Hidden dynamics in a fractional-order memristive Hindmarsh–Rose model
Hidden dynamics in a fractional-order memristive Hindmarsh–Rose model
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DOI：
10.1007/s11071-020-05495-9
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发表时间：
2020-01
影响因子：
5.6
通讯作者：
Y. Yu;Min Shi;Huiyan Kang;Mo Chen;B. Bao
中科院分区：
文献类型：
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作者：
Y. Yu;Min Shi;Huiyan Kang;Mo Chen;B. Bao
通过证明整数阶 Hindmarsh–Rose (H–R) 模型中不存在忆阻器，用分数阶忆阻器对慢离子通道进行重构，其中分数阶导数与记忆效应和离子电流的衰减率有关。然后提出了一种无平衡的分数阶忆阻（FOM）H-R模型。由于参数较小，FOM H–R 模型是一个快慢系统，可以视为快子系统的小扰动系统。利用Lyapunov直接法证明了快速子系统的有界性。借助快速系统的有界性和稳定性，讨论了 FOM H-R 模型的隐藏动力学。由于小参数固定，FOM H-R模型的膜电位在分数阶小于0.5时限制为静止状态，在分数阶大于0.5且小于1时限制为周期性尖峰。在没有小参数的约束的情况下，FOM H-R模型中出现周期性和混沌爆发，这意味着小参数使得膜电位活动更简单。由此可见，衰减速率慢的离子电流对神经元有抑制作用。
By showing that there is no any memristor in the integer-order Hindmarsh–Rose (H–R) model, the slow ion channel is remodeled by a fractional-order memristor, where the fractional-order derivative is related to the memory effect and the decay rate of the ion current. Then a fractional-order memristive (FOM) H–R model without equilibrium is proposed. Due to a small parameter, the FOM H–R model is a fast–slow system which can be taken as the small perturbation system of the fast subsystem. By Lyapunov direct method, the boundedness of the fast subsystem is proved. With the help of the boundedness and stability of the fast system, the hidden dynamics of the FOM H–R model is discussed. As the small parameter is fixed, it is shown that the membrane potential of the FOM H–R model limits to a quiescent state as the fractional order is less than 0.5 and limits to a periodic spiking as the fractional order is greater than 0.5 less than 1. Without the constraint of the small parameter, the periodic and chaotic bursting appears in the FOM H–R model, which implies the small parameter makes the membrane potential activity simpler. It can be drawn a conclusion that the ion current with the slow decay rate can inhibit the neurons.