Applications of Coupled Cell Systems

微分方程组称为细胞。耦合的小区网络是耦合在一起的小区的集合，该网络的体系结构是指示哪些小区是相同的以及哪些小区耦合到哪些小区的图。这一提议解决的主要问题是：耦合系统的动态的哪一部分是由网络体系结构引起的？这些系统的理论是在最近NSF的支持下发展起来的：我们将继续发展它的发展，重点放在前馈网络的分叉和强迫，由分叉得到的解的稳定性，以及耦合系统的爆发。我们还将更直接地专注于应用，包括前庭系统(特别是耳朵中六个半规管神经元和八个颈部肌肉群之间的连接网络)，在一个简单的前馈网络中与同步打破Hopf分叉相关的频率过滤器/放大器的开发，以及从耦合相位振荡器中的不同动力学模式获得的“皮质歌曲”。生物学家J.B.S.霍尔丹，当被问及我们可以通过研究世界了解造物主时，回答说，上帝似乎对甲虫有过度的喜爱。今天的生物学家指出神灵对网络的过度喜爱是情有可原的。网络在生物学中无处不在：例如基因表达、神经回路、生态食物网和疾病传播。网络在许多其他科学分支中也很常见，最近人们对这个话题的兴趣激增。研究文献，包括应用，现在延伸到数千篇论文。我们一直在发展一种网络动力学理论，其中网络中的节点是微分方程组。在这项建议中，我们将重点放在理论的一些概括和具体应用上，包括与前庭系统中称为管颈投影的网络相关的动力学。

A system of differential equations is called a cell. A coupled cell network is a collection of cells that are coupled together and the architecture of that network is a graph that indicates which cells are identical and which cells are coupled to which. The primary question addressed by this proposal is: What part of the dynamics of coupled systems is due to network architecture? The theory of these systems was developed with recent NSF support: we will continue its development by focusing on bifurcations and forcing of feedforward networks, stability of solutions obtained by bifurcation, and bursting in coupled systems. We will also focus more directly on applications including the vestibular system (in particular, the network of connections between neurons of the six semicircular canals in the ears and eight neck muscle groups), the development of a frequency filter/amplifier associated to synchrony breaking Hopf bifurcation in a simple feedforward network, and `cortical songs' obtained from different dynamical patterns in coupled phase oscillators.The biologist J.B.S. Haldane, when asked what we can learn about the Creator by examining the world, replied that God seemed to have an inordinate fondness for beetles. Today's biologists could be forgiven for pointing to the deity's inordinate fondness for networks. Networks are ubiquitous in biology: examples include gene expression, neural circuitry, ecological food webs, and disease transmission. Networks are also common in many other branches of science, and there has been a recent explosion of interest in the topic. The research literature, including applications, now extends to many thousands of papers. We have been developing a theory of network dynamics, where the nodes in the network are systems of differential equations. In this proposal we focus on some generalizations of the theory and on specific applications including the dynamics associated to a network in the vestibular system called the canal-neck projection.