This article focuses on the hybrid effects of memristor characteristics, time delay, and biochemical parameters on neural networks. First, we propose a novel neuron system with memristor and time delays in which the memristor is characterized by a smooth continuous cubic function. Second, the existence of equilibria of this type of neuron system is examined in the parameter space. Sufficient conditions that ensure the stability of equilibria and occurrence of pitchfork bifurcation are given for the memristor-based neuron system without delay. Third, some novel criteria of the addressed neuron system are constructed for guaranteeing the delay-dependent and delay-independent stability. The specific conditions are provided for Hopf bifurcations, and the properties of Hopf bifurcation are ascertained using the center manifold reduction and the normal form theory. Moreover, there exists a phenomenon of bistability for the delayed memristor-based neuron system having three equilibria. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

中文翻译：

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具有忆阻器特性和时间延迟的神经元系统的定性分析和分岔。

本文重点研究忆阻器特性、时间延迟和生化参数对神经网络的混合效应。首先，我们提出了一种具有忆阻器和时间延迟的新型神经元系统，其中忆阻器具有平滑连续三次函数的特征。其次，在参数空间中检查了这种类型的神经元系统是否存在平衡。无延迟地为基于忆阻器的神经元系统给出了保证平衡稳定性和干草叉分叉发生的充分条件。第三，构建了所解决的神经元系统的一些新标准，以保证延迟相关和延迟无关的稳定性。提供了 Hopf 分岔的具体条件，并利用中心流形约简和范式理论确定了Hopf分岔的性质。此外，具有三个平衡的延迟忆阻器神经元系统存在双稳态现象。最后通过数值算例证明了理论结果的有效性。