Connectional Brain Templates (CBTs)

• Connectional Brain Templates (CBTs) are graph-based representations that aggregate individual brain connectomes into a unified template preserving structural, functional, and topological traits.
• Multiple methodologies, including Frobenius mean, edge-wise L1 medoid, and learned aggregation via GNNs, construct CBTs to ensure centeredness, representativeness, and discriminativeness.
• CBTs serve as operational references for normalization, longitudinal forecasting, data augmentation, and cognitive modeling, facilitating enhanced analytic and predictive capabilities.

Connectional Brain Templates (CBTs) are graph-based population representations of brain connectivity, also described as network atlases, average connectomes, or connectional fingerprints. A CBT is typically a single adjacency matrix defined on a fixed set of regions of interest (ROIs) that lies at, or is learned to approximate, the center of a cohort of individual connectomes while preserving shared topological structure and, in several formulations, discriminative traits between populations such as healthy and disordered groups (Chaari et al., 2022, Gurbuz et al., 2020). Across the literature, CBTs function both as descriptive summaries of cohorts and as operational references for normalization, longitudinal forecasting, data augmentation, multimodal integration, federated learning, and cognition-aware generative modeling (Goktas et al., 2020, Chen et al., 2024, Soussia et al., 13 Sep 2025).

1. Core concept and representational scope

In the CBT literature, a subject is represented either by a single connectivity matrix or by a multigraph tensor. For single-view settings, a connectome is an undirected weighted graph with adjacency matrix in $\mathbb{R}^{n_r\times n_r}$, where nodes correspond to ROIs and edge weights encode structural, functional, or morphological relations (Gurbuz et al., 2020). For multi-view settings, each subject is represented by a tensor $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$, where the third mode indexes complementary graph views defined on the same node set (Rekik et al., 2021, Pala et al., 2022).

The recurring design criteria for CBTs are centeredness, representativeness, topological soundness, and discriminativeness. Centeredness denotes low average distance from the template to population graphs, typically measured by Frobenius-type criteria. Representativeness indicates that the template summarizes common connectivity structure rather than idiosyncratic subject noise. Topological soundness requires preservation of graph-organizational traits such as node strength, centrality, clustering, or efficiency. Discriminativeness refers to the ability of class-specific CBTs to reveal group differences, for example between Alzheimer’s disease and late mild cognitive impairment or between autism spectrum disorder and normal control populations (Demirbilek et al., 2021, Rekik et al., 2021, Chaari et al., 2022).

The scope of the concept has broadened substantially. Early formulations emphasize CBTs as normalized graph-based representations of a population of brain networks and as average connectomes (Gurbuz et al., 2020). Subsequent work extends the notion to fixed-reference graphs for residual normalization and nearest-neighbor forecasting (Goktas et al., 2020, Gurler et al., 2020), to longitudinal population templates that evolve over time (Demirbilek et al., 2021), to multimodal and multiresolution universal templates (Cinar et al., 2022), to templates learned under federated non-IID multi-domain constraints (Chen et al., 2024), and to functional or cognitively enhanced templates derived from reservoir-computing models and multi-sensory inputs (Soussia et al., 15 Aug 2025, Soussia et al., 13 Sep 2025).

2. Mathematical formulations of CBTs

A generic CBT objective treats the template as the element minimizing aggregate graph-to-graph dissimilarity. In its simplest Frobenius formulation, one seeks

$C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$

which yields the element-wise mean under squared Frobenius loss (Gurbuz et al., 2020). A closely related formulation appears in functional CBT learning, where for connectomes $M_i\in\mathbb{R}^{R\times R}$ the template

$$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$

has closed-form solution $C=\frac{1}{N}\sum_{i=1}^N M_i$ (Soussia et al., 15 Aug 2025). These formulations define a population-centered normative matrix, but the same papers note that simple linear averaging can be sensitive to outliers or can fail to capture nonlinear variation across subjects and views (Gurbuz et al., 2020).

A distinct formulation defines the CBT edge-wise as an $L_1$ medoid rather than a mean. In the RESNets and gGAN trajectory-prediction frameworks, for each edge $(a,b)$ or $(i,j)$ one collects the scalar edge values across subjects, constructs a high-order distance matrix with entries given by absolute inter-subject differences, computes each subject’s cumulative distance on that edge, and selects the most central subject-specific edge value. Repeating this independently over all edges yields a CBT that is explicitly described as an edge-wise medoid and not a simple arithmetic or spectral average (Goktas et al., 2020, Gurler et al., 2020).

