| In neuroscience, simultaneously recorded spike trains from multiple neurons are increasingly common; however, the computational neuroscience problem of how to quantitatively analyze such data remains a challenge. Gerstein, et al. [5] proposed a gravitational clustering algorithm (GCA) for multiple spike trains to qualitatively study interactions, in particular excitation, among multiple neurons. This thesis is mainly focused on a probabilistic treatment of GCA and a statistical treatment of Gerstein's interaction mode.;For a formal probabilistic treatment, we adopt homogeneous Poisson processes to generate the spike trains; define an interaction mode based on Gerstein's formulation; analyze the asymptotic properties of its cluster index -- GCA distances (GCAD). Under this framework, we show how the expectation of GCAD is related to a particular interaction mode, i.e., we prove that a time-adjusted-GCAD is a reasonable cluster index for large samples. We also indicate possible stronger results, such as central limit theorems and convergence to a Gaussian process.;In our statistical work, we construct a generalized mixture model to estimate Gerstein's interaction mode. We notice two key features of Gerstein's proposal: (1) each spike from each spike train was assumed to be triggered by either one previous spike from one other spike train or environment; (2) each spike train was transformed into a continuous longitudinal curve. Inspired by their work, we develop a Bayesian model to quantitatively estimate excitation effects in the network structure. Our approach generalizes the mixture model to accommodate the network structure through a matrix Dirichlet distribution. The network structure in our model could either approximate the directed acyclic graph of a Bayesian network or be the directed graph in a dynamic Bayesian network. This model can be generally applied on high-dimensional longitudinal data to model its dynamics. Finally, we assess the sampling properties of this model and its application to multiple spike trains by simulation.;Keywords: Poisson process, generalized mixture model, matrix Dirichlet distribution, Bayes network, high-dimensional longitudinal data, multiple spike trains. |
