fhn.qmd

---
title: "FitzHugh-Nagumo Model"
subtitle: "Excitable dynamics with network particle organization"
---

## Overview

The **FitzHugh-Nagumo (FHN)** model is a simplified two-variable system originally developed to describe neuronal excitability. When coupled to diffusiophoresis, it produces qualitatively different dynamics compared to Brusselator and Gray-Scott systems.

::: {.callout-tip}
## Key Discoveries
FHN achieves **hexagonal mode** (~6-7 spots) regardless of particle types (1, 2, or 3). Particles form **network/filamentary structures** rather than tight clusters.

Additionally, the inhibitor diffusion coefficient $D_v$ acts as a **symmetry selector**: $D_v = 0$ produces hexagonal spots, while $D_v \geq 0.1$ produces **square/grid symmetry** — a novel morphology not accessible to any other tested model.
:::

## Mathematical Formulation

The FHN reaction-diffusion equations:

$$
\frac{\partial u}{\partial t} = D_u \nabla^2 u + u - \frac{u^3}{3} - v + I
$$

$$
\frac{\partial v}{\partial t} = D_v \nabla^2 v + \epsilon(u + a - bv)
$$

Where:

- $u$: Fast activator (excitation variable)
- $v$: Slow inhibitor (recovery variable)
- $D_u, D_v$: Diffusion coefficients
- $a, b$: Nullcline parameters
- $\epsilon$: Time-scale separation (small = slow inhibitor)
- $I$: External current/stimulus

## Parameter Documentation

```
Row 0 (u field): [Du, a, b, epsilon, I, time_scale]
Row 1 (v field): [Dv, ...]
Row 2 (coupling): [Pe, consumption, production, influence_radius, boost_exponent, M1]
```

### Regime Classification

| Regime | Condition | Pattern Type |
|--------|-----------|--------------|
| Excitable | $a > 1 - b/3$ | Traveling waves, spirals |
| Oscillatory | $a < 1 - b/3$ | Target patterns, bulk oscillation |
| Turing | $D_v \gg D_u$ | Stationary stripes/spots |
| Bistable | Large $\|a\|$ | Front propagation |

## Exploration Results

### Block 3 Findings (Iterations 17–19)

| Metric | FHN | Brusselator | Gray-Scott |
|--------|-----|-------------|------------|
| Pattern Scale | ~6-7 spots | ~6 spots | Concentric |
| Clustering | -0.2 to 0.3 | 0.6+ | 0.4 |
| Symmetry Breaking | Hexagonal | Hexagonal | None (radial) |

::: {.callout-important}
## Particle-Type Independence Confirmed
FHN hexagonal mode is **NOT** particle-type dependent: 1, 2, and 3 types all achieve ~6-7 hexagonal spots. The finer network in iter 17 may be stochastic variation.
:::

### Configuration

```yaml
mesh_model_name: PDE_Diffusiophoresis_FHN
params_mesh:
  - [0.5, 0.7, 0.8, 0.08, 0.0, 20.0]   # u: Du, a, b, epsilon, I, time_scale
  - [0.0, 0, 0, 0, 0, 0]               # v: Dv (=0, slow inhibitor)
  - [1.0, 180, -180, 0.05, 0, 0]       # Particle-field coupling
n_particle_types: 2
n_frames: 6000
boundary: periodic
```

## Dv Symmetry Selector (Block 6 Discovery)

::: {.callout-important}
## Breakthrough: Inhibitor Diffusion Controls Symmetry Class
The FHN parameter $D_v$ acts as a discrete symmetry selector:

- $D_v = 0$: **Hexagonal** spots (standard Turing)
- $D_v = 0.05$: Disordered transitional
- $D_v \geq 0.1$: **Square/Grid** symmetry (novel!)
:::

### Square Mode Properties

| Property | Hexagonal ($D_v=0$) | Square ($D_v \geq 0.1$) |
|----------|---------------------|--------------------------|
| Symmetry | 6-fold | 4-fold |
| Clustering | 0.26-0.39 | ~0.3 |
| Plateau | 0.00 | 0.00 (intrinsically oscillatory) |
| Multi-type | All types work | Same-sign mobility only |
| Opposing mobility | Works | Destroys square → hexagonal |

