Date of Award

2017

Document Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Thomas Zaslavsky

Abstract

Interaction graphs are graphic representations of complex networks of mutually interacting components. Their main application is in the field of gene regulatory networks, where they are used to visualize how the expression levels of genes activate or inhibit the expression levels of other genes.

First we develop a natural transformation of activation functions and their derived interaction graphs, called conjugation, that is related to a natural transformation of signed digraphs called switching isomorphism. This is a useful tool for the analysis of interaction graphs used throughout the rest of the dissertation.

We then discuss the question of what restrictions, if any, apply to interaction graphs derived from activation functions. Within these restrictions, we then construct activation functions with any desired interaction graph. The specific case of threshold activation functions, a commonly used kind of activation function, is also considered.

We then conclude with some discussion, and new proofs of the conjectures of René Thomas, using the theory of conjugate activation functions. These conjectures relate feedback in the interaction graph to dynamic properties of multi-stationarity and periodic stability. We prove a more general form of Richard and Comet's version of René Thomas' first conjecture. Included is a new counterexample to a local version of Thomas' second conjecture, on only eight components. This is the smallest counterexample that I am currently aware of.

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