HOI-Brain: Signed Higher-Order fMRI Analysis

• HOI-Brain is a framework that models signed higher-order interactions among brain regions using MTD, simplicial complexes, and persistent homology.
• It fuses lower-order edges with quadruplet and void features via a multi-channel Transformer to enhance diagnostic accuracy.
• Empirical results show significant improvements over traditional methods, highlighting the benefits of incorporating signed, temporally resolved neural interactions.

HOI-Brain is a computational framework for brain disorder diagnosis from resting-state fMRI that models signed higher-order interactions (HOIs) among groups of brain regions and extracts their topological organization for classification. In this context, HOIs are not human-object interactions but multi-region neural co-fluctuations, and the framework combines Multiplication of Temporal Derivatives (MTD), signed weighted simplicial complexes, persistent homology, and a multi-channel brain Transformer. Its central claim is that brain disorders are better characterized when analysis moves beyond pairwise connectivity and unsigned topology to signed, temporally resolved, higher-order neural organization (Zhao et al., 27 Jul 2025).

1. Conceptual basis and scope

HOI-Brain addresses several limitations that the higher-order brain-network literature repeatedly identifies. Standard graph neural networks and many Transformer-based brain models operate on ROI-level nodes and pairwise functional connectivity edges, which assumes that interactions are fundamentally dyadic. Hypergraph methods add higher-order structure, but their hyperedges are often formed via $k$-NN or $k$-hop neighborhoods and therefore remain node-centric. Persistent-homology-based brain methods often focus on $H_0$ and $H_1$, characterizing connected components and loops rather than higher-dimensional organization. HOI-Brain is motivated by the view that some triplet interactions can be decomposed into linear combinations of pairwise interactions, which motivates moving to quadruplet-level interactions and $2$-dimensional topological structure (Zhao et al., 27 Jul 2025).

Within this formulation, the sign of a higher-order interaction is biologically meaningful. HOI-Brain distinguishes positively synergistic interactions, in which multiple ROIs simultaneously activate at the current moment relative to the previous one, from negatively synergistic interactions, in which multiple ROIs collectively inhibit at the current moment relative to the previous one. The paper treats this signed distinction as essential for identifying complex coordination, communication, and disease-related disruption of large-scale organization (Zhao et al., 27 Jul 2025).

This orientation places HOI-Brain within a broader higher-order neuroimaging literature, but with a distinctive emphasis. O-information-based frameworks such as MvHo-IB and MvHo-IB++ quantify whether triplets or higher-order tuples are redundancy-dominated or synergy-dominated and fuse pairwise and higher-order views under an information bottleneck (Zhang et al., 3 Jul 2025, Zhang et al., 20 Apr 2026). Topological Signal Processing models the brain as a subject-specific cell complex and analyzes higher-order organization through divergence and curl (Bispo et al., 31 Mar 2026). O-information rate extends higher-order analysis into time and frequency domains for rhythmic physiological and brain processes (Faes et al., 2022). HOI-Brain differs by centering signed co-fluctuation simplicial complexes, persistent-homological extraction of good quadruplets and $2$-dimensional voids, and Transformer-based fusion (Zhao et al., 27 Jul 2025).

2. Construction of signed higher-order interactions

HOI-Brain begins from preprocessed resting-state fMRI, using motion correction, realigning, field unwarping, normalization, bias field correction, and brain extraction. The brain is parcellated with the AAL atlas into 90 ROIs, and each ROI $i$ is represented by a BOLD time series

$\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$

For each ROI, the method computes a first-order temporal derivative

$\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$

and normalizes the derivative by its standard deviation: $q_i(t) = \frac{\dot{x}_i(t)}{\sigma_{\dot{x}_i}}.$

For a group of $k$0 ROIs, the $k$1-order co-fluctuation at time $k$2 is defined as

$k$3

Its sign is

$k$4

The final signed weight is

$k$5

If all possible $k$6-node groups are considered, the number of co-fluctuation series is $k$7. HOI-Brain focuses on up to $k$8, corresponding to quadruplet interactions among four ROIs. At each time point $k$9, these interactions are encoded into a weighted simplicial complex $H_0$0, whose simplices include vertices, edges, triangles, and tetrahedra. The framework then splits this complex into

$H_0$1

representing positively and negatively synergistic structures, respectively (Zhao et al., 27 Jul 2025).

The paper states that it focuses only on concordant signs of positive and not on discordant signs of negative, because discordant patterns may reflect confusing redundancy across regions. This restriction is one of the framework’s explicit design choices and later reappears as a limitation (Zhao et al., 27 Jul 2025).

