复杂网络上信号传输与检测的研究
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摘要
自小世界和无标度网络模型提出以来,复杂网络已逐渐发展成为当今科学界研究的前沿和热点,其研究者分布在各个不同领域。其中,真实系统中的信号传输和系统对信号的检测是目前研究人员最为感兴趣的方向。在本论文中,我们重点在复杂网络上的信号传输、信号检测以及相同步等方面进行了一些研究工作,其安排如下：
本文的第一章是绪论部分,介绍了本文的研究背景、研究综述、并简要介绍了我们的工作。
第二章中,我们给出复杂网络的基本概念和模型,介绍了一些常用且典型的动力学模型和相互作用方式,并从同步的角度讨论了对网络行为的刻画。
在第三章,我们利用一个耦合振子网络模型,研究了弱信号如何在复杂网络上传输。通过理论分析和数值模拟,我们发现节点对弱信号的响应随该节点到信源的距离作指数衰减,并用这一关系检测了相邻节点间的耦合强度。通过选择不同的信源节点,未知网络的拓扑结构就可以逐步地检测出来,如度分布、集群系数、甚至社区结构。我们发现少量数目的信源就可以检测出上述网络的性质。因此,我们的方法可方便地运用到大尺度未知网络的检测。
在第四章,我们针对大脑中化学耦合神经元间存在的不同时间延迟,研究分布时间延迟对bursting神经元相同步的影响。具体考察抑制性化学耦合Hindmarsh-Rose神经元,发现分布时间延迟可以诱导一系列的周期相同步,并且相同步时的时间延迟平均值与耦合强度、平均度均成反比关系。此外,我们发现该现象对神经元的参数失配具有较强的鲁棒性,且不依赖于网络的拓扑结构。利用一个简化模型,我们分析了该现象的形成机制。
在第五章,我们研究了亚阈值周期信号作用下全局耦合可激发神经元的放电行为,其中外部信号的相位无序。区别以往的全同相位,我们发现无序相位对神经元的放电起积极作用,并且其最佳的集体放电行为对应着适当的无序程度。另外,在无序相位的影响下,耦合强度也存在一个类似随机共振的最优值,使得神经元的集体放电处于最佳。最后,我们给出了放电行为的物理机制。该研究可以帮助解释许多生物较强的信号检测能力。
在第六章,我们研究了无标度耦合混沌映象网络中的自组织相同步,其中网络节点的行为用Logistic映象描述。我们发现系统在特定的耦合强度区间可自发形成有序的时空周期图案。该有序行为会随平均度增加而改变,但对网络尺度和参数失配具有很强的鲁棒性。利用一个简化模型,我们计算出自组织现象出现时的耦合强度区间。
第七章是总结与展望。
Since the proposal of small-world network and scale-free network, the com-plex network has become a topic of great interest in various fields. Of these, the signal transmission and the signal detection in realistic system are the most in-teresting research field. Thus, in this thesis, we study the signal transmission in complex network, investigate the signal detection in neural system, and discuss the phase synchronization both in neural system and discrete system. The thesis is organized as follows.
The first chapter is an introduction. In this section, the research background, literature review, and our works are sum-marized.
The second chapter introduces the basic concepts and the models of complex network, provides some typical dynamical equations and presents two models of interaction. The complex network synchronization is also discussed.
In the third chapter, we present a network model of coupled oscillators to study how a weak signal is transmitted in complex networks. Through both the-oretical analysis and numerical simulations, we find that the response of other nodes to the weak signal decays exponentially with their topological distance to the signal source and the coupling strength between two neighboring nodes can be figured out by the responses. This finding can be conveniently used to detect the topology of unknown network, such as the degree distribution, clustering coefficient and community structure, etc., by choosing many fewer nodes as the signal source. Thus our approach to detect the topology of unknown network may be efficient in practical situations with large network size.
In the fourth chapter, we consider the fact that signal transmission time de-lays between synaptically coupled neurons in the brain are different, and study the effects of distributed time delays on phase synchronization of bursting neu- rons. Applying the inhibitory coupled bursting Hindmarsh-Rose neurons, we find that distributed time delays in chemical coupling can induce a variety of phase-coherent dynamic behaviors. The critical mean time delay for the emer-gence of coherent behaviors is inversely proportional to both the coupling strength and the average degree. This phenomenon is robust to nonidentical external in-puts and is independent of network topology. Finally a physical theory is formu-lated to explain the emergence of coherent neuronal activity.
In the fifth chapter, we study the excitations in globally coupled excitable neurons under subthreshold periodic signal, where signal phases are in disor-der. Differing from the identical phase, we find that the disordered phases may play an active part in neuron excitations. And the excitations show a best coher-ence at an optimal level of disorder. Besides, the coherence of excitation exhibit a resonance-like dependence on the coupling coefficient. Finally, an explanation is used to analyze the mechanism of excitation. Our findings may be served as an alternative explanation to understand the robust signal detection in neuron systems.
In the sixth chapter, we study the self-organization of phase synchronization in coupled map scale-free networks with chaotic logistic map at each node and find that a variety of ordered spatiotemporal patterns emerge spontaneously in a regime of coupling strength. These ordered behaviors will change with the in-crease of the average links and are robust to both the system size and param-eter mismatch. A heuristic theory is given to explain the mechanism of self-organization and to figure out the regime of coupling for the ordered spatiotem-poral patterns.
