Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

Ambrosio, Benjamin; Aziz-Alaoui, M. A.; Phan, V L E

doi:10.3934/dcdsb.2018077

Cited by 9 publications

(10 citation statements)

References 19 publications

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“…The functions H i represent excitatory nonlinear coupling (i.e. chemical synaptic coupling) between neurons, see [5,6,8,9] and references therein cited. Without loss of generality, we have set the constant C equal to 1.…”

mentioning

confidence: 99%

Propagation of Bursting Oscillations in Coupled Non-homogeneous Hodgkin–Huxley Reaction–Diffusion Systems

Ambrosio

Aziz-Alaoui

Balti

2017

Differ Equ Dyn Syst

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In this paper, we consider networks of reaction-diffusion systems of Hodgkin-Huxley type. We give a general mathematical framework, in which we prove existence and unicity of solutions as well as the existence of invariant regions and of the attractor. Then, we illustrate some relevant numerical examples and exhibit bifurcation phenomena and propagation of bursting oscillations through one and two coupled non-homogeneous systems.

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confidence: 99%

Propagation of Bursting Oscillations in Coupled Non-homogeneous Hodgkin–Huxley Reaction–Diffusion Systems

Ambrosio

Aziz-Alaoui

Balti

2017

Differ Equ Dyn Syst

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“…The results are verified using Eq. (19). It follows from the study of the amplitude equations that the antispirals exist in the vicinity of the two supercritical Hopf points and that they depend on α and β .…”

Section: B Resultsmentioning

confidence: 99%

“…The mathematical analysis for the emergence of spatial structures is important to understand a wide range of biophysical and pathological phenomena [13][14][15][16][17][18] . The work here is motivated by our earlier work in 12 and other previously studied dif-fusively coupled biophysical excitable systems, such as those in [19][20][21][22][23][24][25][26] . To the best of our knowledge, a clear analytical study describing the dynamics of a diffusively coupled, slow-fast, neuron model in which only the membrane voltage is spatially distributed, has not yet been deeply explored with respect to pattern formation and emergence of spirals.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal characteristics in systems of diffusively coupled excitable slow–fast FitzHugh–Rinzel dynamical neurons

Mondal

Sharma

et al. 2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, we study an excitable, biophysical system that supports wave propagation of nerve impulses. We consider a slow-fast, FitzHugh-Rinzel neuron model where only the membrane voltage interacts diffusively, giving rise to the formation of spatiotemporal patterns. We focus on local, nonlinear excitations and diverse neural responses in an excitable 1-and 2-dimensional configuration of diffusively coupled FitzHugh-Rinzel neurons. The study of the emerging spatiotemporal patterns is essential in understanding the working mechanism in different brain areas. We derive analytically the coefficients of the amplitude equations in the vicinity of Hopf bifurcations and characterize various patterns, including spirals exhibiting complex geometric substructures. Further, we derive analytically the condition for the development of antispirals in the neighborhood of the bifurcation point. The emergence of broken target waves can be observed to form spiral-like profiles. The spatial dynamics of the excitable system exhibits 2-and multi-arm spirals for small diffusive couplings. Our results reveal a multitude of neural excitabilities and possible conditions for the emergence of spiral-wave formation. Finally, we show that the coupled excitable systems with different firing characteristics, participate in a collective behavior that may contribute significantly to irregular neural dynamics.

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“…Still, in practice, the reaction-diffusion phenomenon can not be ignored due to the necessity of describing the behavior of substance in space. Thus reaction-diffusion neural networks have become a research hotspot in recent years [3]- [5]. On the other hand, as an extension of the integral order reaction-diffusion equation, the fractional-order reaction-diffusion equation can model more complex phenomena due to its non-local properties.…”

Section: Introductionmentioning

confidence: 99%

Synchronization of Fractional Reaction-Diffusion Neural Networks With Time-Varying Delays and Input Saturation

Wang

Liu

2021

IEEE Access

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This study is concerned with a synchronization problem of two fractional reaction-diffusion neural networks with input saturation and time-varying delays by the Lyapunov direct method. We extend the traditional ellipsoid method by giving the novel definition of the ellipsoid and linear region of the saturated, which makes our method succinct and effective. First, we linearize the saturation terms by the properties of convex hulls. Then, by using a new Lyapunov-Krasovskii functional, we give the synchronization criteria and estimate the domain of attraction. All the results are presented in the form of linear matrix inequalities(LMIs). Finally, two numerical experiments verify the validity and reliability of our method.

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Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

Abstract: We focus on the long time behavior of complex networks of reactiondiffusion systems. We prove the existence of the global attractor and the L ∞bound for networks of n reaction-diffusion systems that belong to a class that generalizes the FitzHugh-Nagumo reaction-diffusion equations.

Cited by 9 publications

References 19 publications

Propagation of Bursting Oscillations in Coupled Non-homogeneous Hodgkin–Huxley Reaction–Diffusion Systems

Propagation of Bursting Oscillations in Coupled Non-homogeneous Hodgkin–Huxley Reaction–Diffusion Systems

Spatiotemporal characteristics in systems of diffusively coupled excitable slow–fast FitzHugh–Rinzel dynamical neurons

Synchronization of Fractional Reaction-Diffusion Neural Networks With Time-Varying Delays and Input Saturation

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