Patterns of interaction of coupled reaction–diffusion systems of the FitzHugh–Nagumo type

Zhang, Chunrui; Ke, Ai; Zheng, Baodong

doi:10.1007/s11071-019-05065-8

Cited by 17 publications

(4 citation statements)

References 28 publications

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“…This model is commonly referred to as the FHN model. In recent years, many researchers have explored the dynamics using the FHN model [3][4][5][6]. Turing theory, originally proposed by Alan Turing for chemical systems, has been applied in various fields such as ecology [7][8], physics [9] and others [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Turing Instability of the Fitzhugh-Nagumo Model in Diffusive Network

Gao

2024

JAMCS

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This study mainly investigates the dynamical analysis of the FitzHugh-Nagumo (FHN) neuron model. Firstly, it analyzes the equilibrium stability of the system in the absence of network diffusion. Then, it considers two types of network topologies: random networks and higher-order networks. The paper analyzes the Turing instability phenomenon in the presence of network diffusion, identifies the critical diffusion coefficient in the FHN model that leads to Turing instability, and plots the eigenvalue distribution diagram, known as the Turing pattern. The research findings indicate that networks with higher-order connections, as opposed to random networks, display a more intricate interplay among neurons. This heightened interconnection intensifies the Turing instability phenomenon, amplifying its significance within the system. The stability of the dynamical system can be associated with the onset of neurological disorders such as epilepsy, caused by abnormal neuronal firing. This analogy facilitates the transfer of content related to the instability of control systems to the regulation of neurological disorders.

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Section: Introductionmentioning

confidence: 99%

Analysis of Turing Instability of the Fitzhugh-Nagumo Model in Diffusive Network

Gao

2024

JAMCS

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“…In recent years, there exists existing literature that reported the Turing-Hopf bifurcation, please refer to the Refs. [24][25][26][27][28][29][30]. However, there is no literature studies the Turing-Hopf bifurcation and spatiotemporal inhomogeneous pattern of the reaction-diffusion model (1) with chemotaxis and time delay.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal inhomogeneous pattern of a predator-prey model with delay and chemotaxis

Chen

2023

Preprint

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In this paper, we investigate the spatiotemporal inhomogeneous pattern phenomenon of a predatorprey model with chemotaxis and time delay. The precise intervals of the Turing instability are yielded so that we can obtain sufficient conditions to ensure the existence of the Turing instability by adjusting the ranges of the control parameters of time delay and chemotaxis. Also, we perform the occurrence conditions of the Turing-Hopf bifurcation by employing the time delay control parameter and chemotaxis sensitivity coefficient as the bifurcation parameter. Theoretically, there are no Hopf bifurcation, Turing instability, and Turing-Hopf bifurcation as the chemotaxis effect and time delay are absent. Of course, complicated spatiotemporal inhomogeneous patterns are displayed with the help of numerical experiments. Particularly, we find that both the time delay control parameter and chemotaxis sensitivity coefficient can affect the formation of the spatiotemporal inhomogeneous patterns. Certainly, our method could be applied in other reaction-diffusion models with the chemotaxis and full/non-full time delay in reaction kinetics terms.

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“…Recently, properties and optimal control of stochastic FHN (SFHN) models are extensively studied only in the finite dimensional space (see the references [10][11][12][13][14][15][16] ). In specific, the solvability and optimal control of SFHN model is studied by Barbu et al 17 ; optimal control of SFHN model with nonlinear diffusion term is studied by Cordoni and Di Persio 18 via rescaling transformation, Ekeland's variational principle, and big-bang optimal control theory.…”

Section: Introductionmentioning

confidence: 99%

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

Durga

Malik

2023

Math Methods in App Sciences

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This article is devoted to examine the exponential behavior of stochastic FitzHugh-Nagumo equation in Hilbert space. Initially, the proposed model is reformulated into an abstract space, and the existence of mild solution is constructed by employing Mönch fixed-point theorem and Hausdorff measure of non-compactness. Furthermore, suitable conditions are developed to prove the proposed model's exponential behavior which reflects the stable generation and transmission of electric signal in neurons. At last, an example is presented to verify the established theoretical concepts.

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Patterns of interaction of coupled reaction–diffusion systems of the FitzHugh–Nagumo type

Cited by 17 publications

References 28 publications

Analysis of Turing Instability of the Fitzhugh-Nagumo Model in Diffusive Network

Analysis of Turing Instability of the Fitzhugh-Nagumo Model in Diffusive Network

Spatiotemporal inhomogeneous pattern of a predator-prey model with delay and chemotaxis

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

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