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Iterative Implicit Methods for Solving Hodgkin-Huxley Type Systems

Juergen Geiser, Dennis Ogiermann

TL;DR

The paper develops and analyzes iterative implicit time-stepping methods for Hodgkin-Huxley type systems, aiming to preserve limit cycles in regimes that include chaotic dynamics. It introduces a representative 4D nonlinear disease dynamics HH-type model and implements several semi-implicit and iterative schemes (CN, ICN, ISV, MMRK4/AIRK4) with adaptive time stepping controlled by PID and multi-scale step controllers. Through computational bifurcation analysis, Lyapunov spectra, Poincaré sections, and interspike interval distributions, the study identifies Hopf- and chaos-like transitions and assesses how different solvers affect the accessibility and clarity of these structures. The results indicate that structure-preserving, higher-order adaptive methods better retain the qualitative behavior of the system, while some adaptive schemes can blur chaotic features, providing a framework for solver design in HHT-type systems and prompting future stochastic and rigorous ISI-bifurcation work.

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.

Iterative Implicit Methods for Solving Hodgkin-Huxley Type Systems

TL;DR

The paper develops and analyzes iterative implicit time-stepping methods for Hodgkin-Huxley type systems, aiming to preserve limit cycles in regimes that include chaotic dynamics. It introduces a representative 4D nonlinear disease dynamics HH-type model and implements several semi-implicit and iterative schemes (CN, ICN, ISV, MMRK4/AIRK4) with adaptive time stepping controlled by PID and multi-scale step controllers. Through computational bifurcation analysis, Lyapunov spectra, Poincaré sections, and interspike interval distributions, the study identifies Hopf- and chaos-like transitions and assesses how different solvers affect the accessibility and clarity of these structures. The results indicate that structure-preserving, higher-order adaptive methods better retain the qualitative behavior of the system, while some adaptive schemes can blur chaotic features, providing a framework for solver design in HHT-type systems and prompting future stochastic and rigorous ISI-bifurcation work.

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.

Paper Structure

This paper contains 17 sections, 1 theorem, 39 equations, 8 figures, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Hodgkin-Huxley Type Models
  4. A Hodgkin-Huxley Type Nonlinear Disease Dynamics Model
  5. Numerical Methods
  6. Composition with respect to Hamiltonian Systems
  7. Iterative Schemes Based on Finite Difference Schemes
  8. Semi-implicit Integrators
  9. Adaptive Time Step Control of the Iterative CN Scheme
  10. PID-Controller
  11. Classical Time Step Controller for the ICN
  12. Time Step Controller for the Runge-Kutta Methods
  13. Numerical Results
  14. Exploring Structural Properties via Computational Bifurcation Analysis
  15. Convergence Study
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

We deal with 4th order time-integrator methods with tolerance $\varepsilon$. Further, we assume that we have a 4th order numerical solver, which is give as ${\bf u}(t + \Delta t) = A_{\Delta t} \; {\bf u}(t)$ and ${\bf u}(t)$ is the exact solution at time $t$. We apply the $|| \cdot||_p$-norm as a g

Figures (8)

  • Figure 1: Evolution of the system's Jacobian's eigenvalues for some S. The increment between consecutive S is 5.
  • Figure 2: The Lyapunov spectrum of the disease dynamics model for different choices of S.
  • Figure 3: Approximations of the disease dynamics model with the ICN solver (algorithm \ref{['algo:icn']}) and various S. Six approximations for interval $[0,500]$ and initial condition the zero vector, i.e. $u(0) = (0, 0, 0, 0)$, can be seen in pairs of two images, where the left image is the observable $x$ and the left one contains the activation vector ${\bf a}$. We have chosen a tolerance $\varepsilon=10^{-7}$, a time step $\Delta t=0.01$ and a maximum number of iterations $I=10$.
  • Figure 4: The Poincare section for the disease dynamics with the hyperplane $<(1,0,0,0),{\bf u}> = 40$ with increments of 1 on S over the previously mentioned region of interest $[0,400]$.
  • Figure 5: The interspike interval for the disease dynamics with the hyperplane $<(1,0,0,0),u> = 40$ with increments of 1 on S over the previously mentioned region of interest $[0,400]$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof