Dynamical effects of degree correlations in networks of type I model neurons : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Auckland, New Zealand

The complex behaviour of human brains arises from the complex interconnection of the well-known building blocks -- neurons. With novel imaging techniques it is possible to monitor firing patterns and link them to brain function or dysfunction. How the network structure affects neuronal activity is, however, poorly understood. In this thesis we study the effects of degree correlations in recurrent neuronal networks on self-sustained activity patterns. Firstly, we focus on correlations between the in- and out-degrees of individual neurons. By using Theta Neurons and Ott/Antonsen theory, we can derive a set of coupled differential equations for the expected dynamics of neurons with equal in-degree. A Gaussian copula is used to introduce correlations between a neuron’s in- and out-degree, and numerical bifurcation analysis is used determine the effects of these correlations on the network's dynamics. We find that positive correlations increase the mean firing rate, while negative correlations have the opposite effect. Secondly, we turn to degree correlations between neurons -- often referred to as degree assortativity -- which describes the increased or decreased probability of connecting two neurons based on their in-or out-degrees, relative to what would be expected by chance. We present an alternative derivation of coarse-grained degree mean field equations utilising Theta Neurons and the Ott/Antonsen ansatz as well, but incorporate actual adjacency matrices. Families of degree connectivity matrices are parametrised by assortativity coefficients and subsequently reduced by singular value decomposition. Thus, we efficiently perform numerical bifurcation analysis on a set of coarse-grained equations. To our best knowledge, this is the first time a study examines the four possible types of degree assortativity separately, showing that two have no effect on the networks' dynamics, while the other two can have a significant effect

A permutation method for network assembly.

We present a method for assembling directed networks given a prescribed bi-degree (in- and out-degree) sequence. This method utilises permutations of initial adjacency matrix assemblies that conform to the prescribed in-degree sequence, yet violate the given out-degree sequence. It combines directed edge-swapping and constrained Monte-Carlo edge-mixing for improving approximations to the given out-degree sequence until it is exactly matched. Our method permits inclusion or exclusion of 'multi-edges', allowing assembly of weighted or binary networks. It further allows prescribing the overall percentage of such multiple connections-permitting exploration of a weighted synthetic network space unlike any other method currently available for comparison of real-world networks with controlled multi-edge proportion null spaces. The graph space is sampled by the method non-uniformly, yet the algorithm provides weightings for the sample space across all possible realisations allowing computation of statistical averages of network metrics as if they were sampled uniformly. Given a sequence of in- and out- degrees, the method can also produce simple graphs for sequences that satisfy conditions of graphicality. Our method successfully builds networks with order O(107) edges on the scale of minutes with a laptop running Matlab. We provide our implementation of the method on the GitHub repository for immediate use by the research community, and demonstrate its application to three real-world networks for null-space comparisons as well as the study of dynamics of neuronal networks