复杂系统中，个体行为及个体之间的相互作用与关联是系统涌现整体性质与功能的基础。大脑是一个天然的复杂系统，它所涌现出的整体功能毫无疑问是来自于神经元动力学、神经元网络、脑区等之间相互耦合产生的内部信号和外部环境信号共同作用的结果。系统的、整体的研究思路是揭示这些功能涌现机制不可或缺的方法。在大脑的功能行为中，如睡眠和清醒等大脑集体动力学，不仅受到神经系统层面上微观结构的调控还依赖于外部环境昼夜节律的调节，探讨其中的神经动力学，及其结构、环境与功能关系，自然成为系统科学研究的关键科学问题。本文在系统科学的视角下探讨外部环境和网络结构对尖峰神经元集群动力学的作用。其中睡眠和清醒状态下集体动力学标志性的变化，不仅受到黑夜和白天的调控还依赖于大脑底层的网络结构。基于此，我们构建一个既包含内部相互耦合同时又接收外部环境(昼夜节律)控制的尖峰神经元模型来解释实验中睡眠和清醒这一现象，然后从单个神经元水平拓展到神经元集群去探索背后的神经机制；进一步，我们分析了在不同的网络结构和外部环境共同作用下对同步过渡的影响；最后，我们在保留了基本的生物物理学原理的基础上，为探究神经元丰富的尖峰模式提供了一个精简且有效的尖峰神经元模型。本文具体研究的内容如下：

  1.基于大脑温度感受器模型，加入内部的突触耦合电流以及来自昼夜节律和噪声等外部环境的干扰，本文建立了可以描述昼夜节律调控的尖峰神经元模型。该尖峰神经元模型可以在昼夜节律的调控下模拟出单个神经元分别在睡眠和清醒状态下的不同尖峰模式变化情况。
进一步，我们选择与生物实验有对应的局部场电位(LFP)和神经元放电率(MFR)两种指标来将实验现象与模型模拟的结果相对应，结果表明模拟计算得到的结果与生物实验得到的结果是定性一致的。最后，在外部环境干扰下，通过分析不同网络结构分别对局部场电位和平均细胞放电率的影响，以及调节尖峰神经元模型中的参数来观察尖峰模式的变化来判断干扰睡眠和清醒的因素。结果表明网络结构的变化不会影响睡眠与清醒的整体行为，而钙浓度则是影响睡眠清醒的主要因素，钙浓度增加促进清醒。

 2.研究了网络结构和外部环境(黑夜还是白天)共同作用下尖峰神经元网络同步过渡的行为模式。我们使用全局瞬时序参量S和Kuramoto 序参量R作为同步函数，通过改变耦合强度来观察清醒和睡眠时
网络的同步过渡情况，研究了不同网络结构下如规则网络中的晶格网、加入随机长程连接的随机网以及经典的小世界网，对同步过渡的影响。结果显示外部环境输入可以干扰同步过渡类型：在睡眠阶段，无论网络结构如何变化都导致尖峰神经元网络发生受阻的(不连续的)同步。而在清醒阶段，网络结构能够决定该网络的同步过渡类型，产生了不同的同步类型：不连续的同步相变和连续性同步相变。特别是长程连接是影响同步过渡的主要拓扑因素。最后，我们与其他尖峰神经元网络模型(HH 神经元，Izhikevich 神经元等)的同步过渡研究进行了对比。研究结果显示同步过渡十分依赖外部环境，且长程连接是引发网络发生不连续同步相变的关键因素。尤其是神经元的固有频率和尖峰模式的相关性可以导致网络发生受阻性同步且不依赖于网络底层拓扑结构。

 3.本文为分析神经元丰富的动力学行为开发了一个简约高效的尖峰神经元模型：Cubic-Quadratic动力系统，它是通过对原始模型的向上冲程和复极化进行简化数学描述得到的。该模型不仅控制参数的几何图像更为直观，而且只通过两个参数的调整就能够得到丰富的动力学行为，可以表征神经元不同峰值模式。Cubic-Quadratic动力系统，包含一个三次函数，其作用是使尖峰向上冲程，另一个是二次函数，其作用是使尖峰复极化(下冲程)。我们对Cubic-Quadratic动力系统进行了稳定性及其分岔分析，详细讨论了Andronov-Hopf分岔、鞍点分岔和不变环上的鞍点分岔，并证明了该模型存在稳定极限环。进一步，我们研究了该模型在周期输入电流调控下的单个神经元的基本放电模式。随着输入电流频率的变化，该模型能够描述神经元不同峰值模式，如清醒时的单峰、睡眠时的爆发以及睡眠和清醒过渡阶段发生的混沌现象。我们进一步研究了耦合的Cubic-Quadratic动力系统的集体动力学同步与时空模式。首先我们讨论了网络结构对耦合Cubic-Quadratic 动力系统集体动力学在睡眠方面的影响。得到的结果与原始模型的拟合结果有很强的一致性，这表明了Cubic-Quadratic模型是对于大脑睡眠和清醒的底层机制很好的抽象。最后，我们研究了耦合Cubic-Quadratic动力系统集体动力学的时空模式。我们引入有向边的空间耦合模式，通过改变突触连接的方向来观察集体动力学如何随时间演化。我们建立了具有反应扩散项的Cubic-Quadratic动力系统，研究了波的传播方式。模拟结果表明，在一定条件下，随机的初始激励可以导致定性相似的波的传播方式(局部/全局)。网络结构的不同显著影响波的传播方式及其传播时间，特别是长程连接在抑制波的传播时间中起决定性作用，且长程连接的数量大约占比百分之十五时波的传播时间缩短到最小且达到稳定状态。

