Effects of local network topology on the functional reconstruction of spiking neural network models

networks have been used to study complex interactions between many different types of actors (Boccaletti et al. 2006). These actors and interactions represent a range of applications including social connections among people (Holland and Leinhardt 1970; Haynie 2002; Gleiser and Danon 2003), games and economics (Kim et al. 2002; Ebel and Bornholdt 2002; Holme et al. 2003), protein-protein interactions (Barabasi and Oltvai 2004; Maslov and Sneppen 2002; Jeong et al. 2001) and cellular signaling (Weng et al. 1999; Bhalla and Iyengar 1999; Stożer et al. 2013). In neuroscience, abstract complex networks have been implemented to model (a) structural and (b) functional neural networks. The former model the structure of neural tissue by representing either brain regions at the macroscopic level or individual neurons and synapses at the microscopic scale (Downes et al. 2012; Shimono and Beggs 2015; Sporns 2010). Structural networks in © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Akin et al. Applied Network Science (2017) 2:22 Page 2 of 22 this paper represent the microscopic level as much of the brain’s information-processing and storage capacity is thought to arise from its synaptic connections and the structure they determine (Sporns 2010). It is therefore necessary to identify important features of this structure and their roles in information-processing and storage. Functional networks, by contrast, are constructed from correlations of activity between neural regions or neurons. Functional networks of neuronal microcircuitry have been studied using various methods, such as microelectrode arrays, and have been shown to exhibit many nonrandom features such as small-worldedness (Downes et al. 2012; Watts and Strogatz 1998), well-defined community structure (Shimono and Beggs 2015), hubs (Shimono and Beggs 2015; Timme et al. 2016), and motifs (Song et al. 2005; Perin et al. 2011). The preand postsynaptic roles of connected neurons imply that information flow is directed in a neural network, so we only consider the directed case for both structural and functional networks. We note that directed functional connections, which signify causal influence, are frequently referred to as “effective connections.” In spite of our focus on directed connections in this paper, we use “functional” in place of “effective” throughout as our methods could also be applied to the undirected case. While it is known that the functional network is influenced by the synaptic structure, it is not known whether the underlying synaptic connectivity has the same features of the functional network. A few computational studies have attempted to answer how closely the functional networkmaymatch the synaptic network fromwhich it arises. In (Garofalo et al. 2009; Ito et al. 2011), the authors used random networks of Izhekevich neurons and various correlation measures to investigate the one-to-one matching of functional and structural networks. While (Garofalo et al. 2009) showed fairly poor matching between structural and functional networks, (Ito et al. 2011) used higher-order transfer entropy to functionally infer up to 80% of existing synaptic connections (true-positives) at low rates of false-positive occurrence (functional connections where no synaptic connection exists). In (Kobayashi and Kitano 2013), the authors furthered this direction by studying regular and small-world networks. The authors showed that as the probability for the creation of small-world connections increased, the one-to-one matching of the structural network and functional network decreased. These articles demonstrated that functional networks do not match exactly the underlying synaptic structure, but rather may include false-positives and false-negatives (an absence of functional connection where a synaptic connection exists). We investigate the nonrandomness of these false-positive (FP) and false-negative (FN) features to address the influence of structural on functional neural networks. A possible influence of the location of FP’s and FN’s may exist in the local synaptic connectivity, which can be represented as subgraphs of the larger network. Subgraphs that occur more often in a given network than in random models have been of particular interest. These overrepresented subgraphs are called motifs and may play an important role in network function (Milo et al. 2002; Alon 2007). In directed networks, motifs are directed subgraphs whose occurrence counts have high positive Z-scores when compared to a suitable random null model. Subgraph motifs are classified by the number of vertices they contain. Two types of subgraphs have been found to be particularly interesting for neural networks: dyadic (2-vertex) and triadic (3-vertex) (Shimono and Beggs 2015; Song et al. 2005; Perin et al. 2011; Guo and Li 2009; Li 2008; Vasilaki and Giugliano 2014). All nonisomorphic dyadic and triadic subgraphs are shown in Figs. 1 and 2 respectively. For Akin et al. Applied Network Science (2017) 2:22 Page 3 of 22 Fig. 1 Dyadic Subgraphs. All nonisomorphic directed graphs consisting of two vertices. Dyad 3 represents recurrent connections which are thought to be important in information storage and processing in neural