Hopf-GAE

Physics-Informed Graph Neural Network for Normative Brain Dynamics

Anomaly Detection in Major Depressive Disorder via Stuart-Landau–Grounded Denoising Graph Autoencoder with Two-Level Fisher LDA Scoring

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Overview

This repository contains the Hopf-GAE, a physics-informed deep learning architecture that detects depression-related dynamical abnormalities without ever training on depressed brains. Rather than framing MDD detection as binary classification (which fails at $n = 19$), the model learns a normative manifold of healthy brain dynamics and scores MDD subjects by how far they deviate from it.

The key innovations:

1. Biophysically grounded node features — every node carries the per-region bifurcation parameter $a_j$, natural frequency $\omega_j$, and goodness-of-fit $\chi^2_j$ estimated by the Stuart-Landau / Hopf bifurcation framework via the UKF-MDD pipeline, plus Yeo 7-network one-hot encodings (11 features per ROI).

2. Multi-relational graph attention — three edge types (PLV phase synchrony, MVAR Granger causality, SC structural connectivity) with learned per-relation attention weights.

3. Denoising graph autoencoder — Gaussian noise injection ($\sigma = 0.1$) on encoder input and dropout ($p = 0.3$) on the latent code replace the variational bottleneck, which collapsed in all tested configurations due to low within-HC variance of bifurcation parameters.

4. Expanded reconstruction targets — 7-dimensional per-node targets (3 physics + 4 connectivity-derived: PLV node strength, MVAR in/out-strength, within-network PLV) force the bottleneck to encode richer per-ROI structure.

5. Two-level Fisher Linear Discriminant scoring — data-driven combination of node dynamics and edge connectivity anomalies, with per-relation weighting at level 1 and node-vs-edge weighting at level 2. No manual tuning.

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The $n = 19$ Problem

Classification (insufficient data) Binary classification: active NF vs. sham Trained on 18 subjects per fold Memorized subject identity → Cohen's $d$ of $-6$ to $-11$ Implausibly large vs. UKF reference $d = -0.835$ Verdict: classification fails at this sample size

Normative anomaly detection (this work) Train exclusively on healthy controls ($n = 295$ sessions) MDD subjects are test-only — never seen during training Overfitting eliminated by construction HC vs MDD: $d = +4.04$, $p = 3.9 \times 10^{-12}$ All 4 intervention scales survive FDR correction HC holdout false positive rate: 0/36 (0.0%) Verdict: "how far from healthy?" not "which group?"

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Architecture

Node Features (11-dimensional per ROI)

Feature	Dim	Source	Meaning
$a_j$	1	UKF-MDD	Bifurcation parameter — distance from critical point
$\omega_j$	1	Hilbert phase	Natural oscillation frequency (Hz)
$\chi^2_j$	1	UKF fit	Goodness-of-fit (model–data agreement)
Network one-hot	8	Yeo 7 + Subcortical	Functional network membership

Reconstruction Targets (7-dimensional per ROI)

Feature	Weight	Source	Purpose
$a_j$	2.0	UKF	Primary clinical marker
$\omega_j$	1.0	Hilbert	Oscillation dynamics
$\chi^2_j$	1.0	UKF fit	Model–data agreement
PLV node strength	0.5	Edge aggregation	Phase synchrony profile
MVAR in-strength	0.5	Edge aggregation	Directed input connectivity
MVAR out-strength	0.5	Edge aggregation	Directed output connectivity
Within-network PLV	0.5	Edge aggregation	Intra-network coherence

Edge Types (3 relations)

Relation	Type	Source	Encoder Weight (Conv₁)
PLV	Undirected	Phase Locking Value	0.771
SC	Undirected	$\exp(-d/40\text{mm})$	0.185
MVAR	Directed	Lasso-MVAR	0.044

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Model Components

Multi-Relational Graph Attention Convolution

Each GAT layer maintains separate learnable projections $W_r$ and attention vectors $\mathbf{a}_r$ for each edge relation $r \in {\text{PLV}, \text{MVAR}, \text{SC}}$:

$$h_j^{(l+1)} = \text{ELU}!\left( \frac{1}{|R|} \sum_{r \in R} \sum_{i \in \mathcal{N}_r(j)} \alpha_{ij}^{(r)} , W_r , h_i^{(l)} \right)$$

