Uncover Governing Law of Pathology Propagation Mechanism Through A Mean-Field Game
Authors: Tingting Dan
Zhihao Fan
Guorong Wu
Venue: NeurIPS 2025 poster ()
Keywords: physics-informed deep learning
reaction–diffusion model
mean field game
symbolic regression
Alzheimer’s disease
Primary area: neuroscience_and_cognitive_science

Abstract: Alzheimer’s disease (AD) is marked by cognitive decline along with the widespread of tau aggregates across the brain cortex. Due to the challenges of imaging pathology spreading flows \textit{in vivo}, however, quantitative analysis on the cortical pathways of tau propagation and its interaction with the cascade of amyloid-beta (A$\beta$) plaques lags behind the experimental insights of underlying pathophysiological mechanisms.
To address this challenge, we present a physics-informed neural network, empowered by mean-field theory, to uncover the biologically meaningful spreading pathways of tau aggregates between two longitudinal snapshots. 
Following the notion of `prion-like' mechanism in AD, we first formulate the dynamics of tau propagation as a mean-field game (MFG), where the spread of tau aggregate at each location (aka. agent) depends on the collective behavior of the surrounding agents as well as the potential field formed by amyloid burden. Given the governing equation of propagation dynamics, MFG reaches an equilibrium that allows us to model the evolution of tau aggregates as an optimal transport with the lowest cost in \textit{Wasserstein} space. 
By leveraging the variational primal-dual structure in MFG, we propose a \textit{Wasserstein}-1 Lagrangian generative adversarial network (GAN), in which a Lipschitz critic seeks the appropriate transport cost at the population level and a generator parameterizes the flow fields of optimal transport across individuals. 
Additionally, we incorporate a symbolic regression module to derive an explicit formulation capturing the A$\beta$-tau crosstalk. 
Experimental results on public neuroimaging datasets demonstrate that our explainable deep model not only yields precise and reliable predictions of future tau progression for unseen new subjects but also provides a new window to uncover new understanding of pathology propagation in AD through learning-based approaches.

Uncover Governing Law of Pathology Propagation Mechanism Through A Mean-Field Game
Tingting Dan
Zhihao Fan
Guorong Wu'
Departments of Psychiatry and Computer Science University of North Carolina Chapel Hill
Chapel Hill, NC 27599
[Tingting_ Dan, grwu)mmeddd unc. edu zhi www.haaffa@@gaal.ccmmm com
Abstract
Alzheimer's disease (AD) is marked by cognitive decline along with the widespread of tau aggregates across the brain cortex. Due the challenges of imaging pathol- ogy spreading flows in vive, however, quantitative analysis on the cortical pathways tau propagation and its interaction with the cascade of amyloid-beta (A,3) plaques lags behind the experimental insights of underlying pathophysiologica mecha- nisms. To address this challenge, we present physics-informet neural network, empowered by reeanfieeld theory, to uncover the biologically meaningful spreading pathways of tau aggregates between two longitudinal snapshots. Following the notion of 'prion-like' mechanism in AD, we first formulate the dynamics of tau propagation as a mean-field game (MFG), where the spread of tau aggregate at each location (aka. agent) depends the collective behavior of the surrounding agents as well as the potential field formed by amyloid burden. Given the governing equation of propagation dynamics, MFG reaches equilibrium that allows us model the evolution of tau aggregates as an optimal transport with the lowest cost in Wasserstein space. By leveraging the variational primal-dual structure in MFG, we propose a Wasserstein- Lagrangian generative adversarial network (GAN), in which a Lipschitz critic seeks the appropriate transport cost at the population level and a generator parameterizes the flow fields of optimal transport across individuals. Additionally, we incorporate symbolic regression module derive an explicit formulation capturing the A,--tuu crosstalk. Experimental results on public neuroimaging datasets demonstrate that our explainable deep model not only yields precise and reliable predictions of future tau progression for unseen new subjects but also provides a new window uncover new understanding of pathology propagation in AD through learning-based based approaches.
Introduction
Alzheimer's 's disease (AD) is marked by a progressive decline in cognition accompanied by widespread accumulation of tau aggregates across the cortex. Mounting evidence suggests that tau spreads in 'prion-like' fashion: once a small number of molecules misfold, they act as seeds that affect neigh- boring neurons, propagating through neural circuits much like a contagion. In parallel, extracellular amyloid, beta (A,3) plaques, often accumulating years before symptom onset, are known to prime neural tissue by promoting tau hyperphosphorylation and enhancing trans-synaptic spread [3 [5] Together, these two hallmarks of AD form toxic synergy that accelerates protein aggregation, neuronal damage, and memory loss. Yet, the precise cortical pathways along which tau propagates,
*Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

and how A3 modulates or accelerates those flows, remain open questions in neurodegeneration 17 36].
research 3 24
18
Early theoretical work inspired by epidemiology and chemical kinetics formulates this process as reaction-diffusion system, in which tau both drifts along concentration gradients and undergoes nonlinear local amplification. For instance, IturriaMMedina et al. [23] demonstrated that a partial differential equation (PDE) with an explicit reaction term accurately reproduces the spatiotemporal patterns seen in longitudinal tau-PET scans. Crucially, extracellular amyloid-3 plaques, often present years before clinical onset, have been shown to "prime" neural cireuits, enhancing tau phosphorylation and accelerating its trans- synaptic spread [S]. This synergistic interplay drives vicious feed-forwalt loop of protein aggregation and neuronal damage, underscoring the necessity of models that capture the interaction between amyloid plaques and tau aggregates.
In general, there are five popular approaches capture these intertwined dynamics. (1) Reac- tion-Diffusion Models (RDM). Building on prion-like hypotheses, continuous reaction-diffusion equations capture both the drift of tau along spatial gradients and its local nonlinear accumulation [23]. Such models assume homogeneous kinetics and ignore complex network geometry. (2) Connectome- Based Network Diffission. By projecting tau as density on the structural connectome, Raj al. 35 used a linear graph-diffusion operator to simulate tau transport along white-matter tracts. Their network diffusion model (NMD) accurately predicted regional atrophy patterns across cohorts but treats the brain as passive conduit without explicit reaction kinetics. (3) Graph Reaction Citettin-Diiffsionn Extending network diffusion, Vogel et al. [39 introduced nonlinear reaction terms on graph Lapla- cians to jointly model diffusion and local tau-amyloic interactions on anatomical networks. While more expressive, these methods still rely on hand-tuned reaction laws and lack end-to-end learning of reaction kinetics. (4) Drive-riven Deep Learning. Recent work harnesses convolutional neural networks (CNNs) to learn tau progression directly from imaging data. Lee et al. [29] trained CNNs on positron emission tomography (PET) sequences to forecast future tau maps but found these black-box models often overfit and offer limited mechanistic insight. (5) Graph Neural Networks (GNN). Recent works 2 leveraged GNNs to capture both network topology and nonlinear interactions, showing improved regional predictions. Due to the 'black-bx' nature, however, it is challenging to generate interpretable governing laws through GNN only. From system-level perspective, current approaches simply assume the evolution of tau propagation following a pre-defined physics model, without actively identifying or optimizing the most suitable governing principle for the underlying dynamics.
Notably, nearly all existing tau-propagation models [40 21 16 10 operate at the level of coarse anatomical regions or volumetric parcels, effectively "down-sampling" the cortex into handful of nodes and treating each as spatially homogeneous. Although region-based graphs offer computational efficiency from a modeling perspective, such oversimplification of the cortical sheet's fine-grained geometry, such as folds, sulci, and gyri, limits their ability to accurately capture how misfolded proteins diffuse and interact at the voxel resolution. In contrast, surface- based PET studies 12 15
have shown the potential to address this limitation. For example, Xia et al. [42) projected [18F]-AV- 1451 uptake onto individual FreeSurfer surfaces and demonstrated that tau spreads in waves across temporal and parietal gyri, following the cortex's 's folds rather than simple volumetric adjacency. Cho et al. [9] extended this to two-year longitudinal analysis with (fottfiott, revealing concentrated tau accumulation in medial, basal, and lateral temporal regions and clear propagation trajectories along the surface. In light of this, our proposed model is built directly on the cortical surface mesh, with >100,000 vertices that faithfully trace the brain's 's highly convoluted topology.
Taken together, we propose physics-informee deep learning framework that unites biophysical modeling and data-driven discovery to reconstruct tau propagation dynamics from longitudinal tau- PET scans. First, we formulate tau spread as mean mean-field game (MFG), where each cortical location (agent) evolves under the influence of a local tau- amyloid interaction field and the collective behavior of neighboring regions. This variational formulation naturally induces an optimal transport process in Wasserstein space, capturing both reaction and diffusion within a theoretically grounded structure. Second, the within a min-max optimization framework [] As shown in Fig. [ we design a Wasserstein-1 Lagrangian generative adversarial network (GAN), where a generator learns subject-specific tau velocity fields and a Lipschitz critic estimates population-level transport costs. Third, we incorporate symbolic regression to learn an explicit, interpretable tau-A,3 reaction law directly from data. Unlike previous -ased or volumetric models, our model operates directly on the cortical surface mesh leveraging over 100k vertices to capture fine-grained geodesic flows along sulci and gyri. This
structure in MFG naturally leads to saddle point formulation

---

anatomical fidelity enables our model to uncover biologically plausible propagation pathways and mechanistic insight into tau-amyloid interplay. Our experiments demonstrate that our MFG-based deep model not only delivers precise and reliable predictions of future tau accumulation but also reveals interpretable dynamics aligned with neuropathological staging and recent imaging studies.
(a)
HJB
FP
(b)
Reaction-Diffusion
amylaid
Tau
Two-Player Game
Mean Field Game
Potential MFG
sup
inf
Φ
P
Agent
Population
Flow field V
Distribution p
Value function
Population density
sup
Flow maximizer
Density minimizer
inf
p
GAN
strategy +
strategy p (for Player 1)
"Critic"
"Generator"
(for Player 2)
sup
GAN
inf "Generator" G (population update)
sup
inf G
D "Discriminator"
(agent best response)
(flow optimization)
(density update)
Schematic sketch for the methodological connection between RDM. MFG, GAN in model. (a) potential mean field game is obtained by coupling the Hamilton- Bellman (HJB) and Fokker Planckk (FP) equations into a two-player saddle-point formulation: Player 1 (population density p) minimizes while Player 2 (value function Φ) maximizes via sup6- This variational game underpins zero-sum optimal transport dynamics. (b) By casting cortical tau spreading as reaction-diifusion process on the brain cortex, recovers equivalent deterministic MFG between the flow field and density distribution p. Using the Kantorovich- Rubinstein dual of Earth-Mover (Wasserstein-.. Wi) distance, we formulate Wagganganninn GAN: the Critic (Flow Maximizer, sup,) learns the transport cost, and the Generator (Density Minimizer, inf,) predicts the next-time tau distribution Pc+1. Alternating these updates unifies PDE-basee flow optimization and data-driven density forecasting.
Figure I:
inf
2 Methods
Data Description. We can organize brain cortex data as a graph = (X, D). where X {a;
1, ...... N) represents set of N predefined cortical locations (e.g., surface parcels mesh vertices), and D = [d4]] contains the Euclidean distances between all vertex pairs, with di- = ||:; x; At each cortical site Vi, we obtain two time-varying scalar standardized uptake value ratios (SUVR): u(f) = [u (t),... (t)+ utt [v (t). un (f)] and u(t+1) 1)= =
corresponding to longitudinal measurements of tau and amyloid concentrations, respectively.
2.1 How Brain Proteins Travel: Insights from Diffusion and Game Theory
Reaction Diffusion Model. Tau protein propagation in Alzheimer's 's disease often resembles the way drop of ink spreads in water, but constrained by the intricate folding of the cortex. To capture this, we postulate a reaction reaction-diffusion model (RDM) (partial differential equation, PDE) on our cortical graph:
Qu(t) Öt
(u(tt) + R(u(t) u(t),,
S(u) is the diffission operator, modeling pure tau spread along the network topology (brain cortex);
R(u, u) the reaction operator, capturing the interaction between tau and amyloid.
Physically, S(u) governs how tau "leaks" between neighboring regions (much like heat conduction),
while R(u, v) governs how amyloid burden might accelerate inhibit tau accumulation.
In typical machine-leaming implementation, the diffusion term S(u) can be instantiated by graph neural network (GNN) that learns to approximate the action of the Laplacian -V -Vu)). However,

