Mathematical analysis of synchronization in complex networks

同步是多种物理、生物和技术网络中集体行为的重要模式。在许多应用中，局部动力学是用微分方程组来建模的，而相互作用方案是用加权图来定义的。本研究旨在提出图上微分方程耦合系统的同步和模式形成的数学理论。在网络结构的一般假设下，考虑具有不同类型局部动力学的网络，例如由极限环、混沌吸引子或由噪声引起的网络。首席研究员（PI）开发了数学上严格但实际上有效的同步条件，研究了同步对噪声的鲁棒性，并分析了间隙连接耦合神经元网络中的电活动模式。强调了分析结果的图论解释。PI寻求通过将组合技术集成到动力系统分析中来量化网络拓扑对耦合振荡器动力学的贡献的系统方法。同步是一种普遍现象，在科学技术领域有着广泛的应用。电网安全、信息网络的有效通信和无人驾驶车辆的协调只是同步至关重要的众多技术领域中的三个。此外，同步在许多重要的生理和认知过程（如呼吸、睡眠和注意力）以及一些严重的神经退行性疾病（如帕金森病和癫痫）的机制中发挥着突出作用。PI开发了新的数学工具，并使用它们来研究生物物理模型中的同步，包括蓝斑神经网络，这是哺乳动物脑干中参与认知表现和行为调节的一组神经元。这项研究增强了我们对如何在一类重要的模型中实现、控制或破坏同步的理解。这项调查促进了动力系统、随机过程和代数图理论之间的界面研究。这项研究的结果将被整合到由德雷塞尔大学PI教授的动力系统和数学神经科学研究生课程中。该基金资助一名研究生和两名本科生的暑期研究。

Synchronization is an important mode of collective behavior in diverse physical, biological, and technological networks. In many applications, local dynamics is modeled by systems of differential equations and the interaction schemes are defined by weighted graphs. This research is aimed at advancing the mathematical theory of synchronization and pattern formation in coupled systems of differential equations on graphs. Networks with different types of local dynamics, such as those generated by limit cycles, chaotic attractors, or induced by noise, are considered under general assumptions on the network architecture. The principal investigator (PI) develops mathematically rigorous yet practically efficient conditions guaranteeing synchronization, studies robustness of synchrony to noise, and analyzes patterns of electrical activity in gap-junctionally coupled neuronal networks. The graph-theoretic interpretation of the analytical results is emphasized. The PI seeks systematic ways of quantifying the contribution of the network topology to the dynamics of coupled oscillators by integrating combinatorial techniques into dynamical systems analysis.Synchronization is a universal phenomenon with abundant applications across science and technology. Power grid safety, effective communication in information networks, and coordination of unmanned vehicles are just three of many areas of technology where synchronization is crucial. Furthermore, synchronization plays a prominent role in the mechanisms of many vital physiological and cognitive processes such as respiration, sleep, and attention, as well as in the mechanisms of several severe neurodegenerative disorders such as Parkinson's Disease and epilepsy. The PI develops new mathematical tools and uses them to study synchronization in biophysical models including that of the Locus Coeruleus network, a group of neurons in the mammalian brainstem involved in the regulation of cognitive performance and behavior. This study enhances our understanding of how to achieve, control, or destroy synchrony in an important class of models. This investigation fosters research at the interface between theories of dynamical systems, stochastic processes, and algebraic graph theory. The results of this research will be integrated into graduate courses in dynamical systems and mathematical neuroscience that are taught by the PI at Drexel University. This grant supports one graduate student and sponsors summer research for two undergraduate students.