FitzHugh-Nagumo 1D (`fitzhugh_nagumo_1d`)

Excitable media: the model of neurons and cardiac tissue, and the model of threshold behaviour generally. A small perturbation decays; a perturbation past the threshold launches a travelling pulse that propagates without decay and leaves a refractory tail behind it. That threshold is a genuine discontinuity in the input-to-output map, and it is what makes the operator task interesting rather than a smoothing exercise.

Equation

\[\frac{\partial u}{\partial t} = D_u\,\frac{\partial^2 u}{\partial x^2} + u - u^3 - v\]

\[\frac{\partial v}{\partial t} = D_v\,\frac{\partial^2 v}{\partial x^2} + \epsilon\,(u - \gamma v + \beta)\]

with periodic boundary conditions. Here \(u\) is the fast activator, standing for membrane potential, and \(v\) is the slow inhibitor, standing for recovery.

Operator learning task

\[u_0 \mapsto (u_T, v_T)\]

The inhibitor starts from its rest state rather than being drawn, so the activator field is the only varying input.

Parameters

Parameter	Default	Range	Description
`epsilon`	0.08	(0.01, 0.5)	Timescale separation; smaller gives sharper pulses and slower recovery
`diffusivity_u`	1.0	(0.01, 10.0)	Activator diffusion \(D_u\), which sets the wave speed
`time_end`	50.0	(1.0, 200.0)	Final time

Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="fitzhugh_nagumo_1d",
    n_samples=500,
    resolution={"x": 256},
    params={"epsilon": 0.08, "diffusivity_u": 1.0, "time_end": 50.0},
    seed=42,
)

Behaviour

The timescale separation \(\epsilon\) is what makes the medium excitable. When \(\epsilon \ll 1\) the inhibitor lags far behind the activator, so a pulse has time to form and travel before recovery catches up. Raising \(\epsilon\) towards 0.5 removes that separation and the pulses stop being sharp.

An excitation seed has to be wide enough to launch a pulse at all: roughly 8 units of \(\sqrt{D_u}\). Narrower stimuli decay whatever their amplitude, which is worth knowing when designing an input measure, since a measure that mostly produces sub-threshold perturbations will produce a dataset of decays.

Data shapes

dataset.inputs.shape   # (n_samples, nx)      u0
dataset.outputs.shape  # (n_samples, nx, 2)   u_T, v_T on a trailing axis

The inhibitor is initialised internally rather than drawn, so only \(u_0\) comes back as the input; both components are returned.

fitzhugh_nagumo_2d: spirals become possible once a front can break.
allen_cahn_1d: the same cubic nonlinearity without the inhibitor, giving stationary interfaces instead of travelling pulses.
gray_scott_2d: the other two-component reaction-diffusion system.

FitzHugh-Nagumo 1D (fitzhugh_nagumo_1d)