非线性系统理论在Chialvo神经元模型中的应用

发布时间：2018-01-29 19:53

本文关键词： 非线性系统理论吸引子 Neimark-Sacker分岔神经网络耦合强度　出处：《北京交通大学》2017年硕士论文　论文类型：学位论文

【摘要】：非线性动力学是研究非线性系统运动规律的一门学科.其主要研究内容包括系统中的结构的稳定性、轨道的稳定性、分岔、混沌等.对非线性动力系统的研究不仅在理论上具有重要的价值,而且在科研应用方面也具有广泛的应用前景.非线性系统几乎涉及自然科学和社会科学的各个领域.它的理论和方法已经被广泛地应用到物理、化学、生物、生态、医疗以及经济学等领域.本文主要考虑非线性系统理论在神经系统中的应用.本文首先基于Chialvo神经元模型,研究系统在不同参数组合下的二维参数平面图的分岔结构,给出Neimark-Sacker分岔曲线的表达式,讨论系统吸引子的最终状态,包括周期吸引子、无界吸引子、拟周期和混沌吸引子,并且详细分析嵌入拟周期或混沌中的周期吸引子的分岔规律和动力学特征.其次,建立由混沌峰放电(chaotic spiking)的Chialvo神经元构成的网络模型,定性分析和数值模拟当数目众多的无规则的个体通过线性耦合方式紧密联系在一起形成的高维数的神经网络表现出的动力学行为.验证了随着耦合强度的改变,形成的网络系统有可能会表现为稳定的同步不动点,并且达到完全相同步.最后,总结本文的工作,为全面分析单个离散神经元模型以及神经元网络系统的动力学行为奠定基础.
[Abstract]:Nonlinear dynamics is a discipline to study the law of nonlinear system motion. Its main research contents include the stability of structure, the stability of orbit, and bifurcation of the system. Chaos and so on. The study of nonlinear dynamical system is not only of great value in theory. Nonlinear systems involve almost all fields of natural science and social science. Its theories and methods have been widely used in physics, chemistry and biology. This paper focuses on the application of nonlinear system theory in the nervous system. Firstly, this paper is based on the Chialvo neuron model. In this paper, the bifurcation structure of two-dimensional parametric planar graph with different parameter combinations is studied, the expression of Neimark-Sacker bifurcation curve is given, and the final state of the attractor of the system is discussed. It includes periodic attractor, unbounded attractor, quasi periodic attractor and chaotic attractor, and the bifurcation law and dynamic characteristics of periodic attractor embedded in quasi periodic or chaotic system are analyzed in detail. A network model consisting of chaotic spiking Chialvo neurons was established. Qualitative analysis and numerical simulation show the dynamic behavior of high-dimensional neural networks formed by a large number of irregular individuals through linear coupling. The network system may be stable synchronization fixed point, and achieve complete phase synchronization. Finally, summarize the work of this paper. It lays a foundation for analyzing the dynamic behavior of single discrete neuron model and neural network system.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19

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