STCBN-EC: A Spatio-Temporal Constrained Bayesian Causal Network for Multimodal Brain Effective Connectivity Learning

1 Introduction

Brain effective connectivity (EC) characterizes causal and directional interactions among brain regions and is fundamental for understanding cognition, behavior, and the neural mechanisms of neurological disorders [Mamoon et al., 2025; Yu et al., 2025]. Traditional statistical approaches, including structural equation modeling (SEM), dynamic causal modeling (DCM), and Granger causality analysis (GCA), have been widely used for EC learning. However, these methods rely on strong assumptions such as linearity and stationarity [Friston, 2011], which limit their ability to capture the nonlinear and dynamic properties commonly observed in real brain networks [Marinazzo et al., 2011; Stramaglia et al., 2014].

∗Corresponding Author

With the advancement of machine learning [Tank et al., 2021; Zhao et al., 2022], data-driven approaches have sub[Chen and

Zhang, 2023; Dai et al., 2025]. Graph neural networks (GNNs) leverage brain network topology to model causal dependencies, recurrent neural networks (RNNs) capture temporal dynamics and nonlinear lag effects, and generative models such as GANs and diffusion models have been explored for data augmentation and causal structure discovery [Liang et al., 2024; Ji et al., 2025; Chen et al., 2025]. Despite these advances, most existing studies remain focused on unimodal data, typically functional magnetic resonance imaging (fMRI), and fail to fully exploit the complementary information provided by multimodal neuroimaging.

Functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) provide highly complementary representations of brain activity [Ciccarelli et al., 2023; Warbrick, 2022], forming a critical basis for accurate EC modeling [Wei et al., 2025]. fMRI offers high spatial resolution for characterizing large-scale inter-regional interactions, but its temporal resolution is limited by hemodynamic effects. In contrast, EEG directly captures neural activity with millisecond-level temporal precision [Herwig et al., 2003], albeit with relatively limited spatial localization accuracy [Zhang et al., 2024a; Sakkalis, 2011]. Coordinating these modalities within a unified modeling framework that explicitly accounts for their spatio-temporal differences enables the simultaneous preservation of fine-grained spatial organization and fast neural dynamics [Xiong et al., 2025a].

However, existing multimodal EC methods still struggle to fully exploit this complementarity at the modeling level [Li et al., 2024; Su et al., 2025]. Some approaches directly apply Granger causality to multimodal data without enforcing consistent spatial or temporal representations across modalities [Muthuraman, 2016]. Other methods employ linear statespace models [Yun et al., 2025] to integrate temporal information [Valdes-Sosa et al., 2009; Abdullah and Rahim, 2026], but their linear formulations restrict the modeling of complex nonlinear causal dependencies [Tu et al., 2019]. In addition, several cross-modal mapping strategies rely primarily on feature concatenation or linear transformations, neglecting anatomical correspondence between EEG electrodes and brain regions as well as the intrinsic temporal structure of neural activity [Liu et al., 2024]. These designs often decouple multimodal representation learning from causal structure inference, reducing interpretability and robustness, and highlighting the need for principled multimodal EC frameworks that explicitly enforce spatio-temporal consistency and model nonlinear cross-modal causal interactions.

To address these limitations, we propose STCBN-EC, a spatio-temporal constrained Bayesian causal network framework for multimodal effective connectivity learning from EEG-fMRI data. STCBN-EC consists of three key components. First, the Spatio-Temporal Fusion Module constructs a unified spatio-temporal representation by explicitly accounting for anatomical correspondence and temporal alignment across modalities. Second, the Bayesian EC Learning Module enables uncertainty-aware exploration of candidate causal graphs in the discrete and non-differentiable structure space. Third, the Structure-Guided Joint Optimization Module couples multimodal representation with EC learning through a structure-guided optimization strategy. By integrating the Spatio-Temporal Fusion Module, the Bayesian Neural Surrogate for EC Search, and the Structure-Guided Joint Optimization Module into a unified framework, STCBN-EC enables stable and directionally consistent EC learning under heterogeneous and noisy multimodal observations.