3. Persistent homology and higher-dimensional topological features

A central contribution of HOI-Brain is the conversion of signed simplicial complexes into topological descriptors through two filtrations. The general filtration form is

$H_0$2

The first filtration is applied to $H_0$3 and $H_0$4 by sorting the weights of edges, triangles, and quadruplets in ascending order. At a filtration step $H_0$5, the method identifies signed “good” quadruplets satisfying

$H_0$6

These are interpreted as quadruplet-level interaction signatures whose strength exceeds that of their lower-dimensional faces. They are stored as

$H_0$7

$H_0$8

All lower-order edges are also retained as

$H_0$9

The second filtration sorts edges, triangles, and quadruplets in descending order and targets $H_1$0, extracting signed $H_1$1-dimensional voids. The resulting persistence structures are stored as

$H_1$2

$H_1$3

Persistent homology yields persistence diagrams with points

$H_1$4

where $H_1$5 and $H_1$6 are the birth and death times of void $H_1$7, and persistence is

$H_1$8

To convert these topological outputs into network-form features, the method uses a homological scaffold. If an edge $H_1$9 belongs to multiple $2$0-dimensional void generators $2$1, its scaffold weight is

$2$2

The framework also projects good quadruplets back to ROI pairs by assigning each edge $2$3 a weight equal to the average sum of the weights of quadruplets containing that edge.

At each time point, these operations produce five weighted ROI-by-ROI networks: lower-order edge features, positive good quadruplets, negative good quadruplets, positive $2$4-dimensional voids, and negative $2$5-dimensional voids. These matrices are then averaged over time to stabilize the representation and reduce fMRI noise (Zhao et al., 27 Jul 2025).

This topological program has clear affinities with adjacent work. Topological Signal Processing also treats higher-order structure as geometrically organized, but uses cell complexes, Hodge Laplacians, divergence, and curl to model edge flows and circulatory motifs (Bispo et al., 31 Mar 2026). Theoretical work on the “Homological Brain” frames neural computation in terms of simplicial complexes, closure, and homology classes, though at a more speculative level (Li, 3 Dec 2025). HOI-Brain is more operational and data-driven: it derives task features directly from signed co-fluctuation complexes and persistent homology (Zhao et al., 27 Jul 2025).

4. Multi-channel brain Transformer architecture

The final classifier in HOI-Brain is a multi-channel brain network Transformer. Although five topological matrices are first constructed, the model ultimately uses three channels:

1. a lower-order edge matrix $2$6,
2. a fused quadruplet matrix $2$7,
3. a fused void matrix $2$8.

The positive and negative channels for quadruplets and voids are fused by a signed high-order features decoupling mechanism. For quadruplets, the two signed matrices $2$9 are flattened in their upper triangular parts to $2$0, and then fused by learned weights: $2$1

$2$2

The same mechanism is applied to positive and negative void matrices: $2$3

$2$4

Each channel is then processed independently by an $2$5-layer multi-head self-attention module: $2$6 Within a layer,

$2$7

$2$8

To capture modular organization, HOI-Brain adds an orthonormal clustering readout. Given cluster centers $2$9, the soft assignment of node $i$0 to cluster $i$1 is

$i$2

The channel-level graph embedding is

$i$3

Finally, the model uses attention-guided feature fusion across channels. For each channel $i$4, it computes

$i$5

followed by normalized channel weights

$i$6

The final fused feature is

$i$7

which is sent to a three-layer MLP and optimized with cross-entropy loss (Zhao et al., 27 Jul 2025).

The paper reports that some architectural details remain unspecified, including the exact number of Transformer layers $i$8, number of heads $i$9, hidden dimensions, dropout values, and final number of clusters used for each dataset. This partial underspecification is one of the framework’s stated limitations (Zhao et al., 27 Jul 2025).

5. Empirical performance and interpretability

HOI-Brain is evaluated on four resting-state fMRI datasets spanning three disorder families: ADNI for Alzheimer’s disease, TaoWu and PPMI for Parkinson’s syndrome or disease, and ABIDE for autism spectrum disorder. The reported cohorts are:

Dataset	Cohort	HOI-Brain performance
ADNI	AD: 90, MCI: 76, NC: 96	Accuracy 75.9 ± 8.6, Precision 80.0 ± 7.9, Recall 75.6 ± 9.1, F1 75.5 ± 9.2
TaoWu	PD: 20, NC: 20	Accuracy 77.5 ± 12.3, Precision 82.4 ± 9.3, Recall 77.5 ± 12.3, F1 75.9 ± 13.9
PPMI	PD: 53, prodromal: 53	Accuracy 66.1 ± 4.0, Precision 69.0 ± 4.0, Recall 66.3 ± 4.2, F1 64.7 ± 4.7
ABIDE	ASD: 488, NC: 537	Accuracy 65.6 ± 3.5, Precision 66.2 ± 3.8, Recall 65.6 ± 3.4, F1 65.3 ± 3.5

Across these datasets, the paper reports that HOI-Brain outperforms 20 baselines spanning traditional machine learning, GNNs, Transformers, HGNNs, and PH-based methods. The paper also reports average relative improvements over PH-based methods of accuracy +17.4%, precision +21.4%, recall +14.5%, and F1 +16.2% (Zhao et al., 27 Jul 2025).

