# ⚛  L1 Principle — FitzHugh-Nagumo Excitability Model

**ID:** `L1-397` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Neural excitability*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** FHN parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555199

---

## 🧠 1. Introduction

**FitzHugh-Nagumo Excitability Model** is a **parameter-estimation problem** whose unknown lives in **FHN parameter vector** space, within the **Neural excitability** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **action potential recording** sensing mechanism.

The forward operator applies, in order: time evolution of the state; O · chi2 · phase portrait operator; S · continuation · bifurcation operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered FHN_parameter_vector 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 ~= 50); noise_amplitude dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Time derivative → O · chi2 · phase portrait → S · continuation · bifurcation → **y** (detector).

```
y = `S.continuation.bifurcation` `O.chi2.phase_portrait` ∂_t x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.chi2.phase_portrait` | O · chi2 · phase portrait operator |
| `S.continuation.bifurcation` | S · continuation · bifurcation operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Neural excitability |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | FHN_parameter_vector |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered FHN_parameter_vector 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 ~= 50); noise_amplitude dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | phase_portrait_RMSE |
| Secondary | bifurcation_point_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `action_potential_recording`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| B recovery | — | 0.8 |
| A threshold | — | 0.7 |
| Noise sigma | — | 0 |
| Epsilon timescale | — | 0.08 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| B recovery | — | 0.5 – 1.5 |
| A threshold | — | 0.5 – 1.5 |
| Noise sigma | — | 0.0 – 0.5 |
| Epsilon timescale | — | 0.01 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.05 phase_portrait_RMSE

| Metric | Range |
|---|---|
| Phase portrait rmse | 0.005 – 0.3 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **phase_portrait_RMSE**, with κ = `1000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0f0efe82f19e8d237278873f54b39dce08570a944fd6e45f0603e19bd0b77e09`
- **Chain tx hash:** `0x3b1a85302d1679b27651f4b43eaa418a7328c603470d6d53948b62e99fb1717b`
- **Chain block:** `41555199`

---

## File Mapping

This bundle consists of: `L1-397.md`, `L1-397.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-397.md` | Source of truth — edit this | Human or LLM |
| `L1-397.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 `L1-397`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._