The main contributions of this work are summarized as follows:

• We introduce anatomical geometric constraints and slice-level temporal alignment to construct a unified spatio-temporal representation for EEG-fMRI EC learning under heterogeneous observations.
• We design learnable ROI-wise fusion weights α to adaptively integrate EEG and fMRI representations, improving robustness under multimodal noise and temporal mismatch.
• We employ a Bayesian neural surrogate for EC structure scoring to model nonlinear causal dependencies with uncertainty awareness, enhancing the stability of causal direction identification.

2 Preliminary and RelatedWork

2.1 Brain Effective Connectivity

EC describes directed causal influences among brain regions and reflects the directionality of neural information flow [Cai et al., 2018; Huang et al., 2018; Liu et al., 2022]. EC is commonly modeled as a directed graph G = (V,E,W), where V denotes regions of interest (ROIs), E represents directed causal edges, and W is an asymmetric adjacency matrix encoding causal strengths. Given multivariate neural time series X = {x(t)}Tt=1 , EC learning aims to recover the latent causal structure G, which can be viewed as a causal structure learning problem over directed acyclic graphs.

2.2 Multimodal Effective Connectivity

Multimodal effective connectivity extends EC by integrating complementary information from multiple neuroimaging modalities, such as EEG and fMRI, to infer a shared causal brain network [Daunizeau et al., 2011]. Let X(m) = {x(m) (t)}Tm t=1 , m ∈ {EEG, fMRI}, denote modality-specific observations with distinct temporal resolution, noise characteristics ε(m) , and observation operators O(m) (·). Multimodal EC assumes that these heterogeneous observations are generated from a common latent neural process governed by a shared directed causal structure G, and aims to infer a crossmodality consistent connectivity pattern from multi-source data [Liu et al., 2024; Li et al., 2025].

3 Motivations

Achieving reliable and stable inference of directed EC from multimodal data faces core challenges. The spatial mismatch between EEG electrodes and fMRI regions complicates anatomically-grounded representation. The temporal resolution gap, compounded by hemodynamic response, obscures causal timing. Modality-specific noise also renders fixed fusion strategies ineffective, often producing unstable connectivity estimates. Critically, many methods decouple representation learning from causal search, leading to direction identification that is sensitive to noise. Our framework addresses this by explicitly modeling anatomical constraints, enabling adaptive temporal alignment, and jointly optimizing fusion with discovery for robust directional EC learning.

4 Method

We present STCBN-EC, a spatio-temporal constrained Bayesian causal network for learning multimodal EC from EEG-fMRI data. The framework incorporates three modules: a spatio-temporal fusion module, a Bayesian EC learning module, and a structure-guided joint optimization module. Together, these modules enable stable and directionally consistent EC learning. By combining the spatial information from fMRI and the temporal data from EEG, STCBN-EC overcomes the challenges of their differences in resolution and noise characteristics, providing a more reliable solution for EC learning.

4.1 Spatio-Temporal FusionModule

To enable stable and identifiable EC learning from multimodal data, we begin by constructing a spatio-temporal constrained representation that explicitly aligns EEG and fMRI at the ROI level. Let XEEG(t) ∈ RE denote EEG observations from E electrodes, and let {pe} and {qr} denote the 3D coordinates of EEG electrodes and fMRI ROIs, respectively. The EEG-to-ROI mapping is defined as

where σ controls spatial decay and W is row-normalized. This yields ROI-level EEG representations ˜HEEG(t) = WXEEG(t).

