背景神经网络的多稳定性研究

The background neural network is a class of recurrent neural network related to background. This network provides a good interpretation of the impact of background on the activities of neurons and deeply depicts the mechanism of brain. Multistability of the background neural network is one of the most important properties of dynamics. However, because of the complexity of original background neural network model, it is very difficult to completely discuss multistability of that model. Inspired by Oja's seminal work, we shall propose a series of modified background neural network models and study on their multistability and related properties of dynamics. The main strategies we adopt are mathematically theoretical analysis and proof, computer simulation, and numerical analysis and so on. In this project, we shall mainly study on multistability of these modified background neural network models and related problems. The main contents include: the study of multistability of modified background neural networks which are continuous, the study of multistability of modified background neural networks which are discrete, and the study of their bifurcations. These studies on the above mentioned problems are higher value to both in theory and in application.

背景神经网络是一类与背景相关的回复式神经网络。这类神经网络很好地解释了背景对神经元活动的影响，并且深刻地揭示了大脑的工作机制。背景神经网络的多稳定性是其最重要的动力学性质之一。然而，由于原始的背景神经网络模型的复杂性，我们很难对其多稳定性进行深入的探讨。受Oja开创性工作的启发，我们将提出一系列改进的背景神经网络模型，并研究其多稳定性等相关动力学性质。我们将主要采用严格的数学理论分析与证明、计算机仿真与数值分析等研究手段，对改进的背景神经网络模型的多稳定性及相关问题进行深入研究。主要研究内容包括：改进的连续型背景神经网络模型的多稳定性分析；改进的离散型背景神经网络模型的多稳定性分析；改进的背景神经网络的分支问题研究。这些问题的研究具有较高的理论价值和应用前景。

背景神经网络是一类与背景相关的回复式神经网络。这类神经网络很好地解释了背景对神经元活动的影响，并且深刻地揭示了大脑的工作机制。背景神经网络的多稳定性是其最重要的动力学性质之一。受Oja开创性工作的启发，我们提出了一系列改进的背景神经网络模型，并研究其多稳定性等相关动力学性质。我们将主要采用严格的数学理论分析与证明、计算机仿真与数值分析等研究手段，对改进的背景神经网络模型的多稳定性及相关问题进行深入研究。本课题主要研究改进的背景神经网络的多稳定性问题及相关问题。完成情况概述如下：(1) 用结式消去法，罗尔定理等最终证明了网络平衡态的稳定性、具体个数及所在位置。(2) 通过向前欧拉方法，得到了一维离散型型背景神经网络模型，讨论了网络的稳定性。(3) 数值模拟的结果表明了随着背景输入的增加，一维离散型背景神经网络产生分支和混沌行为，计算了网络的李雅普诺夫指数。取得的研究成果总结如下：发表和录用科研论文11篇，其中SCI检索8篇。

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