PHINN-EEG: Topological Time-Series Analysis of Dream-State EEG -- Dynamic Betti Curves for Dream Content Classification and Topology-Conditioned Neural Signal Synthesis

Technical Breakdown§

Core Methodology§

PHINN-EEG (Persistent Homology Inspired Neural Network for EEG) introduces topological time-series analysis to dream-state EEG classification. The key idea is to extract topological invariants (Betti numbers) from sliding-window Takens delay embeddings of multichannel EEG epochs, capturing geometric structure beyond spectral power.

Pipeline: 1. Sliding-window Takens embedding: For each EEG epoch (e.g., 30s pre-awakening), apply time-delay embedding to each channel: $$X_i(t) = [x_i(t), x_i(t+\tau), \dots, x_i(t+(m-1)\tau)] \in \mathbb{R}^m$$ where $\tau$ is the delay and $m$ the embedding dimension. 2. Vietoris-Rips filtration: Build a point cloud from the embedded vectors across channels. Compute persistent homology on the resulting simplicial complex, tracking birth and death of homology groups $H_0$ (connected components) and $H_1$ (loops). 3. Dynamic Betti Curves (DBCs): Extract the evolution of Betti numbers over the filtration parameter $\epsilon$: $$\beta_k(\epsilon) = \text{rank}(H_k(\text{VR}_{\epsilon}))$$ These curves are used as topological features. 4. Classification: DBCs are fed into a neural network (or combined with spectral features) for binary classification (dream vs. no-dream). The paper claims AUC improvement from 0.70 (PSD baseline) to 0.82-0.90. 5. Topology-conditioned flow matching: For synthesis, a rectified flow model is conditioned on topological features (DBCs) to generate synthetic dream-state EEG. A spectral-conditioned baseline isolates the value of topological conditioning.

Implementation Details§

• Database: Open-access subset of DREAM database (1,462 awakenings from 3,191 total, 263 participants, 20 labs).
• EEG preprocessing: Bandpass filter 0.5-45 Hz, artifact removal, epoching (30s pre-awakening).
• Embedding parameters: $\tau$ = 10 ms, $m$ = 5 (empirically chosen).
• Filtration: Vietoris-Rips on combined point cloud from all channels (dimensionality reduction via PCA optional).
• Flow matching: Rectified flow (Liu et al., 2022) conditioned on DBCs or PSD. Training with noise-to-data mapping.

Code Snippet (PyTorch-style):

def compute_dbc(epoch, tau=10, m=5):
    # epoch: (T, C) numpy
    embedded = []
    for c in range(epoch.shape[1]):
        embedding = takens_embedding(epoch[:, c], delay=tau, dim=m)
        embedded.append(embedding)
    point_cloud = np.concatenate(embedded, axis=0)  # (C*(T-(m-1)*tau), m)
    # Vietoris-Rips filtration (using giotto-tda or ripserer)
    dgm = ripser.ripser(point_cloud)['dgms']
    # Compute Betti curves for H0 and H1
    bc0 = betti_curve(dgm[0], max_eps=1.0)
    bc1 = betti_curve(dgm[1], max_eps=1.0)
    return np.concatenate([bc0, bc1])

class PHINNClassifier(nn.Module):
    def __init__(self, input_dim, hidden=128):
        super().__init__()
        self.fc = nn.Sequential(
            nn.Linear(input_dim, hidden),
            nn.ReLU(),
            nn.Linear(hidden, 1),
            nn.Sigmoid()
        )
    def forward(self, dbc):
        return self.fc(dbc)

# Training loop
for epoch, (batch_dbc, labels) in enumerate(dataloader):
    preds = model(batch_dbc)
    loss = F.binary_cross_entropy(preds, labels)
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

Potential Impact§

If validated, this approach shifts EEG analysis from spectral energy to phase-space geometry, applicable to rare-event neural detection. The synthesis model may enable data augmentation for imbalanced dream datasets.