L1 ⚛ L1-399 ⊙ Testnet

Cardiac Electrophysiology (Monodomain)

Computational Biology · Cardiac modeling

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⬇ L1-399.md

# ⚛ L1 Principle — Cardiac Electrophysiology (Monodomain) **ID:** `L1-399` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Computational Biology — *Cardiac modeling* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D transmembrane potential > **📡 Carrier:** N/A · **🌫 Noise:** observation gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555217 --- ## 🧠 1. Introduction **Cardiac Electrophysiology (Monodomain)** is a **nonlinear inverse problem** whose unknown lives in **3D transmembrane potential** space, within the **Cardiac modeling** sub-domain of **Computational Biology**. Measurements consist of N/A via a **ecg body surface potential** sensing mechanism. The forward operator applies, in order: gradient / divergence with respect to position; S · eikonal · fast marching operator; O · ecg · inverse body surface operator. Observations are corrupted by observation gaussian. Existence of the recovered 3D_transmembrane_potential is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 2000); torso_conductivity_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Observation gaussian sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → Spatial derivative → S · eikonal · fast marching → O · ecg · inverse body surface → **y** (detector). ``` y = `O.ecg.inverse_body_surface` `S.eikonal.fast_marching` ∇ x ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `D.space` | Gradient / divergence with respect to position | | `S.eikonal.fast_marching` | S · eikonal · fast marching operator | | `O.ecg.inverse_body_surface` | O · ecg · inverse body surface operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Computational Biology | | Sub domain | Cardiac modeling | | Carrier | N/A | | Problem class | nonlinear_inverse | | Solution space | 3D_transmembrane_potential | | Noise model | observation_gaussian | | Integration axis | 3d_cardiac_volume | | Difficulty delta | 5 | | L dag | 4.5 | ## 📡 4. Measurement Model Existence of the recovered 3D_transmembrane_potential is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 2000); torso_conductivity_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Observation gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | activation_time_error_ms | | Secondary | body_surface_potential_RMSE_mV | ## 📏 5. Operating Range (Ω) **Center problem class:** `nonlinear_inverse` · **Forward operator:** `ecg_body_surface_potential` **Center point:** | Parameter | Unit | Value | |---|---|---| | N mesh nodes | — | 10000 | | Noise snr db | dB | 30 | | N electrodes bsp | — | 128 | | Heart torso distance mm | mm | 50 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N mesh nodes | — | 1000 – 100000 | | Noise snr db | dB | 10 – 60 | | N electrodes bsp | — | 32 – 512 | | Heart torso distance mm | mm | 20 – 100 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 5 activation_time_error_ms | Metric | Range | |---|---| | Activation time error ms | 1 – 30 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **activation_time_error_ms**, with κ = `100000.0` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x781340fe947f4a5e7eb4920a752ea2c505bdbcedb5928b5df5e8afd99b236717` - **Chain tx hash:** `0x8d057901026adf6d57bcb86b8bc050e92aeaed8b462e04aef8991d80536894e2` - **Chain block:** `41555217` --- ## File Mapping This bundle consists of: `L1-399.md`, `L1-399.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-399.md` | Source of truth — edit this | Human or LLM | | `L1-399.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 `L1-399`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-399.md).
Regenerate L1-399.json so every field matches the Markdown.
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L2-399-001 Cardiac Electrophysiology (Monodomain) — Nominal + Mismatch Specs

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📋 JSON metadata

{
  "artifact_id": "L1-399",
  "chain_block": 41555217,
  "chain_hash": "0x781340fe947f4a5e7eb4920a752ea2c505bdbcedb5928b5df5e8afd99b236717",
  "chain_tx_hash": "0x8d057901026adf6d57bcb86b8bc050e92aeaed8b462e04aef8991d80536894e2",
  "domain": "Computational Biology",
  "hardness_fn": {
    "delta": 5,
    "kappa": 100000.0,
    "metric": "activation_time_error_ms",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "activation_time_error_ms",
    "regime": "Existence of the recovered 3D_transmembrane_potential is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 2000); torso_conductivity_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Observation gaussian sets the irreducible data-fidelity floor.",
    "secondary": "body_surface_potential_RMSE_mV"
  },
  "physics_fingerprint": {
    "L_DAG": 4.5,
    "carrier": "N/A",
    "difficulty_delta": 5,
    "domain": "Computational Biology",
    "integration_axis": "3d_cardiac_volume",
    "noise_model": "observation_gaussian",
    "primitives": [
      "D.space",
      "S.eikonal.fast_marching",
      "O.ecg.inverse_body_surface"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "ecg_body_surface_potential",
    "solution_space": "3D_transmembrane_potential",
    "sub_domain": "Cardiac modeling",
    "title": "Cardiac Electrophysiology (Monodomain)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "ecg_body_surface_potential"
    ],
    "allowed_omega_dimensions": [
      "N_electrodes_BSP",
      "N_mesh_nodes",
      "noise_SNR_dB",
      "heart_torso_distance_mm"
    ],
    "allowed_problem_classes": [
      "nonlinear_inverse"
    ],
    "center_spec": {
      "epsilon_fn_center": "5 activation_time_error_ms",
      "forward_operator": "ecg_body_surface_potential",
      "input_format": "measurement_only",
      "omega": {
        "N_electrodes_BSP": 128,
        "N_mesh_nodes": 10000,
        "heart_torso_distance_mm": 50,
        "noise_SNR_dB": 30
      },
      "problem_class": "nonlinear_inverse"
    },
    "epsilon_bounds": {
      "activation_time_error_ms": [
        1,
        30
      ]
    },
    "omega_bounds": {
      "N_electrodes_BSP": [
        32,
        512
      ],
      "N_mesh_nodes": [
        1000,
        100000
      ],
      "heart_torso_distance_mm": [
        20,
        100
      ],
      "noise_SNR_dB": [
        10,
        60
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Cardiac modeling",
  "title": "Cardiac Electrophysiology (Monodomain)"
}

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