Second Order Networks with spatial structure

Abstract

Synchronization of spiking activity across neurons plays a role in many processes in the brain. Using the framework of Second Order Networks (SONETs) paired with a global ring structure, we looked at the relationships between the connectivity statistics and two key eigenvalue quantities related to the synchrony of the network - the largest eigenvalue of the connectivity matrix and the variance of the eigenvalues of the Laplacian. Previously, Zhao et al. (2011) examined these relationships in the case of homogeneous SONETs, in which there is no spatial variation in the network. In this work, we broaden our view to SONETs where we allow the connection probabilities to depend on the spatial structure of the network. First, we develop an algorithm to generate SONETs which allows us to specify both the global and local geometry of the network. We then randomly generated a wide range of SONETs to examine the relationships between the connectivity statistics and the eigenvalue quantities of the resulting networks. We find that two of the second order statistics, namely those corresponding to the frequency of convergent connections and to the frequency of chain connections, primarily influence the values of the two eigenvalue quantities. Our results are remarkably similar to those of the homogeneous case, indicating that the qualitative relationship we see between synchrony and second order statstics should extend to a larger class of networks. We also find that for the networks we considered, the parameters used to describe the overall geometry of the network had a minimal influence on the two key eigenvalue quantities.