Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

Wu, Kaijun; Luo, Tianqi; Lu, Hailong; Wang, Yang

doi:10.1007/s00521-015-1892-1

Cited by 48 publications

(17 citation statements)

References 26 publications

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“…In the past three decades, numerous simplified neuron models had been fantastically extended from the classical Hodgkin-Huxley model [1] to reconstruct the main dynamical characteristics of neuronal electrical activities [2][3][4][5][6][7][8], among which the two-and three-dimensional Hindmarsh-Rose (HR) neuron models are effective and available for dynamical analysis in electrical activities of biological neurons [9,10]. In the last few years, a wide variety of the HR neuron models, such as original three-dimensional HR models [10][11][12][13][14][15][16], extended or nonlinear feedback coupled HR models [17][18][19][20], time delayed HR models [20][21][22], fractional-order HR models [23,24], and memristor based HR models under electromagnetic radiations [9,[25][26][27], have been proposed and further studied by bifurcation analysis methods for understanding the dynamics of electrical activities among neurons [8]. For this reason, bifurcation analysis theory plays an essential role in describing mode transitions between spiking and bursting in the neuronal electrical activities [9][10][11][12][13][14][15][16][17][18]…”

Section: Introductionmentioning

confidence: 99%

“…In the last few years, a wide variety of the HR neuron models, such as original three-dimensional HR models [10][11][12][13][14][15][16], extended or nonlinear feedback coupled HR models [17][18][19][20], time delayed HR models [20][21][22], fractional-order HR models [23,24], and memristor based HR models under electromagnetic radiations [9,[25][26][27], have been proposed and further studied by bifurcation analysis methods for understanding the dynamics of electrical activities among neurons [8]. For this reason, bifurcation analysis theory plays an essential role in describing mode transitions between spiking and bursting in the neuronal electrical activities [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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Three‐Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Bao

et al. 2018

Complexity

128

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Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three-dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Three‐Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Bao

et al. 2018

Complexity

128

View full text Add to dashboard Cite

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“…For example, Hindmarsh and Rose (HR) [16,17] discovered the periodic spiking firing and periodic bursting firing in a 2-dimension (2D) and a 3-dimension (3D) nerve impulse models, respectively. In [18], the generating mechanisms of periodic bursting firing and chaotic bursting firing are revealed in a simplified HR neuron model. And Lakshmanan and his team found that the periodic and chaotic spiking, and the periodic and chaotic burst-ing can be observed based on the HR neuron with time delays [19].…”

Section: Introductionmentioning

confidence: 99%

Firing multistability in a locally active memristive neuron model

et al. 2020

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The theoretical, numerical and experimental demonstrations of firing dynamics in isolated neuron are of great significance for the understanding of neural function in human brain. In this paper, a new type of locally active and non-volatile memristor with three stable pinched hysteresis loops is presented. Then a novel locally active memristive neuron model is established by using the locally active memristor as a connecting autapse, both firing patterns and multistability in this neuronal system are investigated. We have confirmed that, on the one hand, the construced neuron can generate multiple firing patterns like periodic bursting, periodic spiking, chaotic bursting, chaotic spiking, stochastic bursting, transient chaotic bursting and transient stochastic bursting. On the other hand, the phenomenon of firing multistability with coexisting four kinds of firing patterns can be observed via changing its initial states. It is worth noting that the proposed neuron exhibits such firing multistability previously unobserved in single neuron model. Finally, an electric neuron is designed and implemented, which is extremely useful for the practical scientific and engineering applications. The results captured from neuron hardware experiments match well with the theoretical and numerical simulation results.

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“…During the last ten years, many researchers have studied the complex dynamics of a Hindmarsh-Rose-type system with the help of bifurcation theory. Wu et al [8] discussed the bifurcation pictures of a kind of the modified Hindmarsh-Rose model as one or two parameters vary. Period-adding bifurcation, period-doubling bifurcation, and intermittent chaotic phenomena were computed numerically and plotted to be observed clearly.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of a two-dimensional discrete Hindmarsh–Rose type model

2019

Adv Differ Equ

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In this paper, bifurcation analysis of a discrete Hindmarsh-Rose model is carried out in the plane. This paper shows that the model undergoes a flip bifurcation, a Neimark-Sacker bifurcation, and 1 : 2 resonance which includes a pitchfork bifurcation, a Neimark-Sacker bifurcation, and a heteroclinic bifurcation. The sufficient conditions of existence of the fixed points and their stability are first derived. The flip bifurcation and Neimark-Sacker bifurcation are analyzed by using the inner product method and normal form theory. The conditions for the occurrence of 1 : 2 resonance are also presented. Furthermore, the sufficient conditions of pitchfork, Neimark-Sacker, and heteroclinic bifurcations are derived and expressed by implicit functions. The numerical analysis shows us consistence with the theoretical results and exhibits interesting dynamics, especially symmetric and invariant closed orbits. The dynamics observed in this paper can be used to mimic the dynamical behaviors of one single neuron and design a humanoid locomotion model for applications in bio-engineering and so on.

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Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

Cited by 48 publications

References 26 publications

Three‐Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Three‐Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Firing multistability in a locally active memristive neuron model

Bifurcation analysis of a two-dimensional discrete Hindmarsh–Rose type model

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