To reconcile temporal resolution differences, EEG signals are aggregated within each fMRI repetition time (TR), while fMRI ROI time series are convolved with a canonical hemodynamic response function (HRF) to account for temporal smoothing effects. Based on the temporally aligned EEG and fMRI representations, we construct a unified multimodal representation via adaptive ROI-wise fusion:

where α ∈ (0, 1)R denotes learnable fusion weights parameterized by a sigmoid function, which adaptively balance the contributions of EEG and fMRI across brain regions under varying noise conditions.

The resulting fused sequence H = H(k)Tk=1 offers a unified spatio-temporal representation that maintains anatomical structure, temporal consistency, and multimodal complementarity, providing a stable foundation for EC learning.

4.2 Bayesian EC LearningModule

Given the fused multimodal representation H, we proceed to learn EC by modeling directed interactions among ROIs within a structural causal framework.

We model the activity of each Region of Interest (ROI) using a Structural Causal Model (SCM). The causal relationships among ROIs are represented as an EC graph G = (V, E), where V = {1, . . . , R} represents the set of R ROIs. In this EC, an edge (i → j) ∈ E indicates that ROI i has a direct causal influence on ROI j. The parent set of node j in this graph is denoted by paG(j) = {i ∈ V | (i → j) ∈ E}, representing all the ROIs that causally influence ROI j.

The data-generating process for the activities of ROI j at time t is modeled as follows:

where fj(·) represents the causal mechanism governing the evolution of ROI j over time, and ϵ(t) j is an exogenous noise term.

Score-based causal discovery. We adopt a score-based formulation and seek

where GDAG is the space of DAGs over R nodes and S is a consistent score. In practice, we instantiate S using a BICstyle score that decomposes over nodes. Let ̂ fj be the fitted predictor/regressor for node j given paG(j), and define the node-wise mean squared error

Then a common non-equal-variance BIC form is

where |G| is the number of edges in G. This score rewards better predictive fit while penalizing dense graphs.

Parametrized DAG generation with low-rank potentials. Directly searching GDAG is combinatorial and nondifferentiable. To enable efficient global exploration, we map an unconstrained continuous representation to a DAG through a DAG generator τ(·). Following recent constraint-free DAG parameterizations, we associate each node with (i) a scalar node potential vector p ∈ RR that induces an ordering, and (ii) a low-rank embedding matrix R ∈ RR×k with k ≪ R that controls connectivity. Define the gradient-flow operator

and let H(·) be the entry-wise Heaviside step function. We construct a binary adjacency matrix by

where ⊙ denotes the Hadamard product. The first term enforces acyclicity by permitting edges only from lower to higher potential; the second term provides a low-rank connectivity prior. We denote by z ∈ RR(1+k) the concatenation of p and the flattened R, and write τ(z) for brevity. The structure learning objective (4) becomes

Bayesian optimization for discrete-structure search. The score SBIC(D, τ(z)) is expensive to evaluate (it requires fitting node-wise predictors for each candidate DAG) and is non-smooth due to H(·) in (8). We therefore employ BO, which iteratively: (i) proposes candidates using an acquisition function, (ii) evaluates the true score for selected candidates, and (iii) updates a probabilistic surrogate model.

Bayesian neural surrogate as the BO surrogate model. Instead of Gaussian processes, we adopt a Bayesian neural network (BNN) surrogate to improve scalability with the growing number of evaluations. The surrogate models the posterior over the unknown score function

using a parameter distribution p(θ | H) conditioned on the evaluation history

With approximate Bayesian inference (e.g., Monte-Carlo dropout), for any candidate z we draw M stochastic forward passes

and compute predictive mean and epistemic uncertainty:

This uncertainty is critical for robust exploration in the highly non-convex, partially observed structure landscape.

Uncertainty-aware acquisition. We employ an exploration exploitation trade-off via an upper confidence bound (UCB) acquisition:

or Thompson sampling as a stochastic alternative:

At each BO iteration, we sample C preliminary candidates {z(j) }Cj=1 in a trust region around the current best zbest, compute AF(z(j) ), select the top-B candidates, and evaluate their true scores.

