Intelligence (XAI) tool to identify the most discriminative sub-networks. Using fMRI data from healthy and Autism Spectrum Disorder (ASD) subjects, we build multilayer network representations from a multi-frequency decomposition of the signals and provide robust evidence that the machine-learned sub-systemic co-activation patterns significantly improve the identification of affected individuals. Our results demonstrate how functional pruning — which is based on collective, rather than individual, co-activation patterns — provides mechanistic insights that can be reliably used to characterize brain disorders, at variance with statistical pruning. Our approach is general and can be applied to find biomarkers for other brain diseases and, more broadly, to analyze complex systems where only the activity of individual units can be measured.

Introduction

The human brain, with its approximately neurons and synapses, represents a formidable complex system. Understanding this system can be approached through complex network analysis, specifically by examining structural and functional connectivity. Structural connectivity refers to the physical links between neurons, while functional connectivity provides a representation for the patterns of co-activation among different brain regions^1,2. However, reconstructing functional connectivity networks poses significant challenges, as it is an inverse problem that is not easily tackled directly^3–5.

To address this, researchers typically use indirect methods, which involve calculating pairwise cross-correlation or cross-similarity measures for each potential link between ROIs. Accordingly, an arbitrary threshold is applied to filter out less significant links, retaining only those above a certain value, thus transforming the weighted fully-connected network into a (often binarized) sparser one. This approach is known as statistical pruning.

Despite its widespread use, statistical pruning has several inherent problems. On the one hand it can often lead to spurious network reconstructions⁶, an issue that can be partially mitigated by comparing the reconstructed networks against suitable null models and employing reconstruction methods based on network ensembles⁷. On the other hand, another major drawback of standard statistical pruning is its reliance on treating pairwise correlations as independent of one another, ignoring the potential for emergent sub-networks characterized by specific collective co-activation patterns. In fact, it is plausible that specific co-activation patterns within these sub-networks could serve as better biomarkers for distinguishing healthy from affected individuals. In such sub-networks, a highly significant link may co-appear with less significant links, forming distinct patterns. Thus, analyzing whole co-activated sub-systems could offer a more reliable means of identifying brain disorders than traditional network descriptors.

Recent studies attacked this problem by means of ad hoc methods accounting for the higher-order organization of such networks^8–10 (see¹¹ for a review). In particular, information-theoretic analyses have used multivariate measures to characterize redundant and synergistic dependencies among groups of regions in resting-state fMRI and to track how these higher-order interactions reorganize across the lifespan or in disease (e.g.^12–14). Building on related developments in multivariate information decomposition, these works have identified synergistic “cores” and maximally synergistic subsystems that are largely invisible to pairwise functional connectivity, highlighting the relevance of high-order structure for cognition. Complementary approaches have modeled higher-order organization by means of machine-learned embeddings and topological descriptors: for example, connectome embeddings derived from structural connectivity have been shown to capture network-level relations that improve prediction of resting-state fMRI connectivity and lesion effects⁸, while simplicial-complex representations of fMRI co-fluctuations uncover k-node co-activation motifs beyond dyadic links¹⁰. While existing higher-order fMRI studies focus on directly quantifying redundancy/synergy or enumerating k-order motifs, our contribution is complementary: we leverage machine-learned, layer-resolved subnetworks as emergent higher-order units that are optimized for classification and then subjected to network-science analysis, providing mechanistic insight into ASD-related alterations in a way that, to our knowledge, has not been explored in previous higher-order fMRI work.

Here, we propose an intermediate approach to functional connectivity analysis that focuses on functional pruning. Instead of considering individual co-activation signals separately, we examine the collective patterns of co-activation within entire sub-networks. This method, powered by geometric deep learning and a suitable explainer, aims to provide more robust and mechanistic insights into brain disorders.

Note that the method fundamentally differs from existing higher-order ones, since we neither directly disentangle high-order mechanisms in terms of redundant and synergistic signals¹⁵ nor analyze specific high-order interaction patterns¹¹. Instead, we use collective pairwise co-activation patterns to identify sub-networks – with size not fixed a priori – that maximize the distinction between healthy and affected subjects. In the following, we refer to this approach as functional pruning, to distinguish it from the standard statistical pruning.

To demonstrate the ability of our approach to detect sub-networks relevant for characterizing the human brain in health and disease, we use fMRI data from a healthy control group and from a group of patients affected by ASD. ASD represents a particularly suitable test case, as prior work has shown that functional connectivity at specific frequency bands (particularly in the slow-4 and slow-5 bands) can discriminate individuals with ASD from controls¹⁶, suggesting that frequency-specific co-activation patterns may be especially relevant for this disorder. Note, however, that our approach is generalizable and offers a robust framework for analyzing brain networks from other brain disorders and, overall, for all those complex systems where only the activity of individual units can be measured while the underlying structure remains unknown.

Results

The analysis of the networks after our functional pruning approach, summarized in Fig.  1, yielded several key findings, with the main results reported below. For more details on the overall pipeline, including pre-processing, processing and analytics steps, we refer to Materials and Methods.