Frozen Encoder (5,485 parameters)

Two multi-relational GAT layers ($11 \to 32 \to 32$) with a masked residual connection produce per-ROI embeddings $\mathbf{h}_j \in \mathbb{R}^{32}$. The masked residual projects the input through input_proj ($11 \to 32$) but zeros the physics features $(a_j, \omega_j, \chi^2_j)$ — forcing the encoder to reconstruct dynamics through graph message passing rather than shortcutting via identity. A physics head ($32 \to 16 \to 1$) validates the encoder during pre-training ($R^2 = 0.983$). The encoder is frozen after pre-training.

Trainable Denoising GAE (1,962 parameters)

Denoising: During training, Gaussian noise ($\sigma = 0.1$) is injected on the encoder input $\mathbf{x}$, preventing the bottleneck from learning identity-like mappings.

Node path: Deterministic projection $\mathbf{h}j \to z_j \in \mathbb{R}^8$ → dropout ($p = 0.3$) → linear decoder ($8 \to 7$) → reconstructed $(a, \omega, \chi^2, s\text{PLV}, s_\text{MVAR-in}, s_\text{MVAR-out}, \text{PLV}_\text{within})$.

Edge path: Three MLP edge decoders predict edge existence from $|\mathbf{h}_i - \mathbf{h}_j|$ for PLV, SC, and MVAR independently. Each MLP is $32 \to 16 \to 1$ with ELU activation. Edge decoders use frozen $\mathbf{h}$ (not $z$), making them independent of the bottleneck.

Graph-level loss: Per-graph mean and standard deviation of the bifurcation parameter $a$ are reconstructed, ensuring the decoder preserves population-level distributional properties.

Component	Shape	Parameters	Status
$f_z$ (bottleneck)	$32 \to 8$	264	Trainable
Linear decoder	$8 \to 7$	63	Trainable
Edge decoders (PLV, SC, MVAR)	$32 \to 16 \to 1$ each	1,635	Trainable
Total trainable		1,962

Two-Level Fisher LDA Scoring

Level 1 — Per-relation edge weighting: Each edge type gets a signed Fisher weight $w_r = d_r / \sum|d_r|$ derived from its individual HC-vs-MDD effect size. The composite edge score is $\sum w_r \cdot z_r$.

Level 2 — Node-vs-edge weighting: The composite edge score and node reconstruction error are combined with signed Fisher weights: $S = w_\text{node} \cdot z_\text{node} + w_\text{edge} \cdot z_\text{edge}$.

Both levels are fully data-driven. Signed weights handle reversed signals naturally.

Loss function (GAE training):

$$\mathcal{L} = \underbrace{\sum_{f} w_f (x_f - \hat{x}_f)^2}_{\text{feature-weighted node recon}} + ; \lambda_g \cdot \underbrace{(\text{MSE}(\mu_a, \hat{\mu}_a) + \text{MSE}(\sigma_a, \hat{\sigma}_a))}_{\text{graph-level } a \text{ statistics}} + ; \lambda_e \cdot \underbrace{\sum_{r} \text{BCE}(\hat{A}_r, A_r)}_{\text{3-relation edge recon}}$$

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Parameter Budget

Total parameters:                            7,447
├── Frozen encoder:                          5,485  (74%)
│   ├── conv1 (3-relation GAT, 11→32):       1,286
│   ├── conv2 (3-relation GAT, 32→32):       3,302
│   ├── input_proj (masked residual, 11→32):   352
│   └── physics_head (32→16→1):                545
└── Trainable GAE:                           1,962  (26%)
    ├── fc_z (32→8):                            264
    ├── linear_decoder (8→7):                    63
    └── edge_decoders (3 × MLP 32→16→1):     1,635

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Data Isolation

┌────────────┬─────────────────┬──────────────┬──────────────┬──────────────┬──────────────┐
│  Synthetic │    HC train     │  HC holdout  │   HC test    │  MDD rest1   │  MDD rest2   │
│  n = 200   │ 24 subj (199s)  │ ~5 subj (36s)│ 6 subj (60s) │   19 subj    │   18 subj    │
│  Stage 1   │    Stage 2      │  Test only   │  Test only   │  Test only   │  Test only   │
└────────────┴─────────────────┴──────────────┴──────────────┴──────────────┴──────────────┘
 Synthetic + HC train = train  |  HC holdout + HC test + MDD = never trained on

The HC train/test split is by subject (not session) to prevent leakage. HC holdout subjects (~15%) provide an unbiased false positive rate estimate (0/36 = 0.0%). MDD subjects are never seen during any training stage. HC train vs. test overfitting check: $p = 0.81$.

