FAST-Brain: A Flow-Aligned Spatio-Temporal Surrogate Brain Model

1 Introduction

• The brain's interconnected regions jointly support cognition and behavior, and rs-fMRI provides a non-invasive view of BOLD dynamics and inter-regional dependencies.Data-driven surrogate models aim to reproduce neural dynamics and functional dependencies and infer effective interactions computationally.Source: §1 · p. 1
• Existing surrogate models struggle with long-range temporal structure and often use RNNs or multilayer perceptrons that fail to model high-dimensional, long-horizon BOLD dynamics.Such temporal models can smooth variability and accumulate errors over long horizons.Source: §1 · pp. 1–2
• Many existing approaches use a single spatial constraint, while ambient-space noise or velocity targets can obscure BOLD signals' low-dimensional structure and reduce sample efficiency.Brain dynamics also reflect stochastic influences from latent cognitive states and intrinsic noise.Source: §1 · pp. 1–2
• FAST-Brain directly predicts clean BOLD signals and pairs this flow-aligned framework with a Transformer and a graph convolutional network using multiple spatial priors.The Transformer models long-range dependencies; spatial priors include structural connectivity, fiber lengths, and functional network membership.Source: §1 · p. 2, Figure 1 · p. 2
• On HCP, FAST-Brain raises functional connectivity correlation from 0.726 to 0.938 and reduces MAE from 0.199 to 0.073 versus the strongest baseline.The paper also reports state-of-the-art recovery of effective connectivity and intrinsic low-dimensional signal geometry on synthetic and HCP datasets.Source: §1 · p. 2
• Under a low-dimensional subspace assumption, the model's approximation error scales with intrinsic dimension d rather than ambient ROI dimension N.The Bayes-optimal denoiser depends on the noisy observation through its low-dimensional projection, unlike noise- or velocity-based targets.Source: §1 · p. 2

Generative Models for Time Series

• Time-series generation progressed from VAEs and GANs to diffusion and flow matching, but these models parameterize noise or velocity fields in the full ambient space.Flow matching learns a continuous velocity field and can provide faster, more direct deterministic sampling than SDE-based diffusion; FlowTS applies it to conditional time series.Source: §2 · p. 3

Direct Data Prediction for Image Generation

• Image-generation methods report gains from predicting clean data because clean targets inherit low-dimensional structure that noise and velocity targets do not.Direct data prediction has received little attention in structured time-series generation and had not been explored for anatomically constrained fMRI signals.Source: §2 · p. 3

Spatio-temporal Modeling of Brain Dynamics

• Existing data-driven brain surrogates either neglect anatomical spatial priors or fail to exploit fMRI's intrinsic low-dimensional structure.Classical simulators and connectivity-inference methods offer interpretability but rely on predefined generative assumptions or linear dependencies that limit nonlinear spatio-temporal modeling at scale.Source: §2 · p. 3

Problem Setup

• The time-invariant spatial priors G comprise structural connectivity, fiber lengths, and each ROI's macroscopic functional-network assignment.Structural connectivity and fiber lengths are normalized to [0, 1].Source: §3 · p. 3
• The task predicts future BOLD windows from recent history and spatial priors, learning p(Y_t | H_t, G) and rolling out autoregressively for sequences of arbitrary length.The input is an N-by-T matrix of BOLD signals across N ROIs and T acquisition times; downstream functional and effective connectivity can be derived from generated sequences.Source: §3 · pp. 3–4

Flow Matching

• Flow matching linearly interpolates noise and the future BOLD target, then learns a velocity field to match the target velocity.At inference, integrating the learned ODE from noise yields a clean sample approximating the future target.Source: §3 · p. 4
• FAST-Brain retains the flow-matching framework but replaces velocity prediction with direct clean-signal prediction.Its clean-signal parameterization uses the noisy interpolant, flow time, history window, and spatial priors as inputs.Source: §3 · p. 4

4.1 Flow-Aligned Direct Data Prediction

• FAST-Brain predicts clean future BOLD from an interpolated noisy state, then analytically recovers the flow velocity for training.The interpolation retains the flow-matching path, while the prediction is conditioned on history and spatial priors.Source: §4.1 · p. 4

