Allen, Genevera I2022-10-202022-10-202022-122022-09-29December 2Chang, Andersen. "Graphical Models for Functional Neuronal Connectivity." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113740">https://hdl.handle.net/1911/113740</a>.https://hdl.handle.net/1911/113740With modern calcium imaging technology, activities of thousands of neurons can be recorded in vivo. These experiments can potentially provide new insights into intrinsic functional neuronal connectivity, defined as contemporaneous correlations between neuronal activities. As a common tool for estimating conditional dependencies in high-dimensional settings, graphical models are a natural choice for estimating functional connectivity networks. However, raw neuronal activity data presents several statistical challenges when applying graphical models. In this project, we develop new methods to estimate scientifically meaningful functional neuronal connectivity networks using the graphical model framework. One unique facet of calcium imaging data is that the important information lies in rare extreme value observations that indicate neuronal firing, rather than in the observations near the mean. Thus, a graphical modeling technique which finds conditional dependencies between the extreme values of features is required in order to estimate scientifically meaningful functional connectivity networks from calcium imaging data. To address this, we develop a novel class of graphical models, called the Subbotin graphical model, which can be used to find sparse conditional dependency structures for extreme values. We first derive the form of the Subbotin graphical model and show the conditions under which it is normalizable. We then study the empirical performance of the Subbotin graphical model on simulations as well as real-world data. Additionally, in many modern calcium imaging data sets, the complete data set is often comprised of multiple individual recording sessions of partially overlapping subsets of neurons. Thus, in order to estimate a graph on the full data, conditional dependencies in the missing portion of the covariance must be inferred; this is known as the graph quilting problem. We introduce several graph quilting methods that can be applied to for calcium imaging data, which utilize a low-rankness assumption to impute the full covariance matrix. Through several empirical studies, we investigate the efficacy of these methods for estimating graphical models for functional connectivity in the presence of missing joint observations. We also develop new methods for covariate and dynamic latent variable adjustment for functional neuronal data, which can arise from the presence of stimuli, unobserved neurons, and physical activity. We first introduce two models to infer functional connectivity from neuronal activity data after adjusting for dynamic latent brain states, and we use simulation studies to compare their performance to traditional, unconditional graphical models. We then propose a new method for sparse high-dimensional linear regression for extreme values, called the Extreme Lasso. We prove consistency and variable selection consistency for our regression method, and we analyze the theoretical impact of extreme value observations on the model parameter estimates using the concept of influence functions. We then study the empirical performance of the Extreme Lasso for selecting features associated with extreme values in high-dimensional regression. In our work, we demonstrate the applicability of each of our developed methods to finding functional connectivity networks through studies on several real-world calcium imaging data sets. In particular, we compare these network estimates to those from existing methods from both the graphical model and neuroscience literature, and we show that our methods can provide more scientifically sensible functional connectivity estimates.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Graphical modelsfunctional connectivitycalcium imagingThesis2022-10-20