Symmetric bursting behaviors in the generalized FitzHugh–Nagumo model

Abbasian, A. H.; Fallah, Haniyeh; Razvan, M. R.

doi:10.1007/s00422-013-0559-1

Cited by 21 publications

(8 citation statements)

References 38 publications

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“…Abstractly, one may think of our (effective) “third equation” as the integrated effects resulting from coupling the center cell to the rest of the cable. Unlike other conventional models for bursting [ 37 , 38 ], we find bursting to arise in this intrinsically two-variable system through coupling of the cell to an extended system. Interestingly, it has previously been pointed out that within a three-variable model bursting may emerge as a similarly transient phenomenon before the system settles into periodic spiking.…”

Section: Resultscontrasting

confidence: 73%

Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium

2015

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The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed.

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Section: Resultscontrasting

confidence: 73%

Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium

2015

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“…A more convenient and complete classification of bursters was proposed by Izhikevich, who suggested to name bursters according to the two bifurcations occurring in the fast subsystem that lead to the appearance and disappearance of repetitive spiking in the review [Izhikevich, 2000], where 12 codimension-1 bifurcations of a quiescent state that lead to repetitive spiking and 10 codimension-1 bifurcations of a spiking attractor that lead to quiescence were summed up, which thus gave rise to 120 different types of codimension-1 fast-slow bursters. By now, Izhikevich's classification method has been widely used in bursting research, see [Abbasian et al, 2013;Duan et al, 2008;Lu et al, 2008;Lu et al, 2009;Straube et al, 2006;Wagemakers et al, 2006;Zhang et al, 2013] for example.…”

Section: Introductionmentioning

confidence: 99%

Delayed Bifurcations to Repetitive Spiking and Classification of Delay-Induced Bursting

Han

Zhang

et al. 2014

Int. J. Bifurcation Chaos

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Three new routes to repetitive spiking, i.e. the delayed transcritical bifurcation, the delayed supercritical pitchfork bifurcation and the delayed subcritical pitchfork bifurcation, are revealed in this paper. We use bifurcation theory to classify bursting patterns related to three such delayed bifurcations. Then many new bursting patterns are obtained, including 24 new bursting patterns of point-point type, 27 new bursting patterns of point-cycle type and three new bursting patterns of point-torus type. Our study suggests that the classification of bursting remains to be further explored, since many new bursting patterns may be obtained based on new routes to repetitive spiking, even though we just consider codimension-1 bifurcations.

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“…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%

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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Symmetric bursting behaviors in the generalized FitzHugh–Nagumo model

Cited by 21 publications

References 38 publications

Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium

Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium

Delayed Bifurcations to Repetitive Spiking and Classification of Delay-Induced Bursting

Pattern formation in the FitzHugh–Nagumo model

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