4.2 Spatio-Temporal Network Architecture

• The temporal decoder uses a history-conditioned Transformer, with the noisy future as queries and historical signals as keys and values.Self-attention also lets future time points interact across the forecast horizon; multi-step generation reduces exposure bias relative to step-by-step autoregression.Source: §4 · p. 4, §4.2 · p. 5
• The spatial decoder adds a GCN residual correction using connectivity strength, shorter fiber pathways, and shared functional-network membership.The temporal and spatial branches are additively decomposed, with the spatial branch supplying time-invariant anatomical structure.Source: §4.2 · p. 5
• Zero-initializing the spatial decoder’s final projection lets training begin with the temporal trend before progressively adding structural corrections.The additive design is intended to keep the spatial branch from destabilizing the temporal branch early in optimization.Source: §4.2 · p. 5

4.3 Theoretical Guarantees on Low-Dimensional Subspaces

• Under a low-dimensional subspace assumption, FAST-Brain’s approximation error scales with intrinsic dimension d rather than ambient ROI dimension N.The Bayes-optimal denoiser depends on the noisy observation only through its low-dimensional projection; noise- and velocity-based targets retain dependence on the ambient space.Source: §4.3 · p. 6
• The theoretical result assumes the future signal lies almost surely in a d-dimensional subspace, with d much smaller than N.The factorization and approximation result also rely on regularity conditions for the latent denoiser and its growth and tails.Source: §4.3 · p. 6

Data and Baselines

• The evaluation covers RNN (20 ROIs, 8,000 time points), SDDEs (66 ROIs, 3,000 time points), and HCP (360 ROIs, 4,280 time points), using the first 80% for training.RNN and SDDEs are synthetic datasets; HCP is real rs-fMRI data.Source: §5 · p. 7

Metrics and Evaluation

• FC is evaluated with Pearson correlation and MAE, while synthetic RNN effective connectivity is assessed with correlation, AUROC, and AUPRC; the supplied passages do not report EC outcomes.The experiments also include an RNN conditional-generation comparison and unconditional-generation metrics.Source: §5 · p. 7, Table 2 · p. 8, Table 3 · p. 8

Functional Connectivity

• On HCP, FAST-Brain raises the best baseline correlation from 0.726 to 0.938 and lowers MAE from 0.199 to 0.073 for functional connectivity.The reported gains cover both global connectivity topology and absolute FC strengths.Source: Table 1 · p. 7, §5 · p. 8

Conditional Generation

• FAST-Brain achieves the lowest CRPS, QICE, ProbCorr, and Conditional FID among compared methods on RNN conditional generation.The authors associate these results with more accurate probabilistic forecasts, better-calibrated intervals, more faithful correlation structure, and closer alignment with the conditional distribution.Source: §5 · p. 8, Table 3 · p. 8

Ablation Study

• Across RNN, SDDEs, and HCP, the full model performs best in FC recovery; clean-signal prediction and graph-based spatial priors each contribute to accuracy.Replacing clean-signal prediction with ϵ- or v-prediction particularly worsens SDDEs and HCP, and removing the GCN also degrades performance, especially on those datasets.Source: §5 · p. 8, Table 4 · p. 8

Conclusion

• FAST-Brain combines direct clean-BOLD prediction, graph-based anatomical priors, and long-range temporal modeling for resting-state fMRI generation.The conclusion reports recovery of functional connectivity, effective connectivity, and low-dimensional signal geometry.Source: §6 · p. 9

Limitations and Future Work

• The paper focuses exclusively on neuroimaging and raises, rather than answers, whether the approach benefits other structurally constrained spatio-temporal systems.Examples named for future investigation include traffic, climate, weather, and power-grid dynamics.Source: §6 · p. 9
• The theoretical analysis covers denoiser structure and approximation error, but finite-sample statistical learning analysis remains open.The authors identify nonasymptotic estimation-error bounds as future work.Source: §6 · p. 9

A Proof of Theorem 1

• The proof concludes that the Bayes-optimal denoiser depends on the noisy observation only through its low-dimensional projection A⊤Z.It reduces the approximation problem to the latent-space denoiser and uses compact-region approximation with tail control to establish the result.Source: A Proof of Theorem 1 · pp. 14, 16

B Qualitative FC Visualization

• Figure 3 places RNN, SDDEs, and HCP in columns and empirical FC matrices above FAST-Brain-generated matrices.The reported visual comparison finds localized and network-structured recovery on RNN and preservation of major FC organization on SDDEs and HCP.Source: Figure 3 · p. 17, B Qualitative FC Visualization · p. 17

