Point Processes and Time Series: Networks and Wavelets

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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.