On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

Ringkvist, Mattias; Zhou, Yishao

doi:10.1016/j.na.2009.01.149

Cited by 19 publications

(23 citation statements)

References 19 publications

(33 reference statements)

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“…(iii) Assuming that δ(k 2 ) < 0 for some k 2 ̸ ≡ 0, we can achieve a diffusion-driven instability and also get the minimum value of δ k about k 2 to be negative, and get critical value…”

Section: Lemma 2 (The Intermediate Value Theorem Of Continuous Functimentioning

confidence: 99%

“…Some mathematical models for biological neurons which represent neuronal behavior in terms of membrane potentials have been developed such as Hodgkin-Huxley model (1952), FitzHugh model (1969), Morris-Lecar model (1981), Hindmarsh-Rose model (1984), especially Hodgkin-Huxley model which is the motivation for the FitzHugh-Nagumo equation that extracts the essential behavior in a simple form [1][2][3]. A. Yazdan, G. Mehrdad and M. Ghasem have used the cellular automata method to simulate the pattern formation of FitzHugh-Nagumo model and considered the effects of different parameters of the FitzHugh-Nagumo model on changing the initial pattern [4].…”

Section: Introductionmentioning

confidence: 99%

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Pattern formation in the FitzHugh–Nagumo model

Zheng

Shen

2015

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, we investigate the effect of diffusion on pattern formation in FitzHughNagumo model. Through the linear stability analysis of local equilibrium we obtain the condition how the Turing bifurcation, Hopf bifurcation and the oscillatory instability boundaries arise. By using the method of the weak nonlinear multiple scales analysis and Taylor series expansion, we derive the amplitude equations of the stationary patterns. The analysis of amplitude equations shows the occurrence of different complex phenomena, including Turing instability Eckhaus instability and zigzag instability. In addition, we apply this analysis to FitzHugh-Nagumo model and find that this model has very rich dynamical behaviors, such as spotted, stripe and hexagon patterns. Finally, the numerical simulation shows that the analytical results agree with numerical simulation.

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Section: Lemma 2 (The Intermediate Value Theorem Of Continuous Functimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pattern formation in the FitzHugh–Nagumo model

Zheng

Shen

2015

Computers & Mathematics with Applications

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“…From literature review, it is noticed that, most of the articles studied the system under some parameters being zeros, for instance see 4,5,6,7,9,10,11 . Luo Ding Jun in 8 investigated the particular of case 0 = ) (1   , and proved the uniqueness of limit cycle.…”

Section: R I mentioning

confidence: 99%

“…Luo Ding Jun in 8 investigated the particular of case 0 = ) (1   , and proved the uniqueness of limit cycle. In 12 there is a general analysis of the system for bifurcation of limit cycles from Hopf-bifurcation. In 13 , we studied the system (1.2) with all parameters not zeros and proved the uniqueness of limit cycle.…”

Section: R I mentioning

confidence: 99%

A study on limit cycle and non-homoclinic orbits for FitzHugh-Nagumo System

Saeed¹

2017

jusps-A

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“…Therefore the cusp bifurcation of equilibrium cannot occur. In papers [30,31], the authors studied the existence and number of limit cycles in the FitzHugh-Nagumo system: the main result of [31] is that the FitzHughNagumo system has at most two limit cycles bifurcated from equilibrium via Hopf bifurcation. In this paper, we consider the two-dimensional HindmarshRose model, which is a modification of FitzHughNagumo system, and give a rigorous mathematical analysis of codimension-2 bifurcations of this model.…”

Section: Remarks and Conclusionmentioning

confidence: 99%

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

Liu

2011

Nonlinear Dyn

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In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh-Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov-Takens bifurcation which includes a saddle-node bifurcation, an Andronov-Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.

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On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

Cited by 19 publications

References 19 publications

Pattern formation in the FitzHugh–Nagumo model

Pattern formation in the FitzHugh–Nagumo model

A study on limit cycle and non-homoclinic orbits for FitzHugh-Nagumo System

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

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