---

using a standard multilayer perceptron (MLP) for R(u, v) often yields a black-box model with limited interpretability. To address this, we replace the MLP with a symbolic regression module [4] which discovers explicit algebraic formula for R. The outcome is hybrid framework that: (1) Leverages GNNs for accurate, geometry-aware diffusion S()); and (2) Uses symbolic regression to yield transparent, human-readable reaction law R(u,,), v).
From Reaction-Diftusion to Mean Field Games. Interestingly, the same reaction-diffusion PDE (Eq. T can be obtained as the optimality condition of a potential Mean Field Game (MFG) [. In that viewpoint, each infinitesimal "particle" of tau chooses a trajectory to minimize transport cost (diffusion) while experiencing local rewards or penalties from amyloid (reaction). Formally, this formulation boils down
problem:
pdx inf sup {1o [0,0 [0,6+ [(V)+ H(Vo) pdx dt ((:0))=(()) &
Se F(p(e, (), t)) dr dt
(2)
where we choose H(p) 1 ||pl? v) R(s, v) ds, so that o,F(p, v) -R(p,
The corresponding Fallan-e conditions are
(HJB): - - + H(V6) = = R)),
(FP): 0,p V- V, = 0.
Since HpH(p) = p, the uniform density p(x; t) = 1 and identify the value function & with the tau concentration Then
equation becomes 0pp- V (p Vo) = 0. Now assume
8 (Vu) R(u,e),
which exactly reproduces the couction-dffussee PDE (Eq. ① Thus, tau spreading the cortex be viewed both as a network-constrained reaction-diffusion process and as the Nash equilibrium of a deterministic potential MFG. Because such MFGs admit the Kantonvicc-RRbiintein dual (38], this duality naturally connects our reaction-diffusion formulation to the GAN-driven optimal transport framework.
2.2 GAN-Driven Flow Field Evolution Using Wasserstein-1 Metries and Lagrangian Principles
1=1 C Rdd denote cortical coordinate domain (e-g., cortical mesh). At each time f, we observe the tau concentration vector u(t), which defines an empirical distribution Pr E Prob(X). Our goal is to learn both: (1) a flow field v(x, t) on the mesh that drives tau transport, (2) and the predicted tau density ú(t + 1), so that ú(t + 1) ss Pr+1, the true next-step distribution. The classical optimal mass transport (OMT) formulation, under the squared-l2 cost (Wasserstein-2), is given by
Problem Formulation. Let {xx))
[CJx |1a(2,,)) s)12 p(x,s)
inf p(-0)=pt p(:1))p(113 p.12
dx ds, s.t. Oap + v=
where q(x, s) p(x, v(z s) is the flux field. This yields W2(Pe. Pt+1) but requires discretizing (Wasserstein-I) distance
the "pseudo-ting tondoottmmee E [0, 1]. By contrast, the
inf Wi p(+1) 7EIl(p.PP41))
B(aa)>> v/l
(6)
admits the Menihroovicc-Rubisser dual [38]:
Wi
sup [[C]
Eenp [C(x) [ B)>oe[[(()))
This dual form can (1) avoid discretizing an extra "time" variable, (2) provide Lipschitz-constrained critic that yields smoother, more stable gradients, (3) remain well-posed even when Pt and pe+1 have disjoint support.

---

-aasserrtiinn-agrannnnn GAN for Flow Evolution. Building on the dual formulation (Eq. 7). we cast tau spreading as a two-player adversarial game in which one network infers the flow field that transports the current tau distribution Pr. into the next distribution pt+11
Generator Go (Density Predictor). Imagine "pulses" of tau flowing across the brain's surface under the combined influence of diffusion and local biochemical reactions (as shown in Fig, 2 a). To do so, G9 is reaction-diffusion engine (see Sec. 2.3 for details), which, given the current tau/amyloid state (u(t), v(t),, computes the flow field v = Go (u(t,, e(t) and advances tau by one Lagrangian step of size At. The result ú(t + At) induces the "push-forward measure on the mesh
Pet1 = (I vv ā(t + Δt)) Pc.
Critic Co (Flow Maximizer). Let C. : X + IR be a neural network with parameters s, constrained so that its Lipschitz constant satisfies ||C,ple I 1. The critic's objective is to maximize the estimated Earth-Mover gap (xp Ee- (x))
Byr-p.r [c)] (c)))) By pushing Cpp to increase this difference under the I- I-Lipschitz constraint, the critic approximates the Wasserstein-1 distance Wi (pt, Pt+1) between the generated and true tau distributions.
The generator then minimizes this critic score on its own prediction: La( = E 11 [xpp)]
Together, they play the saddle-point game
inf sup 11
Eyyns 1 (C((w))3
(8)
GAN critic solves a deterministic MFG, we interpolate between Pt and pr+11 over "pseudo-time" s [o, 1]. Let p(z, s), E [0, 1], satisfy the continuity equation 0,p+ V- (pv)- O, p(-; 0) the velocity (flow) field. potential MFG formulation then the saddle-point problem
Connections to MFGs. To see why our Wasserstein
p(-, 1) Pi+1, where v(x, 8)
[ inf sup p(:s) &
[0.0 + HVV)) pdz ds ())(,
(9)
with terminal constraint enforced by T(p(-, 1)) so that p(:, PM+1. Here: o(x, s) plays the role
of the critic or value-function. The Hamiltonian is the indicator (p) 0, +o0, || > 1,
||l| 1,
which
corresponds to the Lagrangian A(v) = ||</|- The optimality (Fuler LagraneL) conditions are
(HJB): - + H(V)) = 0,
(10)
(FP): 0.p+ + ( = 0.
Since (p) = 0 whenever l|el| < 1, the HJB equation implies 0. = 0 and hence ő(x, s) = ((x).
Substituting back and using only the terminal constraint p(;, P++1 reduces the saddle point
sup S
Φ(x) Pc+ (z) dx
(11)
Ber (d(xr)]
Be [d(xr)
which exactly the Kantorovich- dual for (i Pt+1). our GAN, the critic Cp approximates this optimal and the generator Go seeks the flow field for which the forward Pit1 minimizes this same dual objective. Thus, the adversarial Wasserstein, -aagsteinnn-Lrraagiaann----- GAN training directly implements the equilibrium of potential MFG, with the critic maximizing the Natth-Mover& gap and the generator minimizing it. For clarity, Fig. Tillustrates how the RDM, its equivalent potential MFG formulation, and the resulting Wasserstein Vlssersteinn-Larraniiaa GAN are formally connected.
2.3 MFG4AD: A Physics-informed GAN for Modeling Tau Propagation AD
Building on the above link between reaction diffun-in, mean field games, and Waaserstein-LLgraagiian GANs. The framework integrates reaction-diffusion modeling, symbolic regression, and GAN into a unified architecture, coined MFG4AD. We now drill into the Genera- for network architecture that powers MFG4AD. At the heart of MFG4AD, our generator Go is reaction-difuusin engine tailored to the cortical mesh G (X, D):
Network Architecture.
5

---

MF64AD: Governing Law of Pathology Propagation Mechanism Through A Mean-Field Game
diffusion
Cortical coordinate domain
(a)
amyloid
di
flow
reaction
tau
Pt+1
vertex
S 4
Network topology
PetAe
W1
Ut+1
MFG4AD
flow v Ho(:)
diffusion
G
u
Rg Oc)
reaction
U
Generator
Critic Cpp
4in
og
Density map
(b)
V
G8
(U+1
V
Figure 2: MFG4AD: A physics-informed deep learning framework for modeling tau propagation and amyloid- tau interactions in Alzheimer's disease. (a) We conceptualize the cortical surface graph, where vertices represent cortical locations and edges encode distances (d4,). The concentration defines the initial and terminal density distributions, Pr and P14-1, respectively, while amyloid acts as an external modulator influencing the evolution of tau through a reaction term. (b) We propose a MFG4AD, consisting of a generator Go that predicts the next-time density (ūc+1), and a critic C.P that evaluates the Wasserstein- I distance between the predicted (Pt+1) 1) and truc distributions (pt+1). The optimal transport flow field guides the evolution of tau density, (c) The generator Go combines GNN model network-constrriied diffusion (tau spreading along cortical pathways) a symbolic regression module for explicit, interpretable tau-amyloid reaction dynamics.
(1) Graph- diffusion. We first leverage a graph neural network S. (Fig. 2 (c), top) to approximate the Laplacian operator - V. (Vu). Concretely, each vertex æ; pools tau values utt) from its neighbors weighted by the geometry-D and computes a discrete diffusion flux P. (u((),, which captures how tau "leaks" along cortical folds.
(2) Symbolic reaction. Next, we account for tau-an auu-amyloid crosstalk, which captures how amyloid cat- alyzes or inhibits tau. A symbolic regression module (Rş) (Fig. (c). bottom) ingests the tau aan-adyloid pair (t,, (f)) at each vertex and outputs an explicit reaction rate Q: = R& (u, (t), (t)).
(3) Infer the flow field. We bundle each vertex's current tau level and its two physics-drive driven quantities into a state descriptor F; = (t). P,, Qi]. A lightweight neural subnetwork He then turns these descriptors into a movement vector E IR: 14: [Ho(F)], Finally, we let each bit of tau ride this flow field via one Lagrangian step of size At:
(t + At) (t) + Ar VIV (u(t) + p2 P. + its Q.).
(12)
where Mi, f12 ni are the learnable scalars that let the model automatically tune the relative strengths of diffusion, reaction, and source contributions to best match longitudinal tau data. By doing so, Go is not black box but reaction-diffasion engine that explicitly computes diffusion, reaction, and advection to predict the next tau map ū(t + At). Then a ppush-forward" derives empirical measure Ω:ạ1 on the mesh which the Critic evaluates against the true distribution P++1 under the
Wasserstein-I metric. Training alternates between optimizing these two networks (Fig. 2 the generator to learn biologically meaningful, accurate predictions of tau propagation, while the critic ensures stable convergence by evaluating the quality of the generated densities.
b), allowing
Training Phase. We summarize our training procedure in Algorithm I (as shown in Appendix). Each generator update is preceded by nc = 5 critic updates, with learning rates set to 10 for the generator. To enforce the 1-Lipschitz constraint the critic C.p.
x 10-5 for
the critic and πạ we apply spectral normalization to every layer (32]. The generator's loss combines the adversarial term with an l [ + 1) ut + 1)1, reconstruction penalty weighted by 10, ensuring both realistic and accurate predictions.
3 Experiments
Data Preprocessing. Tau/AB SUVR Generation. We process each subject's T1-weighted (TIW) MRI with FreeSurfer to reconstruct the cortical surfaces (white, pial, and mid-thickness) and to define

