# ⚛  L1 Principle — Joint MEG-EEG Source Imaging

**ID:** `L1-507` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Multi-modal bioelectromagnetic neural source localization (multi-physics joint inverse)*
> **🎯 Problem class:** linear inverse underdetermined · **🧮 Solution space:** 5D neural source spatiotemporal
> **📡 Carrier:** neural_current · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553359

---

## 🧠 1. Introduction

**Joint MEG-EEG Source Imaging** is a **linear inverse underdetermined** whose unknown lives in **5D neural source spatiotemporal** space, within the **Multi-modal bioelectromagnetic neural source localization (multi-physics joint inverse)** sub-domain of **Medical Imaging**.

Measurements consist of neural current sources reconstructed from external sensors via a **joint magnetic and electric with volume conductor** sensing mechanism.

The forward operator applies, in order: L · neural source operator; L · cortical mesh constraint operator; L · head volume conductor operator; L · maxwell magnetostatic operator; L · poisson electrostatic operator; L · megsensor detection operator; L · eegsensor detection operator; pixel-level spatial averaging on the detector; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. 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.

## ⚙ 2. Forward Model

Physical chain: **x** → L · neural source → L · cortical mesh constraint → L · head volume conductor → L · maxwell magnetostatic → L · poisson electrostatic → L · megsensor detection → L · eegsensor detection → Spatial integration → Temporal integration → **y** (detector).

```
y = ∫_t dt ∫_A dA `L.eegsensor_detection` `L.megsensor_detection` `L.poisson_electrostatic` `L.maxwell_magnetostatic` `L.head_volume_conductor` `L.cortical_mesh_constraint` `L.neural_source` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.neural_source` | L · neural source operator |
| `L.cortical_mesh_constraint` | L · cortical mesh constraint operator |
| `L.head_volume_conductor` | L · head volume conductor operator |
| `L.maxwell_magnetostatic` | L · maxwell magnetostatic operator |
| `L.poisson_electrostatic` | L · poisson electrostatic operator |
| `L.megsensor_detection` | L · megsensor detection operator |
| `L.eegsensor_detection` | L · eegsensor detection operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Multi-modal bioelectromagnetic neural source localization (multi-physics joint inverse) |
| Carrier | neural_current |
| Problem class | linear_inverse_underdetermined |
| Solution space | 5D_neural_source_spatiotemporal |
| Noise model | gaussian |
| Integration axis | spatial_temporal |
| Difficulty delta | 5 |
| L dag | 10 |

## 📡 4. Measurement Model

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.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | spatial_localization_error_mm |

## 📏 5. Operating Range (Ω)

**Center problem class:** `meg_eeg_joint_source_imaging` · **Forward operator:** `meg_eeg_joint_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 20 |
| N dipoles | — | 8000 |
| T samples | — | 1000 |
| N eeg channels | — | 64 |
| N meg channels | — | 306 |
| Head layer count | — | 3 |
| Sampling rate hz | Hz | 1000 |
| Sensor position error | — | 0 |
| Head segmentation error | — | 0 |
| Conductivity uncertainty | — | 0 |
| Reference electrode drift | — | 0 |
| Source orientation assumption | — | cortically_constrained |
| Magnetic artifact contamination | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0.0 – 35.0 |
| N dipoles | — | 1000 – 30000 |
| T samples | — | 100 – 100000 |
| N eeg channels | — | 0 – 256 |
| N meg channels | — | 0 – 306 |
| Head layer count | — | 3 – 7 |
| Sampling rate hz | Hz | 256 – 5000 |
| Sensor position error | — | 0.0 – 0.2 |
| Head segmentation error | — | 0.0 – 0.3 |
| Conductivity uncertainty | — | 0.0 – 0.5 |
| Reference electrode drift | — | 0.0 – 0.3 |
| Magnetic artifact contamination | — | 0.0 – 0.4 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 20.0

| Metric | Range |
|---|---|
| Psnr db | 3.0 – 38.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `1000` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xc5b5823cf02645ff781e015fc071cd01cb138c2ec39adf26564ee8f0716fc417`
- **Chain tx hash:** `0xdebc47053da2a66ee606952ad7d1f1115aa01f0e65e5d73470794d4ce0f74263`
- **Chain block:** `41553359`

---

## File Mapping

This bundle consists of: `L1-507.md`, `L1-507.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-507.md` | Source of truth — edit this | Human or LLM |
| `L1-507.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-507`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._