Periodic Oscillation Analysis for a Coupled FHN Network Model with Delays

Lin, Yuanhua

doi:10.1155/2013/276972

Cited by 4 publications

(4 citation statements)

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“…Recently, some criteria to determine the periodic oscillation were provided in the multiple delayed FHN neural system with three nonidentical cells [19]. In [20], the Hopf bifurcation and Bogdanov-Takens bifurcation were investigated in a coupled FHN neural system with gap junction.…”

Section: Abstract and Applied Analysismentioning

confidence: 99%

“…The simple zero singularity (both pitchfork and transcritical bifurcations) was studied employing the center manifold reduction and normal form method [17]. Further, by applying the Bautin bifurcation theorem, the coexistence of rest state and periodic spiking was reported in the synchronous solution of a coupled FHN neural system with delay [18].Recently, some criteria to determine the periodic oscillation were provided in the multiple delayed FHN neural system with three nonidentical cells [19]. In [20], the Hopf bifurcation and Bogdanov-Takens bifurcation were investigated in a coupled FHN neural system with gap junction.…”

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Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays

Yao

2014

Abstract and Applied Analysis

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A FitzHugh-Nagumo (FHN) neural system with multiple delays has been proposed. The number of equilibrium point is analyzed. It implies that the neural system exhibits a unique equilibrium and three ones for the different values of coupling weight by employing the saddle-node bifurcation of nontrivial equilibrium point and transcritical bifurcation of trivial one. Further, the stability of equilibrium point is studied by analyzing the corresponding characteristic equation. Some stability criteria involving the multiple delays and coupling weight are obtained. The results show that the neural system exhibits the delay-independence and delay-dependence stability. Increasing delay induces the stability switching between resting state and periodic activity in some parameter regions of coupling weight. Finally, numerical simulations are taken to support the theoretical results.

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Section: Abstract and Applied Analysismentioning

confidence: 99%

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confidence: 99%

Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays

Yao

2014

Abstract and Applied Analysis

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“…In the actual nerve conduction process, there is a certain time delay in signal transmission, which caused a lot of research on the FHN model with time delay. Wang et al studied bifurcation and synchronization ( 1 ), bifurcation structure ( 2 ), Fold-Hopf bifurcation ( 3 ), periodic oscillation ( 4 ), and global Hopf bifurcation ( 5 ) of coupled FHN model with time delay. Yu et al ( 6 ) found that the noise level can change the signal transmission performance in the FHN network, and the delay can cause multiple stochastic resonances.…”

Section: Introductionmentioning

confidence: 99%

Turing instability mechanism of short-memory formation in multilayer FitzHugh-Nagumo network

Wang

Shen

2023

Front. Psychiatry

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IntroductionThe study of brain function has been favored by scientists, but the mechanism of short-term memory formation has yet to be precise.Research problemSince the formation of short-term memories depends on neuronal activity, we try to explain the mechanism from the neuron level in this paper.Research contents and methodsDue to the modular structures of the brain, we analyze the pattern properties of the FitzHugh-Nagumo model (FHN) on a multilayer network (coupled by a random network). The conditions of short-term memory formation in the multilayer FHN model are obtained. Then the time delay is introduced to more closely match patterns of brain activity. The properties of periodic solutions are obtained by the central manifold theorem.ConclusionWhen the diffusion coeffcient, noise intensity np, and network connection probability p reach a specific range, the brain forms a relatively vague memory. It is found that network and time delay can induce complex cluster dynamics. And the synchrony increases with the increase of p. That is, short-term memory becomes clearer.

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“…The stability and oscillation of the solutions is investigated by constructing of Liapunov functional and applying the Chafees limit cycle criterion. Zhen and Xu [42] and also Lin [50] generalized the FHN models to a three coupled FHN neurons and Feng and Plamondon [51] to a four coupled FHN neurons with time delay. They derived some criteria to determine the periodic oscillation were provided in the multiple delayed FHN neural system with nonidentical cells.…”

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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Periodic Oscillation Analysis for a Coupled FHN Network Model with Delays

Cited by 4 publications

References 4 publications

Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays

Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays

Turing instability mechanism of short-memory formation in multilayer FitzHugh-Nagumo network

Bifurcation structure of two coupled FHN neurons with delay

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