Fig. 1.

Schematic overview of the proposed pipeline. (A) Preprocessing of fMRI data²⁶ to extract ROI time series. (B) Construction of multilayer functional networks for each subject, where each layer represents a frequency band and edges indicate statistically significant coherence between ROI pairs. (C) Training of a Geometric Deep Learning model to classify subjects based on network topology, capturing collective co-activation patterns. (D) Subject classification using Geometric Deep Learning. (E) Application of GNNExplainer, an Explainable AI (XAI) tool, to identify the most discriminative sub-networks contributing to classification. (F) Threshold selection to identify maximally pruned sub-networks that preserve or improve classification accuracy. (G) Network analysis at ROI and subsystem scales to extract interpretable features and gain insights into disease mechanisms.

The metrics evaluated reveal differences in the distributions of values between ASD subjects and neurotypical controls (i.e., healthy subjects) across all the different slow sub-bands. In the mesoscale representation of the brain functional network, obtained by aggregating ROIs into larger functional subsystems, we find that ASD subjects exhibit higher values of average degree and average local clustering, with a greater mean compared to neurotypical subjects for both measures, as shown in Fig.  2. This can be explained by hyper-connectivity (a higher number of connections between different brain areas) that has often been detected in ASD patients (see later in the text). The average degree measures the average number of connections per node (brain subsystem in our case), while the average local clustering quantifies the tendency of brain regions to form closed triads (i.e., triangles, interconnected groups of three nodes), indicating a higher level of local interconnectedness. This measure is particularly relevant in the study of brain networks because it can reflect how neuronal connections are organized in individuals with ASD compared to neurotypical controls. Variations in the clustering coefficient can indicate significant differences in brain connectivity, which are often associated with the symptoms of ASD. We also note that the degree assortativity (Degree assortativity is a measure of the tendency of nodes in a network to connect to other nodes that have a similar degree. In this context, it reflects how brain regions with similar levels of connectivity tend to be interconnected) does not show any significant difference between healthy and affected subjects. On the other hand, differences in the distributions k-core values (The k-core of a network is a subgraph in which each node is connected to at least k other nodes within the same subgraph (computed, e.g., by recursively removing nodes with degree less than k neighbors), which helps identify densely connected regions in the brain’s functional network) are found between affected subjects and controls. Specifically, k-core measures are higher in affected subjects across all the studied sub-bands and may reflect a higher level of connectivity between different subsystems in patients. We investigate the robustness of these findings by performing a two-sided Mann-Whitney U test to compare the distributions of average degree, average local clustering, and k-core values between ASD subjects and neurotypical controls across all frequency bands. The results shown in Fig.  3 confirm that the observed differences are meaningful (i.e., with a considerable effect size, as measured by the Rank-Biserial Correlation with values and statistically significant even after controlling for the False Discovery Rate Benjamini-Yekutieli (FDR-BY) corrected p value) for four out of five metrics in each frequency band, reinforcing the reliability of our findings. On the other hand, the degree assortativity does not show statistically significant differences between the two groups across any frequency band, and the RBC values should not be considered meaningful. We note that while we assume a corrected p-value threshold of 0.05 for statistical significance, the actual values are much lower in most cases, indicating strong evidence against the null hypothesis, as reported in the full results tables in the Supplementary Information. We also compare the differences and their significance with the non-functionally pruned networks in Fig.  3 using the same statistical analysis described above, finding that no statistically significant differences are observed between ASD subjects and neurotypical controls for any of the network metrics considered.

Fig. 2.

Metrics distribution of subsystem-level networks at explainer threshold . We compute local, first order and global topological metrics of nodes, and compare them across groups. The blue and orange colors represent control and affected subjects respectively, and are shown in the upper and lower quadrants for sake of comparison. The plots show the Kernel Density Estimate, while the inner of the curve (the box-plot) contains the median (white color) and the inner quartiles (gray box) while the whiskers extend to show the rest of the distribution, except for the detected outliers.

Fig. 3.

Rank-Biserial Correlation (RBC) effect sizes from Mann-Whitney U tests comparing network metrics between ASD and Healthy groups at subsystem level, showing the impact of functional pruning. Each cell shows the RBC value for a metric-layer combination (20 tests total per threshold, given 5 metrics and 4 layers). RBC values range from -1 to 1, where positive values (red) indicate Healthy subjects tend to have higher metric values; negative values (blue) indicate ASD subjects tend to have higher values. Significance marker: FDR-BY . Left: unpruned networks (); Right: pruned networks (). Degree assortativity cannot be tested in unpruned networks due to lack of variance across subjects, hence marked in gray. Functional pruning substantially increases the number of significant differences, supporting its use for improving signal-to-noise ratio in group comparisons. In fact, unpruned networks show no significant differences even before correction, while pruned networks show 16/20 significant differences even after FDR-BY correction, indicating that the machine-learned sub-systemic co-activation patterns significantly enhance the detection of group differences.

