Bifurcation analysis and diverse firing activities of a modified excitable neuron model

Mondal, Argha; Upadhyay, Ranjit Kumar; Ma, Jun; Yadav, Binesh Kumar; Sharma, Sanjeev Kumar

doi:10.1007/s11571-019-09526-z

Cited by 77 publications

(20 citation statements)

References 50 publications

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“…In fact, the critical value of equilibrium's number corresponds to a static bifurcation [19,20]. So, in Section 3, we will describe the change of the number of equilibria by bifurcation analysis.…”

Section: Saddle-node Bifurcation Of Nontrivial Equilibriummentioning

confidence: 99%

Complexity Induced by External Stimulations in a Neural Network System with Time Delay

Zhen

Zhang

Song

2020

Mathematical Problems in Engineering

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Complexity and dynamical analysis in neural systems play an important role in the application of optimization problem and associative memory. In this paper, we establish a delayed neural system with external stimulations. The complex dynamical behaviors induced by external simulations are investigated employing theoretical analysis and numerical simulation. Firstly, we illustrate number of equilibria by the saddle-node bifurcation of nontrivial equilibria. It implies that the neural system has one/three equilibria for the external stimulation. Then, analyzing characteristic equation to find Hopf bifurcation, we obtain the equilibrium’s stability and illustrate periodic activity induced by the external stimulations and time delay. The neural system exhibits a periodic activity with the increased delay. Further, the external stimulations can induce and suppress the periodic activity. The system dynamics can be transformed from quiescent state (i.e., the stable equilibrium) to periodic activity and then quiescent state with stimulation increasing. Finally, inspired by ubiquitous rhythm in living organisms, we introduce periodic stimulations with low frequency as rhythm activity from sensory organs and other regions. The neural system subjected by the periodic stimulations exhibits some interesting activities, such as periodic spiking, subthreshold oscillation, and bursting-like activity. Moreover, the subthreshold oscillation can switch its position with delay increasing. The neural system may employ time delay to realize Winner-Take-All functionality.

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Section: Saddle-node Bifurcation Of Nontrivial Equilibriummentioning

confidence: 99%

Complexity Induced by External Stimulations in a Neural Network System with Time Delay

Zhen

Zhang

Song

2020

Mathematical Problems in Engineering

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“…The dynamics of these dynamical systems provide theoretical insight into the biological mechanisms of brain functions. In recent decades, lots of neuron models have been intensively studied, and their equilibria and bifurcations are used to explain the formation and transition modes of the neuronal firing patterns [1][2][3][4][5][6]. However, because of the high dimension and complexity, the large-scale network models which consist of many neurons and synapses are difficult to study by dynamical methods [7].…”

Section: Introductionmentioning

confidence: 99%

“…The blue dashed lines identify the maxima and minima of the unstable limit cycles. The orange circles indicate the Hopf bifurcation points 4. Neural Plasticity E i = −75, η = −5, Δ = 1, and V th = 50.…”

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confidence: 99%

Dynamics of a Large-Scale Spiking Neural Network with Quadratic Integrate-and-Fire Neurons

2021

Neural Plasticity

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Since the high dimension and complexity of the large-scale spiking neural network, it is difficult to research the network dynamics. In recent decades, the mean-field approximation has been a useful method to reduce the dimension of the network. In this study, we construct a large-scale spiking neural network with quadratic integrate-and-fire neurons and reduce it to a mean-field model to research the network dynamics. We find that the activity of the mean-field model is consistent with the network activity. Based on this agreement, a two-parameter bifurcation analysis is performed on the mean-field model to understand the network dynamics. The bifurcation scenario indicates that the network model has the quiescence state, the steady state with a relatively high firing rate, and the synchronization state which correspond to the stable node, stable focus, and stable limit cycle of the system, respectively. There exist several stable limit cycles with different periods, so we can observe the synchronization states with different periods. Additionally, the model shows bistability in some regions of the bifurcation diagram which suggests that two different activities coexist in the network. The mechanisms that how these states switch are also indicated by the bifurcation curves.

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“…The ML neuron model was improved [ 27 – 29 ] along with the dynamical properties being explored, like the memristor synapse-based ML model with abundant periodic and chaotic bursting [ 27 ], the random dynamical behavior driven by Gaussian white noise [ 28 ], and the stochastic hybrid ML model with the extended parameter regime of oscillations related to noise [ 29 ]. In particular, many researchers paid much attention to the well-known HR model and revised it utilizing various methods [ 30 – 38 ], including impulsive effect and period-adding bifurcation in a hybrid HR model [ 30 ]; alternating current-induced coexisting behaviors of asymmetric busters in an external alternating current injected HR model [ 31 ]; diversity of firing patterns in a memristor-coupled HR model [ 32 ]; multiple firing patterns dependent on the complex electrophysiological condition found in a memristor HR model [ 33 ]; coherence resonance induced by phase noise in a 3D memristor-based HR model [ 34 ]; the relationship between the energy and the firing pattern in a type of memristor-based HR model [ 35 ]; the coexisting phenomenon of diverse firing patterns in a 5D memristor-coupled HR neuron model [ 36 ]; hyperchaotic behavior in a kind of neuron model with discontinuous magnetic induction [ 37 ]; and different types of firing patterns in a modified Hindmarsh-Rose model, such as square-wave bursting, chattering, fast spiking, periodic spiking, and mixed-mode oscillations [ 38 ]. Additionally, a novel neuron model, the Wang-Zhang model, built directly at the neuron level was brought forth [ 39 ].…”

Section: Introductionmentioning

confidence: 99%

Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor

Shi

Wang

2020

Neural Plasticity

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A model is introduced by coupling two three-dimensional Hindmarsh-Rose models with the help of a nonsmooth memristor. The firing patterns dependent on the external forcing current are explored, which undergo a process from adding-period to chaos. The stability of equilibrium points of the considered model is investigated via qualitative analysis, from which it can be gained that the model has diversity in the number and stability of equilibrium points for different coupling coefficients. The coexistence of multiple firing patterns relative to initial values is revealed, which means that the referred model can appear various firing patterns with the change of the initial value. Multiple firing patterns of the addressed neuron model induced by different scales are uncovered, which suggests that the discussed model has a multiscale effect for the nonzero initial value.

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Bifurcation analysis and diverse firing activities of a modified excitable neuron model

Cited by 77 publications

References 50 publications

Complexity Induced by External Stimulations in a Neural Network System with Time Delay

Complexity Induced by External Stimulations in a Neural Network System with Time Delay

Dynamics of a Large-Scale Spiking Neural Network with Quadratic Integrate-and-Fire Neurons

Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor

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