# Iterative Implicit Methods for Solving Hodgkin-Huxley Type Systems

## TL;DR

This paper investigates implicit numerical methods for solving Hodgkin-Huxley type models, demonstrating their ability to preserve system dynamics and improve accuracy through adaptive time stepping, especially in chaotic regimes.

## Contribution

It introduces implicit methods tailored for Hodgkin-Huxley models, analyzing their bifurcation behavior and effectiveness in capturing complex dynamics.

## Key findings

- Implicit methods preserve limit cycles in the model.
- Adaptive time stepping enhances accuracy in chaotic regions.
- The approach aids in understanding bifurcation patterns.

## Abstract

We are motivated to approximate solutions of a Hodgkin-Huxley type model with implicit methods. As a representative we chose a psychiatric disease model containing stable as well as chaotic cycling behaviour. We analyze the bifurcation pattern and show that some implicit methods help to preserve the limit cycles of such systems. Further, we applied adaptive time stepping for the solvers to boost the accuracy, allowing us a preliminary zoom into the chaotic area of the system.