To investigate whether the observed differences in network metrics between ASD subjects and neurotypical controls are consistent across all metrics and frequency bands, we performed a global significance test using Stouffer’s method, by combining the p-values from individual Mann-Whitney U tests across all metrics and frequency layers, across layers (i.e., per-metric significance) and across metrics (i.e., per-layer significance).

For pruned networks (), the global test yielded , , with mean absolute effect size . Since the unpruned networks did not show any significant differences in any of the individual tests, the global test yielded non-significant results, with , , and mean absolute effect size .

In summary, these results indicate that the differences in network metrics between ASD subjects and neurotypical controls are statistically significant and meaningful only after applying our functional pruning approach, highlighting its effectiveness in uncovering relevant patterns in brain connectivity.

The full results of the statistical analysis are reported in Tables S2, S3, and S4, while additional details on the statistical analysis are provided in Materials and Methods.

We repeated the analysis of the network metrics at the ROI scale, and — as shown in Supplementary Fig.  S1 — the results are consistent with those obtained at the subsystem level, confirming the robustness of our findings across different scales of analysis. Furthermore, the connectivity analysis at a finer resolution allows us to also assess the difference between affected subjects and controls in an additional metric, namely the number of connected components. A connected component indicates groups of brain regions interconnected with each other, thus exhibiting (at least in pairs) statistically significant spectral coherence compared to the null model. In patients with ASD, lower means and variances are observed across all analyzed frequency bands, particularly in the slow-4 and slow-5 bands, compared to controls. This can be attributed to an increase in the number of connections between different areas present in the affected individuals.

To further investigate the differences in functional connectivity between ASD subjects and controls, we analyzed the co-activation matrices at the subsystem level across different frequency bands. In particular, Fig.  4 reports the differences in the co-activation matrix at the subsystem level between affected subjects and controls for each frequency band. A value greater than zero means that there are more links in the average ASD-affected subject than in the average control subjects. That is, the delta matrix is computed as , the difference of the average number of links in each of the two groups (A and H respectively for affected and healthy subjects). Our results show that the greatest differences are found in the slow-3 and slow-4 bands, often used to discriminate individuals with ASD from controls and correlate with structural measures (white matter) in fMRI experiments in human brain at rest¹⁷. However, in all bands, significant atypical connections are found in the Default Mode Network, Fronto-parietal, Unimodal areas (Visual and Somatosensory), and Thalamus (which is part of Subcortical in our classification). A more detailed inspection of Fig. 4 shows that the most pronounced differences are concentrated in the slow-3 and slow-4 bands. In the slow-3 band, the strongest atypical increases in ASD involve intra-network connectivity within the Default Mode Network (DMN) () and Somatosensory–Somatomotor (), as well as connections between the DMN and Visual () and Somatosensory Hand regions (). In the slow-4 band, similarly elevated differences occur between Visual and Somatosensory Hand (), DMN and Visual (), and DMN and Somatosensory regions (). Moderate but consistent increases are observed in the slow-2 band, while only small differences appear in the slow-5 band. It is worth specifying that, although low-frequency bands can be partially affected by physiological noise around 0.03 Hz (see for example¹⁸), the most pronounced group differences in our study occur in the slow-3 and slow-4 ranges, i.e., frequencies that lie well outside the region most influenced by aliasing. Overall, these patterns indicate that atypical connectivity in ASD is driven by a restricted set of subsystem pairs, primarily involving DMN, Visual, Somatosensory, and Fronto-parietal regions, showing reproducibly heightened co-activation in ASD subjects. These findings are in agreement with the extensive literature reporting that stronger functional connectivity within and between multiple brain areas is often observed in subjects with ASD compared with neurotypical controls^16,19–21. It is important to clarify that although it is well-established that ASD is related to altered communication between brain regions, the experimental evidence regarding connectivity alterations is sometimes conflicting.

Fig. 4.

Connectivity differences between affected subjects and control ones. That is, each value is computed as the difference between the average value of affected and healthy subjects for each pair of subsystems i and j. represents higher average connectivity in the pruned network of affected subjects. Only values with p-value are shown. The p-value is calculated as the probability of observing a delta value larger than the one obtained, based on 100, 000 random shufflings of the subject groups. Moreover, only the upper triangle is masked being the matrix symmetric.

Figure 5 illustrates the analysis of functional brain connectivity in affected subjects (left column) and healthy controls (right column) using the slow-4 frequency band, whereas results for the other frequency bands are shown in the Supplementary Information. The heatmaps in the upper panel display the functional connectivity between different brain systems. In healthy subjects, connectivity appears more evenly distributed across systems, suggesting balanced integration between networks. In affected subjects, higher connectivity is observed in specific systems, particularly the default mode network and the fronto-parietal task control network, which exhibit increased inter- and intra-network connections. This suggests a possible over-recruitment of these networks in the affected group.

Fig. 5.

Slow 4-band clustered connectivity and tanglegram, generated using Ward clustering with Euclidean distance. The left column represents affected subjects, while the right column represents healthy subjects. Clustering is a method of grouping subjects based on their similarities, which helps in identifying patterns within the data. Ward clustering minimizes the variance within each cluster. A tanglegram is a visualization tool that displays the relationships between two hierarchical clusterings, helping to compare the structure of the clusters.

