Extreme Graphical Models with Applications to Functional Neuronal Connectivity

With modern calcium imaging technology, the activities of thousands of neurons can be recorded simultaneously in vivo. These experiments can potentially provide new insights into functional connectivity, defined as the statistical relationships between the spiking activity of neurons in the brain. As a commonly used tool for estimating conditional dependencies in high-dimensional settings, graphical models are a natural choice for analyzing calcium imaging data. However, raw neuronal activity recording data presents a unique challenge: the important information lies in the rare extreme value observations that indicate neuronal firing, as opposed to the non-extreme observations associated with inactivity. To address this issue, we develop a novel class of graphical models, called the extreme graphical model, which focuses on finding relationships between features with respect to the extreme values. Our model assumes the conditional distributions a subclass of the generalized normal or Subbotin distribution, and yields a form of a curved exponential family graphical model. We first derive the form of the joint multivariate distribution of the extreme graphical model and show the conditions under which it is normalizable. We then demonstrate the model selection consistency of our estimation method. Lastly, we study the empirical performance of the extreme graphical model through several simulation studies as well as through a real data example, in which we apply our method to a real-world calcium imaging data set.