geometric-brain-dynamics-mapping-gbf

name	geometric-brain-dynamics-mapping-gbf
description	Geometric Basis Functions (GBF) methodology for noninvasive whole human brain dynamics mapping using participant-specific cortical eigenmodes. Reconstructs whole-brain spatiotemporal dynamics from EEG/MEG with anatomically-constrained source imaging. Activation - geometric basis functions, GBF, brain dynamics, source imaging, cortical geometry, EEG/MEG reconstruction.

Geometric Brain Dynamics Mapping with GBF

Overview

Geometric Basis Functions (GBF) framework for accurate reconstruction of whole-brain spatiotemporal dynamics from non-invasive electrophysiology (EEG/MEG). Uses participant-specific eigenmodes derived from individual cortical surfaces to resolve the inverse problem with anatomically-consistent priors.

Core Methodology

1. Geometric Basis Functions Construction

Cortical Surface Eigenmodes:

• Extract individual participant's cortical surface mesh
• Compute geometric eigenmodes (vibration modes) of the cortical surface
• These modes form a compact, anatomically-meaningful basis set
• Typically hundreds of modes capture whole-brain dynamics

Mathematical Formulation:

Source activity s(r,t) = Σ_i α_i(t) * φ_i(r)

Where:
- φ_i(r) = i-th geometric eigenmode at location r
- α_i(t) = time-varying coefficient for mode i
- r = position on cortical surface
- t = time

2. EEG/MEG Forward Model

Forward Solution:

Measurements m(t) = L * s(t) + noise

Where L is the lead field matrix mapping sources to sensors

GBF-Forward Projection:

m(t) = L * Φ * α(t)

Where:
- Φ = [φ_1, φ_2, ..., φ_n] (geometric basis matrix)
- α(t) = [α_1(t), α_2(t), ..., α_n(t)]^T

3. Source Reconstruction

Inverse Problem Solution:

α̂(t) = argmin_α ||m(t) - L*Φ*α||² + λ * ||α||²

Source estimate ŝ(t) = Φ * α̂(t)

4. Key Advantages

Feature	Traditional Methods	GBF Method
Prior	Generic (LORETA, MNE)	Individual geometry
Dimension	~10k sources	~100-500 modes
Anatomical	Weak	Strong constraint
Noise	Voxel-wise correlated	Mode-wise decorrelated
Interpretation	Location-based	Geometry-based

Implementation Workflow

Step 1: Data Preparation

# Required inputs
subject_mri = "t1w.nii.gz"  # Individual T1-weighted MRI
cortical_surface = "pial.surf"  # Cortical surface mesh
sensor_locations = "electrode_xyz.csv"  # EEG/MEG sensor positions
conductivity_profile = "conductivity.json"  # Head conductivity model

Step 2: Compute Geometric Eigenmodes

import numpy as np
from scipy.sparse.linalg import eigsh

def compute_geometric_modes(surface_mesh, n_modes=500):
    """
    Compute geometric eigenmodes of cortical surface.
    """
    # Compute cotangent Laplacian
    L = compute_cotangent_laplacian(surface_mesh)
    M = compute_mass_matrix(surface_mesh)
    
    # Solve generalized eigenvalue problem
    eigenvalues, eigenvectors = eigsh(L, k=n_modes, M=M, sigma=0)
    
    return eigenvalues, eigenvectors

Step 3: Build GBF Forward Model

def build_gbf_forward_model(lead_field, geometric_modes):
    """
    Project lead field to geometric basis.
    """
    return lead_field @ geometric_modes

Step 4: Source Reconstruction

def reconstruct_gbf_sources(measurements, gbf_forward, lambda_reg=0.01):
    """
    Reconstruct source activity in GBF space.
    """
    n_modes = gbf_forward.shape[1]
    I = np.eye(n_modes)
    
    source_modes = np.linalg.solve(
        gbf_forward.T @ gbf_forward + lambda_reg * I,
        gbf_forward.T @ measurements
    )
    
    # Project back to source space
    source_space = geometric_modes @ source_modes
    
    return source_modes, source_space

Validation Benchmarks

Meta-Source Benchmark

• 10,000+ source configurations
• Multiple cortical regions
• Quantitative localization error metrics

Task-Evoked Validation

• Auditory oddball tasks
• Motor paradigms
• Visual processing

Resting-State Networks

• Default mode network
• Visual network
• Sensorimotor network
• Dorsal attention network

Clinical Applications

• Intracranial stimulation validation
• Epilepsy source localization
• Pre-surgical planning

Applications

1. High-Resolution Source Imaging

• Improved spatial resolution over traditional methods
• Better localization accuracy
• Reduced spatial smearing

2. Dynamic Brain Network Analysis

• Track network evolution over time
• Study connectivity patterns
• Analyze frequency-specific dynamics

3. Clinical Diagnostics

• Epileptic focus localization
• Pre-surgical mapping
• Disease biomarker extraction

4. Cognitive Neuroscience

• Study cognitive processes
• Brain-behavior relationships
• Individual differences

References

• Wang et al. (2026): "A geometry aware framework enhances noninvasive mapping of whole human brain dynamics"
• arXiv:2604.25592
• Categories: q-bio.NC, eess.SP

Activation Triggers

Use this skill when working with:

• EEG/MEG source localization
• Brain dynamics reconstruction
• Cortical surface modeling
• Anatomically-constrained inverse problems
• Individual-specific brain imaging