---

cerebellar gray matter as our reference region. Next, we rigidly register the motion-correctio tau-PET and amyloid-PET volumes to the TIW image, resample them into MRI space, and compute voxel-wiss SUVR by dividing each voxel's uptake by the mean signal in the cerebellar reference. These SUVR volumes are then projected onto the subject's 's pial surface (~~100k vertices) via trilinear interpolation and lightly smoothed along the mesh. To facilitate group analysis, each surface SUVR map is warped into the MNI template (fsaverage in MNI space), resampled onto the same 223.842-vrrex mesh and z-score normalized across cortical vertices. The resulting high-resolution, surface-based SUVR profiles at times t and t++ 1 for tau, and at time for amyloid, constitute the inputs u(t), v(t), u(t + 1) for our MFG4AD. Network Topology Construction. To capture the anatomically faithful geometric relationships among cortical vertices, we construct a sparse undirected graph directly from the native pial-surface mesh generated by FreeSurfer [33 Each vertex is treated graph node, and edges are defined according to the triangular tessellation of the cortical surface: two nodes are connected if they share an edge in the mesh. This results in biologically grounded graph structure with an average node degree of approximately 6, preserving submilimeter-scall geometry while adhering to the true cortical topology. Rather than computing full pairwise distance matrix, we leverage the mesh's 's intrinsic sparsity, storing only the anatomical edges defined by surface adjacency. This allows diffusion operations scale (kNV time, where k is the average vertex degree. To simulate tau propagation, we implement the reaction-diffusion step via vectorized sparse-tensor operations over this mesh-defined graph, enabling full forward Euler step across >100k cortical vertices in milliseconds. The full mresh-based construction process is described in Appendix A.
Experimental Setup. We evaluate the performance of MFG4AD using two longitudinal tau PET datasets: the Alzheimer's 's Disease Neuroimaging Initiative (ADNI) (25) and the Open Access Series of Imaging Studies (OASIS) (8). The ADNI dataset includes 134 participants with both tau and amyloid PET scans, each with 2-6 longitudinal visits, resulting in a total of 631 scan pairs. Subjects in ADNI are categorized into five diagnostic groups: cognitively normal (CN), subjective memory complaint (SMC), early mild cognitive impairment (EMCI), late mild cognitive impairment (LMCI), and AD. The OASIS dataset comprises 77 participants, each with two longitudinal PET scans, and includes two diagnostic groups: CN and AD. Together, these two datasets provide diverse and representative sample across the Alzheimer's disease spectrum, enabling comprehensive evaluation of our predictive framework across multiple disease stages. Comparative methods span five categories: (I) Connectome- Based Network Diffusion Model, NDM [35]. (2) Graph- Based Methods: vanilla GCN 41 and the advanced GCNII 77. (3) Deep Learning Models: deep neural networks (DNN) composed of MLPs, deep symbolic model (DSM) (26 and vanilla GAN [19. (4) PDE-Based
Methods: graph neural diffusion (GRAND) (,, Neuro-ODE S and graph neural reaction-diffii diffusion
networks (GREAD) [T]. (5) Traditional Regression Model: Ridge regression (a regularized linear regression model). For all experiments, we conduct 5-fold cross-validation. The evaluation metrics for testing results include: mean absolute error (MAE) and root mean squared error (RMSE), between the predicted tau burden and the observed tau SUVR from follow-up PET scans. All models are trained for 1,000 epochs with Adam (27 optimizer.
ADNI
2
3.1 Model Behavior and Ablation Study
Prediction Performance of Future Tau Accu- mulation. To evaluate the predictive performance of different computational approaches, we used
11)+1)
e1
erro
D,1
dendity
baseline tau concentration and combined tau + amyloid as two types of input, with follow-up tau
SUVR measurements serving as the ground truth. Prediction errors for each method are summarized in Table D with results from ADNI and OASIS shown on the left and right, respectively. Exper- imental findings demonstrate that our proposed method consistently outperforms all competing approaches. This superior performance stems from the integration of a Wasserstein-1 1 Lagrangian GAN, which improves the fidelity of synthesized tau patterns through distributional alignment in the prediction space, and reaction-diffusioon framework that explicitly captures the biophysical dynamics of tau propagation modulated by amyloid interaction. Representative predictions generated by MFG4AD are visualized in Fig. 3 where the absolute difference between observed and predicted
AD Figure 3: The representative examples (reconstruc- tion error e < 0.01) between the observed u(t + 1) and predicted ū(l + 1) tau SUVRs generated by MFG4AD (left: CN. right: AD). Cognitively normal (CN), Alzheimer's 's discase (AD).
n (t + 1) CN
7

---

tau SUVR remains below 0,01. i.e., L = 11 - < 0.01. Additional visualizations are provided in Appendix A.. We further compared prediction errors (MAE and RMSE) across diagnos- tic groups in the ADNI and OASIS cohorts (Fig. 4p.. In ADNI, both metrics are low, particularly in the EMCI and LMCI groups, with the lowest MAE observed in EMCI (0.0600 = 0.0253) and the lowest RMSE in LMCI (0.0787 0.0072). In contrast, OASIS exhibits higher errors in both CN and AD groups, especially in AD (RMSE = 0.7040 = 0. 0.130), likely reflecting greater cohort heterogeneity. Nevertheless, our method maintains the best overall predictive accuracy across all settings. To further assess spatial modeling fidelity, we visualized cortical maps vertex-wise absolute prediction errors on ADNI and OASIS datasets (Fig. 4b). Both datasets exhibit localized discrepancies, with larger deviations concentrated in temporal and medial regions known for high tau variability. Overall, ADNI shows lower error magnitudes compared to OASIS, supporting the robustness of our model across heterogeneous populations. Finally, we evaluated model performance by comparing predicted and ground-truth mean tau SUVR values for each diagnostic subgroup (Fig. (p). A simple linear regression was performed to quantify the agreement between predicted and observed values across subjects. In ADNI (left), strong linear relationships were observed across all disease stages, with a mean slope of 0.98 = 0.06 and 1t2 of 0.91 + 0.07 (CN/SMC: 1.052 0.822; EMCI: 0.975 0..53;; LMCI: 1.001 0.895; AD: 1.125 0.982). In OASIS (right), the mean slope was 0.90 + 0.16 with R2 of 0.660 + 0.17 (CN: 1.007 0.785; AD; 0.784 0.535). These results indicate strong agreement between predicted and observed tau SUVR values, particularly in earlier disease stages and in the ADNI cohort, highlighting the stability and generalizability of our predictive framework.
Table 1: Prediction performance (MAF/RMSE) on ADNI and OASIS. ** denotes the significant improvement (p-value <0.01, paried t-test.)
Model
OASIS
ADNI
OASIS
MAE
MAE
MAE
RMSE
MAE
DNN
0.1507.o002 0.4767.0.002 0.2057
0.1487 0.6381..111
0.2377-0.063
0.49770..02 $0.029
0.668
GCN
0.15810.017 0.158 0.000
0.637 200587
200064
0.157 0.15720.015
0.219 020
0.159 20.058
0.6-11 066
GCNII
0.17840.092 0.200
0.4884..055
0.608240.052
0.127
0.1892...016
0.4694
0.6492-0.08s
NDM
0.1081o.02 0.13510.o29
0.181go.212 212
0.101 0.613
0.1321
0.159
0.159 2o.394
0.589 0.589".u.26
Neuro-DEE
0.12710.014 0.19011.020
0.486_o-os
0.6562.0.009
0.127
0.189[2o.016
0.487
#0.054
0.664"
GRAND
0.181fa.001 0.3055.o.oos
0..055...057
0,667
40.079
0.2147.0.02s
0.3400
0.169 0.1694...om 40.000
0.6419
20.087
GREAD
0. 0.163 0.269" 20.051
0. 48.54000044
0.685
0.685$0.098
#0.098
0,195
20.0520.024
0.344
0.3442-0.086
0.163
200000
0.653 0.02
Ridge
0. 0.090
0 0.155500053 0.1024
0.620+0.082
0.088
0 0.1327-..027
0.451 0.4514+..05 +0.051
0 0.6233=-.oo
DSM
0.087 10.010
0.163320.o2
can
0.168 20.00
0.727 0.7073
0.083 10.0000000
10.99
10.037
0.702 o6s
GAN
0. 247 40.010
0.3433...00
0.490 00.051
0.6633...oom
0.36154o.oo.
0.51 7 +0.071
0.694
MFG4AD
0. 0.0644 004
0.093
0.435 10 000
0.61940.08)
Input
Tau
Tau-amylold
0.16
ADNI
RMSE
0.8
OASIS
b
ADNI
OASIS
0.14
0.1282
MAE
0.7
0.6100
0.7040
RMSEEMME
0.12
0.0727
0.0790
919
0.0757
T0.05000.0787
0.0617
0.5
4289
0.08
0.0628
0.4
0.06
0.3
0.04
0.02
0.1
0
0.00
0.0
CN/SMC
EMCI
LMCI
AD
CN
AD
Ground truth mean
Kbye 83
Stapa 1,07
p' 20022
n' cass
k2
2.
CN/SMC
LMCI
15
Predicted mean
Predicted mean
Figure 4: (a) Comparison of prediction errors across diagnostic subgroups in the ADNI (left) and OASIS (right)
cohorts. (b) Vertex-wise absolute error maps show spatial patterns of prediction errors, with larger deviations
temporal and medial regions. (c) Predicted vs. observed mean tau SUVR values for each subgroup.
8

---

Ablation Study. To tease apart the roles of the two mechanistic terms in our model, we ablate the diffusion component (S) and reaction term (R), the results are summarized in Fig [ Removing either component results in a noticeable degradation in performance, confirming that both processes are necessary for precise tau proppagattiin modelling. Eliminating the reaction term produces the larger error increase, underscoring amyloid burden as a key modulator of tau dynamics, whereas suppressing diffusion breaks the spatial continuity required to capture network- -aasor--based spread. The learned scaling factors {ur 0.94, μ22 0.97, #3 1.33) make this balance explicit: advection (11) and diffusion (122) contribute almost equally, while the amyloid-tau interaction term μ4 arries the greatest weight. This slightly elevated interaction coefficient, paired with near-unity transport coefficients, yields a physiologically plausible picture in which A "hot-spots" seed local tau build- up. then the advection diffus-oo machinery conveys pathology throughout the connectome. Such dynamics mirror Braak staging (3] patterns and recent multimodal PET observations, and support the biological view that AB deposition primes regions for accelerated tau spread once both pathologies
co-localise, an effect repeatedly reported in experimental studies (20).
0.08
S
R R+S
0.07
I
MAE
3.2 Iollogically-Infommes Interpretation of Tau Propagation
0.06
Tau Propagation Pathways on the Cortical Surface. Fig. illus- trates the modeled evolution of tau pathology across different stages of AD in both the ADNI (top) and OASIS (bottom) datasets. For each group (CN, EMCI, LMCI, AD), we present the initial tau distribution Po. the follow-up accumulation Pt. and the estimated propagation field v. The flow fields (last column of each group) reveal the direction and magnitude of tau propagation, where red colors indicate stronger spread flux. The color intensity in p maps indicates the vopulation-averagel tau accumulation, while the flow field v captures the dominant direction and strength of tau transport across the cortical surface. A clear progression in tau flow strength is observed across disease stages: CN individuals exhibit minimal propagation, while tau flow becomes progressively more prominent in EMCI, LMCI, and reaches its peak in AD. This trend is consistent in both datasets and reflects the escalating spatial spread of tau pathology as the disease advances. Notably, the temporal lobe (indicated in red regions) shows strong and persistent involvement, serving a key hub for tau diffusion in later stages. Importantly, these mod- eled propagation patterns align with established neuropathological findings, where tau pathology is known to originate in the transentorhinal and entorhinal cortex, before spreading to the hippocampus and neocortex in a stereotyped fashion [3) This consistency with Braak [3] staging reinforces the biological plausibility of our model in capturing disease-relevant tau dynamics.
0.05
0.04
Figure 5: Ablation of diffu- sion (S) and reaction (Ii) com- ponents on performance.
New Insights into A Interactions in AD. To probe the mechanistic basis of A,9-tau interac- tions, we analyzed the symbolic reaction functions (u, v) learned by our model, where each term encodes how amyloid burden Uy at region j contributes to tau accumulation at region By systemati- cally scanning across all cortical vertices, we generated spatial map (Fig. 7 in which darker shading highlights regions whose amyloid load most strongly drives downstream tau propagation. Two princi- pal epicenters emerge: (1) Medial Temporal Lobe (pink dashed region) encompassing the entorhinal cortex and parahippocampal gyrus, the canonical nidus of early tauopathy (Braak I-II) [3. Elevated amyloid levels in this region precipitate cascade of tau spreading into neighboring isocortical terri- tories. (2) Medial Prefrontal Cortex (blue dashed region) consistent with Thal A,8 phases 1-2 (77, where surpassing an amyloid threshold triggers accelerated propagation from the medial temporal hub into anterior cortical areas, reinforcing the pathological cascade. To illustrate a representative sym- bolic reaction law, we selected vertex in the oc-temp Satttmppeed-aaaaapppppa region. Its update is given by u((+1)= = -0 -0009 + 0.02#0g1 - 0.2 sin (2.2 +1.9 (e.pp3.xxxxpppppeeee (3.5+x+)+++)) where 141 cor- responds to the s_ carinn region, U8i to S_ cingul-Mid Ant, Ug to G_ _ccnnnnnnllPp Sosulll-Pstddorsssa and U16 to G_front_ patptttssuupppaaappppaaaaa
These interconnected nodes form the entorhi- nal-binpocamppll-cnnuuuku loop-a circuit widely recognized as the earliest site of tau accumulation in AD (Braak [-III). As downstream hub of this loop, cempemp demp__eedPeeeeeeeeeeeeeeeee med-Parahip exhibits tau increase at tit+ ++1 that is jointly driven by local A tau interactions and diffusion flux from upstream medial nodes. In contrast, remote regions such as G_orbital (orbital frontal cortex) typically become involved only later Braak stages and do not participate in this early propagation network. For exam- ple, -0.09m44: The negative coefficient from the calcarine cortex (a primary visual region affected in late Braak stages V-VT) may reflect a dampening influence on tau accumulation in early-stage cigions-poeentiallyy capturing long-range regulatory effeects in network dynamics. Constant term