A third family of methods learns subject-specific CBTs and then aggregates them. In DGN, MGN-Net, ReMI-Net, MultigraphGNet, M2GraphIntegrator, and CogGNN, a graph neural network maps each subject’s multigraph input to a single-view template $\mathbf{C}_s$ or $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$0, after which the final population-level CBT is obtained by element-wise median across subjects (Gurbuz et al., 2020, Rekik et al., 2021, Demirbilek et al., 2021, Pala et al., 2022, Cinar et al., 2022, Soussia et al., 13 Sep 2025). In ReMI-Net, this median is computed at each timepoint to form a population template trajectory $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$1 (Demirbilek et al., 2021).

Formulation	Defining operation	Representative papers
Frobenius mean	Minimize $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$2; closed-form element-wise mean	(Gurbuz et al., 2020, Soussia et al., 15 Aug 2025)
Edge-wise $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$3 medoid	Per edge, choose subject value minimizing total absolute distance to all others	(Goktas et al., 2020, Gurler et al., 2020)
Learned subject templates + median	Predict $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$4 from each subject, then take element-wise median	(Rekik et al., 2021, Demirbilek et al., 2021, Cinar et al., 2022)

This diversity of formulations is central to the topic. A common misconception is that a CBT is necessarily an arithmetic average. The literature instead uses at least three distinct constructions: a mean under Frobenius loss, an edge-wise medoid under absolute distance, and a learned set of subject-biased templates aggregated by median (Goktas et al., 2020, Gurbuz et al., 2020, Soussia et al., 15 Aug 2025).

3. Graph-learning architectures for estimating CBTs

The transition from analytic aggregation to learned CBT estimation is dominated by geometric deep learning. Deep Graph Normalizer (DGN) formalizes CBT estimation as a mapping from a multi-view tensor $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$5 to a single adjacency matrix, using edge-conditioned graph convolutions. After $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$6 layers, the final node embeddings are converted into a CBT by pairwise $$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$7 distances,

$$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$8

and training uses a randomized weighted Subject Normalization Loss that compares each generated template to a small random subset of training graphs, with view-specific normalization weights to counter scale differences across views (Gurbuz et al., 2020). Post-training, DGN produces a final unbiased CBT by taking the element-wise median over all subject-specific outputs (Gurbuz et al., 2020).

MGN-Net retains the same general many-to-one normalization principle but makes the objective explicitly topology-constrained. Its total loss combines a centeredness term with a symmetric Kullback–Leibler divergence on node-strength distributions:

$$\mathcal{T}_s\in\mathbb{R}^{n_r\times n_r\times n_v}$$9

Its architecture uses three edge-conditioned layers, all-ones initial node features, and a readout defined by absolute differences between final node embeddings (Rekik et al., 2021). This directly links template estimation to node-level topological fidelity rather than only graph proximity.

ReMI-Net extends CBT learning to longitudinal multigraph populations. It applies learnable view normalization, cyclic recurrent graph message passing, and a time-dependent loss

$C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$0

combined with a centeredness loss based on random training-subject subsets (Demirbilek et al., 2021). Node embeddings are updated through double-RNN pairwise message passing, and each subject-specific CBT is reconstructed from pairwise $C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$1 distances between node hidden states. The final population template at time $C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$2 is again the element-wise median over subjects (Demirbilek et al., 2021).

M2GraphIntegrator generalizes the problem to multimodal and multiresolution populations. It first aligns resolutions using resolution-specific graph autoencoders, maps minimal-resolution graphs through a self-mapper, and integrates latent representations into subject-based CBT vectors. Training combines reconstruction, topology, and centeredness losses, and the centeredness term is computed on a heterogeneous subset selected by clustering rather than random sampling (Cinar et al., 2022). This architecture is explicitly designed for heterogeneous graphs across modalities and ROI granularities.

MetaFedCBT places CBT learning in a federated setting where data remain local to hospitals or domains. The framework uses local DGN-based learning, server-side FedAvg aggregation,

$C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$3

and a metadata-driven loop that predicts synthetic-domain statistics from local/global weight residuals and generates informative connectivity augmentations to mitigate non-IID drift (Chen et al., 2024). The underlying CBT notion remains that of a population center, but the estimation procedure is explicitly conditioned by domain heterogeneity.

4. CBTs as references for prediction, augmentation, and cognitive modeling

In trajectory-prediction methods, the CBT is not merely a summary output but a fixed operational reference. RESNets first learn graph embeddings through an adversarial connectome embedding network whose encoder is a two-layer GCN and whose decoder reconstructs edges via sigmoid inner products (Goktas et al., 2020). Once the CBT $C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$4 is embedded to $C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$5, each training and test graph is converted into a residual embedding,

$C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$6

Similarity between test and training subjects is then computed by cosine similarity in residual space, top-$C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$7 neighbors are selected, and the follow-up graph is predicted by averaging those neighbors’ true follow-up graphs (Goktas et al., 2020). The related gGAN framework uses the CBT as the adversarial normalization target and computes residuals from the CBT embedding, with similarity defined by a residual dot product (Gurler et al., 2020). In both cases, the CBT supplies a common frame in which deviation patterns become comparable across subjects.

