Joint MEG-EEG Source Imaging L1-507

Medical ImagingMulti-modal bioelectromagnetic neural source localization (multi-physics joint inverse)δ=5 · advancedL_DAG = 10📋 Stub — not mineable

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Unclaimed Principle — open for contribution

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Forward model E

B (r_{s} e n sor, t) = (m u_{0} /4 \cdot p i) \cdot in t e g r a l [J (r^{'}, t) x (r_{s} e n sor - r^{'}) /∣ r_{s} e n sor - r^{'} ∣^{3}] d r^{'} (B i o t - S a v a r t, M E G f or w a r d); V (r_{e} l ec t r o d e, t) = in t e g r a l G_{E} (r_{e} l ec t r o d e, r^{'}; σ_{h} e a d) \cdot d i v (J (r^{'}, t)) d r^{'} (P o i sso n, E E G f or w a r d); j o in tl y in v er t y (t) = [B (t); V (t)] = [G_{M}; G_{E}] \cdot J (t) + n (t) \to J (r, t) g i v e nh e a d m o d e l σ_{h} e a d (r)

Joint MEG-EEG Source Imaging: joint multi-physics forward couples (i) Maxwell magnetostatic Biot-Savart law for the magnetic field from intracranial neural currents B(r_sensor, t) = integral G_M(r_sensor, r'; sigma_head) * J(r', t) dr'; (ii) Poisson electrostatic equation for the scalp potential V(r_electrode, t) = integral G_E(r_electrode, r'; sigma_head) * J(r', t) dr' in a layered head-volume-conductor model (brain, CSF, skull, scalp); (iii) shared anatomical model parameterized by head conductivity profile sigma_head(r) derived from MR-derived segmentation. The forward DAG has 8 primitives with two coupling constraints (n_c = 2): (i) joint observation — the same neural source J(r, t) is simultaneously observed by MEG sensors (magnetic field B) and EEG electrodes (scalp potential V); (ii) shared head conductivity model — both Green's functions G_M and G_E depend on the same sigma_head(r), creating a structural coupling that improves identifiability when both modalities are jointly inverted. Recovery is posed as the joint linear inverse problem y(t) = [B(t); V(t)] = [G_M; G_E] * J(t) + n(t) where the stacked operator [G_M; G_E] has lower effective condition number than either G_M or G_E alone. Difficulty tier delta = 5 with raw condition number kappa ~ 1000 (the inverse problem is fundamentally underdetermined — finite sensors, infinitely many sources) and effective kappa_eff ~ 100 after Tikhonov / minimum-norm / sLORETA / beamformer regularization. Mismatch parameters: head_segmentation_error, conductivity_uncertainty, sensor_position_error, source_orientation_assumption, reference_electrode_drift, magnetic_artifact_contamination. Additive Gaussian thermal noise sets the data-fidelity floor. See forward_model field for the closed-form joint imaging equation.

L-DAG

L.neural_source -> L.cortical_mesh_constraint -> L.head_volume_conductor -> L.maxwell_magnetostatic -> L.poisson_electrostatic -> L.megsensor_detection -> L.eegsensor_detection -> int.spatial -> int.temporal

L.neural_sourceL.cortical_mesh_constraintL.head_volume_conductorL.maxwell_magnetostaticL.poisson_electrostaticL.megsensor_detectionL.eegsensor_detectionint.spatialint.temporal

Well-posedness W

Existence:: true
Uniqueness:: conditional
Stability:: conditional
κ:: 1000

Existence of recovered neural source distribution J(r, t) is guaranteed within the cortical-mesh constraint and the declared Omega bounds. Uniqueness is fundamentally conditional — the joint inverse problem is underdetermined (~15000 source dipoles with constrained orientation vs ~400 sensors typical), so unique solutions require regularization (minimum-norm, sLORETA, beamformer, sparse priors, dynamical / temporal constraints). Stability is moderately conditioned (kappa_eff ~ 100 after L2 regularization) — head_segmentation_error dominates source-localization bias; conductivity_uncertainty contributes scaling factor; sensor_position_error contributes a few-millimeter localization shift; source_orientation_assumption (free vs cortically-constrained vs surface-normal) contributes prior bias. Joint Hadamard well-posedness for the coupled MEG+EEG forward (with regularization) is established by Mosher et al. 1992, Hamalainen-Ilmoniemi 1994, Pascual-Marqui 2002 (sLORETA), Sharon et al. 2007, Henson et al. 2009, and Huang et al. 2014.

Solvability C

Solver class:: linear-operator + convex optimisation [minimum-norm estimator MNE; sLORETA; eLORETA; LCMV beamformer; mixed-norm priors] | nonlinear [dynamic causal modeling DCM; particle filtering] | linear-operator + deep neural prior [DeepSourceNet]
Convergence rate q:: 2
Complexity:: O(N_dipoles * (N_MEG + N_EEG) * T_samples) per iteration via direct Green's-function evaluation; FEM-based BEM approaches O(N_dipoles * N_BEM_nodes^2) per forward; learned variants O(N_dipoles * F_theta_cost) per forward pass

Specs (0)

No L2 specs registered yet for this principle.