In this paper, dynamical behaviors of fraction-order Hopfield neuron network are investigated. Firstly, Mittag-Leffler stability analysis is carried out and some sufficient conditions are obtained. On the basis of theoretical analysis, two criteria for determining the stability of fraction-order Hopfield neuron network are presented and comparison between them is given by theoretical analysis along with numerical simulation. According to the proposed criteria, by selecting suitable system parameters, it can be obtained that fraction-order Hopfield neuron network can stabilize to the equilibrium point or an attractor, which can be a periodic orbit or two points. Secondly, considering the inevitable noise in the complex environment of neuron network, the effect of noise on the dynamics of fraction-order Hopfield neuron network is discussed via calculating coefficient of variation and numerical simulations. Results suggest that random noise can cause coherence resonance in fraction-order Hopfield neuron network for certain noise intensity.

中文翻译：

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分数阶Hopfield神经元网络的稳定性分析和噪声诱导的相干共振

本文研究了分数阶Hopfield神经元网络的动力学行为。首先，进行了Mittag-Leffler稳定性分析，并获得了一些充分的条件。在理论分析的基础上，提出了确定分数阶Hopfield神经元网络稳定性的两个标准，并通过理论分析和数值模拟对它们进行了比较。根据提出的标准，通过选择合适的系统参数，可以得到分数阶Hopfield神经元网络可以稳定到平衡点或吸引子，该吸引子可以是周期轨道或两个点。其次，考虑到神经元网络复杂环境中不可避免的噪声，通过计算变异系数和数值模拟，讨论了噪声对分数阶Hopfield神经元网络动力学的影响。结果表明，在一定的噪声强度下，随机噪声会在分数阶Hopfield神经元网络中引起相干共振。