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Key Design Decisions

Denoising autoencoder (not variational) — The variational bottleneck ($\mu, \log\sigma^2$, reparameterization, KL divergence) was tested across five architectural configurations. KL collapsed to $< 0.001$ nats in all cases because within-HC variance of $(a, \omega, \chi^2)$ is too low for variational regularization. Gaussian noise injection ($\sigma = 0.1$) and dropout ($p = 0.3$) on $z$ serve as regularizers instead, preventing the encoder from learning identity-like mappings through the bottleneck.

Linear decoder (63 parameters) — An MLP decoder ($8 \to 32 \to 16 \to 7$, ~867 params) can learn a mean-output shortcut: memorize the HC population mean and output it regardless of $z$, achieving low reconstruction loss without $z$ carrying per-ROI information. A linear layer ($8 \to 7$, 63 params) cannot learn this shortcut — it must use $z$ to reconstruct per-ROI variation.

Expanded reconstruction targets (7 features) — Reconstructing only $(a, \omega, \chi^2)$ allows the bottleneck to ignore connectivity structure. Adding PLV/MVAR-derived features forces $z$ to encode both dynamical and connectivity information per ROI, producing richer anomaly scores.

Node-level (not graph-level) bottleneck — Graph-level pooling into a single $z$ vector for the whole graph could be bypassed by the frozen encoder embeddings. Node-level bottleneck gives each ROI its own $z_j$, forcing per-ROI dynamical information through the bottleneck.

Edge decoders on $\mathbf{h}$ (not $z$) — MLP decoders on $|\mathbf{h}_i - \mathbf{h}_j|$ use the rich 32-dim frozen encoder output directly, decoupled from the bottleneck.

Absolute difference $|\mathbf{h}_i - \mathbf{h}_j|$ (not concatenation) — Concatenation $[\mathbf{h}_i | \mathbf{h}_j]$ gives the decoder access to individual node magnitudes, allowing it to exploit the fact that MDD $\mathbf{h}$ vectors have different norms — a confound rather than a connectivity signal. Absolute difference isolates the pairwise relationship.

Feature-weighted reconstruction — Weights $[2, 1, 1, 0.5, 0.5, 0.5, 0.5]$ on the 7 reconstruction targets emphasize the physics features (the UKF pipeline's primary clinical markers) while still requiring accurate connectivity reconstruction.

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Key Results

Metric	Value	95% CI	UKF Reference
HC vs MDD separation	$d = +4.04$, $p = 3.9 \times 10^{-12}$	—	—
Permutation null (10,000)	$p < 0.0001$	—	—
HC holdout FP rate	0/36 (0.0%)	—	—
HC holdout vs MDD	$d = +4.74$	—	—
Overfitting check	$p = 0.81$	—	—
Whole-brain intervention	$d = +1.69$, FDR $p = 0.011^*$	$[0.86, 3.19]$	$d = -0.84$, $p = 0.080$
Circuit intervention	$d = +1.51$, FDR $p = 0.011^*$	$[0.72, 2.83]$	$d = -1.09$, $p = 0.027$
Limbic intervention	$d = +1.52$, FDR $p = 0.011^*$	$[0.77, 2.63]$	—
Subcortical intervention	$d = +1.92$, FDR $p = 0.004^*$	$[1.18, 3.31]$	—
Circuit enrichment (top-10)	2.50× (8/10), hypergeom $p = 0.002$	—	—
Circuit enrichment (top-20)	2.03× (13/20), hypergeom $p = 0.002$	—	—
Circuit vs non-circuit	$d = +0.37$, $p = 0.023$	—	—
Heterogeneity (raw $a$, circuit)	$d = +1.44$, $p = 0.008$	—	$d = +1.01$, $p = 0.042$
#1 anomalous ROI	RH Default PFCdPFCm₄	—	Converges with Ch. 5 cluster
#1 anomalous network	Limbic	—	—

All four intervention scales survive Benjamini-Hochberg FDR correction. Active group moves away from HC (increased anomaly), sham moves toward HC (decreased anomaly).

