复杂网络上的同步问题及相关模型在脑科学中的应用研究

发布时间：2018-11-26 10:02

【摘要】：以图为基础的模型为人们研究复杂的系统动态提供了重要的方法,许多复杂系统的动态问题都可以使用图上的动力系统进行建模.图上的平均协议是一个被国内外学者广泛研究的动力系统,它在控制论等领域中有重要的应用.本文研究了图上平均协议的同步问题.图上同步问题的研究目的是得到对图的结构而言尽量宽泛而容易验证的充分条件,使图上的通过平均协议定义的动力系统可以随着时间渐进地达到同步.本文将图上的同步问题与马尔可夫链的遍历性等价起来,并以此观点出发,通过估计马尔可夫链的Dobrushin系数,来研究其遍历性的问题.这样就发展出一套处理同步问题的方法,将静态和各种随时间变化的图可以统一看待,进而自然地得到了图上的平均协议达到同步的充分必要条件以及一个对时变(或随机)网络的极为宽泛的,且对控制论有实际意义的充分条件.本文还研究了在二维网格上和捷径模型(小世界模型)上的带有随机噪声的平均协议.研究这个模型的目的是为了研究人的大脑在静息态时的活动规律.人的大脑在静息态时,通过fMRI扫描,可以得到大脑中每个体素的活动强度.考虑到大脑中存在神经元的大量连接,这样就构成了一个复杂的图.实验表明大脑的神经元之间拥有远程的连接,而且连接的图具有小世界效应.我们使用了小世界网络上带有噪声的平均协议来描述大脑在静息态时的活动,并且通过这个模型构造了空间相关函数和空间熵等统计量,意图通过大脑的活动数据来反映底层神经连接的不同.以此观点来分析真实的fMRI数据,我们得到了正常人和精神分裂症患者的大脑活动数据之间的差异,并得出了一些具有生物学意义的结论.
[Abstract]:Graph-based models provide an important method for people to study the dynamics of complex systems. Many dynamic problems of complex systems can be modeled using dynamic systems on the graph. The average protocol on the graph is a dynamic system which is widely studied by domestic and foreign scholars. It has important applications in cybernetics and other fields. In this paper, we study the synchronization of average protocols on graphs. The purpose of this study is to obtain sufficient conditions for the structure of graphs to be as broad as possible and easy to verify so that the dynamic systems defined by the average protocol on graphs can gradually achieve synchronization over time. In this paper, the synchronization problem on graphs is equivalent to the ergodicity of Markov chains, and from this point of view, the ergodicity of Markov chains is studied by estimating the Dobrushin coefficients of Markov chains. In this way, we have developed a method to deal with the problem of synchronization, in which the static and all kinds of graphs that change over time can be treated in a unified way. A sufficient and necessary condition for the average protocol on a graph to achieve synchronization and a sufficient condition for the time-varying (or random) network to be extremely broad and of practical significance to cybernetics are obtained. In this paper, we also study the average protocol with random noise on two-dimensional meshes and short-cut models (small-world models). The purpose of this model is to study the pattern of brain activity in a resting state. The activity of each voxel in the brain can be obtained by fMRI scanning while the brain is resting. Given the large number of connections in the brain, this creates a complex picture. Experiments have shown that there are remote connections between neurons in the brain and that the graph of the connections has a small world effect. We use an average protocol with noise on a small-world network to describe the brain's activity in a resting state, and construct statistics such as spatial correlation functions and spatial entropy through this model. The intention is to reflect differences in underlying neural connections through brain activity data. From this point of view, we analyze the real fMRI data, we get the difference between normal people and schizophrenic patients' brain activity data, and we draw some biological conclusions.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O157.5

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