# ⚛  L1 Principle — Hodgkin-Huxley Neuron Model Parameter Estimation

**ID:** `L1-396` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Computational neuroscience*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** HH parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555199

---

## 🧠 1. Introduction

**Hodgkin-Huxley Neuron Model Parameter Estimation** is a **parameter-estimation problem** whose unknown lives in **HH parameter vector** space, within the **Computational neuroscience** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **patch clamp voltage measurement** sensing mechanism.

The forward operator applies, in order: time evolution of the state; O · chi2 · voltage trace operator; S · mcmc · bayesian parameter operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered HH_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 ~= 1000); channel_noise_stochastic 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 · voltage trace → S · mcmc · bayesian parameter → **y** (detector).

```
y = `S.mcmc.bayesian_parameter` `O.chi2.voltage_trace` ∂_t x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.chi2.voltage_trace` | O · chi2 · voltage trace operator |
| `S.mcmc.bayesian_parameter` | S · mcmc · bayesian parameter operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Computational neuroscience |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | HH_parameter_vector |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Existence of the recovered HH_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 ~= 1000); channel_noise_stochastic dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | voltage_trace_RMSE_mV |
| Secondary | spike_timing_error_ms |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| G na ratio | — | 120 |
| I ext ua cm2 | — | 10 |
| Temperature c | — | 6.3 |
| N voltage samples | — | 10000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| G na ratio | — | 50 – 500 |
| I ext ua cm2 | — | 0 – 50 |
| Temperature c | — | 0 – 37 |
| N voltage samples | — | 1000 – 100000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 2.0 voltage_trace_RMSE_mV

| Metric | Range |
|---|---|
| Voltage trace rmse mv | 0.2 – 20.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **voltage_trace_RMSE_mV**, with κ = `50000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0074d8715b7c2c36083cafb128edee3a0f95e23f753fe1bdd627106c1bebb2c3`
- **Chain tx hash:** `0xf8caebe1341877f30445bda4241cd04ad18b051ea44cf87f060b15574f17369b`
- **Chain block:** `41555199`

---

## File Mapping

This bundle consists of: `L1-396.md`, `L1-396.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-396.md` | Source of truth — edit this | Human or LLM |
| `L1-396.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 `L1-396`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._