How Structure Determines Correlations in Neuronal Networks

Networks are becoming a ubiquitous metaphor for the understanding of complex biological systems, spanning the range between molecular signalling pathways, neural networks in the brain, and interacting species in a food web. In many models, we face an intricate interplay between the topology of the network and the dynamics of the system, which is generally very hard to disentangle. A dynamical feature that has been subject of intense research in various fields are correlations between the noisy activity of nodes in a network. We consider a class of systems, where discrete signals are sent along the links of the network. Such systems are of particular relevance in neuroscience, because they provide models for networks of neurons that use action potentials for communication. We study correlations in dynamic networks with arbitrary topology, assuming linear pulse coupling. With our novel approach, we are able to understand in detail how specific structural motifs affect pairwise correlations. Based on a power series decomposition of the covariance matrix, we describe the conditions under which very indirect interactions will have a pronounced effect on correlations and population dynamics. In random networks, we find that indirect interactions may lead to a broad distribution of activation levels with low average but highly variable correlations. This phenomenon is even more pronounced in networks with distance dependent connectivity. In contrast, networks with highly connected hubs or patchy connections often exhibit strong average correlations. Our results are particularly relevant in view of new experimental techniques that enable the parallel recording of spiking activity from a large number of neurons, an appropriate interpretation of which is hampered by the currently limited understanding of structure-dynamics relations in complex networks. Author Summary: Many biological systems have been described as networks whose complex properties influence the behaviour of the system. Correlations of activity in such networks are of interest in a variety of fields, from gene-regulatory networks to neuroscience. Due to novel experimental techniques allowing the recording of the activity of many pairs of neurons and their importance with respect to the functional interpretation of spike data, spike train correlations in neural networks have recently attracted a considerable amount of attention. Although origin and function of these correlations is not known in detail, they are believed to have a fundamental influence on information processing and learning. We present a detailed explanation of how recurrent connectivity induces correlations in local neural networks and how structural features affect their size and distribution. We examine under which conditions network characteristics like distance dependent connectivity, hubs or patches markedly influence correlations and population signals.