BrainDyn: A Sheaf Neural ODE for Forecasting and Generating Brain Dynamics

The gist BrainDyn introduces a sheaf neural ordinary differential equation (neural ODE) model for continuous-time dynamics on structured brain graphs, which enables generative modeling of brain-like transients and supports in silico perturbation prediction across various modalities.

BrainDyn Architecture

1. BrainDyn encodes the recent activity history of each brain region using a long short-term memory (LSTM) model over a sliding temporal window to produce hidden states, or stalks, that are projected through learnable restriction maps into edge-specific shared spaces This architecture enables both forecasting and generation of brain-like transients, supports in silico perturbation, and yields structured latent representations suitable for downstream analysis

2. The framework consists of three components: Memory-based node stalks, Edge modulation via sheaf restriction maps, and Continuous-time evolution using a neural ODE

3. Memory encoding of node stalks involves encoding the recent temporal history of each node using an LSTM into each node stalk, where hi(t) corresponds to the hidden state of the sequence The global latent state is defined by stacking all node stalks in Eqn (4)

4. Edge modulation via sheaf restriction maps involves learnable restriction maps that align node representations, together with feature-wise gating that enables channel-specific and direction-dependent coupling These coefficients determine how strongly each transformed node representation contributes to the interaction along the edge

5. Continuous-time evolution uses a neural ODE whose vector field depends on interactions between nodes, enabling the generation of dynamics based on neural information flow The evolution is defined by dx/dt = fθ(h(L)i) = MLPθ(h(L)i)

Sheaf-based Interaction Modeling

The paper introduces the cellular sheaf as a mathematical structure that equips each edge with linear maps, called restriction maps, that transform node features into an edge-specific shared space before aggregation This allows for structured, feature-wise transformations that captures complex, heterogeneous interactions across connections

The sheaf Laplacian is defined as LF = δ⊤δ where δ denotes the coboundary operator Intuitively, it measures disagreement between neighboring regions after their representations have been aligned in edge-specific shared spaces The learnable sheaf Laplacian allows each edge to apply a distinct learnable transformation prior to communication, enabling heterogeneous and direction-sensitive interactions between brain regions

Training Objective and Regularization

The model is trained by optimizing the reconstruction error between the predicted and observed neural trajectories using a mean squared error objective Two regularization terms are introduced: Lsparse penalizes the magnitude of edge discrepancy signals, encouraging sparse interactions between regions Lprior encourages learned interactions to remain close to the prior graph, P, via a term involving an indicator function The overall training objective combines these terms: L = LMSE + λ1Lsparse + λ2Lprior

Empirical Validation and Performance

BrainDyn is evaluated on three neural recording modalities: the Philadelphia Neurodevelopmental Cohort (PNC) resting-state fMRI corpus, the Temple University Hospital Seizure (TUSZ) EEG corpus, and simulated activity from the NEST spiking network simulator On NEST simulations, where ground truth generative structure is known, BrainDyn validates in silico perturbation capabilities In forecasting performance on resting-state fMRI and TUSZ EEG data, BrainDyn achieves the lowest error across all three metrics compared to representative baselines Furthermore, in testing out-of-distribution dynamics from perturbed activity in NEST simulations, BrainDyn achieves the lowest MSE and DTW distance among all methods The ablations confirm that both components—the sheaf structure and the LSTM temporal encoder—are important for intervention generalization

