In this paper, the effects of low and fast response speeds of neuron activation gradient of a simple 3D Hopfield neural network are explored. It consists of analyzing the effects of low and high neuron activation gradients on the dynamics. By considering an imbalance of the neuron activation gradients, different electrical activities are induced in the network, which enable the occurrence of several nonlinear behaviors. The significant sensitivity of nontrivial equilibrium points to the activation gradients of the first and second neurons relative to that of the third neuron is reported. The dynamical analysis of the model is done in a wide range of the activation gradient of the second neuron. In this range, the model presents areas of periodic behavior, chaotic behavior and periodic window behavior through complex bifurcations. Interesting behaviors such as the coexistences of two, four, six and eight disconnected attractors, as well as the phenomenon of coexisting antimonotonicity, are reported. These singular results are obtained by using nonlinear dynamics analysis tools such as bifurcation diagrams and largest Lyapunov exponents, phase portraits, power spectra and basins of attraction. Finally, some analog results obtained from PSpice-based simulations further verify the numerical results.

中文翻译：

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低和高神经元激活梯度对简单 3D Hopfield 神经网络动力学的影响

在本文中，探索了一个简单的 3D Hopfield 神经网络的神经元激活梯度的低和快响应速度的影响。它包括分析低和高神经元激活梯度对动力学的影响。通过考虑神经元激活梯度的不平衡，在网络中引发不同的电活动，从而导致几种非线性行为的发生。据报道，非平凡平衡点对第一个和第二个神经元的激活梯度相对于第三个神经元的激活梯度具有显着的敏感性。模型的动力学分析是在第二个神经元的激活梯度的大范围内完成的。在这个范围内，模型通过复杂的分岔呈现周期性行为、混沌行为和周期性窗口行为的区域。报道了有趣的行为，例如两个、四个、六个和八个断开的吸引子的共存，以及并存的反单调现象。这些奇异结果是通过使用诸如分岔图和最大李雅普诺夫指数、相图、功率谱和吸引力盆地等非线性动力学分析工具获得的。最后，从基于 PSpice 的模拟中获得的一些模拟结果进一步验证了数值结果。