The dendrograms depict hierarchical clustering of brain systems based on the similarity of their connectivity profiles and the tanglegram in the central panel compares the dendrograms of the two groups, illustrating differences in clustering patterns. In more general terms, the tanglegram provides a compact way to compare the hierarchical network organization of functional subsystems between affected and healthy individuals. Hierarchical clustering defines how brain systems group together across multiple levels of organization, from the closest pairs of networks to larger, multi-system communities. When the two dendrograms are aligned in a tanglegram, crossed links reveal subsystems whose relative positions in the hierarchy differ between groups, indicating that the similarity structure of their connectivity profiles has been reorganized. Results show that the fronto-parietal network is grouped differently in affected individuals compared to healthy controls. The salience network also exhibits notable reorganization in affected individuals, where it clusters closely with the subcortical system. In contrast, in healthy controls, the salience network is more closely associated with the fronto-parietal network, reflecting a different pattern of functional integration

These clustering differences suggest significant alterations in the functional integration and hierarchical organization of brain networks in the affected group, possibly reflecting compensatory mechanisms or network dysfunctions²². In other words, these findings indicate that the alterations extend beyond individual connections to involve the multi-level architecture governing how functional systems cluster into larger organizational units. Such reconfigurations have previously been interpreted as markers of altered functional integration or compensatory reorganization in neurodevelopmental conditions^23–25. Together, these patterns indicate that ASD is associated with both strengthened connections in specific networks and a broader restructuring of mesoscale hierarchy, where the organisational relationships between functional systems are rearranged.

Fig. 6.

Inverse percolation analysis of the network across increasing thresholds. The vertical dashed red line marks the threshold chosen for the primary analysis (see also Fig. 7), hypothesized to approximate the onset of the structural percolation threshold. At this point, we observe the disintegration of the Largest Weakly Connected Component (LWCC), whose size is shown on the y axes in both panels (for affected and healthy subjects, respectively). The second-largest connected component, which increases near the percolation threshold, is tracked to better capture the network’s transition into a disconnected phase. The selection of this threshold aims to maximize the stability of inferred network properties as it might correspond to an optimal point of the underlying connectivity.

Discussion

Using reconstruction techniques to build robust functional representation of the human brain is a delicate task. From the choice of the correlation metric to the choice of the pruning approach to drop non-significant links, every step contributes to introduce a new level of uncertainty in the network reconstruction⁶.

Here, we tackle this challenge and present a framework to analyze the functional features of human brain networks in health and disease. We develop a novel method capable of better characterizing the functional connectivity of such networks in terms of sub-systems of size not fixed a priori. The analysis of such sub-networks allows us to better distinguish between healthy and ASD subjects. Our approach exploits collective co-activation patterns to perform functional pruning as opposed to statistical pruning, which is based on dropping out links under an arbitrarily chosen correlation threshold, while considering each link independently from the others, thus neglecting potentially relevant high-order correlations. Our approach is motivated by the essential significance of the selected links for classification purposes.

Specifically, we select the critical edges crucial for classification. The first step of the proposed methodology consists in transforming the fMRI scans into functional co-activation networks. To this aim, we use a statistically sound algorithm that de-noises the cross-coherence of the signals from the various ROIs and only keeps the most significant ones with respect to a suitable null model. In this setting, each ROI is a node of the network, whereas a link in each layer corresponds to a correlation between the two regions at a given frequency band. Thus, a multilayer network is built for each subject, with layers corresponding to a predefined set of frequency bands. While this pipeline builds on existing literature^27,28, the networks are only used so far to compute non-learnable topological descriptors at node level that do not provide mechanistic insights on the disease in terms of sub-systemic co-activation patterns. Instead, here we feed such networks to a Geometric Deep Learning model^29,30, a Machine Learning algorithm that can process graphs, and that can be leveraged to extract the most important co-activations needed to classify each subject, corresponding to links in their multilayer network representation. Unlike other methods that utilize a fixed number of edges (see¹⁶), our method enables the association of a “classification score”, quantifying the degree of importance of the edge for classification, overcoming the standard limitations of deep learning models, often used as a black box. The resulting sub-networks — i.e., the subset of relevant co-activations (links) — can then be analyzed by domain scientists, such as neuroscientists, and/or with statistical and network science tools to find relevant information to distinguish between healthy and affected subjects. The benefits of the approach are threefold: first, we provide a data-driven tool that can be trained on subject data, aiding the expert diagnosis; second, the insights can lead to new diagnostic tools and to a better understanding of the disease itself; third, such understanding can provide new information useful to drive targeted therapies. Furthermore, similarities in brain sub-networks among individuals with ASD can provide valuable insights for diagnosis and treatment. It has been shown that it is possible to identify subgroups of individuals based on inter-subject deviation in functional connectivity³¹, and that connectivity patterns correlate with the severity of ASD symptoms, providing insights into personalized symptom prediction and management. Similarly, Zerbi et al.²⁰ applied clustering to 16 mouse models of ASD, discovering four distinct subtypes and indicating varied network dysfunctions tied to different etiologies of ASD.