Indirect surrogate modeling via node-wise local scores. A key practical enhancement is to exploit the decomposability of (6). Rather than modeling f(z) directly, we model node-wise local quantities and recombine them. Specifically, for each node j, define the local target

which is observable once G is evaluated. We train R independent dropout networks {DropoutNNj}Rj=1 to predict ℓj(G) from a binary parent-indicator vector uj(G) ∈ {0, 1}R, where uj,i(G) = I{i ∈ paG(j)}. For a candidate G = τ(z), we sample

and combine them to obtain a sampled global score:

Repeating (19) for m = 1, . . . ,M yields µ(z) and σ(z) used in (15). This local-to-global surrogate improves sampleefficiency and reduces spurious correlations by avoiding a single shared network across nodes.

Surrogate update. After evaluating the selected batch, we update the experience set:

and additionally store node-wise local targets {ℓj(τ(z(j) ))}Rj=1 to update DropoutNNj with replay-based training. This continual refinement yields an increasingly accurate uncertainty-aware surrogate, enabling efficient recovery of a high-scoring DAG, which we take as the estimated EC graph ̂G for Sec. 4.3.

4.3 Structure-Guided Joint Optimization

Let H = {H(k)}Tk=1 denote the fused multimodal representation obtained via Eq. (2), parameterized by fusion weights α. Given a candidate causal graph G∗ ∈ {0, 1}R×R inferred by the Bayesian neural surrogate, our goal is to update α such that the resulting representation H is maximally consistent with the causal dependencies encoded by G∗ .

Treating G∗ as fixed, we impose a linear structural equation model on the fused representation:

where PaG∗ (j) denotes the parent set of node j under G∗ , and εj(k) is a noise term. The regression coefficients wj = {wij} are estimated independently for each node.

To ensure numerical stability and differentiability with respect to H, we estimate wj via ridge regression:

whose closed-form solution

EC-guided surrogate loss. Using the fitted regression model, we define a structure-consistency loss that penalizes violations of the inferred causal dependencies:

where ˆHj denotes the ridge-based prediction induced by G∗ . This loss measures how well the current fused representation admits a dynamical factorization consistent with the EC.

The fusion parameters α are optimized by minimizing the following composite objective:

where Lalign enforces multimodal consistency, Lsmooth regularizes temporal dynamics, and Lα prevents degenerate fusion solutions. Importantly, Ldag provides causal supervision without requiring gradients through the DAG search process.

Alternating optimization scheme. Training proceeds via block-coordinate descent: (i) update fusion parameters α with G∗ fixed by minimizing Eq. (25); (ii) update the causal structure by re-running Bayesian neural surrogate-guided search on the updated representation H. This alternating procedure progressively aligns multimodal representations with structurally stable causal graphs, yielding robust and directionally identifiable effective connectivity estimates.

5 Experiments

We evaluate STCBN-EC on simulated and real EEG-fMRI datasets to assess its performance. Three simulated datasets validate EC recovery against ground truth, while two real datasets test robustness. As real data lacks ground-truth EC, inferred networks are evaluated via brain-state classification. All methods use the same feature extraction and a 10-fold cross-validated k-Nearest Neighbors (KNN) classifier.

5.1 Baseline Methods and EvaluationMetrics

We compare STCBN-EC with a range of classical and stateof-the-art EC learning methods, including Patel [Patel et al., 2006], pwLiNGAM [Hyvarinen, 2010], lsGC [DSouza et al., 2017], DiffAN [Sanchez et al., 2023], MetaRLEC [Zhang et al., 2024b], MCAN [Liu et al., 2024], FSTA-EC [Xiong et al., 2025b], and DrBO [Duong et al., 2025].

For all baselines, we follow the parameter settings recommended in the original literature and further tune hyperparameters on a validation subset.

We report standard EC evaluation metrics throughout the experiments, including Accuracy (ACC), F1-score (F1), Structural Hamming Distance (SHD), Precision (Prec), Recall (Rec), and ROC-AUC.