---

ADNI
CN
LMCI
high
density
flow
EMCI
AD
ow
OASIS
CN
AD
Po
P1
P1
Figure 6: Visualization of tau propagation across stages of AD progression ADNI (top) OASIS (bottom).
(+1.9): Captures the intrinsic baseline accumulation of in the medial temporal lobe, consistent with spontancous age-related tauopathy in entorhinal and parahippocampal areas [4). Last term (ex((x3 v16) 1): A sigmoid-shaped A3 term, indicative of saturating dose-response, reflect known dynamics where tau is more sensitive to A, at subthreshold levels but less responsive once amyloid burden becomes extensive [24]. Taking together, these results show that MFG4AD learns biologically meaningful reaction laws, linking local A3 burden and network connectivity to tau spread [33]. The derived symbolic equations reveal early epicenters and propagation trajectories consistent with Braak I-Ill staging, offering interpretable and predictive insights into AD progression.
Frontal
ADNI
OASIS
Temporal
Enterdind
Entorhinal
Entorhinal
Enterhinal
Parahiypscampal
Parahignncmppal
low
high
Figure 7: The brain surface mapping of A,f-tau interaction. Dark color indicates active involvement of amyloid cascade in the tau propagation.
4 Conclusion
In this work, we introduced MFG4AD, a unified, physics-informed deep learning framework that: (1) models tau spread as law for tau-am aau-amyloid crosstalk; (2) casts this system as an equivalent potential mean field game, linking classical PDE theory to tau propagation; and (3) employs Wasserstein-1 Lagrangian GAN learn optimal transport flows accurate tau forecasting. delivers bate-of-the-ar predictions for unseen subjects and resolves tau-flow directions, pinpointing peak-flux flux hotspots, while also uncovering an explicit, interpretable reaction law, offering powerful combination of predictive performance and mechanistic insight into Alzheimer's pathology.
reaction diffusion process with data-driven symbolic
ADNI and OASIS cohorts. MFG4AD
10

---

Acknowledgement
This work was supported by the National Institutes of Health (AG091653. AG068399, AG084375) and the Foundation of Hope. The views and conclusions contained in this document those of the authors and should not interpreted as representing the official policies, either expressed or implied, of the NIH.
References
[1] Achdou, Y., Camilli, F., Capuzzo-Dolcettt, Dolcetta, I.: Mean field games: convergence of a finite difference method. SIAM Journal on Numerical Analysis 51))), 2585-2612 (2013)
[2] Balaji, V., Song, T.A., Yang. F., Jacobs, Johnson, K., Dutta, J.: A graph neural network model for the prediction of longitudinal tau aggregation. Journal of Nuclear Medicine 63sppplemen
2), -233-2233 (2022)
[3] Braak, H., Braak, E.: Neuropathological stageing of alzheimer-related changes. Acta Neu- ropathologica 82(4), 239-259 (1991)
[4] Braak, H., Del Tredici, K.: The preclinical phase of the pathological process underlying sporadic Alzheimer's 's disease. Brain 138(10), 2814-2833 (2015)
[5] Busche, M.A., Hyman, B.T.: Synergy between amyloid-3 and tau in Alzheimer's disease. Nature Neuroscience 23(10), 1183-1193 (2020)
[6] Chamberlain, B., Rowbottom, J., Gorinova, M.I., Bronstein, M., Webb, S., Rossi, E.: GRAND: Graph neural diffusion. In: International Conference Machine Learning. pp. 1407-1418. PMLR (2021)
[7] Chen, M., Wei, Z., Huang, Z., Ding, B., Li, Y:: Simple and deep graph convolutional networks. In: ICML. Proceedings of Machine Learning Research, vol. 119, pp. 1725-1735. PMLR (2020)
[8] Chen, R.T., Rubanova, Y., Bettencourt, J., Duvenaud, D.K.: Neural ordinary differential equations. Advances in Neural Information Processing Systems 31 (2018)
[9] Cho, H., Choi, J.Y. Lee, H.S., Lee, J.H., Ryu, Y.H., Lee, M.S., Jack, C.R., Lyoo, C.H.: Progressive tau accumulation in alzheimer disease: 2-year follow-up study. Journal of Nuclear Medicine 60(11), 161 1-11-1621 (2019)
[10] Cho, H., Wei, Z., Lee. S., Dan, T., Wu, G., Kim, W.H.: Conditional diffusion with ordinal re- gression: Longitudinal data generation for neurodegenerative disease studies. In: The Thirteenth International Conference on Learning Representations (2025)
[11] Choi, J., Hong, S., Park, N., Cho, S.B.: Gread: Graph neural reaction-diffusia networks. In: International Conference Machine Learning. pp. 5722-5777. PMLR (2023)
[12] Clavaguera, F., Bolmont, T., Crowther, R.A., Abramowski, D.,, Frank, S., Probst, A., Fraser, G., Stalder, A.K., Beibel, M., Staufenbiel, M., et al.: Transmission and spreading of tauopathy in transgenic mouse brain. Nature Cell Biology 11(7), 909-913 (2009)
[13] Dale, A.M., Fischl, B., Sereno, M.I.: Cortical surface-based analysis: I. segmentation and surface reconstruction. Neuroimage 9(2), 179 194 (1999)
[14] Dan, T., Dere, M., Kim, W.H., Kim, M., Wu, G.: Tauflownet: Revealing latent propagation mechanism tau aggregates using deep neural transport equations. Medical Image Analysis 95, 103210 (2024)
[15] Dan, T., Huang, Y., Yang, Y., Wu, G.: Revealing cortical spreading pathway of neuropathological events by neural optimal mass transport. IEEE Transactions on Medical Imaging 44(7), 3100- 3109 (2025)

---

[16] Dan, T., Kim, M., Kim, W.H., Wu, G.: Tauflownet: Uncovering propagation mechanism of tau aggregates by neural transport equation. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp- 77-86. Springer Nature Switzerland, Cham (2023)
[17] Dan, T., Kim, M., Kim, W.H., Wu, G.: Enhance early diagnosis accuracy of Alzheimer's s disease by elucidating interactions between amyloid cascade and tau propagation. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 66-76. Springer Nature Switzerland, Cham (2023)
[18] Dan, T., Wu, G.: Explainable deep model for understanding neuropathological events through neural symbolic regression. In: International Conference on Information Processing in Medical Imaging. pp. 37-50. Springer Nature Switzerland, Cham (2025)
[19] Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial networks. Communications of the ACM 63(11), 139-144 (2020)
[20] He, Z., Guo, J.L., McBride, J.D., Narasimhan, S., Kim, H., Changolkar, L... Zhang. B.,, Gathagan, R.J., Yue, C., Dengler, C., al.: Amyloid-B plaques enhance alzheimer's 's brain tau-seeded pathologies by facilitating neuritic plaque tau aggregation. Nature Medicine 24(1). 29-38 (2018)
[21] Huang, H., Wang, Y., Dan, T., Yang, Y., Wu, G.: A multi-layer neural transport model for char- acterizing pathology propagation neurodegenerative diseases. In: International Conference on Information Processing in Medical Imaging. pp. 51-64. Springer Nature Switzerland, Cham (2025)
[22] Huang, Y., Dan, T., Kim, W.H., Wu, G.: Uncovering cortical pathways of prion-like pathology spreading in Alzheimer's disease by neural optimal mass transport. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 498-508. Springer Nature Switzerland, Cham (2024)
[23] Iturria-Medina, Y., Sotero, R.C., Toussaint, P..., Evans, A.C., Initiative, A.D.N.: Epidemic spreading model characterize misfolded proteins propagation in aging and associated neu- rodegenerative disorders. PLoS Computational Biology 10(11), e1003956 (2014)
[24] Jack, C.R., Knopman, D.S., Jagust, W.J., Petersen, R.C., Weiner, M.W., Aisen, P.S., Shaw, L.M., Vemuri, P., Wiste, H.J., Weigand, S.D., et al.: Tracking pathophysiological processes in alzheimer's disease: an updated hypothetical model of dynamic biomarkers. The Lancet Neurology 12(2), 207-216 (2013)
[25] Jack Jr, C.R., Bennett, D.A., Blennow, K., Carrillo, M.C., Dunn, B., Haeberlein, S.B., Holtzman, J.M.Js W., Jessen, F., Karlawish, J., et al.: Nia-aa research framework: toward biological definition of alzheimer's disease. Alzheimer's & dementia 14(4), 535-562 (20188))
[26] Kim, S., Lu, P.Y., Mukherjee, S., Gilbert, M., Jing. L., Čeperié, V., Soljačić, M.: Integration of neural network-basee symbolic regression in deep learning for scientific discovery. IEEE Transactions on Neural Networks and Learning Systems 32(9), 4166-4177 (2020)
[27] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization (2017), https://arxx agg/abs/1412.c 6980
[28] LaMontagne, P.J., Benzinger, T.L., Morris, J.C., Keefe, S., Hornbeck, R., Xiong, C., Grant, E., Hassenstab, J., Moulder, K., Vlassenko, A.G., et al.: Oasis-3; longitudinal neuroimaging, clinical, and cognitive dataset for normal aging and alzheimer disease. MedRxiv pp. 2019-12 (2019)
[29] Lee, J., Burkett, B.J., Min, H.K., Senjem, M.I.,, Dicks, E.,, Corriveau-Lecavallee N., Mester, C.T., Wiste, H.J., Lundt, E.S., Murray, M.E., et al.: Synthesizing images of tau pathology from cross-modal neuroimaging using deep learning. Brain 147(3), 980-995 (2024)
[30] Lee, W.J., Brown, J.A., Kim, H.R., La Joie, R., Cho, H.J., Lyoo, C.H., Seeley, W.W., et al.: Re- gional A,3-tau interactions promote onset and acceleration of Alzheimer's disease tau spreading. Neuron 110(12), 1932-1943 (2022)
12

---

[31] Li, W., Lee. W., Osher, S.: Computational mean mean-field information dynamics associated with draction-diffusse equations. Journal of Computational Physics 466, 111409 (2022)
[32] Miyato, T., Kataoka, T., Koyama, M., Yoshida, Y:: Spectral normalization for generative
adversarial networks. In: International Conference on Learning Representations (2018)
[33] Palmqvist, S., Schöll, M., Strandberg, O., Mattsson, N., Stomrud, E.,, Zetterberg, H., Blennow, K., Landau, S., Jagust, W., Hansson, O.: Earliest accumulation of 3-amyloid occurs within the default-mode network and concurrentyy affects brain connectivity. Nature Communications 8(1, 1214 (2017)
[34] Petersen, B.K., Larma, M.L., Mundhenk, T.N., Santiago, C.P., Kim, S.K., Kim, J.T.: Deep symbolic regression: Recovering mathemaatica expressions from data via risk-secking policy gradients. In: International Conference on Learning Representations (2021)
[35] Raj. A., Kuceyeski, A., Weiner, M.: A network diffusion model of disease progression in dementia. Neuron 73(6), 1204-121: (2012)
[36] Roemer-Cassiano, S.N., Wagner, F., Evangelista, L., Rauchmann, B.S., Dehsarvi, A.. Steward, A., Dewenter, A., et al.: Amyloid-associatee hyperconnectivvty drives tau spread across con- nected brain regions in Alzheimer's disease. Science Translational Medicine 17(782), cadp2564 (2025)
[37] Thal, D.R., Rub, U., Orantes, M., Braak, H.: Phases of A,8-depoiiionn in the human brain and its relevance for the development of ad. Neurology 58(12) (2), 1791-1800 (2002)
[38] Villani, C.: Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (Jun 2008)
[39] Vogel, J.W., Iturria-Medina, Y., Strandberg, O.T., Smith, R., Levitis, E., Evans, A.C., Hansson, O.: Spread of pathological tau proteins through communicating neurons in human Alzheimer's disease, Nature Communications 11(1), 2612 (2020)
[40] Wang. Y., Huang, H., Dan, T., Yang, Y., Wu, G.: Understanding the spreading mechanism of Tau propagation in alzheimers disease through multi-layer transport. In: 2025 IEEE 22nd International Symposium on Biomedical Imaging (ISBI). pp. 1-5 (2025)
[41] Wu, F., Jing. X., Wei, P., Lan, C., Ji, Y., Jiang, G., Huang, Q.: Semi-supervised multi-view graph convolutional networks with application to webpage classification. Information Sciences 591, 142-154 (2022)
[42] Xia, C., Makaretz, S.J., Caso, C., McGinnis, S., Gomperts, S.N., Sepulcre, J., Gomez-Isla, T., Hyman, B.T., Schultz, A., Vasdev, N., et al.: Association of vivo [18f] AV-1451 tau PET imaging results with cortical atrophy and symptoms in typical and atypical alzheimer disease. JAMA Neurology 74(4), 427-436 (2017)
13