The ablations are especially informative. Combining higher-order and lower-order features improves over using edge features alone. “Good quadruplets” outperform violating triangles and $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$0-dimensional loops. Signed good quadruplets and signed $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$1-dimensional voids outperform their unsigned variants. In ADNI, edge-only accuracy is 68.4, edge + good quadruplets + $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$2D voids reaches 74.7, and edge + signed good quadruplets + signed $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$3D voids reaches 75.9. In ABIDE, edge-only accuracy is 58.3, edge + $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$4D loops + $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$5D voids reaches 64.3, and edge + signed good quadruplets + signed $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$6D voids reaches 65.6 (Zhao et al., 27 Jul 2025).

Architectural ablations show that removing signed high-order feature decoupling, attention-guided fusion, or orthonormal clustering degrades performance. Sensitivity analysis on the number of clusters $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$7 shows that performance improves as $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$8 increases to around 10 or 20, then declines, which the paper links to prior neuroscience observations that the number of functional modules is typically below 25 (Zhao et al., 27 Jul 2025).

The interpretability analysis is organized around channel attention, sign attention, ROI saliency, and higher-order biomarkers. In ADNI and ABIDE, quadruplet-level interaction signatures receive the highest average channel attention, often exceeding 0.5. The paper reports that negative synergistic quadruplets are often more important than positive ones, whereas positive synergistic voids are often more important than negative voids. This suggests that direct higher-order interactions and higher-dimensional cavities may be perturbed differently by disease (Zhao et al., 27 Jul 2025).

At the biomarker level, the paper highlights specific disease-related regions and quadruplets. In Alzheimer’s disease, prominent ROIs include Right Caudate (CAU.R), Left Hippocampus (HIP.L), Right Parahippocampal Gyrus (PHG.R), and Left Amygdala (AMYG.L), and quadruplets such as $\mathbf{x}_i = [x_i(1), x_i(2), \ldots, x_i(T)].$9 and $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$0 show significant differences from CN to MCI. In Parkinson’s disease, top ROIs include Right Precentral Gyrus (PCG.R), Left Medial Orbitofrontal Cortex (ORBsupmed.L), Right Inferior Parietal Lobule (IPL.R), and Right Thalamus (THA.R), with significant quadruplets such as $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$1 and $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$2. In autism spectrum disorder, top ROIs include Right Inferior Temporal Gyrus (ITG.R), Left Supramarginal Gyrus (SMG.L), Left IFG triangular part (IFGtriang.L), and bilateral caudate, with several ITG-centered quadruplets interpreted as involving a “visual-auditory-semantic-action-control” closed-loop system (Zhao et al., 27 Jul 2025).

The paper further reports disease-specific progression trends. In AD and ASD, positive higher-order interactions weaken while negative higher-order interactions strengthen. In PD, the positive/negative progression trend is reported to be the opposite, and the authors speculate that this may relate to hallucination-related enhanced sensory-higher-order integration (Zhao et al., 27 Jul 2025).

6. Relation to adjacent higher-order brain-network frameworks

HOI-Brain belongs to a broader methodological transition from pairwise connectomics to explicit higher-order modeling. The nearby literature reveals several distinct formalizations.

Information-theoretic HOI models: MvHo-IB defines HOIs through $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$3-information, using

$\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$4

where positive values are redundancy-dominated and negative values are synergy-dominated. It combines a pairwise mutual-information view and a triplet HOI tensor with a GIN encoder, Brain3DCNN, and a multi-view information bottleneck, achieving reported accuracies of 83.12 ± 5.74 on UCLA, 73.23 ± 4.37 on ADNI, and 82.13 ± 6.96 on EOEC (Zhang et al., 3 Jul 2025). MvHo-IB++ extends this to third- and fourth-order $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$5-information, adds Brain4DCNN, and reports over a 30-fold computational speedup using Gaussian and randomized Rényi estimators, with accuracies including 83.36 ± 5.63 on UCLA and 72.61 ± 2.68 on ABIDE (Zhang et al., 20 Apr 2026).

Dynamic HOBC forecasting: DCHO addresses a limitation that HOI-Brain leaves open, namely explicit temporal prediction of higher-order connectivity. DCHO first infers latent HOBC from fMRI and FC, then predicts future latent trajectories and decodes them into future HOBC, using a dual-view encoder, latent combinatorial learner, and LSTM predictor. It focuses on triplet interactions encoded in a third-order tensor and is evaluated on HCP task and resting-state data as well as synthetic Lorenz and Hindmarsh–Rose systems (Li et al., 27 Aug 2025). This suggests that a future extension of HOI-Brain could integrate signed topological descriptors with explicit latent forecasting.

