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EEG Inverse Problem & Source Localization Studio

Electrophysiology · Underdetermined Ill-Posed Inverses, Minimum Norm Estimation (MNE), Depth Weighting & LCMV Beamforming

The Core Inverse Challenge: While the forward model \(V = L J + \epsilon\) is deterministic, reconstructing intracranial current dipoles \(J\) from scalp potentials \(V\) is mathematically ill-posed and underdetermined (\(M\) electrodes \(\ll P\) cortical sources). Without regularization, unconstrained pseudoinverses suffer catastrophic noise amplification. Comparing MNE, depth-weighted wMNE, and LCMV spatial beamformers reveals the fundamental trade-offs between point-spread dispersion, depth bias, and noise sensitivity.

Ground Truth Dipole (Click to place)

Estimated Current Density Ĵ

Scalp Electrodes (10-20)

Localization Error

Peak-to-True Distance

Point-Spread (PSF FWHM)

Spatial Dispersion

Goodness of Fit (R²)

Scalp Field Explained

Tikhonov Regularizer (λ) 0.10

Under-reg (Noise spikes) Over-reg (Over-smoothed)

Scalp Sensor Noise (SNR) 15 dB

Ground Truth Sources 1 Dipole

Preset Dipole Locations:

Interactive Guidance:

Click anywhere on the gray cortical mantle to position the active dipole. Watch how unweighted MNE systematically pulls deep sources toward superficial electrodes, while Depth-Weighted MNE and LCMV Beamforming recover correct deep anatomical coordinates.

Mathematical Formulations of Inverse Solvers

1. The Discrete Leadfield Matrix:

The forward problem discretizes the cortex into \(P = 48\) target locations, mapping dipole moments \(J \in \mathbb{R}^P\) to \(M = 16\) scalp electrode measurements: \(V = L J + \epsilon\). The leadfield column \(L_{:,p}\) represents the scalp potential pattern generated by a unit dipole at location \(p\). Because \(M \ll P\), the nullspace \(\text{null}(L)\) is non-empty: infinitely many silent source configurations produce exactly \(0\text{ V}\) on the scalp.

2. Minimum Norm Estimation (MNE):

MNE resolves ambiguity by penalizing total source energy via Tikhonov regularization:

\hat{J}_{\text{MNE}} = \arg\min_J \left( ||V - LJ||_2^2 + \lambda^2 ||J||_2^2 \right) = L^T (L L^T + \lambda^2 I)^{-1} V

While numerically stable, leadfield magnitude falls off rapidly with distance from the scalp (\(||L_p|| \propto 1/r^2\)), inherently biasing unweighted MNE to place spurious activity in the most superficial cortical layers.

3. Depth-Weighted wMNE:

To counteract superficial bias, depth-weighting scales each source by its leadfield norm:

W = \text{diag}(||L_{:,p}||^{-\gamma}) \quad (\gamma \approx 0.8)

The regularized solution becomes:

\hat{J}_{\text{wMNE}} = W^{-1} L^T (L W^{-1} L^T + \lambda^2 I)^{-1} V

This provides uniform sensitivity across both deep sulcal structures and superficial gyral crowns.

4. LCMV Spatial Beamforming:

The Linearly Constrained Minimum Variance beamformer constructs an adaptive spatial filter \(w_p\) for each location \(p\) that passes unit gain from \(p\) (\(w_p^T L_p = 1\)) while minimizing overall variance from all other brain regions:

w_p = \frac{C_v^{-1} L_p}{L_p^T C_v^{-1} L_p} \quad \text{where } C_v = \text{Cov}(V)

Beamformers achieve remarkably sharp Point-Spread Functions when multiple uncorrelated sources are active simultaneously.