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Please refer to the abstract and introduction parts.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results
be expected to generalize other settings.
is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Please refer to Appendix A.3
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low images are taken in low lighting. Or speech-to-text system might be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question:
For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
14

---

Justification:
Please refer to Sec.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification: Please refer Appendix A.2
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm.
(b) the contribution is primarily a new model architecture, the paper should describe the architccture clearly and fully.
(c) If the contribution is new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reprodueibility. the case of closed-source models, may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
15

---

Answer: [Yes]
Justification:
Please refer to
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.c
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
/ /pipps.cc/pullc//uuui wwwpc//ggue////eeeeeoooc CoodeSubeiiie ssi ssionnolli onnolicy)))) for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Please refer to Appendix A.2 and Sec.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: Please refer to Table 1 and Fig.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, at least for the experiments that support main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
should be clear whether the error bar is the standard deviation the standard error of the mean.
16

---

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of enrors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: Please refer Appendix A.2
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g.. preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: http:/::////ups/eeeeeeeee http:/ssscc...cccccccccccceeeeea
Answer: [Yes]
Justification: Please refer the whole manuscript.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is a special consid- eration due to laws or regulations their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: Please refer Appendix A.4
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to
17

---

generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative socictal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: The paper poses such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: Please refer to Appendix A.2
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, wwppppswpptseeee whworee.wthhhe....c. com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
this information is not available online, the authors are encouraged to reach out to the asset's is creators.
13. New assets
Question: Are new assets introduced in the paper well documented and the documentation provided alongside the assets?
18

---

Answer: [Yes]
Justification:
We uploaded the code of our model into GitHub.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, other labor should be paid at least the minimum wage in the country of the data collector.
Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Answer: [NA]
19

---

The core method development in this research does not involve LLMs as any important, original, or non-standard components.
Justification:
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non- standad components.
Please refer to our LLM policy hattps://neurips... 200ps//ee///0ooeeeeeee////////// for what should or should not be described.
20

---

A Technical Appendices and Supplementary Material
A.1 Brain Network Construction from Cortical Surface Mesh
To capture anatomically faithful pathways for tau propagation along the cortical mantle, we con- structed sparse, geometry-awar graph based on the native cortical surface topology. Specifically, we utilized the 1h/rh. pial surface mesh of the left/right hemisphere generated by FreeSurfer [13]. which represents the cortical sheet triangular mesh composed of N 163,842 vertices and approximately 327, ,680 faces.
Each triangular face defines three local connections between mesh vertices. We constructed graph G = (x, &, D) by treating each vertex as a node E X, and adding an undirected edge (x,, xy & the vertices 3i and j are connected by least triangle. This results in a
adjacency matrix D E RNxx encoding binary connectivity that reflects local anatomical continuity.
To integrate geometric information relevant spatial diffusion, we assigned edge weights based on the Euclidean distance between connected vertices. For each edge (2i, Iz)) the weight was defined as:
Dij 1|= xzl12 = V(a: a,) (b, b,)² + (a: c,)
(13)
where 2; = (a,, by, dj = (aj, bj. 5)) IR denote the 3D coordinates of vertices 2; and zy (Note, æı represent the index and coordinates of the node uniformly for simplicity). The resulting weighted adjacency matrix provides a biologically plausible scaffold for modeling local propagation dynamies constrained the cortical surface, the illustration is shown in Fig.
face
vertex
0
d
Brain surface
Figure 8: An illustration of constructing brain network topology.
The constructed graph exhibits an average node degree of approximately 6, consistent with the local connectivity induced by the triangular tessellation of the cortical sheet. In contrast to conventional eneearest-neighhee (kNN) graphs that are built solely based on Euclidean proximity, this approach ensures geometric and topological consistency by avoiding spurious long-range connections that may cross sulcal boundaries or violate the anatomical folding patterns.
To support downstream spectral and learning based analyses, we further derived the symmetrically normalized graph Laplacian: Lyym = I - A 1/2 1/2 DA 1/2 where A is the diagonal degree matrix with A, Σj Dij.
Finally, for compatibility with graph learning frameworks such as PyTorch Geometric, exported the graph an edge list edge index E 22x and corresponding edge weight vector edge_wer weight e
Our approach leverages the native triangular mesh of the cortical surface, where each node is con- nected to its immediate neighbors based on surface topology. This results a fixed and biologically grounded neighborhood structure, typically with average node degree of 6. The mesh graph strictly adheres to the geometry the cortical sheet, preserving anatomical continuity and avoiding noc-l shortcuts.
This anatomically faithful structure is especially important for modeling prion-like tau propagation, which is believed to follow trans-neurondl transmission along physically connected pathways. Clava-
21

---

guera et al. [12], for example, demonstrated that tau pathology spreads from the injection site to anatomically connected regions, supporting the need for realistic graph representations that reflect underlying biological constraints.
A.2 Implementation Details and Experimental Results
The pseudocode for our method is presented in Algorithm | The full implemennlaioo--nnlldunn all hyperparameter settings -is available from our anonymous GitHub repository: https://gittm:
col//Daady572//MF4AD20000
Algorithm Training MFG4AD for Tau Dynamics
Require:
For each subject,
u(t), u(t 1) E R (tau at times t and t + 1) v(t) IR (amyloid
at time t), the coordinates of each vertex 2; C X, learning rates r1c, na 0, clip threshold
> O, critic steps nc. the weight of reconstruction À
1: while not converged do
2: Build graph 9 = (X. with distances dis = || - 2zy1-
3:
for k- 1, ................................ nC do
4:
- CRITIC UPDATE
5:
Obtain flow field E NRd by Go
6:
Predict (t + 1) via Eq. (12)
Define the loss of Critic: Lo + 1 E Op(ūx((+1)) - x/ En Cp + 1))
7:
N
8:
Gradient-ascent step yp v + QJG VoLc
Spectral Norm enforces I-Lipschitz
9:
end for
10:
GENERATOR UPDATE
11:
+ (u(u(t), v(t)
Gg is composed of Sg, Rg. and Ho
12:
1) Eq.
12
Generator loss: La Li=l (ū, + Σ/ (1+ 1) (u + 1)
N
14:
Gradient-descem step 0 f 0 - 7G VoLG
end while
More visualization results generated by proposed MFG4AD on ADNI and OASIS datasets in Fig.
u(t
ADNI
OASIS
ON
EMCI
LMCE
AD
CN
AD
Figure 9: The representative examples (reconstruction error L < 0.02) hetween the observed u(t + 1) and predicted ū(t + 1) tau SUVRs generated by MFG4AD (left: ADNI, right: OASIS). Cognitively normal (CN). early mild cognitive impairment (EMCI), late mild cognitive impairment (LMCI).
All experiments were conducted on RTX A5000 GPU. The corresponding inference times are reported Table 2 exponential operation.
The main cost of NMD is the diffusion kernel e- which involves matrix
Table 2: The inference time for each model.
Model Time(s)
DNN 0.05
GCN 0.09
NDM GCNII 0.18
Neuro-ODE 0.04
0.63
Model Time (s)
GRAND 0.14
GREAD 0.36
DSM 0.26
GAN 0.05
Ridge 0.01
MFG4AD 0.27
22

---

A.3 Discussion and Limitation
As proof-of-concept. we leverage the analytic reaction diffusion laws discovered by our model to ask fundamental question: Does amyloid drive tau aggregation locally within the same region, or remotely across distinct cortical areas? By fitting symbolic expressions every vertex, we observe hybrid interaction: amyloid deposits both amplify tau buildup in their own region and "prime" downstream nodes for accelerated spread. In addition, our current framework fits an independent reaction law at each of the 100K surface vertices, which greatly increases computation and memory costs. To address this, we distill these per-vertex laws into a single, regionally parameterized global reaction function defined over cortical subdomains of 1,000 vertices each preserving interpretability while enabling fast, large-scale prediction.
The surface-basee mesh graph constructed from cortical triangular tessellation provides an anatom- ically faithful substrate for modeling prion-like like-liieeda tau propagation along the cortical mantle. This structure is especially suited for simulating local trans-neuronal spread that adheres to physical cortical continuity, which characterizes the early stages of pathological tau aggregation. To cap- ture long- range propagation, future extensions may incorporate structural connectivity data (e.g.. tractography- based inter-regional projections).
In the future, we will cross-validate our framework on additional AD and AD-related cohorts, extend the symbolic module to capture interactions with other biomarkers (e.g. neuroinflammation, synaptic loss), and perform disease simulations driven by our reaction-diffusion engine to test hypothetical interventions before clinical trials.
A.4 Impact Statement
From machine-learning perspective, our work introduces a physics-informed Wasserstein La- grangian GAN combined with symbolic regression learn interpretable, PDE-like reaction-diffusee diffusion dynamics directly on irregular cortical graphs, bridging black-box GNNs and white-box biophysical models and yielding reusable "reaction-diifusionn engines" for spatiotemporal forecasting. From a neuroscience standpoint, the same framework uncovers explicit amyloid-tau interaction kernels and cortical propagation pathways, quantitatively reproducing tau-spread patterns consistent with Braak staging and pinpointing vulnerable hub regions whose amyloid burden drives downstream aggregation, thereby providing data-driven foundation for mechanistic hypothesis testing and targeted intervention in Alzheimer's 's disease. Ultimately, by translating these mechanistic insights into personalized predictive tools, our approach paves the way for earlier diagnosis and more effective, tailored therapies in clinical practice.
23