Moreover, there are various advantages with respect to standard approaches: in fact, to the best of our knowledge, this is the first example of the application of interpretability tools (XAI) to Geometric Deep Learning models for functional brain networks, and the first application of such models to trainable algorithms in this context. While this methodology is general and can be applied to any fMRI and EEG scans, and also supports subject metadata — such as age, sex, number of comorbidities, etc. — here we demonstrate it with ASD subjects. Also, while the discrimination performance for such subjects is already high (0.84 ROC), better results can be achieved using larger datasets.

All our findings are consistent with a substantial amount of the available literature on ASD, according to which affected subjects exhibit stronger functional connectivity within and between multiple brain areas. In our case, the hyper-connectivity pattern is not uniform across the brain but is concentrated in specific large-scale systems. As shown in our subsystem-level analyses (see Figs. 4 and 5), ASD subjects display markedly increased intra- and inter-network connectivity in the Default Mode Network and the Fronto-parietal control network, together with atypically strong couplings involving Visual and Somatosensory regions as well as Subcortical/Thalamic structures. These system-specific alterations are most prominent in the slow-3 and slow-4 frequency bands and indicate that hyper-connectivity in ASD arises from a reproducible set of circuits rather than from a global, unspecific increase in overall functional coupling. It is nonetheless important to emphasize that both hypo-connectivity^32,33 and hyper-connectivity, defined respectively as a decrease and an increase in the number of connections in affected individuals compared to neurotypical controls, have been observed. The heterogeneity in results depends on the specific study and the brain areas involved. It has been recently hypothesized that this heterogeneity can be attributed to two major subtypes of brain connectivity profiles, one expressing hypo-connectivity and the other hyper-connectivity. These subtypes do not differ in structural metrics or behavioral scores but solely in the expression of either a decrease or an increase in the number of connection within and between areas. Notably, only the subgroup expressing greater functional connectivity exhibits an excitation-inhibition imbalance, a well-known primary mechanism in the pathophysiology of ASD³⁴.

The validity of our model in reflecting the behavioral and symptomatic dimensions of ASD can be supported by the correlation analysis between the scores from our classification model and several clinical measures, specifically three scores that are widely used to evaluate the clinical presentation and severity of ASD symptoms. These measures are: the ADOS (Autism Diagnostic Observation Schedule) TOTAL, which represents the overall severity of autism-related symptoms across multiple domains; the ADOS SOCIAL, focusing on deficits in social communication; and the ADOS GOTHAM RRB, a specific score targeting Restricted and Repetitive Behaviors (RRBs), which are core features of ASD. As shown in Supplementary Fig. S6, the Spearman correlation coefficients indicate significant negative associations for all measures. These findings suggest that lower scores obtained with our classification approach (indicative of affected subjects, as they range between 0 and 1, where 0 represents an affected subject and 1 represents a healthy individual) are associated with higher ADOS clinical scores. This pattern is consistent with the expected relationship between clinical severity and classification outcomes.

Besides applications, in the near future it will also be interesting to further investigate why our approach performs so well at a given threshold. As we show in the Supplementary Information, a preliminary analysis suggests that the optimal threshold is at the onset of the percolation threshold. If confirmed by other analyses, based on other data sets, and possibly other diseases, this finding raises the interesting, yet speculative, hypothesis that the optimal reconstruction of connectomes might happen close to the structural percolation threshold that could be related to criticality and dynamical scaling³⁵. However, the validation of this hypothesis is beyond the scope of this work, and will require a neuronal-level dynamics analysis to be confirmed.

Materials and methods

Overview of the approach

In the following, we provide a minimal overview of the steps required to put our results in the correct context. The overall pipeline, including pre-processing, processing and analytics steps is schematically summarized in Fig. 1, while we refer to the rest of this section for details on each step.

Our approach starts with analyzing the time series data of signals recorded from various ROIs using functional Magnetic Resonance Imaging (fMRI) scans. These signals are collected from multiple subjects, providing a detailed representation of neural activity in different brain regions over time.

Subsequently, we build the functional multilayer network representation of the brain of each subject, where each layer corresponds to a specific frequency band and two ROIs are connected if the coherence between their signals is statistically significant, and prune it using the proposed approach, which extracts the most relevant sub-networks in a data-driven way by considering the collective activity instead of just pairwise correlations, as detailed in Fig.  1 and in the Methods section.

We therefore computed various network metrics using these most-relevant sub-networks and conducted the analysis both at the subsystem level and at a finer scale of individual ROIs. This is because subsystems and ROIs refer to different levels of analysis in the study of brain connectivity and function, where the former are defined as large, functionally integrated networks of the brain that consist of multiple regions working together to perform specific processing and can therefore provide insights at the macroscale, while the latter are smaller brain areas that allow for a detailed analysis of activity or connectivity at a finer resolution, ranging from the microscale to the mesoscale.

