Abundant bursting patterns of a fractional-order Morris–Lecar neuron model

Shi, Min; Wang, Zaihua

doi:10.1016/j.cnsns.2013.10.032

Cited by 60 publications

(28 citation statements)

References 31 publications

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“…A few prior studies have investigated fractional-order V m dynamics in other excitable cell models. Shi and Wang demonstrated that the fractional-order Morris-Lecar neuron model can exhibit a wider range of bursting behavior than can be reproduced by the original, first-order model [ 40 ]. Jun et al similarly show that fractional V m dynamics in the Hindmarsh-Rose neuronal model alters spiking patterns and also found a larger applied current threshold for repetitive spiking for smaller fractional-order [ 41 ], in agreement with our findings in the Hodgkin-Huxley model (Fig 6B and 6C ).…”

Section: Discussionmentioning

confidence: 99%

Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model

Weinberg

2015

PLoS ONE

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Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

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Section: Discussionmentioning

confidence: 99%

Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model

Weinberg

2015

PLoS ONE

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“…A "fold/Hopf (homoclinic)" burster means that a fold bifurcation causes a neuron to change from a quiescent state to repetitive spiking, and a Hopf (homoclinic) bifurcation causes a neuron to change from a spiking attractor to a quiescent state. Usually, the two bifurcations can be analyzed by nullclines of fast subsystems and phase planes [26] . Our goal is to make u 2 − V (:= u d ) → 0 as t → +∞, as far as possible.…”

Section: Simulationmentioning

confidence: 99%

Controlling a neuron by stimulating a coupled neuron

Song

Wang

2018

Appl. Math. Mech.-Engl. Ed.

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Despite the intensive studies on neurons, the control mechanism in real interactions of neurons is still unclear. This paper presents an understanding of this kind of control mechanism, controlling a neuron by stimulating another coupled neuron, with the uncertainties taken into consideration for both neurons. Two observers and a differentiator, which comprise the first-order low-pass filters, are first designed for estimating the uncertainties. Then, with the estimated values combined, a robust nonlinear controller with a saturation function is presented to track the desired membrane potential. Finally, two typical bursters of neurons with the desired membrane potentials are proposed in the simulation, and the numerical results show that they are tracked very well by the proposed controller.

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“…In [18,35], a three-dimensional fractional-order Hindmarsh-Rose model is considered with fixed numerical values of the system parameters, and extensive numerical simulations are carried out to exemplify the dynamical characteristics of the model, without focusing on theoretical analysis. Recently, a fractionalorder Morris-Lecar neuron model with fast-slow variables has been investigated in [28], revealing some bursting patterns that do not exist in the corresponding integer-order model. This paper is devoted to the theoretical analysis of the two-and three-dimensional fractional-order Hindmarsh-Rose neuronal models, focusing on stability properties and occurrence of Hopf bifurcations, choosing the fractional order of the system as bifurcation parameter.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Two- and Three-Dimensional Fractional-Order Hindmarsh-Rose Type Neuronal Models

Kaslik

2017

FCAA

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A theoretical analysis of two-and three-dimensional fractional-order Hindmarsh-Rose neuronal models is presented, focusing on stability properties and occurrence of Hopf bifurcations, with respect to the fractional order of the system chosen as bifurcation parameter. With the aim of exemplifying and validating the theoretical results, numerical simulations are also undertaken, which reveal rich bursting behavior in the three-dimensional fractional-order slow-fast system.

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Abundant bursting patterns of a fractional-order Morris–Lecar neuron model

Cited by 60 publications

References 31 publications

Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model

Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model

Controlling a neuron by stimulating a coupled neuron

Analysis of Two- and Three-Dimensional Fractional-Order Hindmarsh-Rose Type Neuronal Models

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