### Square Mode Constraints

- **Uniform mobility direction required**: All particle types must have the same sign for $M_1, M_2$. Opposing mobilities revert square → hexagonal
- **Intrinsically oscillatory**: Plateau is always 0.00 regardless of feedback parameters; non-convergence is inherent to FHN square mode
- **Robust to particle count**: $D_v = 0.15$ also produces square; not a narrow parameter island

### Robustness Hierarchy

Hexagonal (survives multi-type, reversed feedback) > Square (same-sign multi-type, fails opposing mobility) > Stripe (1-type only, Brusselator)

---

## Comparison: Four PDE Models

```{mermaid}
flowchart TD
    A[Reaction-Diffusion Model] --> B[Brusselator]
    A --> C[Gray-Scott]
    A --> D[FitzHugh-Nagumo]
    A --> E[Schnakenberg]

    B --> B1[Hexagonal + Stripes]
    B --> B2[Tight clusters]

    C --> C1[Radial only]
    C --> C2[Moderate clustering]

    D --> D1[Hexagonal + Square]
    D --> D2[Network structure]

    E --> E1[Radial only]
    E --> E2[Genuine steady state]

    style B fill:#c8e6c9
    style C fill:#ffcdd2
    style D fill:#fff9c4
    style E fill:#e1bee7
```

## Exploration Gallery

### Iter 17 (7/10)

![C1/C2 fields evolve from noise through concentric rings into multi-lobed flower/mandala structure with 4--5 branching arms. Three particle types stratify --- blue core, orange intermediate, green outer shell. Late frames show elaborate branching with crescent/yin-yang sub-structures at dispersed Turing spots. Boundaries between types are slightly sharper than parent due to stronger adhesion.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_017.png){.lightbox group="fhn"}

### Iter 18 (5/10)

![Fields develop smooth concentric ring patterns that maintain circular symmetry throughout --- no flower/labyrinthine breakup. Particles form extremely compact concentric bullseye disc (blue core, orange ring, green outer ring) that barely changes shape across all frames. Late frames show tiny satellite spots beginning to appear at edges, but main pattern stays as smooth layered disc. Flow field shows uniform radial convergence/divergence.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_018.png){.lightbox group="fhn"}

### Iter 45 (8/10)

![Hexagonal Turing spots develop in both fields at 150x150 resolution. Particles self-organize into elaborate flower/mandala with radial arms and type-specific petal sorting. 3-type segregation creates concentric core-ring evolving into multi-armed rosette with radial spokes.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_045.png){.lightbox group="fhn"}

### Iter 48 (4/10)

![Fields show expanding rings with increasingly extreme contrast. Particles initially form concentric rings but scatter outward in later frames. 96% retention indicates significant particle escape. Fields are diverging with the highest pattern growth ever recorded.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_048.png){.lightbox group="fhn"}

## Key Insights

1. **Hexagonal symmetry**: Like Brusselator, FHN achieves hexagonal mode (unlike Gray-Scott/Schnakenberg)
2. **Square symmetry**: $D_v \geq 0.1$ switches to 4-fold square/grid patterns — unique to FHN
3. **Network organization**: Particles form filamentary/networked structures instead of tight clusters
4. **Particle-type independent**: 1, 2, or 3 particle types all achieve hexagonal spots
5. **$D_v$ is a symmetry selector**: Continuous parameter produces discrete symmetry transitions
6. **Square mode constraints**: Requires uniform mobility direction; opposing mobilities destroy square → hexagonal

## References

1. FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. *Biophys. J.*, 1(6), 445-466.

2. Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. *Proc. IRE*, 50(10), 2061-2070.

3. Ermakova, E. A., et al. (2009). On propagation of excitation waves in moving media: The FitzHugh-Nagumo model. *PLoS ONE*, 4(2), e4454.