Point Processes and Time Series: Networks and Wavelets

本文主要研究两个基本问题：1)节点上具有时间过程的网络研究；2)涉及时空点过程的小波分析变体。应用于了解大脑听觉系统的相互联系，并在破坏性事件发生后创建地震的等震图。网络的工作涉及图形建模，而小波工作可以被视为局部加权程序的竞争对手，用于处理非平稳性。这些主题在评估学习是否在生物神经元网络中发生的问题上结合在一起。在网络研究的情况下，主要的应用是从猫和老鼠死后确定的位置的生物神经元中获得的同时的尖峰序列，目的是推断潜在的接线图。在小波研究的情况下，利用小波方法处理非平稳性的能力，地震学和神经生理学都得到了应用。同时考虑了时间和空间两种情况。例如，提出这项研究的动机是生物学和地震学领域中出现的问题，并且相信这些问题可以通过涉及网络概念的当代研究工作得到很好的解决。涉及新颖数学表达式的工作也与这项研究有关，这些数学表达式允许在某种情况下发生时间或空间上的一些变化。将涉及理论发展和经验模型拟合和验证。有可能应用于联邦战略利益的生物技术（BIOT）领域。在生物学和地震学等不同领域工作的价值在于，为其中一个领域考虑的分析技术可以被抽象出来并应用于另一个领域。

The proposed research concerns two basic topics: a) the study of networks with temporal processes at the nodes and b) variants of wavelet analysis involving temporal and spatial point processes. Applications are made to understanding the interconnections of the auditory system of the brain on the basis of recorded spike trains and to creating isoseismal maps of earthquakes in the aftermath of a destructive event. The network effort relates to graphical modeling while the wavelet work may be seen as a competitor to local weighting procedures, for handling nonstationarity. The topics come together in the problem of assessing whether learning is taking place in a biological neuron network.In the case of the network study, principal applications are to contemporaneous spike trains obtained from biological neurons at locations, determined post mortem in cats and rats, having in mind the goal of inferring a latent wiring diagram. In the case of the wavelet research, applications are made to both seismology and neurophysiology seizing on the ability of the wavelet approach to handle nonstationarity. Both the temporal and the spatial cases are considered.The research proposed is motivated by problems arising in the fields of biology and seismology, for example, and the belief that they may be well-addressed by contemporary research work involving the concept of networks. Work involving novel mathematical expressions, which allow some change in time or space to take place in a situation, also relates to the research. Both theoretical development and empirical model fitting and verification will be involved. There may be applications to the Biotechnology (BIOT) area of Federal strategic interest. The value of working in fields as diverse as biology and seismology is that analytical techniques thought of for one may be abstracted and applied in the other.