5.2 Dataset Details

Simulated Datasets. We evaluate STCBN-EC using three simulated datasets with varying network sizes (R ∈ {5, 10, 30}). Each dataset includes EEG and fMRI signals, with ground-truth effective connectivity structures for controlled EC recovery tests. The fMRI data is simulated using DCM, and EEG signals are generated via a state-space model.

Sleep-Rest Dataset. This dataset 1 includes EEG and fMRI data from 33 healthy participants, total 248 samples [Gu et al., 2022]. EEG was recorded from 32 channels, with fMRI data acquired on a 3T Siemens Prisma scanner (TR = 2.1s, TE = 25 ms, resolution = 3 × 3 × 3 mm3 ). Time series were extracted using the AAL3 atlas and DPABI preprocessing.

XP2 Motor Imagery Neurofeedback Dataset. The XP2 dataset 2 [Lioi et al., 2020] contains simultaneous EEG-fMRI data from 40 dNF and 28 MI participants. EEG was recorded from 64 channels, and fMRI was obtained using an EPI sequence on a 3T scanner (TR = 1s, TE = 23 ms, resolution = 3 × 3 × 3 mm3 ). Time series were extracted using the AAL atlas and DPABI preprocessing.

5.3 Results on Simulated Datasets

Parameter Sensitivity Analysis

We evaluate the sensitivity of STCBN-EC to its key hyperparameters on the Sim-1 setting, including the fusion learning rate, number of training evaluations, and surrogate hidden size. As shown in Table 1, the model demonstrates stable performance across a broad range of parameter values. For the fusion learning rate, 5×10-3 achieves the best overall performance, yielding the highest ACC and F1 as well as the lowest SHD. Increasing the training evaluation budget substantially improves EC recovery compared to the low-budget setting, highlighting the importance of uncertainty-aware structure exploration. In addition, a medium hidden size consistently outperforms both smaller and larger configurations. Overall, these results indicate that STCBN-EC is robust to hyperparameter variations and does not require extensive tuning to achieve strong performance.

Ablation Study

Table 3 presents ablation results for STCBN-EC. Removing joint optimization (w/o Joint) degrades ACC and F1 while increasing SHD, showing that decoupled representation learning harms causal discovery. Excluding EEG (w/o EEG) also reduces performance, confirming that multimodal fusion adds complementary temporal information beyond fMRI. Disabling HRF modeling (w/o HRF) or shuffling EEG signals (EEG shuffled) further impairs results, underscoring the need for physiological alignment and cross-modal consistency. The full model achieves the best performance across all metrics, indicating each component's contribution to stable EC learning.

1 https://openneuro.org/datasets/ds003768/versions/1.0.13

2 https://openneuro.org/datasets/ds002338/versions/2.0.2

Comparison with Baseline Methods

We evaluate STCBN-EC on three simulated datasets with increasing network sizes (Sim-1, Sim-2, and Sim-3) to assess its ability to recover ground-truth effective connectivity (EC) structures. Table 2 reports the EC learning results of all compared methods. Overall, STCBN-EC consistently achieves superior performance across all simulation settings. In Sim-1, STCBN-EC attains the highest F1-score and the lowest SHD, indicating accurate edge identification and strong structural consistency. While several baseline methods achieve comparable accuracy, their substantially larger SHD values suggest inferior recovery of the underlying EC.

In Sim-2, STCBN-EC clearly outperforms all competing approaches, achieving the best accuracy, F1-score, and SHD

simultaneously. As network complexity increases, classical and recent learning-based baselines exhibit degraded structural recovery, whereas STCBN-EC remains robust, demonstrating good scalability to moderately sized networks.

In the most challenging Sim-3 setting, all methods experience increased SHD due to the difficulty of large-scale EC. Nevertheless, STCBN-EC maintains competitive accuracy and achieves the best F1-score, indicating stable performance under high-dimensional conditions.