In longitudinal population modeling, ReMI-Net addresses a different forecasting problem: instead of predicting an individual trajectory from a single observation, it infers a baseline CBT for an input multigraph population and predicts its evolution at later timepoints (Demirbilek et al., 2021). This reframes the CBT from a static fingerprint into a dynamic population trajectory.

In graph augmentation, CBTs serve as compact generators or seeds. MultigraphGNet uses a many-to-one GNN to collapse a multigraph population into a CBT and a reverse one-to-many U-Net to reconstruct the multigraph population from that CBT, with the two components jointly trained by a cyclic loss (Pala et al., 2022). Once trained, a global CBT is obtained by element-wise median, perturbed with Gaussian noise, and decoded into new multigraphs for one-shot augmentation (Pala et al., 2022). A related one-shot framework trains a graph-based GAN on a single class-specific CBT and synthesizes new connectomes around that template, improving classification metrics relative to template-only one-shot learning (Özgür et al., 2022).

In cognition-oriented models, the CBT is reused as a recurrent substrate. mCOCO first derives individual functional connectomes from BOLD signals via reservoir computing, averages them into a group-level CBT, and then uses that CBT as the recurrent weight matrix of a cognitive reservoir receiving multi-sensory inputs such as text, audio, and visual data (Soussia et al., 15 Aug 2025). CogGNN similarly feeds each learned CBT into an Echo State Network and introduces a visual-memory loss so that CBT optimization is coupled to memory reconstruction performance (Soussia et al., 13 Sep 2025). These works explicitly argue that structure- and topology-only CBT learning overlooks cognitive capacity and that the template itself can be endowed with cognitive traits (Soussia et al., 15 Aug 2025, Soussia et al., 13 Sep 2025).

5. Evaluation criteria and empirical behavior

The comparative survey of multigraph integration methods systematizes CBT evaluation into five criteria: centeredness, biomarker-reproducibility, node-level similarity, global-level similarity, and distance-based similarity (Chaari et al., 2022). Centeredness is usually computed as average Frobenius distance between the CBT and held-out graphs. Biomarker reproducibility compares discriminative ROIs extracted from class-specific CBT differences with an independent MKL-SVM ranking. Node-level similarity uses centrality distributions, local efficiency, or participation coefficient; global-level similarity evaluates modularity and global efficiency; distance-based similarity uses normalized Hamming and weighted Jaccard distances between templates (Chaari et al., 2022).

On multi-view CBT estimation, DGN is repeatedly reported as a strong baseline or state-of-the-art reference. Its original study found that DGN templates were significantly closer to held-out test graphs than netNorm and produced higher overlap rates with disease-related ROI rankings in several settings (Gurbuz et al., 2020). The later survey reports that DGN significantly outperforms other multi-graph and all single-view integration methods in centeredness, reproducibility, and preservation of local and global graph-level traits across the reviewed datasets (Chaari et al., 2022).

Subsequent architectures refine different aspects of CBT quality. MGN-Net yields the smallest average Frobenius distance to held-out test graphs across folds, the lowest $C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$8 for PageRank, effective size, and clustering coefficient distributions, and higher SVM accuracies from top-$C^*=\arg\min_C \sum_{i,v}\|C-T_i^v\|_F^2,$9 discriminative edges in several tasks (Rekik et al., 2021). ReMI-Net achieves the lowest average Frobenius distances at both timepoints and reduces MAE in node strength from DGN left hemisphere $M_i\in\mathbb{R}^{R\times R}$0 to $M_i\in\mathbb{R}^{R\times R}$1 and right hemisphere $M_i\in\mathbb{R}^{R\times R}$2 to $M_i\in\mathbb{R}^{R\times R}$3 for AD / LMCI, while improving biomarker reproducibility versus GMKL (Demirbilek et al., 2021). In trajectory prediction from a single observation, RESNets achieves the lowest MAD and MSE for all tested $M_i\in\mathbb{R}^{R\times R}$4 and reports roughly $M_i\in\mathbb{R}^{R\times R}$5–$M_i\in\mathbb{R}^{R\times R}$6 relative error reduction against ablated variants and traditional approaches (Goktas et al., 2020), while CBT-gGAN achieves the lowest MAE on OASIS-2 and reports improvements on the order of $M_i\in\mathbb{R}^{R\times R}$7 absolute connectivity units (Gurler et al., 2020).