The seventh chapter summarizes the thesis and gives an outlook of the future study.
本文的第一章是绪论部分,介绍了本文的研究背景、研究综述、并简要介绍了我们的工作。
第二章中,我们给出复杂网络的基本概念和模型,介绍了一些常用且典型的动力学模型和相互作用方式,并从同步的角度讨论了对网络行为的刻画。
在第三章,我们利用一个耦合振子网络模型,研究了弱信号如何在复杂网络上传输。通过理论分析和数值模拟,我们发现节点对弱信号的响应随该节点到信源的距离作指数衰减,并用这一关系检测了相邻节点间的耦合强度。通过选择不同的信源节点,未知网络的拓扑结构就可以逐步地检测出来,如度分布、集群系数、甚至社区结构。我们发现少量数目的信源就可以检测出上述网络的性质。因此,我们的方法可方便地运用到大尺度未知网络的检测。
在第四章,我们针对大脑中化学耦合神经元间存在的不同时间延迟,研究分布时间延迟对bursting神经元相同步的影响。具体考察抑制性化学耦合Hindmarsh-Rose神经元,发现分布时间延迟可以诱导一系列的周期相同步,并且相同步时的时间延迟平均值与耦合强度、平均度均成反比关系。此外,我们发现该现象对神经元的参数失配具有较强的鲁棒性,且不依赖于网络的拓扑结构。利用一个简化模型,我们分析了该现象的形成机制。
在第五章,我们研究了亚阈值周期信号作用下全局耦合可激发神经元的放电行为,其中外部信号的相位无序。区别以往的全同相位,我们发现无序相位对神经元的放电起积极作用,并且其最佳的集体放电行为对应着适当的无序程度。另外,在无序相位的影响下,耦合强度也存在一个类似随机共振的最优值,使得神经元的集体放电处于最佳。最后,我们给出了放电行为的物理机制。该研究可以帮助解释许多生物较强的信号检测能力。
在第六章,我们研究了无标度耦合混沌映象网络中的自组织相同步,其中网络节点的行为用Logistic映象描述。我们发现系统在特定的耦合强度区间可自发形成有序的时空周期图案。该有序行为会随平均度增加而改变,但对网络尺度和参数失配具有很强的鲁棒性。利用一个简化模型,我们计算出自组织现象出现时的耦合强度区间。
第七章是总结与展望。
Since the proposal of small-world network and scale-free network, the com-plex network has become a topic of great interest in various fields. Of these, the signal transmission and the signal detection in realistic system are the most in-teresting research field. Thus, in this thesis, we study the signal transmission in complex network, investigate the signal detection in neural system, and discuss the phase synchronization both in neural system and discrete system. The thesis is organized as follows.
The first chapter is an introduction. In this section, the research background, literature review, and our works are sum-marized.
The second chapter introduces the basic concepts and the models of complex network, provides some typical dynamical equations and presents two models of interaction. The complex network synchronization is also discussed.
In the third chapter, we present a network model of coupled oscillators to study how a weak signal is transmitted in complex networks. Through both the-oretical analysis and numerical simulations, we find that the response of other nodes to the weak signal decays exponentially with their topological distance to the signal source and the coupling strength between two neighboring nodes can be figured out by the responses. This finding can be conveniently used to detect the topology of unknown network, such as the degree distribution, clustering coefficient and community structure, etc., by choosing many fewer nodes as the signal source. Thus our approach to detect the topology of unknown network may be efficient in practical situations with large network size.
In the fourth chapter, we consider the fact that signal transmission time de-lays between synaptically coupled neurons in the brain are different, and study the effects of distributed time delays on phase synchronization of bursting neu- rons. Applying the inhibitory coupled bursting Hindmarsh-Rose neurons, we find that distributed time delays in chemical coupling can induce a variety of phase-coherent dynamic behaviors. The critical mean time delay for the emer-gence of coherent behaviors is inversely proportional to both the coupling strength and the average degree. This phenomenon is robust to nonidentical external in-puts and is independent of network topology. Finally a physical theory is formu-lated to explain the emergence of coherent neuronal activity.
In the fifth chapter, we study the excitations in globally coupled excitable neurons under subthreshold periodic signal, where signal phases are in disor-der. Differing from the identical phase, we find that the disordered phases may play an active part in neuron excitations. And the excitations show a best coher-ence at an optimal level of disorder. Besides, the coherence of excitation exhibit a resonance-like dependence on the coupling coefficient. Finally, an explanation is used to analyze the mechanism of excitation. Our findings may be served as an alternative explanation to understand the robust signal detection in neuron systems.
In the sixth chapter, we study the self-organization of phase synchronization in coupled map scale-free networks with chaotic logistic map at each node and find that a variety of ordered spatiotemporal patterns emerge spontaneously in a regime of coupling strength. These ordered behaviors will change with the in-crease of the average links and are robust to both the system size and param-eter mismatch. A heuristic theory is given to explain the mechanism of self-organization and to figure out the regime of coupling for the ordered spatiotem-poral patterns.
The seventh chapter summarizes the thesis and gives an outlook of the future study.
引文
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