5.4 Results on Real EEG-fMRI Datasets

We further evaluate STCBN-EC on two real-world EEGfMRI datasets: XP2 and Sleep-Rest. Quantitative classification results are summarized in Table 4.

XP2 Dataset

Table 4 shows that STCBN-EC achieves consistently superior classification performance on the XP2 dataset, with marked improvements in ACC, F1, and AUC compared to baseline methods. Notably, the gain is most pronounced in precision and overall F1-score, indicating that the learned EC networks provide highly discriminative and stable features for distinguishing MI and dNF states. This improvement is attributed to the ability of STCBN-EC to recover directional effective connectivity, encoding asymmetric information flow not captured by undirected or weakly directed models.

Beyond quantitative classification performance, the learned EC patterns exhibit clear physiological interpretability. As illustrated in Fig. 3, MI and dNF states are characterized by distinct large-scale interaction profiles at the functional-group level.

In the MI state, strong directed interactions are predominantly observed among parietal and sensorimotor-related systems, particularly involving the parietal group (e.g., Superior underlined.

Parietal Gyrus and Precuneus) and frontal-parietal pathways. These regions are known to support sensorimotor integration and spatial attention, suggesting that MI relies on coordinated information integration across association cortices.

In contrast, the dNF state shows enhanced EC strength within subcortical and temporal systems, including thalamic, basal ganglia, and auditory-related regions. The prominence of directed interactions involving subcortical groups indicates increased thalamo-cortical coordination, which is associated with global regulation and state monitoring. Meanwhile, stronger temporal interactions may reflect sensory processing during neurofeedback. Compared with MI, the dNF EC exhibits a more distributed interaction pattern, suggesting reduced centralization and a shift toward global modulation.

These findings show that STCBN-EC improves classification accuracy and learns physiologically meaningful directional interaction patterns differentiating cognitive states. The explicit directed EC modeling captures state-dependent reorganization of information flow, explaining its superior performance in both quantitative metrics and neuroscientific interpretability.

Sleep-Rest Dataset

Connection Strength We compared the learned EC between the sleep and restingstate conditions. Across both states, common hub regions were observed, indicating a stable network backbone. Frontal and temporal cortices consistently appeared as dominant hubs, suggesting their central role in large-scale information integration. Thalamic nuclei also prominently featured in both states, highlighting the importance of thalamocortical interactions in network coordination.

In the sleep condition, the network showed stronger involvement of frontal and orbitofrontal regions, suggesting internally driven regulatory processes. The cerebellum also emerged as a key hub, contributing to large-scale cortical coordination. This pattern reflects the role of the cerebellum in sleep-dependent synchronization and information processing.

In contrast, the resting-state network revealed a stronger focus on posterior regions such as the Precuneus, indicating a self-referential processing mode associated with the default mode network. Additionally, subcortical regions related to neuromodulatory control, including the brainstem and basal ganglia, were more prominent, suggesting that resting-state EC is influenced by global arousal regulation. This indicates that, while sleep emphasizes frontal and cerebellar modulation, the resting state is shaped by posterior integration and neuromodulatory regulation.

These results show that sleep and rest share a common fronto-temporo-thalamic network, while exhibiting statespecific reorganization of hub regions.

6 Conclusion and Limitations

We propose STCBN-EC that learns EC from complementary EEG and fMRI data while capturing physiologically meaningful differences across cognitive states. Experimental results on two real-world datasets of different scales demonstrate STCBN-EC's reliability in identifying state-dependent EC patterns, highlighting its practical value for multimodal neuroimaging analysis.

Despite these strengths, the method's computational efficiency remains limited, particularly for large-scale networks and multimodal data. Future work will focus on enhancing efficiency and extending the framework to additional neuroimaging modalities.

Acknowledgements

This work was sponsored by the Beijing Nova Program (20240484635).