Asymptotic behaviors and application of nonlinear networks

Abstract

We study the asymptotic behavior of networks with discrete quadratic dynamics. While single-map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in an oriented network, generates new, interesting questions and applications to the life sciences. We extend results from single-node dynamics to the more general case of networks, and present novel, network-speci c results. We then consider two existing models from the dynamic networks literature: threshold-linear networks and a reduced model of inhibitory neural clusters. We search for graph features which lead to robust dynamics under minor perturbations within our model, as well as between the three di erent models; in other words, we search for possible features of universality and the conditions under which they hold. We create a classi cation system of large-scale networks. This classi cation system is based on network dimensionality reduction (i.e. treating a large group of nodes as a single node). Additionally, we present conditions under which reducing network dimensionality is permittable. This has important implications for applications to the study of natural networks (such as biological systems), which are often extremely large (composed of many coupled nodes). Finally, we explore possible applications of the techniques used in these three network models (complex quadratic networks, threshold-linear networks, and inhibitory clustering neural networks) to other problems in the natural sciences: a chemical oscillator model and a neural clustering model.