Dataset and fMRI data preparation

For demonstration of our method we used the dataset from Autism Brain Imaging Data Exchange (ABIDE)³⁶ initiative that contains resting-state fMRI scans of subjects with ASD and neurotypical controls, and that was preprocessed and publicly shared by the Preprocessed Connectome Project (PCP) (see²⁶ for preprocessing pipeline). The average time-series were extracted from each of 264 ROIs provided by Power et al.³⁷. For the usage of our method, we did not apply a high-pass filter to the time-series. In particular, we use 162 subjects from five sites (Caltech, CMU, Stanford, Trinity, USM) which use the same sampling period of 2 seconds in order to produce the same frequency bands. This pre-processing builds the part of the pipeline represented in Fig. 1A. The subjects are divided into two groups: 85 ASD affected subjects (mean age years, range 7.53–50.22 years, median age 18.42 years) and 77 neurotypical controls (mean age years, range 7.75–56.20 years, median age 18.14 years). The sex distribution is as follows: in the ASD group, there are 6 females and 79 males, while in the control group, there are 10 females and 67 males. A Mann-Whitney U test revealed no significant difference in age between the two groups (, ) and a Fisher’s exact test showed no significant difference in sex distribution (). However, there are significant differences in cognitive scores: Full-Scale IQ (FIQ) (, ), Verbal IQ (VIQ) (, ), and Performance IQ (PIQ) (, ), with the ASD group showing lower scores compared to controls. We refer the reader to Figs.  S9, S10 and S11 and to Table S1 in the Supplementary Information for detailed demographic and cognitive score summaries.

Multilayer network generation

Frequency bands

Low-frequency components (generally below 0.5 Hz) of the oscillations observed in fMRI signals are thought to be associated with modulation of cortical excitability and long-distance synchronization^38,39. In this work, we analyze frequencies in the range 0.01–0.25 Hz. The lower bound of 0.01 Hz is the conventional choice to eliminate low-frequency drifts, while the upper bound of 0.25 Hz corresponds to the Nyquist frequency for our data. Specifically, the Nyquist frequency — defined as half the sampling frequency²³ — represents the maximum frequency that can be accurately represented in a sampled signal; given that the fMRI scans in the ABIDE dataset have a sampling period (repetition time, TR) of 2 seconds, the Nyquist frequency is Hz for our data. Following previous work²⁷, we subdivide this range into 4 frequency sub-bands: slow-5 (0.01–0.027 Hz), slow-4 (0.027–0.073 Hz), slow-3 (0.073–0.198 Hz), and slow-2 (0.198–0.25 Hz), obtained by grouping the original 12 sub-bands of fixed width 0.02 Hz. This subdivision is motivated by recent studies showing that different brain regions exhibit resting-state activity predominantly within specific sub-bands (slow-4 and slow-5), while activity in the slow-3 band tends to correlate with structural connectivity as measured by the number of fibers obtained through Diffusion Tensor Imaging (DTI) ^17,39.

Multilayer network representation

We use multilayer networks⁴⁰ to represent the multi-frequency decomposition of functional connectivity in the human brain, following current literature^27,41–44. Adopting a similar approach to²⁷, we first compute the coherence between signals from all ROI pairs, which quantifies their similarity in the frequency domain and yields values between 0 and 1 for each pair. Once computed, the coherence values are organized into a multilayer network for each subject, where each layer corresponds to one of the four frequency bands. Each network has 264 nodes (one per ROI) shared across all layers, and the link weights in each layer are given by the coherence values between ROIs at the corresponding frequency band. Then, we reduce the number of links in the network by applying statistical thresholding to the coherence values. To do so, we employ a null-model built by shuffling the time series of the ROIs using the Iterative Amplitude Adjusted Fourier Transform (IAAFT) algorithm⁴⁵, which preserves the power spectrum of the original signals. For each set of shuffled time series, we compute the coherence between the ROIs as described above, and we repeat this process 1, 000 times to build a distribution of coherence values in each frequency band. The coherence values of the original signals are then compared to the distribution of coherence values obtained from the null model using z-scores, and we keep only the links that are significantly different from the null model with a Confidence Interval of . The pruned networks are sparse with a total number of links that is a fraction between and of the total number of possible links. The resulting networks are then used as input to the learning framework. Overall, the aforementioned processing builds the part of the pipeline represented in Fig. 1B. This statistical thresholding step is crucial for building a sparse representation of brain networks that is more suitable for the subsequent learning framework. Networks reconstructed from time series signals are inherently dense and potentially affected by spurious connections arising from indirect correlations and noise. Reducing such spurious correlations is essential, as they can hinder the learning process, particularly given the limited number of subjects typically available in neuroimaging studies. While in principle the learning model could process dense networks directly, this would require substantially larger datasets and more complex models to avoid overfitting. We therefore employ statistical pruning as a preprocessing sparsification step to reduce noise and computational complexity before feeding the data into the deep learning model, allowing the subsequent functional pruning to focus on identifying the most relevant co-activation patterns among the statistically significant connections.

