Tau-BNO: Brain Neural Operator for Tau Transport Model

Nuutti Barron; Heng Rao; Urmi Saha; Yu Gu; Zhenghao Liu; Ge Yu; Defu Yang; Ashish Raj

cs.CEMar 9, 2026

Tau-BNO: Brain Neural Operator for Tau Transport Model

Authors: Nuutti Barron, Heng Rao, Urmi Saha, Yu Gu, Zhenghao Liu, Ge Yu, Defu Yang, Ashish Raj,

Organizations: Department of Radiology and Biomedical Imaging, University of California San Francisco, USA · College of Computer Science and Engineering, Northeastern University, China · Department of Computer Science, Wake Forest University, USA · Department of Computer Science, Hangzhou Normal University, China

Abstract

Mechanistic modeling provides a biophysically grounded framework for studying the spread of pathological tau protein in tauopathies like Alzheimer's disease. Existing approaches typically model tau propagation as a diffusive process on the brain's structural connectome, reproducing macroscopic patterns but neglecting microscale cellular transport and reaction mechanisms. The Network Transport Model (NTM) was introduced to fill this gap, explaining how region-level progression of tau emerges from microscale biophysical processes. However, the NTM faces a common challenge for complex models defined by large systems of partial differential equations: the inability to perform parameter inference and mechanistic discovery due to high computational burden and slow model simulations. To overcome this barrier, we propose Tau-BNO, a Brain Neural Operator surrogate framework for rapidly approximating NTM dynamics that captures both intra-regional reaction kinetics and inter-regional network transport. Tau-BNO combines a function operator that encodes kinetic parameters with a query operator that preserves initial state information, while approximating anisotropic transport through a spectral kernel that retains directionality. Empirical evaluations demonstrate high predictive accuracy ( $R^2\approx$ 0.98) across diverse biophysical regimes and an 89% performance improvement over state-of-the-art sequence models like Transformers and Mamba, which lack inherent structural priors. By reducing simulation time from hours to seconds, we show that the surrogate model is capable of producing new insights and generating new hypotheses. This framework is readily extensible to a broader class of connectome-based biophysical models, showcasing the transformative value of deep learning surrogates to accelerate analysis of large-scale, computationally intensive dynamical systems.

Figures & tables

Figure 1: Tau-Brain Neural Operator (Tau-BNO) Architecture. a. Input consists of initial tau concentration fields derived from experimentally defined mouse injection sites, together with sampled aggregation and fragmentation rate parameters ( $\lambda_{f},\lambda_{\gamma},\lambda_{\delta},\lambda_{\epsilon},\lambda_{\mu}$ ). b. A Query Operator encodes regional initial tau concentrations, and a Function Operator encodes kinetic parameters; their interaction is passed through a Directed Graph Operator defined on the structural connectome to model asymmetric inter-regional transport. c. A learnable projector generates tau spatiotemporal dynamics across all 426 brain regions.

Figure 2: Brain network representations for tau transport modeling. a. Original directed MCA connectome encoding region-to-region tau transport pathways. b–d. Three derived undirected proximity networks capturing complementary topological features: b. first-order symmetric network preserving global connectivity structure; c. second-order indegree network emphasizing convergent (afferent) connectivity; d. second-order outdegree network emphasizing divergent (efferent) connectivity. Edge brightness indicates connection strength. Together, these representations enable graph neural networks to recover directional transport dynamics from symmetrized adjacency matrices.

Figure 3: Performance comparison of Tau-BNO with 11 benchmarking models. a. Quantitative evaluation (RMSE, MAE) across six general-purpose architectures, highlighting the limitations of standard predictive models for PDE-governed tau transport. b. Comparison among five neural operator variants, showing Tau-BNO’s superiority within the operator learning family. c. Training dynamics of Tau-BNO over 1000 epochs, showing log-scale RMSE and stable convergence behavior. d. Temporal evolution of absolute prediction error for Tau-BNO and three representative neural operators, illustrating sustained accuracy across the full trajectory.

Figure 4: Ablation analysis of Tau-BNO architecture. Performance comparison for modular (FO, QO, DGO) and kernel (Fourier, differential) ablations.

Figure 5: Spatiotemporal prediction accuracy across diverse physiological regimes. Brain heatmaps of relative prediction error for Tau-BNO and three neural operator baselines under three representative biophysical regimes. Parameters $[\lambda_{f},\lambda_{\gamma},\lambda_{\delta},\lambda_{\epsilon},\lambda_{\mu}]$ denote tau production rate, aggregate rate, anterograde transport velocity, retrograde transport velocity, and uptake release rate, respectively. a. Transport-only regime ( $\lambda_{f}=0$ ): $[0.0,8\times 10^{-3},100,100,1.20]$ . b. High anterograde regime (anterograde dominant transport): $[9\times 10^{-4},4.8\times 10^{-3},80.6,22.6,0.47]$ . c. High retrograde regime (elevated production, strong retrograde transport): $[9.8\times 10^{-3},4.2\times 10^{-3},58.6,92.5,0.51]$ . Color scale denotes relative error magnitude. Across regimes, Tau-BNO exhibits consistently lower error than competing models.

Figure 6: Prediction accuracy under various and extreme biophysical parameter regimes. a. Tau trajectories comparing ground truth NTM simulations (solid lines) and Tau-BNO prediction (dashed lines) for four biophysical parameter regimes, aggregated from the 426-region MCA atlas to the 44-region DS atlas. Regime specifications are provided in Supplementary Table 2 . b. Predicted versus ground truth spatial tau concentrations at 4-month intervals under four biophysical parameter regimes. Blue dots represent brain regions under the MCA mouse atlas. Insets show glass brain visualizations for ground truth (top left) and Tau-BNO (bottom right), with red/orange intensity reflecting regional tau concentration.

