Dynamics in a Coupled FHN Model with Two Different Delays

Xu, Changjin; Wu, Yusen; Lu, Lin

doi:10.4304/jcp.9.8.1834-1842

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…The Fold-Hopf bifurcation is investigated in a coupled FHN neural system with delay by Zhen and Xu [49]. Fan and Hong [14] considered the stability and Hopf bifurcation of double delay coupled FHN neurons, see also [46]. The steady state bifurcations of two coupled FHN neurons due to coupling strength and small time delay is investigated in [50] and [36].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Tehrani¹,

Razvan²

2016

Preprint

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This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with quadratic term and delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics. Bifurcation diagrams are obtained numerically or analytically from the mathematical model, and the parameter regions of different behaviors are clarified. The neural system exhibits a unique rest point or three ones by employing the saddle-node bifurcation, when strong coupling is applied in the system. Also the trivial rest point shows transcritical bifurcation with one of the new rest points. The asymptotic stability and possible Hopf and Bautin bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Fold cycle, torus, fold-torus, and big homoclinic bifurcations of limit cycles, together with 1 : 1 and 1 : 4 resonances are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf, Hopf-transcritical, and double-zero bifurcation points can be obtained from the intersection of different bifurcation branches. Various patterns of multistability have been observed, both for small and large values of delay. The system may exhibit one synchronous together with one or two anti-phase periodic activities, two synchronous and two anti-phase periodic solutions, one synchronous and one anti-phase periodic solution also one equilibrium, and one anti-phase periodic solution and non-trivial eguilibria, which occur due to Hopf, fold cycle and torus bifurcations. Also one synchronous periodic solution and one torus due to fold cycle, torus and Chenciner bifurcations can be seen. By increasing

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Section: Introductionmentioning

confidence: 99%

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Tehrani¹,

Razvan²

2016

Preprint

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“…However, only codimension-1 bifurcations are discussed in these papers. Fan and Hong [36], and also Xu et al [37], considered the stability and Hopf bifurcation of double delay coupled FHN system with different coupling strength. 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.…”

Section: Introductionmentioning

confidence: 99%

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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Dynamics in a Coupled FHN Model with Two Different Delays

Cited by 2 publications

References 23 publications

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Bifurcation analysis and multistability detection of two delay-coupled FHN neurons

Bifurcation structure of two coupled FHN neurons with delay

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