Reviews:
[{"review_id": "fINuljMLqP", "reviewer": "Reviewer_ACia", "summary": "They propose a physics-informed deep learning framework that unites biophysical modeling and data-driven discovery to reconstruct tau propagation dynamics from longitudinal tau PET scans. This paper formulates the dynamics of tau propagation as a mean-field game (MFG). By leveraging the variational primal-dual structure in MFG, they propose a Wasserstein-1 Lagrangian generative adversarial network (GAN), in which a Lipschitz critic seeks the appropriate transport cost at the population level and a generator parameterizes the flow fields of optimal transport across individuals. Additionally, they incorporate a symbolic regression module to derive an explicit formulation capturing the Aβ-tau crosstalk.", "questions": "When you show the prediction performance in table 1, do you use all of the samples? How is the performance in EMCI, LMCI, and AD, respectively? What is the difference between CN, MCI, and AD? Can you also demonstrate the correlation between the truth and the predicted one? Can we get the same conclusion?\nIs there any reference to support the new insights of Abeta Tau interaction in AD? \nIs there any statistical result to demonstrate similar findings in tau propagation and interaction between ADNI and OASIS datasets?", "limitations": "The paper does not provide performance metrics for specific subgroups such as EMCI, LMCI, and AD. \nThe study does not demonstrate the correlation between predicted and true values.. This may limit the interpretability and robustness of the findings.\nThe manuscript introduces new claims about Abeta-Tau interactions and tau propagation across datasets (ADNI and OASIS) without citing relevant literature or providing statistical evidence to support consistency between datasets.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "This work models tau spread as a network-constrained reaction–diffusion process with a data-driven symbolic law for tau–amyloid crosstalk; casts this system as an equivalent potential mean field game, linking classical PDE theory to tau propagation; and employs a Wasserstein-1 Lagrangian GAN to learn optimal transport flows for accurate tau forecasting. On ADNI and OASIS cohorts, MFG4AD delivers state-of-the-art predictions for unseen subjects and resolves tau-flow directions, while also uncovering an explicit, interpretable reaction law, offering a powerful combination of predictive performance and mechanistic insight into Alzheimer’s pathology. \n\nHowever, the paper does not provide performance metrics for specific subgroups such as EMCI, LMCI, and AD. The study does not demonstrate the correlation between predicted and true values.. This may limit the interpretability and robustness of the findings. The manuscript introduces new claims about Abeta-Tau interactions and tau propagation across datasets (ADNI and OASIS) without citing relevant literature or providing statistical evidence to support consistency between datasets.", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your thoughtful summary and evaluation of our work. We greatly appreciate your recognition of our framework’s strengths—including its principled integration of biophysical modeling and deep learning, the novel MFG-based formulation, and its potential to yield both accurate predictions and interpretable mechanistic insights. Below, we address each of your concerns in detail.**\n\n**Q1: Subgroups' performance**\n\n**A1**: We appreciate the reviewer’s attention to subgroup-level performance. In the original manuscript, we provided visual comparisons of model performance and tau propagation across diagnostic subgroups (CN, EMCI, LMCI, and AD), as shown in Fig. 5  (main text) and Fig. 8 (Appendix). Table 1 summarizes the performance on all samples. Thank you for reminding us and this is a very helpful suggestion. We have now included the quantitative performance metrics (e.g., MAE, RMSE) for each subgroup. These results complement the original table and provide a more rigorous and transparent evaluation of model performance across disease stages. We will incorporate all the results in the final version.\n\n\n|  **ADNI**    | CN\\|SMC            | EMCI                | LMCI                | AD                  |\n| ---- | ------------------ | ------------------- | ------------------- | ------------------- |\n| MAE  | 0.0727 ± 0.033 | 0.0600 ± 0.0253 | 0.0628 ± 0.0086 | 0.0617 ± 0.0295 |\n| RMSE | 0.1282 ± 0.029 | 0.0757 ± 0.0207 | 0.0787 ± 0.0072 | 0.0790 ± 0.0147 |\n\nWe also note that the CN group in our analysis includes subjects with Subjective Memory Complaint (SMC) in ADNI dataset. While clinically categorized as cognitively normal, SMC individuals often present subtle biomarker changes or early cognitive concerns. Their inclusion likely increases the heterogeneity within the CN group and may contribute to the slightly higher prediction error observed in this subgroup. We have clarified this point in the revised manuscript.\n\n|  **OASIS**     | CN                 | AD                 |\n| ---- | ------------------ | ------------------ |\n| MAE  | 0.4289 ± 0.065 | 0.4919 ± 0.075 |\n| RMSE | 0.6104 ± 0.050  | 0.7041 ± 0.130  |\n\n**Q2: Relevant literature about Abeta-Tau interactions and, quantitative cross‑dataset evidence**\n\n**A2**: We appreciate this observation and have revised the manuscript to provide both (i) supporting literature and (ii) quantitative cross‑dataset evidence.\n- We now cite several large‑scale imaging and modelling studies demonstrating that local Aβ burden amplifies connectivity‑constrained tau propagation: [1] Lee et al. “Regional A-tau interactions promote onset and accelertion of Alzheimer’s disease tau spreading”, Neuron, 2022. [2] Roemer-Casiano et al. “Amyloid-associated hyperconnectivity drives tau spread across connected brain regions in Alzheimer’s disease”, Science Translatioal Medicine, 2025\n- To demonstrate cross‑dataset consistency, we computed vertex‑wise Aβ–tau interaction frequency maps for ADNI (64,262 left‑ and 54,128 right‑hemisphere vertices) and OASIS (34,917 LH / 26,898 RH) and then summed these frequencies within the 160 Destrieux regions of interest [Destrieux, C., et al. Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature. Neuroimage]. This procedure identified 67 high‑frequency ROIs in ADNI and 37 in OASIS; 22 regions (24663 vertices)—bilateral entorhinal, parahippocampal, fusiform, inferior/middle temporal, posterior cingulate/precuneus, caudate, and amygdala—were common in both cohorts, encompassing more than 20 000 vertices (Dice = 0.42). Crucially, the Aβ‑related frequencies in these 22 overlapping parcels remained strongly concordant across datasets (Spearman ρ = 0.46). A 10,000‑iteration permutation test that randomly reassigned OASIS ROI labels produced an empirical p‑value of 0.009, confirming that such concordance is exceedingly unlikely to occur by chance and reinforcing the cross‑dataset reproducibility of the spatial pattern of Aβ–tau synergy observed in ADNI and OASIS.\n\nWe will incorporate all the results in the final version.\n\n**Q3:  The correlation between the truth and the predicted one**\n\n**A3:** Thank you for your comment. We have added a dedicated figure (Fig. R1) that visualises predicted vs. observed tau‑SUVR means for every diagnostic subgroup in each cohort. Each dot is an individual subject, the numerical results are summarised here:\n### Predicted vs. Observed Tau-SUVR Fitting Metrics\n\n| Cohort / Subgroup | Slope         | R²             |\n|-------------------|---------------|----------------|\n| **ADNI (overall)**| 0.98 ± 0.06   | 0.91 ± 0.07    |\n| └── CN            | 1.055         | 0.822          |\n| └── EMCI          | 0.975         | 0.953          |\n| └── LMCI          | 1.001         | 0.895          |\n| └── AD            | 1.125         | 0.982          |\n| **OASIS (overall)**| 0.90 ± 0.16  | 0.660 ± 0.17   |\n| └── CN            | 1.007         | 0.785          |\n| └── AD            | 0.784         | 0.535          |\n\nThese metrics demonstrate an agreement between predicted and true values across all subgroups. We have prepared the corresponding fitting curves (Rebuttal_Fig.R1.pdf) to support verification. However, due to the rebuttal policy restricting supplementary uploads during the review process, we are unable to provide this figure at this stage. Nevertheless, we will incorporate these results into the final version of the manuscript, as we agree that they provide important support for the paper. We sincerely appreciate your suggestion.\n\n**Thank you for your time and consideration.**"}, {"review_id": "WsTq69Rqnm", "reviewer": "Reviewer_mzBH", "summary": "This paper proposes MFG4AD, a novel physics-informed deep learning framework to model tau propagation in Alzheimer's disease. It uniquely combines reaction-diffusion modeling, mean-field game theory, and a Wasserstein-1 Lagrangian GAN, operating directly on cortical surface meshes. A symbolic regression module is integrated to derive interpretable Aβ-tau interaction laws. The authors demonstrate superior predictive performance on ADNI and OASIS datasets and claim to uncover biologically plausible propagation pathways and mechanistic insights.", "questions": "a) Please provide more comprehensive technical details on the symbolic regression module, including its search space and an analysis of the robustness and consistency of the discovered symbolic formulas across different runs.\n\nb) How do the biological assumptions about tau propagation align with or necessitate specific adaptations of the standard Mean-Field Game framework, which typically assumes weakly interacting agents?\n\nc) Beyond predictive performance, can the authors provide more direct empirical evidence or analysis demonstrating how the derived reaction laws and propagation pathways genuinely enhance biological understanding or provide novel insights?\n\nd) For a high-resolution model, please provide comprehensive details on computational cost (e.g., training time, GPU memory) and a sensitivity analysis for key hyperparameters to ensure reproducibility and practical understanding.", "limitations": "While the paper details the strengths of the MFG formulation, it lacks a comprehensive discussion on its inherent limitations or underlying assumptions when applied to biological systems like tau propagation in AD. Mean-Field Games typically assume a large population of weakly interacting agents. The biological validity of these specific assumptions, particularly concerning very early-stage disease or highly localized interactions, could be more thoroughly addressed.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\na) The integration of mean-field games, optimal transport, GANs, and symbolic regression into a unified physics-informed framework is novel and innovative for modeling AD pathology. This approach bridges biophysical modeling with data-driven discovery effectively.\n\nb) The model's operation directly on the cortical surface mesh with over 100,000 vertices allows for capturing fine-grained geodesic flows, which is crucial for uncovering biologically plausible propagation pathways. Furthermore, the inclusion of a symbolic regression module to derive an explicit and interpretable Aβ-tau reaction law is a significant advantage, moving beyond \"black-box\" models and offering mechanistic insights.\n\nc) The experimental results on public neuroimaging datasets (ADNI and OASIS) demonstrate that MFG4AD consistently outperforms a range of comparative methods in predicting future tau accumulation, achieving superior mean absolute error (MAE) and root mean squared error (RMSE).\n\nWeaknesses:\n\na) The paper strongly emphasizes the interpretability of the learned tau-Aβ reaction law and the biologically plausible propagation pathways. While predictive performance is well-demonstrated, the main paper offers limited direct qualitative or quantitative evidence to substantiate how interpretable the learned laws are or how the derived pathways specifically align with established biological knowledge or neuropathological staging beyond general statements. Figure 3 visualizes prediction accuracy, but not the \"new understanding of pathology propagation\" claimed.\n\nb) The claim of interpretability hinges on the symbolic regression module. However, the paper provides insufficient technical details regarding its implementation. This includes the specific search space of functions considered, the criteria for selecting the \"best\" equation, the optimization process, or, critically, the robustness and consistency of the discovered formulas across different cross-validation folds or random initializations. \n\nc) The paper lists specific hyperparameters used for training, such as the number of critic updates, learning rates, and the reconstruction penalty weight. However, the paper does not discuss the sensitivity of the model's performance to variations in these hyperparameters. GAN training is often known for its sensitivity to hyperparameter choices.", "was_revised": false, "final_justification": "", "rebuttal": "**We thank the reviewer for recognizing our work. MFG4AD combines physics-based modeling and deep learning in a novel way to track tau spread directly on the brain surface. It not only predicts future tau accurately but also reveals interpretable Aβ–tau interactions, showing strong results on ADNI and OASIS datasets. Below, we address each of your concerns in detail.**\n\n**Q1: limited direct qualitative or quantitative evidence to substantiate how interpretable the learned laws**\n\n**A1:** Thank you for highlighting the need for more explicit evidence supporting the interpretability of the learned Aβ–tau interaction law. In response, we have revised Section 3.2 (line 298-320) to provide a detailed examination of the symbolic expressions $R_j(u,v)$ generated by our model. \n\nTo demonstrate mechanistic interpretability, we present a representative symbolic reaction equation at a vertex in the medial parahippocampal cortex—a region known to initiate tau pathology (Braak I–II) [Braak & Braak, 1991]. The symbolic expression incorporates terms from spatially and functionally linked Aβ sources, including the anterior/posterior cingulate cortex and frontal–parahippocampal junction, consistent with known structural pathways and disease staging trajectories. Each term in the symbolic expression is biologically meaningful:\n\n- Positive coefficients (e.g., $+0.02*v_{81}$​) denote facilitatory effects from local or upstream Aβ burden. For instance, anterior cingulate cortex ($v_{81}$) has been identified as an early Aβ hub in Thal Phase 2 and is tightly linked to the medial temporal lobe via the cingulum bundle [Thal et al., 2002,Neurology; Jacobs et al., 2018,Neuron].