Top 15 Anomalous ROIs

Rank	ROI	Network	Circuit?
1	RH Default PFCdPFCm₄	Default Mode	✓
2	LH Limbic TempPole₁	Limbic	✓
3	RH Vis₁	Visual
4	NAcc-rh	Subcortical	✓
5	LH Limbic TempPole₂	Limbic	✓
6	RH SalVentAttn FrOperIns₁	Salience/VentAttn
7	LH Limbic TempPole₄	Limbic	✓
8	LH Default Temp₅	Default Mode	✓
9	RH Limbic TempPole₁	Limbic	✓
10	RH Limbic TempPole₂	Limbic	✓
11	LH Cont Cing₂	Frontoparietal
12	LH Default PHC₁	Default Mode
13	Thal-rh	Subcortical	✓
14	LH Default Par₁	Default Mode
15	RH Default PFCdPFCm₃	Default Mode	✓

Bottleneck Dimension Sensitivity

Results are robust to the bottleneck dimension $d_z$. Intervention effects are significant across all tested widths:

$d_z$	Params	HC–MDD $d$	WB $d$	WB $p_\text{perm}$	Sub $d$	Sub $p_\text{perm}$	Top-10	hp$_{10}$
3	1,762	+3.68	+1.36	0.020	+1.79	0.002	2.50×	0.002
4	1,802	+3.49	+1.36	0.018	+1.80	0.002	2.50×	0.002
6	1,882	+3.54	+1.26	0.031	+1.75	0.004	2.19×	0.013
8	1,962	+3.44	+1.30	0.026	+1.75	0.005	1.88×	0.059
12	2,122	+3.57	+1.29	0.027	+1.70	0.006	1.88×	0.059

Whole-brain and subcortical intervention effects are significant ($p_\text{perm} < 0.05$) at all five dimensions.

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Upstream Dependencies

The Hopf-GAE consumes outputs from the R biophysical pipeline (UKF-MDD):

Input	File	Format
Bifurcation parameters	results/v3/sl_stage1_results_216roi.csv	CSV (one row per ROI per subject per session)
PLV matrices	results/v3/plv/plv_all_216roi.rds	R list, keyed "subject_id|session"
MVAR matrices	results/v3/s2_mvar_all_216roi.rds	R list, keyed "subject_id|session"
HC comparison data	results/ch5_v4def/ch5_v4def_results.rds	R list

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Requirements

Python packages torch, torch_geometric, numpy, pandas, scipy, statsmodels, pyreadr, scikit-learn, matplotlib

Upstream (R pipeline) pracma, MASS, Matrix, dplyr, tidyr, ggplot2, scales, glmnet, igraph, parallel, zoo

System: Python ≥ 3.9 · PyTorch ≥ 2.0 · PyTorch Geometric ≥ 2.4 · R ≥ 4.2 (for upstream pipeline only)

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Quick Start

# 1. Ensure upstream pipeline has been run
# (github.com/skaraoglu/UKF-MDD)

# 2. Install Python dependencies
pip install torch torch_geometric pyreadr scikit-learn statsmodels

# 3. Run the full pipeline
jupyter execute main_analysis.ipynb

# Pipeline stages:
#   S1–S6:   Data loading, graph construction, quality control
#   S7–S10:  Synthetic pre-training (encoder, 100 epochs)
#   S11–S12: HC data loading + augmentation, GAE training (200 epochs)
#   S13:     Anomaly scoring (two-level Fisher LDA)
#   S14:     Statistical analysis (FDR, permutation tests, enrichment)

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Citation

If you use this architecture or build on this work, please cite:

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Built with PyTorch Geometric · Node dynamics from UKF-MDD · Parcellation: Schaefer 2018 + Melbourne Subcortex