In complex systems, individual behavior and interaction and correlation between individuals are the basis of the overall nature and function of system emergence. The brain is a natural complex system, and its emerging overall function is undoubtedly from the combined effect of the mutual coupling between dynamics of neurons, neural ensemble, encephalic regions, and external environmental signals. A systematic and holistic approach is an indispensable way to reveal these functional emergence mechanisms. In the functional behavior of the brain, such as sleep and wake, are not only regulated by the microscopic structure of the nervous system but also depend on the interference of the external environment. Therefore, the exploration on dynamics of neurons and the relevance between structure, environment and function has naturally become a pivotal scientific issue in system science. In this paper, the effects of external environment and internal structure on the collective dynamics of spiking neuron are investigated from the perspective of systems science. The signature changes in collective dynamics during sleep and wake are regulated not only by night and day but also by underlying internal structure. Consequently, a network model of spiking neurons containing both internal coupling and external environmental control (circadian rhythm) is constructed to explain the phenomenon of sleep and wake during the experiment, and broadened from a single neuron to multiple neurons coupled with each other to illustrate the underlying neural mechanisms. Furthermore, the synchronization transition in spiking neurons model under the combined effects of different network structures and external environments are considered in detail. Finally, a simple and effective model is provided for researching the rich behavior of neurons while preserving the basic biophysical principles. The concrete research in this paper is shown below:

   1. Based on the brain temperature receptor model, adding the internal synaptic coupling current and the interference from the external environment such as circadian rhythm and noise,  a spiking neuron model is built that can describe the regulation of circadian rhythm. The different spiking patterns of individual neurons during sleep and wake can be simulated by the regulation of a circadian rhythm. Furthermore, two indicators corresponding to the biological experiment are selected, local field potential (LFP) and mean firing rate (MFR), to correspond the experimental phenomenon to the simulation results of the model. The results of calculation and simulation can be very good for experimental phenomena. Finally, under the interference of external environment, by analyzing the influences of different network structures on local field potential and mean firing rate, and adjusting the parameters in the model of spiking neurons to observe the changes of membrane potential, the factors that interfere with sleep and wake are determined. The results indicate that the overall behavior of sleep and wake are independent of network structure, and calcium concentration is the main factor affecting sleep and wake, and the increase of calcium concentration promotes wake.

   2. The behavior pattern of synchronization transitions in spiking neurons model is considered under the influence of network structure and external environment (day and night). The two order parameters S and R (Kuramoto) are used as synchronization functions to observe the synchronization transition during wake and sleep by adjusting coupling strength, the influence of different network structures such as lattice networks in regular networks, random networks, and classical WS networks on synchronization transitions is explored. The results show that the external environment input has great influence on the type of synchronization transition. During sleep, spiking neurons networks will produce frustrated (discontinuous) synchronization regardless of network structures. Nevertheless, the synchronization type is closely related to network topology during wake, such as, continuous synchronous phase transitions and discontinuous synchronous phase transitions. Especially, the long-range connections are the primary reason affecting synchronization. Finally, compared with synchronization transitions in other spiking neurons model (Izhikevich neuron, HH neuron, etc.). In this study, the external environment has a significant influence on synchronization transition, and the long-range connections can lead to discontinuous synchronous phase transition. In particular, the correlation between spiking patterns and natural frequency can lead to frustrated synchronization, which is independent of the underlying topology of networks.

   3. A simple and effective model, Cubic-Quadratic dynamical system, is proposed that can reproduce the abundant dynamical behaviors of neurons, and the model is obtained by simplified mathematical description of the upward stroke and repolarization of the original model. Not only is the geometric image of the control parameters more intuitive, but also rich dynamical behaviors can be obtained through the adjustment of only two parameters, which can characterize different spiking modes of neurons. Cubic-Quadratic dynamical system, where the cubic function makes the spikes stroke up, the other is a quadratic function, whose function is to repolarize the spikes (downstroke). The bifurcation and stability analysis of Cubic-Quadratic model are conducted. A lot of bifurcation have been carefully discussed by means of debugging bifurcation parameters. In view of this situation, the basic firing mode of a single neuron under periodic input current is studied. The model can reproduce diversified dynamical behaviors, such as spikes during wake, bursts during sleep, and chaotic phenomena occurring during the transition between sleep and wake. Further, the synchronization and space-time patterns of the coupled Cubic-Quadratic dynamical system are analyzed. Firstly, the sleep and wake of coupled Cubic-Quadratic dynamical system are discussed on the basic of network structure. The results have a strong agreement with those of the original model, which suggests that the Cubic-Quadratic model is a good abstraction of the underlying mechanism of sleep and wake in the brain. Finally, the space-time pattern of collective dynamics of coupled Cubic-Quadratic dynamical system is investigated. The edge-oriented spatial coupling model is introduced to observe how collective dynamics evolve over time by changing the direction of synaptic connections. The Cubic-Quadratic dynamical system with reaction-diffusion term is built to analyze wave propagation. The simulation results demonstrate that random initial excitation can lead to qualitatively similar wave propagation modes (local/global). However, the propagation mode and propagation time of wave are significantly affected by the network structure. In particular, long-range connections can suppress the duration of the wave, and duration of the wave is reduced to a minimum and stable when the number of long-range connections accounts for about 15 percent.