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Taming Epilepsy: Mean Field Control of Whole-Brain Dynamics

Ming Li, Ting Gao, Jingqiao Dua

Abstract

Controlling the high-dimensional neural dynamics during epileptic seizures remains a significant challenge due to the nonlinear characteristics and complex connectivity of the brain. In this paper, we propose a novel framework, namely Graph-Regularized Koopman Mean-Field Game (GK-MFG), which integrates Reservoir Computing (RC) for Koopman operator approximation with Alternating Population and Agent Control Network (APAC-Net) for solving distributional control problems. By embedding Electroencephalogram (EEG) dynamics into a linear latent space and imposing graph Laplacian constraints derived from the Phase Locking Value (PLV), our method achieves robust seizure suppression while respecting the functional topological structure of the brain.

Taming Epilepsy: Mean Field Control of Whole-Brain Dynamics

Abstract

Controlling the high-dimensional neural dynamics during epileptic seizures remains a significant challenge due to the nonlinear characteristics and complex connectivity of the brain. In this paper, we propose a novel framework, namely Graph-Regularized Koopman Mean-Field Game (GK-MFG), which integrates Reservoir Computing (RC) for Koopman operator approximation with Alternating Population and Agent Control Network (APAC-Net) for solving distributional control problems. By embedding Electroencephalogram (EEG) dynamics into a linear latent space and imposing graph Laplacian constraints derived from the Phase Locking Value (PLV), our method achieves robust seizure suppression while respecting the functional topological structure of the brain.
Paper Structure (25 sections, 25 equations, 7 figures, 1 algorithm)

This paper contains 25 sections, 25 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. MFG Modeling for Epilepsy Suppression
  4. APAC-Net Deep Solver: From Variational Principle to Adversarial Generative Architecture
  5. Variational Primal-Dual Structure of Potential MFGs
  6. ValueNet and HJB Value Function Approximation
  7. Generator Network and Density Evolution: Global Spatial Exploration Driven by Real Data
  8. Direct Extraction of Optimal Control Law $u^*(t)$
  9. Network Control with Graphic Constrain and Laplacian Regularization
  10. Construction of Dynamic Brain Networks Based on Phase Locking Value (PLV)
  11. Graph Laplacian Operator and Manifold Regularized Cost
  12. Pinning Control Based on Core Node Theory
  13. Global Linearization of Complex Nonlinear Dynamics Using RC-Koopman Operators
  14. Spatio-Temporal Feature Fusion and Reservoir State Lifting
  15. Closed-Form Ultra-Fast Solution via Ridge Regression
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Brain Network Construction and Topological Structure Exploration: Figure 1(a) is the adjacency matrix A of the brain network connections constructed from PLV, with colors mapped to the weights of the edges. (b) is the Laplacian matrix of the brain network: $L=D-A$, where D is the degree matrix. (c) is the 2-D visualization of the brain network constructed from PLV with an edge weight threshold of 0.4.
  • Figure 2: Solution of Control Matrix B: Figure 2 shows the top 5 key sub-nodes identified by the weighted combination of Degree Centrality, Betweenness Centrality, and Eigenvector Centrality. For the control matrix B, the diagonal elements corresponding to nodes with control inputs are all set to 1, and the remaining elements are all set to 0.
  • Figure 3: Linearization of Stochastic Nonlinear Dynamical Systems via RC Reservoir: Panels 3a-e show the prediction performance of the top 5 key sub-nodes (electrode channels) after approximating the Koopman operator with RC, where the solid black lines are the trajectories of the original data from 1300s to 1500s, and the dashed red lines are the predicted trajectories. Panels 3f-j show the corresponding Probability Density Functions (PDFs), where the gray represents the PDF of the original data from 1300s to 1500s, and the red represents the predicted PDF. Panel 3h is the global (23 channels) RMSE heatmap, where darker colors indicate larger errors. Panel 3i is the global (23 channels) PDF, where the gray represents the PDF of the original data from 1300s to 1500s across 23 channels, and the red represents the predicted PDF of the 23 channels from 1300s to 1500s.
  • Figure 4: Optimal Control $u^*$ Solution via MFG: Panels a-e show the trajectory control performance of the top 5 key sub-nodes (channels), where the red lines are the trajectories of the original uncontrolled data from 1300s to 1500s, and the blue lines are the corresponding controlled trajectories. Panels f-j are the corresponding control input signals. Panels k-o are the corresponding Probability Density Functions (PDFs), where the red represents the amplitude PDF of the original data from 1300s to 1500s, and the blue represents the amplitude PDF after corresponding control. Panel p is the global amplitude PDF, where the red represents the amplitude PDF of the original data (23 channels), the green represents the amplitude PDF of the original healthy state data from 0s to 500s, and the blue represents the amplitude PDF of the 23 channels after control. Panel q is the training loss curve of the $\phi$ function (purple) and the $G_\theta$ function (green) in the APAC-Net.
  • Figure 5: Controlled Probability Density Functions of 23 Channels: Panels a-w show the controlled PDFs of the 23 channels of original data, where the red represents the amplitude PDF of each channel from 1300s to 1500s, and the blue represents the corresponding PDF after control.
  • ...and 2 more figures