## Mathematics, Department of

#### Date of this Version

Fall 12-1-2015

#### Abstract

Sequences of neuronal activation have long been implicated in a variety of brain functions. In particular, these sequences have been tied to memory formation and spatial navigation in the hippocampus, a region of mammalian brains. Traditionally, neuronal sequences have been interpreted as noisy manifestations of neuronal templates (i.e., orderings), ignoring much richer structure contained in the sequences. This paper introduces a new tool for understanding neuronal sequences: the bias matrix. The bias matrix captures the probabilistic tendency of each neuron to fire before or after each other neuron. Despite considering only pairs of neurons, the bias matrix captures the best total ordering of neurons for a sequence (Proposition 3.3) and, thus, generalizes the concept of a neuronal template.

We establish basic mathematical properties of bias matrices, in particular describing the fundamental polytope in which all biases reside (Theorem 3.25). We show how the underlying simple digraph of a bias matrix, which we term the bias network, serves as a combinatorial generalization of neuronal templates. Surprisingly, every simple digraph is realizable as the bias network of some sequence (Theorem 3.34). The bias-matrix representation leads us to a natural method for sequence correlation, which then leads to a probabilistic framework for determining the similarity of one set of sequences to another. Using data from rat hippocampus, we describe events of interest and also sequence-detection techniques. Finally, the bias matrix and the similarity measure are applied to this real-world data using code developed by us.

Advisor: Vladimir Itskov

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Vladimir Itskov. Lincoln, Nebraska: December, 2015

Copyright (c) 2015 Zachary J. Roth