# Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system

## TL;DR

This paper develops an analytical approach to determine the initiation of excitation waves in the FitzHugh-Nagumo model, using singular perturbation theory with two small parameters to derive a closed-form strength-duration curve.

## Contribution

It introduces a semi-analytical method employing singular perturbation with two small parameters to derive explicit ignition criteria in the FitzHugh-Nagumo system.

## Key findings

- Derived analytical expressions for the boundary between resting and excited states.
- Obtained a closed-form strength-duration curve for wave initiation.
- Validated the analytical results with numerical simulations.

## Abstract

We study the problem of initiation of excitation waves in the FitzHugh-Nagumo model. Our approach follows earlier works and is based on the idea of approximating the boundary between basins of attraction of propagating waves and of the resting state as the stable manifold of a critical solution. Here, we obtain analytical expressions for the essential ingredients of the theory by singular perturbation using two small parameters, the separation of time scales of the activator and inhibitor, and the threshold in the activator's kinetics. This results in a closed analytical expression for the strength-duration curve.