Enhancing Brain Source Reconstruction through Physics-Informed 3D Neural Networks

TL;DR

This work proposes the novel hybrid method 3D-PIUNet for EEG source localization that effectively integrates the strengths of traditional and deep learning techniques and significantly improves spatial accuracy, demonstrating superior performance over both traditional and end-to-end data-driven methods.

Abstract

Reconstructing brain sources is a fundamental challenge in neuroscience, crucial for understanding brain function and dysfunction. Electroencephalography (EEG) signals have a high temporal resolution. However, identifying the correct spatial location of brain sources from these signals remains difficult due to the ill-posed structure of the problem. Traditional methods predominantly rely on manually crafted priors, missing the flexibility of data-driven learning, while recent deep learning approaches focus on end-to-end learning, typically using the physical information of the forward model only for generating training data. We propose the novel hybrid method 3D-PIUNet for EEG source localization that effectively integrates the strengths of traditional and deep learning techniques. 3D-PIUNet starts from an initial physics-informed estimate by using the pseudo inverse to map from measurements to source space. Secondly, by viewing the brain as a 3D volume, we use a 3D convolutional U-Net to capture spatial dependencies and refine the solution according to the learned data prior. Training the model relies on simulated pseudo-realistic brain source data, covering different source distributions. Trained on this data, our model significantly improves spatial accuracy, demonstrating superior performance over both traditional and end-to-end data-driven methods. Additionally, we validate our findings with real EEG data from a visual task, where 3D-PIUNet successfully identifies the visual cortex and reconstructs the expected temporal behavior, thereby showcasing its practical applicability.

Figures (11)

• Figure 1: Overview of different approaches for source localization from EEG data. Top The classical (Minimum Norm) approaches uses the pseudo-inverse ${\textbf{L}^\dag}$ of the forward model to estimate the sources $\hat{\textbf{x}}$ from the EEG measurements $\textbf{y}$. Middle: Data-driven end-to-end approaches train some neural network $g_\theta$ on simulated data, where the source activity $\textbf{x}$ is drawn from the data prior $p(\textbf{x})$. During inference, the network predicts the sources $\hat{\textbf{x}}$ directly from measurements without explicitly using the forward model. Bottom: Our approach, 3D-PIUNet, integrates data-driven learning with physics-informed modeling by incorporating the forward model's physical information through the pseudo-inverse. After transforming the EEG measurements into a grid-like source space, a 3D convolutional U-Net $f_\theta$ refines the pseudo-inverse solution according to the learned data prior, resulting in an improved source estimate $\hat{x}$.
• Figure 2: We generate a diverse setting of brain activation. From left to right, we have 1. a single active source with minimal width (10mm), 2. 4 active sources with minimal width, 3. a single medium-sized (20mm) active source, 4. a single large active source (80mm), and 5. a diverse sample with 4 active sources of different size, where two sources are partly overlapping.
• Figure 3: The ground truth activation and reconstruction from 3D-PIUNet , the fully connected Network and eLORETA for top a single active source with SNR of $0$dB and bottom 3 active sources with SNR of $40$dB. 3D-PIUNet incorporates the learned data prior and refines the shape of an active region to match the ground truth. When multiple sources are active, it does focus on the strongest sources, missing weak activations in other parts of the brain.
• Figure 4: Evaluation on the validation set across different Signal-to-Noise Ratios (SNR). The shaded area represents the standard deviations calculated from 5 trained models, emphasizing the consistency and reliability of the observed trends. Left: Normalized EMD decreases as SNR increases, with 3D-PIUNet consistently showing the lowest values, indicating superior performance in minimizing distribution discrepancies. Center: For low SNR, both neural network approaches substantially outperform eLORETA. The discrepancy shrinks as SNR increases, with 3D-PIUNet consistently achieving lower mean squared error. Right: Weighted Cosine Distance remains high across all SNR levels for the neural networks, suggesting robust similarity in orientation, while eLORETA shows a significant improvement with increasing SNR.
• Figure 5: 3D-PIUNet outperforms all baselines for different spatial extent of a single source with $5$dB noise.