Learning framework

The learning framework we adopt is based on Geometric Deep Learning^46,47 and, in particular, on Graph Neural Networks (GNNs). This builds the part of the pipeline represented in Fig. 1C.

Graph Neural Networks and learning task

Graph Neural Networks are a class of Machine (Deep) Learning algorithms that can process graph-structured data and can be trained to produce a latent representation — called an embedding — of each node (i.e., ROI) in the network. Embeddings are learned mappings into a Euclidean space that capture the network structure in a way that is meaningful for the task at hand. Internally, GNNs process networks through message passing: each node aggregates incoming messages (features) from its neighbors and updates its own representation based on the aggregated information. Both the aggregation and update functions are learned during training, with each GNN layer performing one iteration of the message-passing process. While GNNs support feature-rich networks, our goal is to extract the sub-structure of the network that is most relevant for classification based on connectivity patterns alone. Therefore, we adopt a one-hot encoding of nodes as input features, where each node is represented as a vector of zeros with a single one in the position corresponding to its ROI index. This encoding serves two purposes: it allows the model to distinguish between different brain regions (avoiding permutation invariance), and ensures that each node corresponds to a specific, fixed ROI across all subjects according to the Power et al. atlas³⁷. However, we feed the weight (coherence) of the links as an edge feature to the GNN. Since the nodes are regions of the brain and the different network layers represent connections at different frequency bands, we treat the network as edge-colored, i.e., the set of nodes is the same across all layers, but the set of edges is different, and processed by a dedicated GNN for each layer. Lastly, considering that we need a prediction for each subject, we aggregate the node embeddings into a single vector, the graph-embedding, representing the entire network, and we feed it to a multi-layer perceptron to produce the final classification score.

Model architecture

The model architecture, schematically represented in Fig.  S7, is designed to process multilayer brain networks where each layer corresponds to a different frequency band. The architecture consists of a series of Heterogeneous Graph Neural Network (HeteroGNN) blocks, followed by a global pooling layer and a fully connected classifier. Each HeteroGNN block contains multiple parallel GNN layer instances — one for each frequency band — that independently process the connectivity patterns at their respective frequencies. In other words, each block applies a different GNN layer instance to each network layer in parallel: the message-passing process is performed independently for each layer, with the same input features but different edges and layer parameters, as described in⁴⁸. This is meant to capture the different connectivity patterns and dynamics at different frequency bands. The output of each layer in the block is aggregated by sum to produce the output node embeddings of the block. This design choice allows the model to learn frequency-specific representations while maintaining a unified framework for classification. The specific GNN layers used in our experiments are wsGAT⁴⁹, a variant of the Graph Attention Network (GAT)⁵⁰, which is a GNN layer that uses attention mechanisms to weight the incoming messages from the neighbors, effectively assigning different importance to each neighbor. Moreover, residual connections are used to facilitate the training process and to avoid the vanishing gradient problem. The convolutional blocks are followed by a global pooling layer to aggregate the node embeddings, and a fully connected layer to produce the classification score. Regarding the activation functions, we use ELU (Exponential Linear Unit) for the inner layers and sigmoid for the output layer to produce a classification score between 0 and 1, where 0 represents an affected subject and 1 a healthy one.

Training

The networks are divided into training, validation, and test sets, with a , , and split, respectively, using a stratified strategy to ensure that the class distribution is the same across the three sets. That is, we use 162 subjects in total: 103 subjects to train the model, 26 for the validation and 33 (20%) for the test phase. The model is trained using the AdamW⁵¹ optimizer with AMSGrad⁵², a learning rate of 0.01, and weight decay ( regularization) of 0.005. Dropout is used with a probability of 0.3 by GNN layers, and no batch normalization is performed. The loss function is the Binary Cross-Entropy loss with a balanced class weight to account for the slight class imbalance. An early-stopping criterion is adopted to speed up the training process, and the model is trained for a maximum of 10k epochs, at the end of which the model with the best validation performance is selected and evaluated on the test set. We run a grid search to find the best hyper-parameters for the model, including the number of convolutional blocks, the size of the channels the number of attention heads in the GAT layers.

Explainable AI and functional pruning

Once we identify the best-performing model, we use the GNNExplainer⁵³ algorithm to gain insights into how it makes predictions. GNNExplainer is an Explainable AI (XAI) tool for Graph Neural Network models that identifies the most influential sub-network for the model’s predictions. It does so by learning an edge mask — a value between 0 and 1 for each edge — indicating the importance of that edge for the final prediction, as well as a node-feature mask quantifying the importance of node features. The node-feature mask, however, is not informative in our application since we use only one-hot encodings of ROI identifiers as node features. This design choice is intentional: by not providing additional node features beyond ROI identifiers, we ensure that the model learns from connectivity patterns alone rather than from node-specific attributes. We train GNNExplainer to produce an edge mask for each subject, which still leaves us with a thresholding problem: selecting the appropriate threshold to distinguish important from unimportant edges. To address this, we systematically vary the threshold from 0 to 1 in steps of 0.01. For each threshold value, we prune the network by removing edges with mask values below the threshold; nodes are considered removed if they become isolated after edge removal. We then evaluate the classification performance of the model on these pruned networks without retraining. Remarkably, we find that when selecting edges with mask values above 0.51 — which removes more than of edges on average — the classification performance actually improves to approximately 0.9 ROC AUC compared to the original model, as shown in Fig.  7.

