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Continuation and bifurcation analysis of a periodically forced excitable system
1Institut de Physique (B5a), Université de Liège, Allée du 6 Août 17, B-4000 Liège, Belgium. hcroisier@ulg.ac.be
Journal of Theoretical Biology
|March 6, 2007
Summary
The FitzHugh-Nagumo model reveals complex electrical rhythms in excitable cells under periodic stimulation. Numerical analysis shows diverse periodic solutions and connections between stable rhythms, offering insights into cellular behavior.
Area of Science:
- Computational Neuroscience
- Mathematical Biology
- Nonlinear Dynamics
Background:
- Excitable cells exhibit complex responses to periodic electrical stimulation.
- Understanding these responses is crucial for fields like cardiology and neuroscience.
- Previous studies documented various rhythmic behaviors but lacked a unified framework for their connections.
Purpose of the Study:
- To model and analyze the response of an excitable cell to periodic electrical stimulation using the FitzHugh-Nagumo model.
- To investigate the influence of stimulation amplitude and period on cellular rhythms.
- To map the complex dynamics and connections between different stable rhythms.
Main Methods:
- Utilized the FitzHugh-Nagumo (FHN) model for simulating excitable cell dynamics.
- Employed numerical continuation and bifurcation analysis with AUTO97 software.
- Studied responses to gaussian-shaped electrical pacing with varying amplitude and period.
Main Results:
- Identified diverse "M:N" periodic solutions, period-doubling sequences, and bistability.
- Tracked unstable solutions to map connections between stable rhythms in bifurcation diagrams.
- Observed complex bifurcation structures with main trees and isolated loops containing N:1 and Wenckebach rhythms.
Conclusions:
- The FitzHugh-Nagumo system exhibits rich dynamics under periodic stimulation, mirroring experimental observations in cardiac cells and axons.
- Bifurcation analysis reveals intricate relationships between various stable rhythms, including M:N and N:N-1 patterns.
- The study provides a framework for understanding the emergence of complex rhythms and their dependence on stimulation parameters.
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