Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model

Alidousti, Javad; Ghaziani, Reza Khoshsiar

doi:10.1134/s2070048217030036

Cited by 33 publications

(11 citation statements)

References 21 publications

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“…The case (99), with {b = 0.8, = 0.08, = 0.002} has been investigated in Yadav et al (2016) and the case (100), with the same b and , has been investigated in Alidousti and Khoshsiar Ghaziani (2017). The procedures applied in Yadav et al (2016), Alidousti and Khoshsiar Ghaziani (2017), are completely different of the which ones of the present paper. In particular, boundedness and existence of absorbing sets are not considered (Fig.…”

Section: And Hencementioning

confidence: 71%

“…In the present paper, we investigate the longtime behaviour of (1) solutions, the instability of steady states and the oscillatory bifurcations onset via the Instability Coefficients Power (ICP) approach (Rionero 2020(Rionero , 2021a. Because of the neurons oscillatory activity, the onset of oscillatory bifurcation has attracted the attention of many scientists {see Wojcik and Shilnikov 2011;Yadav et al 2016;Alidousti and Khoshsiar Ghaziani 2017;Xie et al 2018;Temam 1988 and the references therein} , but the results obtained-although of sure interest-with respect to the rich dynamics of (1) generated by the seven parameters (I, , a, b, , c, d)-appear to be longely partial. Our aim is to obtain-via the instability coefficient power (ICP) approach (Rionero 2020(Rionero , 2021a-general criteria guaranteeing the existence of oscillatory Hopf bifurcations and an estimate of their locations (in order to obtain-via a simple closed form-the neurons activity frequency).…”

Section: Introductionmentioning

confidence: 99%

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Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

Rionero

2021

Rend. Fis. Acc. Lincei

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The longtime behaviour of the FitzHugh–Rinzel (FHR) neurons and the transition to instability of the FHR steady states, are investigated. Criteria guaranteeing solutions boundedness, absorbing sets, in the energy phase space, existence and steady states instability via oscillatory bifurcations, are obtained. Denoting by $$ \lambda ^{3} + \sum\nolimits_{{k = 1}}^{3} {A_{k} } (R)\lambda ^{{3 - k}} = 0 $$ λ 3 + ∑ k = 1 3 A k ( R ) λ 3 - k = 0 , with R bifurcation parameter, the spectrum equation of a steady state $$m_0$$ m 0 , linearly asymptotically stable at certain value of R, the frequency f of an oscillatory destabilizing bifurcation (neuron bursting frequency), is shown to be $$ f=\displaystyle \frac{\sqrt{A_2(R_\mathrm{H})}}{2\pi } $$ f = A 2 ( R H ) 2 π with $$R_\mathrm{H}$$ R H location of R at which the bifurcation occurs. The instability coefficient power (ICP) (Rionero in Rend Fis Acc Lincei 31:985–997, 2020; Fluids 6(2):57, 2021) for the onset of oscillatory bifurcations, is introduced, proved and applied, in a new version.

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Section: And Hencementioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

Rionero

2021

Rend. Fis. Acc. Lincei

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“…On the other hand, the two-dimensional FitzHugh-Nagumo (FHN) model [53,54] is simplified from the four-dimensional Hodgkin-Huxley model for geometrical explanation of neuronal excitability and spiking under excitation. Several studies have been performed on the significant and complex dynamical aspects of the FHN model [51,[55][56][57]. rough all these studies, the results showed that the nonautonomous memristive chaotic circuits exhibit rich dynamical behaviors, such as chaos and hyperchaos, coexisting multiple attractors, hidden attractors, and hidden extreme multistability [47][48][49][50][51][52][58][59][60][61].…”

Section: Introductionmentioning

confidence: 99%

Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

et al. 2021

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In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is found that for ω = ν , the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8 , the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors.

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“…Recently, some articles have partially studied the fractional-order Fitzhugh-Nagumo (FFHN) model [9][10][11][12], or its modified versions. In this article, we investigate the MMO based on complex canard explosion in the FFHN model.…”

Section: Introductionmentioning

confidence: 99%

Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.

Lozi

Abdelouahab²,

Chen

2020

Applied Mathematics and Nonlinear Sciences

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This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an S−asymptotically T−periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.

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Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model

Cited by 33 publications

References 21 publications

Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.

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