DFG-RSF: Complex dynamical networks: effects of heterogeneity, adaptivity, and delays

迄今为止，文献中主要考虑了两种类型的网络模型：静态网络和动态网络。在静态网络中，只有连接的结构是重要的。相反，在动态网络中，每个单元的状态取决于时间，其动态受到与其他单元相互作用的影响。尽管有大量有趣和有意义的结果，但动态网络的一般理论仅处于早期发展阶段，这是由于该主题的极端复杂性和广泛性。本课题解决了复杂非线性动态网络理论中的一个基本问题。更具体地说，它考虑了复杂性的以下方面：非同构拓扑结构、自适应连接和耦合延迟的存在。要研究的网络类型包括但不限于神经和神经形态网络、无线传感器网络和电子振荡器网络。主要目标包括：1 .发展动态网络术语“复杂性”的定义，以及其复杂性的定性标准和定量措施。这个方向是非常重要的，因为术语“复杂网络”至今还没有严格的定义，尽管它在非线性动力学中的广泛应用长期以来一直要求对其定义进行阐述。具有复杂模块化结构的神经形态网络模型的建立及其分析方法的研究。我们将构建再现真实神经网络结构特性的模型，主要是它们的模块化，并研究连接特性与其功能之间的联系。3。耦合延迟和连接自适应对相位振荡器脉冲耦合网络动力学的影响研究。主要目的是研究抑制全局同步和相位模式和其他复杂动力机制的出现。我们的目的是了解延迟耦合和塑性如何影响脉冲耦合系统中去同步的分岔机制。具有耦合延迟的复杂网络的电子原型的发展和研究，证明了非平凡的动力学机制。特别是，我们计划实现一个振荡网络，通过长链的可激发单元相互作用，并展示新型的嵌合体行为。

Two types of network models have been mostly considered so far in the literature: static and dynamic networks. In static networks, only the structure of connections is of importance. In contrast, in dynamical networks the state of each unit depends on time, and its dynamics is influenced by the interaction with other units. In spite of the large number of interesting and significant results, the general theory of dynamical networks is only at the stage of early development, which is due to the extreme complexity and broadness of the subject. This project addresses a fundamental problem of the theory of complex nonlinear dynamical networks. More specifically, it considers the following aspects of complexity: inhomogeneous topological structure, adaptive connections, and the presence of coupling delays. The types of networks to be studied include but are not limited to neural and neuromorphic networks, wireless sensor networks, and networks of electronic oscillators. The main objectives include: I. Development of the definition of the term ''complexity'' for dynamical networks, as well as the qualitative criteria and quantitative measures for their complexity. This direction is very important since the term ''complex network'' has not been strictly defined so far, although its wide use in nonlinear dynamics has long been calling for elaboration of its definition.II. Development of neuromorphic network models with complex modular structure and analytic methods of study of such models. We will construct models reproducing structural properties of real neuronal networks, primarily their modularity, and study the link between the connectivity characteristics and their function. III. Study of the influence of coupling delays and adaptivity of connections on the dynamics of pulse-coupled networks of phase oscillators. The main objective is to study the suppression of the global synchronization and emergence of phase patterns and other complex dynamical regimes. We aim to understand how delayed coupling and plasticity affect the bifurcation mechanisms for the desynchronization in pulse-coupled systems. IV. Development and study of electronic prototypes of complex networks with coupling delays demonstrating nontrivial dynamical regimes. Particularly, we plan to implement a network of oscillators interacting via long chains of excitable units and demonstrating novel types of chimera-like behavior.