Topological signal processing approaches: Multimodal Higher-Order Brain Networks frames the brain as a subject-specific 2-dimensional cell complex, combining diffusion MRI and rs-fMRI to infer triangles, quadrilaterals, and pentagons, and then decomposes edge flows into gradient, harmonic, and curl-supported components. It reports a DMN-centered gradient backbone, limbic-centered rotational flows, and behaviorally relevant divergence and curl signatures (Bispo et al., 31 Mar 2026). Compared with HOI-Brain, this line emphasizes discrete vector calculus on cell complexes rather than signed simplicial co-fluctuation and persistent-homological classification.

Spectral dynamic HOI analysis: O-information rate extends higher-order analysis to time and frequency domains for multivariate rhythmic processes. In ECoG during anesthesia, it finds redundancy-dominated cortical HOIs with anesthesia-dependent reconfiguration, including increased redundant $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$6 interactions in frontal-containing multiplets and reduced redundant $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$7 interactions in parietal-temporal-visual multiplets (Faes et al., 2022). This suggests a complementary avenue for HOI-Brain: its present topological descriptors are temporally averaged, whereas spectral or dynamical decomposition could preserve band-specific higher-order structure.

Hierarchical organization learning: BrainHO learns latent subgraphs from PCC matrices using node-to-subgraph and subgraph-to-graph attention, orthogonality constraints, and hierarchical consistency, reporting strong results on ABIDE and REST-meta-MDD (Tang et al., 10 Mar 2026). It is closely related conceptually, but it targets hierarchical organization rather than explicitly signed higher-order interaction topology.

Taken together, these neighboring approaches indicate that “HOI-Brain” is best understood not as an isolated method name but as part of a rapidly developing family of higher-order brain-network models. This suggests a taxonomy in which current methods differ along at least four axes: interaction definition (co-fluctuation vs $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$8-information vs topological flow), topological object (simplicial complex vs cell complex vs tensor), learning strategy (Transformer, CNN, IB, forecasting), and temporal treatment (static, time-resolved, spectral, predictive).

7. Limitations and open directions

HOI-Brain states several limitations explicitly. First, discordant negative signs are largely ignored: the method focuses on concordant effects and only limitedly explores discordant negative patterns in interpretation. Second, although the framework extracts timepoint-wise topological features, those features are averaged over time to suppress noise, so it does not model temporal dependencies across timepoints. Third, the experiments use only fMRI, leaving EEG, MEG, and multimodal neuroimaging to future work. Fourth, some implementation details remain underspecified, including Transformer depth, number of heads, hidden dimensions, exact clustering settings, and some details of positive-versus-negative concordant splitting. Fifth, the two filtrations are described algorithmically, but the paper does not provide a full formal algebraic-topology treatment of signed chain complexes or explicit boundary operators (Zhao et al., 27 Jul 2025).

The broader literature sharpens these open questions. O-information frameworks show that moving from third-order to fourth-order interactions yields useful gains but also substantial computational pressure, motivating Gaussian approximations, randomized Rényi estimators, or low-rank strategies (Zhang et al., 20 Apr 2026). DCHO shows that higher-order connectivity can be embedded in a predictive latent space, suggesting one route beyond HOI-Brain’s current temporal averaging (Li et al., 27 Aug 2025). Topological Signal Processing indicates that higher-order topology can also be analyzed as discrete flow geometry with divergence and curl, suggesting an alternative interpretation of higher-order organization not directly captured by persistence diagrams (Bispo et al., 31 Mar 2026). Frequency-domain O-information rate shows that whole-band summaries can obscure band-specific synergy and redundancy, which is a plausible caution for any temporally averaged HOI representation (Faes et al., 2022).

A plausible implication is that future HOI-Brain variants may need to combine three properties that are currently distributed across different frameworks: signed higher-order topology, explicit temporal or spectral dynamics, and scalable multi-order estimation. Another plausible implication is that stronger formalization of signed simplicial operators could bring HOI-Brain closer to the richer topological toolkits already explored in cell-complex and homological approaches.

In its current form, HOI-Brain’s main contribution is clear and specific: it demonstrates that signed quadruplet interactions and signed $\dot{x}_i(t-1) = x_i(t) - x_i(t-1),$9-dimensional voids, extracted from MTD-based co-fluctuation simplicial complexes and integrated by a multi-channel Transformer, yield superior disorder classification and biologically interpretable higher-order biomarkers from resting-state fMRI (Zhao et al., 27 Jul 2025).