C.1 RNN Dataset

• The simulated network has 20 spatially embedded nodes assigned to four functional communities, with fiber lengths based on Euclidean distances adjusted by a cross-hemispheric penalty.Each node's community assignment is denoted c_i.Source: §C.1 · p. 17
• Structural connectivity is generated with a distance-dependent stochastic block model, whose expected connection intensity uses a scaling factor, network affinity, and exponential spatial decay.The resulting SC_ij values follow a Poisson distribution.Source: §C.1 · p. 17
• Neural activity follows stochastic RNN dynamics and is simulated with Euler discretization, adding Gaussian noise to each update.The discretized update uses the structural connectivity matrix and tanh-transformed activity.Source: §C.1 · pp. 17–18

C.2 SDDEs Dataset

• Signal transmission delays and effective coupling weights scaled by global coupling determine network input, alongside conduction velocity, a Hadamard product, and baseline Gaussian white noise.Source: §C.2 · p. 18
• A generic 2D oscillator defines each node's latent activity as a system of stochastic delay differential equations, with excitatory LFP and slow inhibitory recovery variables.The system uses fixed nonlinear parameters.Source: §C.2 · p. 18
• The continuous macroscopic observation is obtained by convolving the high-frequency latent state with a first-order Volterra hemodynamic response function, then sampling at the repetition time yields the BOLD matrix.The BOLD measurement is defined as y_i sampled at m · TR.Source: §C.2 · p. 18

C.3 HCP dataset

• The real rs-fMRI experiments follow the NPI HCP setting, using HCP-MMP's 360-ROI representation and preprocessing protocol.Source: §C.3 · p. 18

D Implementation Details

• Each series is split chronologically, using the first 80% of temporal samples for training and the remainder for rolling evaluation; windows stay within their split.Source: D Implementation Details · p. 18
• Models condition on a history window and predict 24 future steps, with history lengths of 60 for RNN and SDDEs and 10 for HCP.Source: D Implementation Details · p. 18
• RNN and SDDE node signals are min-max normalized to [−1, 1], while HCP discards the first 30 REST frames, retains 360 Glasser ROIs, and standardizes each session independently.Source: D Implementation Details · p. 18
• The structural branch uses normalized graph powers through order 2, and flow ODE sampling uses explicit Euler with 20 steps; experiments run on one NVIDIA H100 GPU.Source: D Implementation Details · p. 18

Conditional generation metrics

• CRPS evaluates probabilistic forecast accuracy using stochastic samples, while QICE assesses whether empirical predictive-interval coverage approaches the nominal uniform target.Lower values indicate better calibrated and sharper forecasts for CRPS and better calibration for QICE.Source: §E.1 · p. 19
• ProbCorr measures differences between predicted and true conditional correlation matrices, and Conditional FID evaluates real-versus-generated conditional sequence distributions in representation space.For both metrics, lower values indicate better alignment with the corresponding real conditional structure or distribution.Source: §E.1 · pp. 19–20

Unconditional generation metrics

• Discriminative Score measures how distinguishable generated samples are from real sequences using a held-out classifier; values closer to zero indicate greater indistinguishability.Source: §E.1 · p. 20
• Predictive Score tests forecasting structure by training a predictor on generated sequences and evaluating its mean absolute error on real sequences, where lower values are better.Source: §E.1 · p. 20
• Context-FID compares real and generated sequences in contextual representation space, while Correlational Score measures deviation between their average ROI-wise correlation matrices.Smaller values indicate closer contextual representations and better-preserved dependence structure.Source: §E.1 · p. 20

• On HCP, FAST-Brain achieves the best CRPS, QICE, ProbCorr, and Conditional FID among conditional-generation metrics.Source: §E.2 · p. 20, Table 6 · p. 21
• On SDDEs, FAST-Brain leads CRPS and QICE, while TimeDiff leads ProbCorr and FlowTS Conditional FID.Source: §E.2 · p. 20, Table 5 · p. 20
• Across RNN, SDDEs, and HCP, FAST-Brain records the best Discriminative Score and Correlational Score.Source: §E.2 · p. 21, Table 7 · p. 21, Table 8 · p. 21
• On HCP unconditional generation, FlowTS slightly exceeds FAST-Brain on Predictive Score and Context-FID, although FAST-Brain outperforms most baselines on Discriminative and Correlational Scores.Source: §E.2 · p. 21, Table 9 · p. 21