Applications of Coupled Cell Systems

一个微分方程组称为一个单元。 耦合单元网络是耦合在一起的单元的集合，并且该网络的架构是指示哪些单元是相同的以及哪些单元耦合到哪个单元的图。 这个建议解决的主要问题是：耦合系统的动力学的哪一部分是由于网络架构？这些系统的理论开发与最近NSF的支持：我们将继续其发展，重点是分叉和前馈网络的强制，分叉获得的解决方案的稳定性，并在耦合系统中爆裂。 我们还将更直接地关注包括前庭系统在内的应用（特别是，耳朵中的六个半规管和八个颈部肌肉群的神经元之间的连接网络），与简单前馈网络中的同步破坏Hopf分叉相关联的频率滤波器/放大器的开发，和“皮层歌曲”从不同的动力模式在耦合相位振荡器。当被问到我们可以通过观察世界来了解造物主的时候，哈尔科夫回答说，上帝似乎对甲虫有着过度的喜爱。 今天的生物学家指出神对网络的过度喜爱是可以原谅的。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，现在扩展到成千上万的论文。我们一直在发展网络动力学理论，网络中的节点是微分方程组。 在这个建议中，我们专注于一些概括的理论和具体的应用，包括动态相关的前庭系统中的网络称为运河颈投影。

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.