Figure 7: Prediction accuracy across various initial seeding conditions. a. Ground truth and Tau-BNO predicted tau trajectories under different seeding configurations, aggregated from the 426-region MCA atlas to the 44-region DS atlas. Region color legend for tau trajectories follows Figures 6 . b. Glass brain visualizations showing the seeded regions (top) and resulting spatial tau distributions 3 months post-seeding (bottom) across three representative NTM parameter regimes.

Figure 8: Tau-BNO predictions illustrate the macroscale effects of varying microscopic transport parameters along the mouse structural connectome. Regional tau trajectories (left) are aggregated from the 426-region MCA atlas to the 44-region DS atlas. Glass brain visualizations (right) display spatial tau distributions at selected time points, with increasing orange intensity indicating higher regional tau concentration. The baseline prediction (top left) uses parameter vector $[\lambda_{f},\lambda_{\gamma},\lambda_{\delta},\lambda_{\epsilon},\lambda_{\mu}]=[5\times 10^{-4},8\times 10^{-3},10,10,2.2]$ . Increased anterograde and retrograde transport conditions are generated by modifying $(\lambda_{\delta},\lambda_{\epsilon})$ to $(100,10)$ and $(10,100)$ , respectively. Decreased aggregation is modeled with $\lambda_{\gamma}=1\times 10^{-3}$ , and increased production with $\lambda_{f}=1\times 10^{-2}$ . The blue dot in the baseline glass brain denotes the initial seeding location.

Type	Method	L-Layers	Hidden Dim	FNO Modes	LR	Dropout	Act.
General Architecture	MLP	3	[32, 128, 256]	—	$2\times 10^{-4}$	0.1	GELU
	KAN	3	[32, 128, 256]	—	$2\times 10^{-4}$	0.0	SiLU
	Neural ODE	2	256	—	$5\times 10^{-4}$	0.0	SiLU
	Transformer	4	256	—	$8\times 10^{-4}$	0.1	SiLU
	Mamba	3	256	—	$8\times 10^{-4}$	0.1	SiLU
	DeepONet	3+3 †	256	—	$8\times 10^{-4}$	0.0	Tanh

Table 1: Architectural specifications and hyperparameters for all evaluated models. L-layer denotes the number of layers; FNO modes indicates the number of Fourier modes retained in frequency domain methods.

	Seed Region(s)	$\lambda_{f}$ $(\times 10^{-4})$	$\lambda_{\gamma}$ $(\times 10^{-3})$	$\lambda_{\delta}$	$\lambda_{\epsilon}$	$\lambda_{\mu}$	Description
Biophysical Regime 1	Field CA1, LH	$100$	$1.0$	$10$	$100$	$2.2$	High production, low aggregation, high retrograde transport bias, high cellular uptake and release
Biophysical Regime 2	Field CA1, LH	$5$	$1.0$	$10$	$10$	$2.2$	Low production, low aggregation, low dibirectional transport, high cellular uptake and release
Biophysical Regime 3	Field CA1, LH	$0$	$8.0$	$100$	$100$	$3.2$	No production, high aggregation, high bidirectional transport, high cellular uptake and release
Biophysical Regime 4	Field CA1, LH	$0$	$8.0$	$100$	$100$	$0.2$	No production, high aggregation, high bidirectional transport, low cellular uptake and release
Biophysical Regime 5	Caudoputamen, Primary motor area; both RH	$1.8$	$7.2$	$81$	$18$	$1.0$	Low production, high aggregation, anterograde transport bias, low uptake and release
Biophysical Regime 6	Medulla, Pons in LH; Amygdala, Auditory cortex, Hypothalamus, Medulla, Pons, Posterior parietal association area, Primary motor area, Restrosplenial area, Thalamic in RH.	$3.3$	$4.4$	$30$	$64$	$0.5$	Low production, intermediate aggregation, mild retrograde transport bias, low uptake and release

Table 2: Description of biophysical regimes used to generate NTM simulations in Figures 6 and 7 (LH: left hemisphere, RH: right hemisphere).

Figure 9: Trajectory predictions across ten diverse physiological regimes. Dashed lines represent Tau-BNO predictions; solid lines indicate ground truth trajectories from high-fidelity PDE solutions. The model demonstrates robust performance across parameter conditions for five representative brain regions, accurately capturing both transient dynamics and steady-state behavior. Each condition represents distinct combinations of production rate ( $\lambda_{f}$ ), aggregation rate ( $\lambda_{\gamma}$ ), transport parameters ( $\lambda_{\delta}$ , $\lambda_{\epsilon}$ ), and uptake release rate ( $\lambda_{\mu}$ ).

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

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SC-TauPath: A Structural Connectivity Attribution Framework for Mapping Tau Propagation Pathways in Alzheimer's Disease

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University of Texas at Arlington, Arlington, TX, USA · Michigan State University, East Lansing, MI, USA · University of Texas Southwestern Medical Center, Dallas, TX, USA

Alzheimer's Disease Neuroimaging Brain Network Analysis

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Understanding how structural connections are associated with tau propagation in Alzheimer's disease (AD) remains a central open question, yet existing computational models either rely heavily on biophysical assumptions or lack neurobiologically interpretable pathway maps. We present SC-TauPath, a structural connectivity (SC) attribution framework that maps tau propagation pathways from in vivo neuroimaging data. SC-TauPath combines a Network Diffusion Model (NDM)-augmented multilayer perceptron with gradient

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