Multimodal and augmentation-oriented frameworks show corresponding gains on their target tasks. M2GraphIntegrator reports statistically significant improvements in centeredness, topological soundness, KL divergence of edge-weight distributions, and pairwise distance between real and synthesized graphs, with best reported values such as centeredness $M_i\in\mathbb{R}^{R\times R}$8 versus $M_i\in\mathbb{R}^{R\times R}$9 and KL divergence $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$0 versus $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$1 against the Ablated(R) comparison (Cinar et al., 2022). MultigraphGNet raises one-shot ASD vs. control accuracy from approximately $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$2 with two global CBTs to approximately $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$3 after augmentation, with precision $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$4, recall $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$5, and F1 $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$6 (Pala et al., 2022). The single-template gGAN augmentation pipeline improves left-hemisphere one-shot AD/LMCI accuracy from $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$7 to $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$8 and increases sensitivity from $$C=\arg\min_{C\in\mathbb{R}^{R\times R}}\sum_{i=1}^N \|M_i-C\|_F^2$$9 to $C=\frac{1}{N}\sum_{i=1}^N M_i$0, albeit with reduced specificity (Özgür et al., 2022).

The cognitive line introduces additional metrics beyond conventional connectomic evaluation. mCOCO reports significantly lower centeredness than a DGN-based CBT across five folds, lower KL divergences for graph-theoretic measures, ASD vs. TD SVM accuracy of $C=\frac{1}{N}\sum_{i=1}^N M_i$1 with F1 $C=\frac{1}{N}\sum_{i=1}^N M_i$2 versus $C=\frac{1}{N}\sum_{i=1}^N M_i$3 and F1 $C=\frac{1}{N}\sum_{i=1}^N M_i$4 for SVM-DGN, and higher modality-specific memory capacities, including auditory memory capacity greater than $C=\frac{1}{N}\sum_{i=1}^N M_i$5 for “Ode to Joy” and greater than $C=\frac{1}{N}\sum_{i=1}^N M_i$6 for Quranic recitation (Soussia et al., 15 Aug 2025). MetaFedCBT, in turn, minimizes centeredness and KL divergence across all simulated hospitals and datasets and yields the highest one-shot SVM performance, reported as up to approximately $C=\frac{1}{N}\sum_{i=1}^N M_i$7, among the compared federated and non-federated baselines (Chen et al., 2024).

6. Conceptual distinctions, limitations, and active directions

Several distinctions structure the modern CBT literature. First, single-view CBT integration differs from multi-view or multigraph CBT integration: the former learns from one graph per subject, whereas the latter fuses stacks of complementary graphs defined on the same ROI set (Chaari et al., 2022). Second, multimodal and multiresolution integration addresses heterogeneity not only in edge semantics but also in graph size, requiring explicit resolution unification before population integration (Cinar et al., 2022). Third, static CBT learning differs from temporal CBT learning, where the object of interest is a sequence of templates constrained to evolve smoothly across time (Demirbilek et al., 2021). Fourth, centralized learning differs from federated CBT learning, where raw data remain local and domain heterogeneity must be handled through model aggregation and metadata-guided augmentation (Chen et al., 2024).

The literature also records several limitations. DGN notes computational complexity, scalability challenges for larger numbers of ROIs or views, and sensitivity to hyperparameters such as layer sizes, subset size for randomized losses, and view weights (Gurbuz et al., 2020). The gGAN one-shot augmentation study states that variability is limited by one template and that single-view generation may underestimate population diversity (Özgür et al., 2022). MetaFedCBT explicitly states that it does not enforce formal differential privacy, relying instead on data locality, aggregation, and noise-driven metadata generation (Chen et al., 2024). The mCOCO paper argues that existing CBT-learning methods based on conventional machine learning and GNNs are hindered by poor interpretability, high computational cost, and exclusive focus on structure and topology while overlooking cognitive capacity (Soussia et al., 15 Aug 2025).

A final conceptual clarification concerns discriminativeness. Some frameworks optimize it indirectly rather than explicitly. ReMI-Net states that discriminativeness is not directly enforced by the loss but emerges when one computes element-wise differences between class-specific CBTs and ranks ROIs by change (Demirbilek et al., 2021). This suggests that “discriminative CBT” can denote either an optimization target or an empirical property measured after training, depending on the method. More broadly, the field has moved from viewing the CBT as a static average connectome toward treating it as a learned, task-bearing population object: a normalization reference, a generator seed, a temporal state, a federated consensus structure, or a cognitively constrained recurrent substrate (Goktas et al., 2020, Demirbilek et al., 2021, Chen et al., 2024, Soussia et al., 13 Sep 2025).