Dynamical implications of network statistics

Abstract

Dynamics on large networks can be highly complex. I present several methods for investigating the effects of network structure/statistics on rate dynamics and spike correlations. The dynamical models under consideration come from computational neuroscience, but these methods may generalize to other contexts. The thesis focuses on both network constructions and dynamics on networks.I present two approaches to network constructions: random networks, and networks with patterns. For random networks, I give a generalization of the expected degree model (EDM) and a formulation of the EDM and its generalization which is invariant of the number of nodes. This generalization allows one to produce random networks which have nontrivial second and third order correlations among edges. I also introduce a method for constructing networks with nontrivial structures/patterns at multiple scales. I investigate the spectral properties of the resulting networks and extend the method to include stochastic elements.The singular value decomposition (SVD) is a powerful tool with many applications. I review its application to network adjacency matrices, including a known result which relates the singular values of an adjacency matrix to a measure of its randomness. Further, I demonstrate that for several random network models the degree sequence is the most significant feature of the connectivity.The primary dynamical model I consider is the Poisson spiking model (PSM). I derive first and second order statistics for the PSM using a path integral formalism.The major contribution of this work is a dimension reduction method for dynamics on a network using the SVD. I demonstrate how one can use these low rank representations of the connectivity, together with the reduced equations to approximately recover node-specific activity. Thus, not only do I present methods that reduce the number of dynamical variables, but I show how the dynamics of the full system may be decomposed into the reduced variables and network structure.