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.

中文翻译：

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时滞神经网络系统中外部刺激引起的复杂性

神经系统的复杂性和动力学分析在优化问题和联想记忆的应用中起着重要作用。在本文中，我们建立了具有外部刺激的延迟神经系统。利用理论分析和数值模拟研究了外部模拟引起的复杂动力行为。首先，我们通过非平凡平衡的鞍节点分叉来说明平衡数。这意味着神经系统对于外部刺激具有一/三平衡。然后，通过分析特征方程找到Hopf分支，我们获得了平衡的稳定性，并说明了外部刺激和时间延迟引起的周期性活动。神经系统表现出具有增加的延迟的周期性活动。进一步，外部刺激可以诱导和抑制周期性活动。系统动力学可以从静态（即稳定的平衡状态）转变为周期性活动，然后随着刺激的增加而变为静态。最后，受活生物体普遍存在的节律的启发，我们从感官器官和其他区域引入周期性的低频刺激作为节律活动。受到周期性刺激的神经系统表现出一些有趣的活动，例如周期性的尖峰，阈下振荡和类似爆发的活动。而且，亚阈值振荡可以随着延迟的增加而切换其位置。神经系统可以采用时间延迟来实现Winner-Take-All功能。稳定的平衡）到周期性活动，然后随着刺激的增加而进入静止状态。最后，受活生物体普遍存在的节律的启发，我们从感官器官和其他区域引入周期性的低频刺激作为节律活动。受到周期性刺激的神经系统表现出一些有趣的活动，例如周期性的尖峰，阈下振荡和类似爆发的活动。而且，亚阈值振荡可以随着延迟的增加而切换其位置。神经系统可以采用时间延迟来实现Winner-Take-All功能。稳定的平衡）到周期性活动，然后随着刺激的增加而进入静止状态。最后，受活生物体普遍存在的节律的启发，我们引入了来自感官器官和其他区域的低频率的周期性刺激作为节律活动。受到周期性刺激的神经系统表现出一些有趣的活动，例如周期性的尖峰，阈下振荡和类似爆发的活动。而且，亚阈值振荡可以随着延迟的增加而切换其位置。神经系统可以采用时间延迟来实现Winner-Take-All功能。我们引入了来自感官器官和其他区域的有规律的低频刺激作为节律活动。受到周期性刺激的神经系统表现出一些有趣的活动，例如周期性的尖峰，阈下振荡和类似爆发的活动。而且，亚阈值振荡可以随着延迟的增加而切换其位置。神经系统可以采用时间延迟来实现Winner-Take-All功能。我们引入了来自感官器官和其他区域的有规律的低频刺激作为节律活动。受到周期性刺激的神经系统表现出一些有趣的活动，例如周期性的尖峰，阈下振荡和类似爆发的活动。而且，亚阈值振荡可以随着延迟的增加而切换其位置。神经系统可以采用时间延迟来实现Winner-Take-All功能。