\n\n- Nonlinear components (e.g., $0.2\\sin(2.2v_9) + 1.9 + 1.9\\exp(3.5v_{16}) + 1$) capture thresholded or saturating influences of Aβ, reflecting empirical findings that Aβ’s effect on tau becomes pronounced only beyond certain levels [Hanseeuw et al., 2019, JAMA Neurology]. These forms also simulate sigmoid-like molecular switch behavior, analogous to pathological tipping points in disease cascades.\n\n- Negative weights (e.g., $-0.09u_{44}$​) arising from regions such as the calcarine (visual cortex) suggest inhibitory or damping effects, consistent with their involvement in later Braak stages (V–VI) and possible negative feedback or non-contributory roles in early tau spread [Busche & Hyman, 2020,Nature Neuroscience].\n\nImportantly, the spatial configuration of contributing terms reflects a biologically coherent propagation route—the entorhinal–hippocampal–cingulate loop, widely regarded as the epicenter of early Alzheimer’s pathology [Braak & Braak, 1991; Jacobs et al., 2018].\n\nTogether, these findings demonstrate that the learned symbolic laws are not only mathematically predictive, but also biologically interpretable, aligning with canonical disease progression stages and known anatomical connectivity.\n\n**Q2: insufficient technical details regarding its symbolic regression module**\n\n**A2** Thank you for this important point. We have provided more detailed implementation of the symbolic regression module. Specifically: \n\n- Architecture: The symbolic module (*SymbolicNetL00*) is composed of multiple *SymbolicLayerL0* layers (as detailed in the anonymous GitHub). Each layer transforms inputs using a collection of interpretable mathematical functions (e.g., identity, sin, exp, product) and learns to select sparse combinations through a differentiable binary mask.\n\n- Search space of functions: The model uses a library of symbolic primitives including: Unary functions: identity, $\\sin(x)$, $\\cos(x)$, $\\exp(x)$, etc. Binary functions: product $x \\cdot y$, quotient $x/y$, etc.\n\n- Optimization & regularization: We employ the hard concrete distribution as in [Louizos et al., 2017] to sample differentiable binary masks $z \\in [0,1]$ applied element-wise to the weight matrix. This enables $L0$ sparsity regularization, encouraging simpler and more interpretable symbolic formulas.\n\n- Selection criteria: After training, symbolic terms with high dropout probabilities (i.e., low expected activation z) are pruned to remove weak or unstable components. The remaining active terms are ranked based on the magnitude of their learned weights, and their corresponding expressions are reconstructed using a predefined set of interpretable mathematical functions (e.g., linear, sine, exponential), ensuring biological and mathematical interpretability.\n\n- Robustness & Reproducibility: We evaluated the learned symbolic expressions using 5-fold cross-validation. Key contributing regions—such as the entorhinal cortex—consistently ranked among the top terms across folds. Fig. 6 summarizes the selection frequency of each Aβ region, highlighting the stability and biological plausibility of the learned interaction patterns. We will make it clear in the final version, thanks.\n\n**Q3: the sensitivity of the model's performance to variations in these hyperparameters**\n\n**A3:** Thank you for your comment, we have included the sensitivity analysis of hyperparameter choices (line 203-208 in the manuscript).\n\n| $η\\_C\\|η\\_G$ | $n\\_C$ | gp | λ | MAE       | RMSE       |\n| ---------- | ---- | -- | ---------- | --------------- | --------------- |\n| 1e-5\\|1e-4 | 5    | 10 | 1          | 0.3911 ± 0.1521 | 0.5505 ± 0.2551 |\n| 1e-5\\|1e-4 | 5    | 10 | 5          | 0.0696 ± 0.0052 | 0.0976 ± 0.0063 |\n| 1e-5\\|1e-4 | 5    | 10 | 10         | **0.0643 ± 0.0044** | **0.0933 ± 0.0094** |\n| 1e-5\\|1e-4 | 5    | 10 | 15         | 0.0664 ± 0.0049 | 0.0945 ± 0.0059 |\n| 1e-5\\|1e-4 | 5    | 10 | 20         | 0.0671 ± 0.0045 | 0.0952 ± 0.0055 |\n| 1e-4\\|1e-4       | 5    | 5  | 10         | 0.0667 ± 0.0043 | 0.0947 ± 0.0053 |\n| 1e-4\\|1e-4       | 5    | 10 | 10         | 0.0675 ± 0.0052 | 0.0955 ± 0.0062 |\n| 1e-4\\|1e-4       | 5    | 15 | 10         | 0.0674 ± 0.0052 | 0.0955 ± 0.0062 |\n| 1e-4\\|1e-4      | 3    | 10 | 10         | 0.0667 ± 0.0044 | 0.0947 ± 0.0054 |\n| 1e-4\\|1e-4       | 7    | 10 | 10         | 0.0675 ± 0.0051 | 0.0955 ± 0.0061 |\n| 5e-5\\|5e-5       | 5    | 10 | 10         | 0.0674 ± 0.0052 | 0.0954 ± 0.0062 |\n| 5e-4 \\|5e-4      | 5    | 10 | 10         | 0.0668 ± 0.0045 | 0.0947 ± 0.0056 |\n| 1e-3\\|1e-3       | 5    | 10 | 10         | 0.0678 ± 0.0048 | 0.0957 ± 0.0058 |\n\n- A smaller λ fails to adequately enforce reconstruction fidelity, leading to suboptimal predictions. The best performance is observed when $λ \\in [10, 15]$, indicating this range strikes a desirable balance between adversarial realism and numerical accuracy.\n- Gradient penalty (gp) shows mild sensitivity across 5–15, suggesting the 1-Lipschitz constraint is sufficiently enforced within this range without overly penalizing the critic’s learning.\n- Critic steps ($n_C$) shows no monotonic trend; moderate values (e.g., 5) suffice to balance critic convergence and training efficiency.\n- Learning rates ($\\eta_C = 1\\text{e-5}, \\eta_G = 1\\text{e-4}$) yield the best result, as they support stable adversarial updates. Higher rates destabilize optimization, while lower rates may slow convergence.\n\nTraining time is 74.02s/it, inference time is 0.23 s, on H100 NVL with 8 GPUs (94 G/per).\n\n**Q4: the biological assumptions about tau propagation, and empirical evidence or analysis show how the derived reaction laws**\n\n**A4:** Thank you for the question. Our model treats each cortical vertex as an agent whose tau level evolves under local interactions, governed by a surface-based brain graph. This graph is constructed directly from the native cortical surface mesh, preserving geodesic continuity and the anatomical folding pattern of the cortex (see Appendix A.1, Fig. 7). Unlike SC/FC-based connectomes, our geometry-aware network ensures that tau propagation follows biologically realistic, anatomically constrained trajectories. \n\nThe weak coupling assumption in MFG aligns well with the biological reality: tau spread in one region is influenced not by a single source but by the average tau level across its neighboring regions. This naturally captures the notion of local accumulation and distributed influence along the cortical mantle. We adapt the MFG framework by integrating spatial connectivity into the interaction terms, such that the propagation dynamics respect surface-based neighborhood structures.\n\nThe learned reaction laws provide not only enhanced predictive capability but also interpretable symbolic expressions, which allows us to quantify how local Aβ burden and spatially adjacent influences drive tau accumulation. These symbolic laws highlight early-stage epicenters (e.g., entorhinal and medial prefrontal cortex) and map out downstream propagation patterns consistent with Braak I–III staging. By coupling biophysically grounded diffusion, geometric network constraints, and interpretable equations, our model offers novel, data-driven insights into the spatial logic of tau spread in Alzheimer's disease.\n\n**Q5: lacks a comprehensive discussion**\n\n**A5:** Thank you for this important question. Beyond predictive accuracy, our model generates interpretable diffusion fields that offer biological insights into tau propagation dynamics. In Fig. 5, we visualize the spatial propagation of tau across clinical stages and datasets (ADNI and OASIS). The inferred flow fields $\\nu$ reveal coherent directional patterns that mirror canonical staging: for instance, early spread originates in the medial temporal lobe (e.g., entorhinal cortex), progressing toward isocortical regions such as parietal and frontal cortices—consistent with Braak stages I–VI.\n\nImportantly, these flow fields were derived purely from data via our model, without imposing priors about direction or staging. The emergence of biologically plausible propagation axes, repeatedly observed across datasets and stages, demonstrates that our framework captures not just the distribution but also the mechanism of tau spread. This provides novel empirical support for network-based propagation hypotheses in AD."}, {"review_id": "e8I1fQTxKA", "reviewer": "Reviewer_5WWy", "summary": "This paper proposes a new model, MFG4AD, to describe how tau pathology spreads in Alzheimer’s disease. It treats tau movement as a mean-field game, combining ideas from physics, optimal transport, and machine learning. The model uses a graph-based cortical mesh, a GAN to learn transport flows, and a symbolic regression module to learn how amyloid affects tau. Experiments on two public datasets show that the model gives accurate predictions and reveals meaningful biological patterns, such as consistent tau spread along known disease pathways.", "questions": "Please see the weaknesses above. \n\n- A difference map would also be informative.\n- A more fine-grained temporal visualization/analysis would be helpful.\n- Figure 1: Mean Filed Game -> Mean Field Game\n\nIn general, along with the weaknesses mentioned above, the lack of temporal analyses would add much value to this work. Also, I am wondering if there could be more subject-level analyses to appreciate the quality of the individualized predictions.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The overall formulation is novel, combining MFGs and Wasserstein GANs in a way I haven’t seen before in medical modeling.\n- The model uses realistic brain geometry rather than simplifying to regions, which is important for tau spread.\n- The symbolic regression adds interpretability, and the reaction terms make biological sense.\n- The experiments are well done, and the method outperforms several strong baselines.\n\n**Weaknesse**\n\nThe weaknesses are relatively minor:\n- The model assumes that amyloid influences tau, but doesn’t explore the reverse, or model amyloid propagation at all.\n- The learned symbolic terms are interesting, but could benefit from more validation or comparison to known biology.\n- The paper doesn’t address uncertainty in predictions, which could be important in clinical settings.", "was_revised": true, "final_justification": "I thank the authors for their response. I have raised my score.", "rebuttal": "**We sincerely thank the reviewer for their thoughtful and encouraging feedback. We are especially grateful for your recognition of the novelty of our formulation, the use of realistic cortical geometry, and the biological interpretability introduced by the symbolic regression module. Your positive evaluation motivates us to further refine and extend this line of research. Below, we address each of your concerns in detail.**\n\n**Q1: The model assumes that amyloid influences tau, but doesn’t explore the reverse, or model amyloid propagation at all.**\n\n**A1:** Thank you for raising this important point. Our model focuses on Aβ → tau interactions because substantial neuropathological and longitudinal imaging studies suggest that amyloid-β (Aβ) accumulation precedes and facilitates tau pathology, rather than the reverse. Aβ and tau exhibit distinct propagation mechanisms: Aβ spreads in a relatively diffuse and spatially global manner—primarily governed by extracellular deposition—while tau propagates transneuronally in a more localized and network-constrained fashion (Braak et al., 1991; Thal et al., 2002; Vogel et al., 2020).\n\nMoreover, Aβ accumulation tends to plateau early in the disease course, while tau continues to spread and correlate more strongly with neurodegeneration and cognitive decline (Jack et al., 2013; Hanseeuw et al., 2019). Given these asymmetries in dynamics and temporal ordering, we adopt the prevailing neuroscience perspective by modeling Aβ as an upstream driver of tau, not the other way around.\n\nFuture extensions could incorporate Aβ propagation using different mechanisms (e.g., global diffusion with clearance dynamics), but doing so would require introducing distinct priors or differential equations, which would make the model significantly more complex. Our current scope prioritizes biological plausibility and model interpretability for tau dynamics, where Aβ acts as an external input shaping regional vulnerability. We will add this discussion to the final version.\n\n**Q2: The learned symbolic terms are interesting, but could benefit from more validation or comparison to known biology.**\n\n**A2:** Thank you for your very helpful comment. We agree that validation against known biology is crucial. To address this, we now provide an explicit interpretation of a representative learned symbolic equation for tau update at the medial parahippocampal region: \n\n-  $u_{44}$ (S_calcarine): The negative coefficient from the calcarine cortex (a primary visual region affected in late Braak stages V–VI) may reflect a dampening influence on tau accumulation in early-stage regions—potentially capturing long-range regulatory effects in network dynamics [Braak & Braak, 1991].  \n- $v_{81}$ (S_cingul-Mid-Ant): The positive linear Aβ contribution from the anterior cingulate cortex—a region known to accumulate Aβ early (Thal Phase I–II)—suggests a modest but biologically plausible driving force for tau spread [Thal et al., 2002; Palmqvist et al., 2017]. \n- $\\sin(2.2 \\cdot v_9)$ (G_cingul-Post-dorsal): A nonlinear oscillatory term, potentially reflecting a thresholded or biphasic Aβ influence, aligns with evidence that Aβ’s impact on tau propagation may saturate or even reverse at high concentrations [Busche & Hyman, 2020]. \n- Constant term (+1.9): Captures the intrinsic baseline accumulation of tau in the medial temporal lobe, consistent with spontaneous age-related tauopathy in entorhinal and parahippocampal areas [Braak & Del Tredici, 2015]. \n- $\\frac{1.9}{\\exp(3.5 \\cdot v_{16}) + 1}$ (G_front_sup–parahippocampal): A sigmoid-shaped Aβ term, indicative of a saturating dose-response, reflects known dynamics where tau is more sensitive to Aβ at subthreshold levels but less responsive once amyloid burden becomes extensive [Jack et al., 2013].\n\nThese regions collectively form the entorhinal–cingulate–frontal circuit, known as the epicenter of early tau pathology (Braak I–III). Importantly, these symbolic terms were not cherry-picked; they consistently emerged as top contributors across folds (see Fig. 6), providing converging evidence that our model captures biologically grounded and interpretable tau–Aβ interaction rules.\n\nWe will add this analysis into the final version. \n\n**Q3: The paper doesn’t address uncertainty in predictions, which could be important in clinical settings.