Fig. 7.

ROC AUC performance on the test set as a function of the edge importance threshold used for network pruning. The vertical dashed red line indicates the selected threshold (), which corresponds to the onset of percolation (see Fig. 6). Notably, this threshold not only maximizes classification performance but also coincides with structural changes in network topology, suggesting that optimal pruning may occur near critical structural transitions. This observation hints at a potential principled approach for threshold selection that balances predictive accuracy with preservation of critical connectivity patterns, though further investigation across different datasets and disorders is needed to establish the generality of this relationship³⁵.

Mesoscale networks

After identifying the most relevant edges for subject classification using the GNNExplainer, we analyze the resulting sub-networks at both the subsystem and ROI levels. To map ROIs to subsystems, we utilize the Power atlas³⁷, which categorizes each of the 264 ROIs into one of 14 functional subsystems. The subsystems include: Auditory, Cerebellar, Cingulo-opercular, Default mode, Dorsal attention, Fronto parietal, Memory retrieval, Salience, Sensory/somatomotor hand, Sensory/somatomotor mouth, Subcortical, Ventral attention, Visual, and Uncertain (regions not clearly assigned to a specific subsystem). This mapping allows us to construct mesoscale networks where nodes represent subsystems, and edges indicate connections between these subsystems based on the presence of edges between their constituent ROIs in the pruned networks.

Statistical analysis

To assess the statistical significance of differences in network metrics between healthy and affected subjects, we performed statistical comparisons of network metrics between groups using two-sided Mann-Whitney U tests (MWU). We chose MWU tests because they are non-parametric and do not assume normality of the data, making them suitable for our analysis, where the distribution of network metrics may not be normally distributed. After computing the MWU test, we also calculated effect sizes to quantify the magnitude of differences between groups through Rank-Biserial Correlation (RBC), providing additional context on the strength of the observed effects. To assess the effect of edge pruning on discriminability, we performed this analysis for both unpruned () and pruned () networks. To control the False Discovery Rate (FDR) across multiple comparisons, we employ the Benjamini-Yekutieli (BY) procedure⁵⁴, valid under arbitrary dependence structures (appropriate given the correlated nature of network metrics). The FDR is controlled at , meaning that the expected proportion of false discoveries among all significant results is limited to . Then, we computed the overall discriminative power of each metric and layer by combining the p-values across all tests using Stouffer’s method, which aggregates p-values by converting them to z-scores, averaging them, and converting back to a combined p-value. This approach provides a comprehensive assessment of each metric’s ability to distinguish between healthy and affected subjects across metrics and layers. We also apply Stouffer’s method separately across layers and across metrics to obtain the overall discriminative power of each metric and each layer.

Regarding the connectivity differences at the mesoscale level between healthy and affected subjects, we performed a non-parametric permutation test with 100,000 random shufflings of the subject groups (i.e., affected vs. controls, preserving the original group sizes). In each permutation, we computed the difference in mean connectivity between the two groups for each pair of subsystems, then compared the observed difference to the null distribution generated from the permutations to obtain a p-value for each pairwise connection.

The very conservative threshold of was chosen to ensure that only highly robust differences are highlighted, effectively serving as a stringent filter rather than a formal hypothesis test across all pairwise comparisons.

Software libraries

The learning framework is implemented in PyTorch and PyTorch Geometric⁵⁵, in particular via Heterogeneous Graphs^48,55 The dataset splitting and metrics computation are performed using the scikit-learn library⁵⁶. The graph-processing is done using graph-tool⁵⁷. The statistical analysis is performed using the SciPy⁵⁸ and the pingouin libraries⁵⁹. Finally, the plots using seaborn⁶⁰, and the brain visualization using nilearn⁶¹.

Author contributions

M.G., M.D.D., and G.M. designed the research; M.G. implemented the software; K.F. collected and pre-processed the data; M.G. generated the network data and performed the experiments; M.G., V.dA., M.D.D. and G.M. analyzed the results; All authors wrote the manuscripts.

Data availability

Original data is available at the Autism Brain Imaging Data Exchange (ABIDE) initiative³⁶ and the Preprocessed Connectome Project (PCP)²⁶. The code used to generate the results will be publicly made available, upon publication, at GitHub. [36] Di Martino, A. et al. The autism brain imaging data exchange: towards a large-scale evaluation of the intrinsic brain architecture in autism. Molecular psychiatry 19, 659– 667 (2014). [26] Nielsen, J. A. et al. Multisite functional connectivity MRI classification of autism: ABIDE results. Frontiers in human neuroscience 7, 599 (2013).

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-47391-z.