MATHEMATICAL THEORY OF OSCILLATORY NEURAL NETWORKS

振荡神经网络和振荡神经元是常见的， 神经系统然而，神经网络的功能， OsCillatory元素在很大程度上是未开发的。拟议的工作旨在 开发一套与处理这种关系有关的技术， 使用这种技术来进一步了解一些特定的 神经网络这项工作将集中在一个脊椎动物和一个无脊椎动物 神经网络，中央模式发生器（CPGS）从事生产 节律性运动输出在每一种情况下，工作将与 神经生理学家，他们的数据将限制建模活动。 研究脊椎动物的CPG是波动运动的CPG; 主要的实验动物是七鳃鳗。这项工作继续进行， 协同努力，制定和调查一个普遍和灵活的 可以从中得出结论的数学框架 关于七鳃鳗的实验数据，了解网络如何运作。工作 在不久的将来，将集中于长程耦合的性质 在局部振荡元件中， 全球涌现行为的局部网络， 神经行为和机械活动之间的联系。 设计并进行相关实验。 无脊椎动物的CPG是甲壳动物的口胃神经节（STG）。这 这项工作是一个涉及建模和实验的更大项目的一部分。我们 该项目的一部分是发展数学来帮助解决这些问题， 问题如下：（一）如何在网络上解释网络的行为？ 组成神经元的性质的基础？(ii)起搏器和 振荡的紧急模式，这两个已知存在于STG， 相互作用和合作，以产生稳定和灵活的系统行为？

Oscillatory neural networks and oscillatory neurons are common in the nervous system. Yet the function of neural networks that contain OsCillatory elements is largely unexplored. The proposed work aims to develop a body of technique relevant to addressing this relationship, and to use this technique to further the understanding of some particular neural networks. The work will focus on one vertebrate and one invertebrate neural network, both central pattern generators (CPGS) engaged in producing rhythmic motor output. In each case, the work will be collaborative with neurophysiologists whose data will constrain the modelling activities. The vertebrate CPG to be studied is that for undulatory locomotion; the main experimental animal is the lamprey. This work continues a collaborative effort to formulate and investigate a general and flexible mathematical framework within which conclusions may be drawn from experimental data on the lamprey about how the network functions. Work in the immediate future will focus on the properties of long range coupling among the ii local oscillatory elements, the implications of the design of local networks for the global emergent behavior, and the interaction between the neural behavior and the mechanical activity that it regulates. Relevant experiments will be designed and carried out. The invertebrate CPG is the crustacean stomatogastric ganglion (STG). This work is part of a larger project involving modelling and experiments. Our part of the project is to develop mathematics to help address such questions as: (i) How can the behavior of the network be explained on the basis of the properties of the component neurons? (ii) How do pacemaker and emergent modes of oscillation, that are both known to exist in the STG, interact and cooperate to produce stable and flexible system behavior?