How Structure Determines Correlations in Neuronal Networks Volker Pernice1,2*, Benjamin Staude1,2, Stefano Cardanobile1,2, Stefan Rotter1,2 1 Bernstein Center Freiburg, Freiburg, Germany, 2 Computational Neuroscience Lab, Faculty of Biology, Albert-Ludwig University, Freiburg, Germany Abstract 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. Citation: Pernice V, Staude B, Cardanobile S, Rotter S (2011) How Structure Determines Correlations in Neuronal Networks. PLoS Comput Biol 7(5): e1002059. doi:10.1371/journal.pcbi.1002059 Editor: Olaf Sporns, Indiana University, United States of America Received November 25, 2010 Accepted April 1, 2011 Published May 19, 2011 Copyright: �� 2011 Pernice et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This work was supported by the German Federal Ministry of Education and Research (BMBF) grant 01GQ0420 to BCCN Freiburg, http://www.bmbf.de/ en/3063.php, and the German Research Foundation (DFG), CRC 780, project C4, http://www.dfg.de/en/. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: pernice@bcf.uni-freiburg.de Introduction Analysis of networks of interacting elements has become a tool for system analysis in many areas of biology, including the study of interacting species [1], cell dynamics [2] and the brain [3]. A fundamental question is how the dynamics, and eventually the function, of the system as a whole depends on the characteristics of the underlying network. A specific aspect of dynamics that has been linked to structure are fluctuations in the activity and their correlations in noisy systems. This work deals with neuronal networks, but other examples include gene-regulatory networks [4], where noise propagating through the network leads to correlations [5], and different network structures have important influence on dynamics by providing feedback loops [6,7]. The connection between correlations and structure is of special interest in neuroscience. First, correlations between neural spike trains are believed to play an important role in information processing [8,9] and learning [10]. Second, the structure of neural networks, encoded by synaptic connections between neurons, is exceedingly complex. Experimental findings show that synaptic architecture is intricate and structured on a fine scale [11,12]. Nonrandom features are induced by neuron morphology, for example distance dependent connectivity [13,14], or specific connectivity rules depending on neuron types [15,16]. A number of novel techniques promise to supply further details on local connectivity [17,18]. Measured spike activity of neurons in such networks shows, despite high irregularity, significant correlations. Recent technical advances like multiple tetrode recordings [19], multielectrode arrays [20���22] or calcium imaging techniques [23,24] allow the measurement of correlations between the activity of an increasingly large number of neuron pairs in vivo. This makes it possible to study the dynamics of large networks in detail. Since recurrent connections represent a substantial part of connectivity, it has been proposed that correlations originate to a large degree in the convergence and divergence of direct connectivity and common input [8] and must therefore strongly depend on connectivity patterns [25]. Experimental studies found evidence for this thesis in a predominantly feed-forward circuit [26]. In another study, only relatively small correlations were detected [27] and weak common input effects or a mechanism of active decorrelation were postulated. In recent theoretical work recurrent effects have been found to be an important factor in correlation dynamics and can account for decorrelation [20,22]. Several theoretical studies have analysed the effects of correlations on neuron response [28,29] and the transmission of correlations [30���34], also through several layers [35]. However, the description of the interaction of recurrent connectivity, correlations and neuron dynamics in a self-consistent PLoS Computational Biology | www.ploscompbiol.org 1 May 2011 | Volume 7 | Issue 5 | e1002059

theory has not been presented yet. Even in the case of networks of strongly simplified neuron models like integrate and fire or binary neurons, nonlinear effects prohibit the evaluation of effects of complex connectivity patterns. In [36,37] correlations in populations of neurons were studied in a linear model that accounted for recurrent feedback. With a similar model, the framework of interacting point processes developed by Hawkes [38,39], we analyse effects of different connectivity patterns on pairwise correlations in strongly recurrent networks. Spike trains are modeled as stochastic processes with presynaptic spikes affecting postsynaptic firing rates in a linear manner. We describe a local network in a state of irregular activity, without modulations in external input. This allows the