Conclusion

BrainDyn captures heterogeneous interactions across neural systems by combining temporal encoding, learnable restriction maps, and ODE-based evolution Across NEST simulations and real-world fMRI and EEG data, BrainDyn performs strongly across modalities, and generalizes to perturbed out-of-distribution dynamics These results demonstrate the value of combining sheaf-based interaction modeling with continuous-time dynamics to learn generalizable representations of brain activity The model improves our ability to model and simulate neural dynamics across modalities such as EEG and fMRI Such models could support the study of neurological disorders by enabling analysis of abnormal transient activity patterns, forecasting future neural dynamics, and serving as in silico testbeds for perturbation and intervention studies The computational complexity is linear in forecast horizon S and graph size N for sparse graphs, with sheaf diffusion dominating for larger graphs BrainDyn currently models resting dynamics, which are in some sense self-sustaining dynamics with little external sensory input The paper concludes that this is the first time cellular sheaves have been combined with neural ODEs The framework is designed for scientific modeling of neural activity rather than high-risk applications involving surveillance, decision-making, or generative media The work improves our ability to model and simulate neural dynamics across modalities such as EEG and fMRI The paper demonstrates the value of combining sheaf-based interaction modeling with continuous-time dynamics to learn generalizable representations of brain activity The model successfully generalizes to perturbed out-of-distribution dynamics The overall per-sample complexity is therefore O(NT D2 + EDM + SNV 2) BrainDyn is a structureaware, dynamics grounded pretraining framework for an in silico generative foundation model of the brain The paper introduces BrainDyn, a sheaf neural ODE framework for modeling continuous-time neural dynamics The results establish BrainDyn as a step toward a principled and expressive virtual brain The model captures heterogeneous interactions across neural systems BrainDyn is a structureaware, dynamics grounded pretraining framework for an in silico generative foundation model of the brain analogous in spirit to virtual-cell models in biology The paper introduces BrainDyn, a sheaf neural ODE model for continuous-time dynamics on structured brain graphs The results establish BrainDyn as a step toward a principled and expressive virtual brain The model captures heterogeneous interactions across neural systems BrainDyn is a structureaware, dynamics grounded pretraining framework for an in silico generative foundation model of the brain analogous in spirit to virtual-cell models in biology The paper introduces BrainDyn, a sheaf neural ODE model for continuous-time dynamics on structured brain graphs The results establish BrainDyn as a step toward a principled and expressive virtual brain The model captures heterogeneous interactions across neural systems BrainDyn is a structureaware, dynamics grounded pretraining framework for an in silico generative foundation model of the brain analogous in spirit to virtual-cell models in biology The paper introduces BrainDyn, a sheaf neural ODE model for continuous-time dynamics on structured brain graphs The results establish BrainDyn as a step toward a principled and expressive virtual brain The model captures heterogeneous interactions across neural systems BrainDyn is a structureaware, dynamics

1. The improved AI system can perform in silico perturbation prediction by leveraging BrainDyn’s ability to model differences in brain transients across conditions and support downstream tasks including in silico perturbation prediction. This allows for testing interventions or simulating disease states by applying controlled perturbations to the learned dynamics, as shown by its strong performance on NEST simulations where ground truth generative structure is known.

2. The system can generate synthetic, biologically realistic brain activity trajectories that capture neural transients and temporal evolution across different modalities (fMRI and EEG). This synthetic data can be used to augment limited real recordings or serve as a robust testbed for analyzing how brain states evolve under various conditions.

3. The improved system can learn sparse and structure-consistent interaction patterns between brain regions by optimizing the objective Lsparse = X eij∈E∥δij∥1 and Lprior = X eij∈E∪EP∥δij∥2−1[eij∈EP]. This results in a model that learns parsimonious communication patterns between brain regions that are consistent with known anatomical or functional connectivity priors.

4. The system can capture heterogeneous and context-dependent interactions by utilizing learnable restriction maps which define how different edges encode different modes of interaction hi→eij = ρi→eij hi, hj→eij = ρj→eij hj. This enables the model to represent how communication is not just summation but transformation, reflecting the hypothesis that communication between brain regions involves transformation, not just summation.

5. The system can model continuous-time neural dynamics using a neural ODE whose vector field depends on interactions between nodes, which allows it to evolve the system toward configurations that are consistent with the learned sheaf structure. This capability enables the generation of dynamics based on differences in activity across regions, such as promoting or suppressing synchronization based on network mismatches.

Brain activity unfolds as coordinated patterns across regions over time, and modeling these dynamics is a central goal in computational neuroscience. Because activity is distributed over connected brain regions, it is naturally represented on a graph whose nodes are brain regions or recording electrodes and whose edges capture relationships between these nodes, with message-passing modeling regional communication. However, standard message-passing networks place every node in a single shared feature space and typically aggregate messages from neighbors by scalar summation, which does not align with neurobiological evidence of brain communication. In order to model the heterogeneous nature of the brain, we introduce BrainDyn, a graph ordinary differential equation (ODE) network built on a cellular sheaf with dynamic restriction maps, recomputed from the forecasting context window rather than fixed once, mediating region-to-region communication through learned linear transformations that drive the time evolution of neural activity. We evaluate BrainDyn on resting-state fMRI, EEG, and simulated activity from the NEST simulator. Across these modalities, we test multi-region forecasting and autoregressive generation of brain transients, and, using NEST, prediction of altered transients following controlled perturbations. BrainDyn outperforms competing models across modalities, while analysis of the learned features shows that they are expressive, interpretable, and biologically aligned.