Simple zero singularity analysis in a coupled FitzHugh–Nagumo neural system with delay

Zhen, Bin; Xu, Jian

doi:10.1016/j.neucom.2009.09.015

Cited by 27 publications

(20 citation statements)

References 19 publications

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“…Thus the periodic solutions are due to coupling of the neurons, or time delay. Synchronized solutions in many works in the literature are caused by the symmetry of the proposed systems, like the study of Zhen and Xu [39], and also Li and Jiang [43]. In this paper, we show the existence of both almost synchronized and almost anti-phase solutions for non-identical neurons in different range of parameters.…”

Section: Introductionmentioning

confidence: 51%

“…Yao and Tu [38] discussed the combined effect of coupling strength and multiple delays on the stability of the rest point, and they obtained stability switches in the coupled FHN neural system with multiple delays. In addition to the study of Hopf bifurcation, Zhen and Xu [39], and also Rankovic [40] discussed the effects of the coupling strength and delay on steady state bifurcations in coupled systems with identical neurons. The Bautin bifurcation due to small delay, was analyzed by Buric and Todorovic [41].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we investigate the effect of the coupling strength and time delay on the stability and bifurcations of the system of two coupled FHN neurons. Note that neurons in the work of Zhen and Xu [39], and also Li and Jiang [43] are assumed identical which leads to a symmetry for their system of equations, that can not be found in our system. Instead the symmetry in our solutions is due to the inherent symmetry of the FHN neurons, namely our system of equations is equivariant under antipodal map.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation structure of two coupled FHN neurons with delay

Tehrani

Razvan

2015

Mathematical Biosciences

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This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.

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Section: Introductionmentioning

confidence: 51%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bifurcation structure of two coupled FHN neurons with delay

Tehrani

Razvan

2015

Mathematical Biosciences

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“…Dhamala et al (2004) studied in detail the effect of the delay on the complete synchronization of the coupled HR systems. Zhen and Xu (2010), Neefs et al (2010) touched respectively on the stability of the stationary point in a non-chemical coupled FHN and HR neural system with delay. All works but not limited to the mentioned above have promoted better understanding of the dynamics of coupled systems with delay.…”

Section: Introductionmentioning

confidence: 99%

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

Duan

Fan

2013

Cogn Neurodyn

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In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh-Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.

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“…Some results on bifurcations for the system (1.1) have been derived. [15] discussed the stability and Hopf bifurcation of double delay coupled FHN system with different coupled strength; [16] gave the results on fold bifurcation of (1.1) with f (x) = tanh(x); [17] studied Bautin bifurcation of completely synchronous FHN neurons.…”

Section: Introductionmentioning

confidence: 99%

Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

Jiang

2010

Nonlinear Dyn

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In this paper, Hopf bifurcation and Bogdanov-Takens bifurcation with codimension 2 in a coupled FitzHugh-Nagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov-Takens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov-Takens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of BogdanovTakens bifurcation indicates that the neuron signals can transit between resting and spiking.

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Simple zero singularity analysis in a coupled FitzHugh–Nagumo neural system with delay

Cited by 27 publications

References 19 publications

Bifurcation structure of two coupled FHN neurons with delay

Bifurcation structure of two coupled FHN neurons with delay

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

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