**\n\n**A3:** This is a great comment. We agree with the reviewer that incorporating uncertainty estimates could be valuable for clinical applications. While our current model provides point estimates, we plan to extend it with Bayesian inference or dropout-based uncertainty quantification in future work. Thanks.\n\n**Q4: A difference map would also be informative.**\n\n**A4:** Thank you for the suggestion. We have added the difference maps for both the ADNI and OASIS datasets (see Fig. R2). These maps highlight regional deviations between predicted and observed tau levels, offering additional insight into model performance. We have prepared the corresponding population-level brain mapping (Rebuttal_Fig.R2.pdf) to support verification. However, due to the rebuttal policy restricting supplementary uploads during the review process, we are unable to provide this figure at this stage. Nevertheless, we will incorporate these results into the final version of the manuscript, as we agree that they provide important support for the paper. We sincerely appreciate your suggestion.\n\n**Q5: In general, along with the weaknesses mentioned above, the lack of temporal analyses would add much value to this work. Also, I am wondering if there could be more subject-level analyses to appreciate the quality of the individualized predictions.**\n\n**A5:** Thank you for this valuable suggestion. In our response, we have added subject-level (16 subjects across different disease stages on ADNI dataset) scatter plots comparing predicted versus observed tau values on ADNI datasets (see Fig. R3, will be public after acceptance), providing a clearer view of individualized prediction quality.\n\n| Group   | Subject | Slope | R²   |\n|---------|---------|-------|------|\n| **CN**      |  #1   | 1.04  | 0.97 |\n|         |  #2   | 1.05  | 0.96 |\n|         |  #3   | 1.06  | 0.95 |\n|         |  #4   | 1.12  | 0.95 |\n| **EMCI**    |  #1 | 0.98  | 0.97 |\n|         |  #2 | 0.93  | 0.97 |\n|         |  #3 | 1.11  | 0.97 |\n|         |  #4 | 0.99  | 0.96 |\n| **LMCI**    |  #1 | 0.97  | 0.98 |\n|         |  #2 | 1.10  | 0.97 |\n|         |  #3 | 0.89  | 0.97 |\n|         |  #4 | 1.14  | 0.95 |\n| **AD**      |  #1   | 1.01  | 0.99 |\n|         |  #2   | 0.97  | 0.98 |\n|         |  #3   | 1.05  | 0.97 |\n|         |  #4   | 0.92  | 0.97 |\n\nAdditionally, we agree that incorporating temporal analyses would further enrich the work. While this current study focuses on baseline-to-follow-up prediction, we are actively extending our framework to model full longitudinal trajectories, which we plan to include in future work.\n\n**Q6: Mean Filed Game -> Mean Field Game**\n\n**A6**: Amended. Thanks.\n\nRefs:\n\n[1] Braak, H., & Braak, E. (1991). Neuropathological stageing of Alzheimer-related changes. Acta neuropathologica.\n\n[2] Thal, D. R., et al. (2002). Phases of Aβ-deposition in the human brain and its relevance for the development of AD. Neurology\n\n[3] Palmqvist, S., et al. (2017). Earliest accumulation of β-amyloid occurs within the default-mode network and concurrently affects brain connectivity. Nature communications\n\n[4] Busche, M. A., & Hyman, B. T. (2020). Synergy between amyloid-β and tau in Alzheimer’s disease. Nature neuroscience\n[5] Braak, H., & Del Tredici, K. (2015). The preclinical phase of the pathological process underlying sporadic Alzheimer’s disease. Brain\n\n[6] Jack, C. R., et al. (2013). Tracking pathophysiological processes in Alzheimer's disease: an updated hypothetical model of dynamic biomarkers. The lancet neurology"}, {"review_id": "nKglCQMCEE", "reviewer": "Reviewer_qm52", "summary": "To address the situation where the quantitative analysis of the cortical pathways of tau propagation and its interaction with the cascade of amyloid-beta (Aβ) plaques lags behind the experimental insights of in vivo imaging pathological mechanisms, and to study the interaction between tau aggregates and amyloid plaques, the authors analyzed the latest methods, focused on identifying and optimizing the dynamics, and proposed a physics-informed neural network empowered by mean-field theory. They built the model directly on the cortical surface mesh to improve the ability to accurately capture the interactions of misfolded proteins. As a result, the modeling formula can induce an optimal transport process in the Wasserstein space. By incorporating symbolic regression, they learned an explicit and interpretable tau–Aβ reaction law from the data, thereby uncovering biologically plausible spreading pathways and mechanistic insights into the tau - amyloid interaction. The authors conducted experiments on public neuroimaging datasets. The results not only provide precise and reliable predictions of future tau accumulation but also reveal interpretable dynamics consistent with neuropathological staging and recent imaging studies.", "questions": "Additional concerns are outlined in the weaknesses above.\n\nQ1: How reliable is the symbolic regression component in inferring biologically meaningful tau–Aβ reaction laws?\n\nQ2: Is the model’s performance limited by the quality or resolution of the neuroimaging datasets?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- The paper combines mean - field game theory, reaction - diffusion models, and generative adversarial networks (GANs) in a physics - informed deep learning framework. This interdisciplinary approach is innovative and provides a unique perspective on modeling tau propagation in Alzheimer's disease (AD). \n- Incorporating symbolic regression to learn explicit, interpretable tau - Aβ reaction laws is a significant strength.\n- The proposed deep learning method overcomes the end-to-end learning of reaction kinetics in previous methods.\n- The paper compares the proposed MFG4AD model with five categories of existing methods, including connectome - based network diffusion models, graph - based methods, deep learning models, PDE - based methods, and traditional regression models. The results clearly show that MFG4AD outperforms these alternative approaches in terms of prediction accuracy, as demonstrated by lower mean absolute error (MAE) and root mean squared error (RMSE).\n- The paper is clearly written and well structured, with detailed methodological and experimental descriptions that support reproducibility.\n\nWeaknesses:\n\n- Fitting an independent reaction law at each of the 100K surface vertices greatly increases computation and memory costs.\n\n- The surface - based mesh graph constructed from cortical triangular tessellation is well - suited for simulating local trans - neuronal spread, but it may have limitations in capturing long - range propagation.", "was_revised": false, "final_justification": "", "rebuttal": "**We sincerely thank the reviewer for their thoughtful and constructive comments. We truly appreciate the recognition of our work’s strengths, including the innovative integration of mean-field game theory, physics-informed modeling, and symbolic regression for modeling tau–Aβ interactions in Alzheimer's disease. We are encouraged by the reviewer’s positive evaluation of the model’s interpretability, methodological clarity, and superior predictive performance compared to a broad spectrum of existing approaches. Below, we address each of the reviewer’s concerns point by point.**\n\n**Q1: Fitting an independent reaction law at each of the 100K surface vertices greatly increases computation and memory costs.**\n\n**A1:** Thank you for raising this important concern. We agree that fitting an independent reaction law at ~100,000 surface vertices is computationally intensive. To address this, we use a GPU-efficient symbolic regression model that enforces shared functional forms with sparse parameterization. Dropout-based sparsification further enhances convergence and reduces overfitting. \n\nTraining is conducted on an **8×NVIDIA H100 NVL GPU system (94GB per GPU)**, enabling parallel computation over large-scale data. While training takes $\\sim$10h, inference is highly efficient ($\\sim$0.2s). Despite the computational demands, this vertex-level modeling is essential to capture the regional heterogeneity of tau–Aβ interactions, which would otherwise be lost in coarser, atlas-based approaches. We will clarify these details in the final version.\n\n**Q2: The surface - based mesh graph constructed from cortical triangular tessellation is well - suited for simulating local trans - neuronal spread, but it may have limitations in capturing long - range propagation.**\n\n**A2:** Thank you for the insightful suggestion. We acknowledge that the current surface-based mesh primarily captures local, cortical trans-neuronal spread. To address long-range propagation mechanisms,we are currently developing a multi-layer diffusion framework that integrates both local (surface-based) and global (connectome-based) long-range pathways. This hybrid approach allows us to simulate tau spread across multiple spatial scales, reflecting both short-range anatomical adjacency and long-range anatomical or functional connectivity. This layered formulation is aligned with recent neurobiological models and will be a key component of our ongoing and future work.\n\n**Q3: How reliable is the symbolic regression component in inferring biologically meaningful tau–Aβ reaction laws?**\n\n**A3:** Thank you for your helpful comment. We agree that validation against known biology is essential. To address this, we now provide an explicit interpretation of a representative learned symbolic equation for tau update in the medial parahippocampal region. The expression includes five interpretable terms:\n\n-  $u_{44}$ (S_calcarine): The negative coefficient from the calcarine cortex (a primary visual region affected in late Braak stages V–VI) may reflect a dampening influence on tau accumulation in early-stage regions—potentially capturing long-range regulatory effects in network dynamics [Braak & Braak, 1991]. \n- $v_{81}$ (S_cingul-Mid-Ant): The positive linear Aβ contribution from the anterior cingulate cortex—a region known to accumulate Aβ early (Thal Phase I–II)—suggests a modest but biologically plausible driving force for tau spread [Thal et al., 2002; Palmqvist et al., 2017]. \n- $\\sin(2.2 \\cdot v_9)$ (G_cingul-Post-dorsal): A nonlinear oscillatory term, potentially reflecting a thresholded or biphasic Aβ influence, aligns with evidence that Aβ’s impact on tau propagation may saturate or even reverse at high concentrations [Busche & Hyman, 2020]. \n- Constant term (+1.9): Captures the intrinsic baseline accumulation of tau in the medial temporal lobe, consistent with spontaneous age-related tauopathy in entorhinal and parahippocampal areas [Braak & Del Tredici, 2015]. \n- $\\frac{1.9}{\\exp(3.5 \\cdot v_{16}) + 1}$ (G_front_sup–parahippocampal): A sigmoid-shaped Aβ term, indicative of a saturating dose-response, reflects known dynamics where tau is more sensitive to Aβ at subthreshold levels but less responsive once amyloid burden becomes extensive [Jack et al., 2013].\n\nThese regions collectively form the entorhinal–cingulate–frontal circuit, known as the epicenter of early tau pathology (Braak I–III). Importantly, these symbolic terms were not cherry-picked; they consistently emerged as top contributors across folds (see Fig. 6), providing converging evidence that our model captures biologically grounded and interpretable tau–Aβ interaction rules.\n\nWe will add this analysis to the final version. \n\nRefs:\n\n[Braak & Braak, 1991] Neuropathological staging of Alzheimer-related changes, Acta Neuropathologica\n\n [Thal et al., 2002] Phases of Aβ-deposition in the human brain and its relevance for the development of AD, Neurology\n\n[Palmqvist et al., 2017] Earliest accumulation of β-amyloid occurs within the default-mode network and concurrently affects brain Synergy between amyloid-β and tau in Alzheimer's disease connectivity, Nature Communications\n\n[Busche & Hyman, 2020] Synergy between amyloid-β and tau in Alzheimer's disease, Nature Neuroscience\n\n[Jack et al., 2013] Tracking pathophysiological processes in Alzheimer's disease: an updated hypothetical model of dynamic biomarkers, Lancet Neurology\n\n[Braak & Del Tredici, 2015] The preclinical phase of the pathological process underlying sporadic Alzheimer's disease\n\n\n**Q4: Is the model’s performance limited by the quality or resolution of the neuroimaging datasets?** \n\n**A4:** This is a good question. The performance of our model does depend on the resolution and quality of the neuroimaging data, as tau and amyloid SUVR values are derived from PET images with limited spatial resolution and potential noise. To mitigate these issues, we model tau propagation on the cortical surface mesh, which preserves anatomical geometry and supports fine-grained modeling of local diffusion and interregional interactions (Fig. 2a). Furthermore, by integrating symbolic regression and connectivity-aware diffusion (via GNNs), our model can partially compensate for noise by learning consistent propagation dynamics across folds and cohorts.\n\nAdditionally, our GAN-based optimal transport formulation operates on tau density distributions, rather than raw voxel-wise intensities, making it more robust to imaging variability. As shown in Table 1, MFG4AD achieves strong predictive performance across both ADNI and OASIS, despite differences in scanner protocols and cohort demographics. Future work may explore incorporating higher-resolution imaging modalities or harmonization techniques to further enhance reliability. Thanks."}]

Author rebuttal:

Meta-review: This paper a novel physics-informed deep learning framework to model tau propagation in Alzheimer's disease. The proposed method adopts a graph-based cortical mesh, a GAN to learn transport flows, and a symbolic regression module to learn how amyloid affects tau. Experiments on public neuroimaging datasets show that the proposed method not only provides precise and reliable predictions of future tau accumulation but also reveals interpretable dynamics consistent with recent imaging studies.

All of the reviewers recognized the novelty and technical contributions of this work. The overall formulation is novel, e.g., combining MFGs and Wasserstein GANs. Also, the idea of incorporating symbolic regression to learn explicit, interpretable tau - Aβ reaction laws is quite interesting. In addition, experiments are comprehensive and convincing, and results show that the proposed method outperforms several strong baselines.

Meanwhile, reviewers raised some questions regarding assumptions, technical details, paper presentation, experiments, etc. The authors have provided detailed responses with additional results, which have addressed the previous concerns from reviewers. The authors are strongly encouraged to incorporate the new results and discussions into the final version of the paper.
Decision: Accept (poster)