Instability of periodic traveling wave solutions in a modified FitzHugh–Nagumo model for excitable media

Gani, M. Osman; Ogawa, Teiichiro

doi:10.1016/j.amc.2015.01.109

Cited by 16 publications

(10 citation statements)

References 44 publications

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“…In this paper, we propose a reaction-diffusion system for excitable media to mimic cardiac electrical activities as reported in [ 37 ]. The model consists of two equations describing fast and slow dynamics of the system and it is given as follows:

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Section: Mathematical Model and Methods Of Computationmentioning

confidence: 99%

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study

Gani

Ogawa

2014

International Scholarly Research Notices

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The determination of the mechanisms of spiral breakup in excitable media is still an open problem for researchers. In the context of cardiac electrophysiological activities, spiral breakup exhibits complex spatiotemporal pattern known as ventricular fibrillation. The latter is the major cause of sudden cardiac deaths all over the world. In this paper, we numerically study the instability of periodic planar traveling wave solution in two dimensions. The emergence of stable spiral pattern is observed in the considered model. This pattern occurs when the heart is malfunctioning (i.e., ventricular tachycardia). We show that the spiral wave breakup is a consequence of the transverse instability of the planar traveling wave solutions. The alternans, that is, the oscillation of pulse widths, is observed in our simulation results. Moreover, we calculate the widths of spiral pulses numerically and observe that the stable spiral pattern bifurcates to an oscillatory wave pattern in a one-parameter family of solutions. The spiral breakup occurs far below the bifurcation when the maximum and the minimum excited states become more distinct, and hence the alternans becomes more pronounced.

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Section: Mathematical Model and Methods Of Computationmentioning

confidence: 99%

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study

Gani

Ogawa

2014

International Scholarly Research Notices

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“…Proof. From Theorem 4.1, the stability and dynamic transition of the system (7) are determined by the reduced equations (23). At the critical value α =α c , we have β 1 1q = 0.…”

Section: 1mentioning

confidence: 99%

“…The system is obtained as a simplification for the Hodgkin-Huxley model describing nerve impulse propagation. Due to the essential feature to describe the initiation and propagation of action potentials in neurons [12], the FN system has been investigated from different angles including bifurcation [12,8,10,17,27], traveling wave solutions [1,3,4,7,25,26] and other dynamic aspects [2,23]. Schonbek [23] studies the local and global existence of solutions for the FN equations.…”

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confidence: 99%

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

Xing¹,

Pan²,

Wang³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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The article aims to examine the dynamic transition of the reactiondiffusion Fitzhugh-Nagumo system defined on a thin spherical shell and a 2Drectangular domain. The mathematical tool employed is the theory of phase transition dynamics established for dissipative dynamical systems. The main results in this paper include two parts. First, for the system on a thin spherical shell, we only focus on the transition from a real simple eigenvalue. More precisely, if the first eigenspace is three-dimensional, the system undergoes either a continuous transition or a jump transition. Besides, a mix transition is also allowed if the first eigenspace is one-dimensional. Second, for the system on a rectangular domain, both the transitions from a simple real eigenvalue and a pair of simple complex eigenvalues are considered. Our results imply that two steady-state solutions bifurcate, which are either attractors or saddle points, and a Hopf bifurcation is also possible in the system on the rectangular domain.

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“…Generally stability change of the PTWs of partial differential equations (PDEs) has two types: Eckhaus type and Hopf type. A change of stability is said to be Eckhaus (sideband) type when the spectrum of the solution changes sign at the origin.However, if a fold is generated in the curvature of the spectrum and crosses the imaginary axis away from the origin is known as a stability change of Hopf type [22], [23].…”

Section: Stability Of Periodic Traveling Wave Solutionsmentioning

confidence: 99%

Numerical Bifurcation Analysis to Study Periodic Traveling Wave Solutions in a Model of Young Mussel Beds

Arif

Gani

2019

GANIT: J. Bangladesh Math. Soc.

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Self-bottomed mussel beds are dominant feature of ecosystem-scale self-organization. Regular spatial patterns of mussel beds in inter-tidal zone are typical, aligned perpendicular to the average incoming tidal flow. In this paper, we consider a two-variable partial differential equations model of young mussel beds. Our aim is to study the existence and stability of periodic traveling waves in a one-parameter family of solutions. We consider a parameter regime to show pattern existence in the model of young mussel beds. In addition, it is found that the periodic traveling waves changes their stability by two ways: Hopf type and Eckhaus type. We explain this stability by the calculation of essential spectra at different grid points in the two-dimensional parameter plane. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 1-10

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Instability of periodic traveling wave solutions in a modified FitzHugh–Nagumo model for excitable media

Cited by 16 publications

References 44 publications

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

Numerical Bifurcation Analysis to Study Periodic Traveling Wave Solutions in a Model of Young Mussel Beds

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