Dynamics of the FitzHugh-Nagumo equation with numerical methods

Research Article
Open access

Dynamics of the FitzHugh-Nagumo equation with numerical methods

Erjia Fu 1*
  • 1 Wuhan University    
  • *corresponding author 2020300002101@whu.edu.cn
Published on 17 November 2023 | https://doi.org/10.54254/2753-8818/10/20230339
TNS Vol.10
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-131-5
ISBN (Online): 978-1-83558-132-2

Abstract

In this paper, our research focuses on investigating the behavior of the FitzHugh-Nagumo (FHN) equation dynamics, a mathematical model that describes the spiking behavior of neurons. We begin by presenting the mathematical formulation of the FHN equation as a system of two coupled ordinary differential equations. We then apply methods from the theory of dynamical systems to analyze the behavior of the system, including stability analysis and phase-plane analysis. We also discuss the bifurcations that occur in the system as its parameters are varied. In addition, we present numerical methods for solving the FHN equation, including explicit and implicit methods. We compare the accuracy and efficiency of these methods and discuss their suitability for different types of problems.Our results show that the FHN equation exhibits a rich range of dynamical behaviors, including periodic and chaotic solutions, bistability, and hysteresis.Additionally, FHN equation can be used to simulate the behavior of networks of excitable cells, such as neural circuits or cardiac tissue,and FHN models can also be used to study the effects of drugs or other interventions on the behavior of excitable cells. The numerical methods we present provide a powerful tool for studying the FHN equation and its variants, allowing us to explore the parameter space and predict the behavior of the system under different conditions.

Keywords:

FitzHugh-Nagumo equation, dynamical systems, numerical methods,bifurcation analysis

Fu,E. (2023). Dynamics of the FitzHugh-Nagumo equation with numerical methods. Theoretical and Natural Science,10,179-185.
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References

[1]. R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6):445–466, 1961.

[2]. Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon[J]. Proceedings of the IRE, 1962, 50(10): 2061-2070.

[3]. M. Morris W. Hirsch, Stephen Smale, Robert L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos, [M]. Oxford, Elsevier, 2013

[4]. M. Jinyan Zhang, Beiye Feng, The Stability Theory of Ordinary Differential Equations and Bifurcation Problems, [M]. Beijing, Beijing University Press, 2008.

[5]. Hoff A, dos Santos J V, Manchein C, et al. Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators[J]. The European Physical Journal B, 2014, 87: 1-9.

[6]. Celestino A, Manchein C, Albuquerque H A, et al. Stable structures in parameter space and optimal ratchet transport[J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(1): 139-149.

[7]. Medeiros E S, Medrano-T R O, Caldas I L, et al. Torsion-adding and asymptotic winding number for periodic window sequences[J]. Physics Letters A, 2013, 377(8): 628-631.

[8]. Liu F, Turner I, Anh V, et al. A numerical method for the fractional Fitzhugh–Nagumo monodomain model[J]. Anziam Journal, 2012, 54: C608-C629.

[9]. Yu Q, Liu F, Turner I, et al. Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013, 371(1990): 20120150

[10]. Bonaventura L, Fernández-García S, Gómez-Mármol M. Efficient implicit solvers for models of neuronal networks[J]. arXiv preprint arXiv:2210.01697, 2022.

[11]. Bandera A, Fernández-García S, Gómez-Mármol M, et al. A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations[J]. Mathematical Modelling of Natural Phenomena, 2022, 17: 11.

[12]. Bonaventura L, Mármol M G. The TR-BDF2 method for second order problems in structural mechanics[J]. Computers & Mathematics with Applications, 2021, 92: 13-26.

Cite this article

Fu,E. (2023). Dynamics of the FitzHugh-Nagumo equation with numerical methods. Theoretical and Natural Science,10,179-185.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN：978-1-83558-131-5(Print) / 978-1-83558-132-2(Online)
Editor：Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.10
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6):445–466, 1961.

[2]. Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon[J]. Proceedings of the IRE, 1962, 50(10): 2061-2070.

[3]. M. Morris W. Hirsch, Stephen Smale, Robert L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos, [M]. Oxford, Elsevier, 2013

[4]. M. Jinyan Zhang, Beiye Feng, The Stability Theory of Ordinary Differential Equations and Bifurcation Problems, [M]. Beijing, Beijing University Press, 2008.

[5]. Hoff A, dos Santos J V, Manchein C, et al. Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators[J]. The European Physical Journal B, 2014, 87: 1-9.

[6]. Celestino A, Manchein C, Albuquerque H A, et al. Stable structures in parameter space and optimal ratchet transport[J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(1): 139-149.

[7]. Medeiros E S, Medrano-T R O, Caldas I L, et al. Torsion-adding and asymptotic winding number for periodic window sequences[J]. Physics Letters A, 2013, 377(8): 628-631.

[8]. Liu F, Turner I, Anh V, et al. A numerical method for the fractional Fitzhugh–Nagumo monodomain model[J]. Anziam Journal, 2012, 54: C608-C629.

[9]. Yu Q, Liu F, Turner I, et al. Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013, 371(1990): 20120150

[10]. Bonaventura L, Fernández-García S, Gómez-Mármol M. Efficient implicit solvers for models of neuronal networks[J]. arXiv preprint arXiv:2210.01697, 2022.

[11]. Bandera A, Fernández-García S, Gómez-Mármol M, et al. A multiple timescale network model of intracellular calcium concentrations in coupled neurons: Insights from ROM simulations[J]. Mathematical Modelling of Natural Phenomena, 2022, 17: 11.

[12]. Bonaventura L, Mármol M G. The TR-BDF2 method for second order problems in structural mechanics[J]. Computers & Mathematics with Applications, 2021, 92: 13-26.

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