具有周期扰动项的混沛Rulkov神经元的动态

发布时间：2018-04-24 21:04

  本文选题：参数平面 + Lyapunov指数　； 参考：《北京交通大学》2017年硕士论文

【摘要】：神经元在中枢神经系统处理信息的过程中有着非常重要的地位,神经元能够加工、处理和传输信息,而这些过程有丰富的非线性特征。近十年来,Rulkov和Izhikevich分别应用离散(map-based)神经元数学模型,成功地对大脑皮层(cortical layer)和丘脑皮层系统(thermostatically systems)神经元的动态行为进行定性分析和数值模拟,并与基于常微分方程组(ODEs)的Hodgkin-Huxley模型进行比较,结果表明:离散神经元网络能够模拟生物神经元的真实行为,离散神经元模型在计算时间、计算算法的透明性、计算资源和数据存储等方面具有明显的优越性。它们不仅可以在大型的数值计算方面具有明显的优越性,而且可以调节混沌动力系统,产生丰富的聚合行为。目前,离散神经元数学模型作为研究大脑神经元系统的一种简化数学形式,已经被广泛地用到大脑数值模拟中。在第一章中,我们介绍了与本文相关的背景知识。在第二章中,讨论了基于Rulkov映射模型中周期扰动的影响。通过固定外部调节振幅σ和参数η,改变外部周期扰动角频率ω,对参数平面的不动点、周期解、拟周期解和混沌进行了详细的论述。在二维参数平面中,我们发现一个虾型周期区域淹没在一个混沌区域当中。此外,在参数平面中我们还可以观察到倍周期分岔结构。混沌可以被看作是抑制周期性窗口嵌入在混沌区域窗口的结果。在第三章中,我们利用统计学相关知识对第二章的数值计算结果进行了进一步的分析,主要讨论第二章出现的结果是否正确。根据数据的最小值、最大值、均值以及方差等分析数据、检验结果。最后,对本文的研究内容进行了总结。
[Abstract]:Neurons play an important role in the processing of information in the central nervous system. Neurons can process, process and transmit information, and these processes have rich nonlinear characteristics. In the last ten years, Rulkov and Izhikevich have successfully carried out qualitative analysis and numerical simulation of the dynamic behavior of cortical layersand thalamic cortical system by using discrete map-based neuron mathematical model, respectively. Compared with the Hodgkin-Huxley model based on ordinary differential equations, the results show that the discrete neuron network can simulate the real behavior of the biological neurons, and the computation time of the discrete neuron model and the transparency of the algorithm are obtained. Computing resources and data storage have obvious advantages. They not only have obvious advantages in large-scale numerical calculation, but also can adjust chaotic dynamical system and produce rich aggregation behavior. At present, as a simplified mathematical form to study the brain neuron system, discrete neuron mathematical model has been widely used in brain numerical simulation. In the first chapter, we introduce the background of this paper. In chapter 2, we discuss the effect of periodic perturbation based on Rulkov mapping model. The fixed amplitude 蟽 and parameter 畏 are adjusted to change the external periodic disturbance angular frequency 蠅. The fixed point, periodic solution, quasi periodic solution and chaos of the parameter plane are discussed in detail. In the two-dimensional parametric plane, we find that a shrimp periodic region is submerged in a chaotic region. In addition, the periodic bifurcation structure can be observed in the parameter plane. Chaos can be seen as the result of suppressing periodic windows embedded in chaotic regions. In the third chapter, we use the relevant knowledge of statistics to further analyze the numerical results of the second chapter, and mainly discuss whether the results in the second chapter are correct or not. According to the data minimum, maximum, mean and variance analysis data, test results. Finally, the research content of this paper is summarized.
【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】：O175
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本文编号：1798237