self-consistent analytical treatment of recurrent feedback and a transparent description of structural effects on pairwise correlations. One application is the disentanglement of the explicit contributions of recurrent input on correlations in spike trains in order to take into account not only effects of direct connections, but also indirect connectivity, see Figure 1. We find that variations in synaptic topology can substantially influence correlations. We present several scenarios for character- istic network architectures, which show that different connectivity patterns affect correlations predominantly through their influence on statistics of indirect connections. An influential model for local neural populations is the random network model [40,41], possibly with distance-dependent connectivity. In this case, the average correlations, and thereby the level of population fluctuations or noise, only depend on the average connectivity and not on the precise connectivity profile. The latter, however, influences higher order properties of the correlation distribution. This insensitivity to fine-tuning is due to the homogeneity of the connectivity of individual neurons in this type of networks. The effect has also been observed in a very recent study, where large-scale simulations were performed [42]. In networks with more complex structural elements, like hubs or patches, however, we find that also average correlations depend on details of the connectivity pattern. Part of this work has been published in abstract form [43]. Methods Recurrent networks of linearly interacting point processes In order to study correlations in networks of spiking neurons with arbitrary connectivity we use the theory derived in [38], which we refer to as Hawkes model, for the calculation of stationary rates and correlations in networks of linearly interacting point processes. We only summarise the definitions and equations needed in the specific context here. A mathematically more rigorous description can be found in [38] and detailed applications in [44,45]. We will use capital letters for matrices and lower case letters for matrix entries, for example G~(gij). Vectors will not be marked explicitly, but their nature should be clear from the context. Fourier transformed quantities, discrete or continuous, will be denoted by ^, : : for example ^(v). a a Used symbols are summarised in Table 1. Our networks consists of N neurons with NE excitatory and NI inhibitory neurons. Spike trains si(t)~ P j d(t{tj(i)) of neurons i~1 . . . N are modeled as realisations of Poisson processes with time-dependent rates yi(t). We have yi(t)~Ssi(t)T, ��1�� where S:T denotes the mathematical expectation, in this case across spike train realisations. Neurons thus fire randomly with a fluctuating rate which depends on presynaptic input. For the population of neurons we use the spike train vector s and the rate vector y. Spikes of neuron j influence the rate of a connected neuron i by inducing a transient rate change with a time course described by the interaction kernel gij(t), which can in principle be different for all connections. For the sake of simplicity we use the same interaction kernels for all neurons of a subpopulation. The rate change due to a spike of an excitatory presynaptic neuron is described by gE(t) and of an inhibitory neuron by gI (t).��The total excitatory synaptic weight can then be defined as��gE : gE(t)dtw0 and the inhibitory weight accordingly as gI : gI (t)dtv0. Connections between neurons are chosen randomly under varying restrictions, as explained in the following sections. For unconnected neurons gij~0. The evolution of the rate vector is governed by the matrix equation y(t)~y0z ��? {? G(t0)s(t{t0)dt0~y0z�� G s��(t): ��2�� The effect of presynaptic spikes at time t{t0 on postsynaptic rates is given by the interaction kernels in the matrix G(t0) and depends on the elapsed time t0. Due to the linearity of the convolution, effects of individual spikes are superimposed linearly. The constant spike probability y0 can be interpreted as constant external drive. We Figure 1. Connectivity induces correlations. A: Activity in a pair of neurons (red) in a network can become correlated due to direct connections (blue) and different types of shared input (cyan). B: For a complete description a large number of indirect interactions (yellow, orange) and indirect common input contributions (green) have to be taken into account. However, not all nodes and connections contribute to correlations (grey). doi:10.1371/journal.pcbi.1002059.g001 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 correla- tions 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 mark- edly influence correlations and population signals. Structure and Correlations in Neuronal Networks PLoS Computational Biology | www.ploscompbiol.org 2 May 2011 | Volume 7 | Issue 5 | e1002059