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

CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains

Recent developments in electrophysiological and optical recording techniques enable the simultaneous observation of large numbers of neurons. A meaningful interpretation of the resulting multivariate data, however, presents a serious challenge. In particular, the estimation of higher-order correlations that characterize the cooperative dynamics of groups of neurons is impeded by the combinatorial explosion of the parameter space. The resulting requirements with respect to sample size and recording time has rendered the detection of coordinated neuronal groups exceedingly difficult. Here we describe a novel approach to infer higher-order correlations in massively parallel spike trains that is less susceptible to these problems. Based on the superimposed activity of all recorded neurons, the cumulant-based inference of higher-order correlations (CuBIC) presented here exploits the fact that the absence of higher-order correlations imposes also strong constraints on correlations of lower order. Thus, estimates of only few lower-order cumulants suffice to infer higher-order correlations in the population. As a consequence, CuBIC is much better compatible with the constraints of in vivo recordings than previous approaches, which is shown by a systematic analysis of its parameter dependence