Network topology

Most studies to date have analyzed the kind of synchronization thought to

coordinate information flow between pairs of neurons, areas, or local populations. Yet, the brain or brain areas are a strong interconnected network on the anatomical as well as functional scale (Berger et al., 2007; Bullmore and Sporns, 2009; Markov et al., 2014), which makes it essential to analyze the functional network structure to understand the

coordination of information flow. However, due to the above-mentioned possibility that aspects of the communication can average out at the level of population signals, it is essential to analyze functional interaction on the level of single neurons to understand the formation of potential ensembles.

1.4.2. Network topology

The ability to record many neuronal signals simultaneously (e.g. with recently developed optical, electrophysiological, and computational tools (Buzsáki, 2004; Sejnowski et al., 2014;

Yuste, 2015; Pnevmatikakis et al., 2016), see above), has allowed investigators to estimate functional networks using functional connectivity measures (Bastos and Schoffelen, 2016).

However, identifying the functional connectivity of hundreds or thousands of neuronal signals presents a problem for analyzing these networks in terms of their structure and their organizational principles, referred to as network topology. Many useful analyses for this purpose were developed by mathematicians from the field of network science, which was only recently established in the late 1990s based on graph theory (Watts and Strogatz, 1998;

Bullmore and Sporns, 2009).

In the first study of the field of network science (Watts and Strogatz, 1998), three important network measures were defined. The first two are the cluster coefficient, which measures interconnectivity between direct neighbors of one node of a network, and the shortest path length, which measures the minimum number of nodes which have to be passed to get to another node. A simple regular network where each node is connected to its four spatial neighbors has a high average cluster coefficient but a long average shortest path length. In contrast, a random network has a small average cluster coefficient and a short average path length. An interesting finding of this study was that, by randomly switching pairs of connections of a regular network, an intermediate state of high average cluster coefficient and small average shortest path length was present before the network became random. Networks that combine both are referred to as small-world, which was the

third defined network measure. Interestingly, the anatomical single neuron network of C.

elegans and who-played-with-whom network of Hollywood actors both turned out to be small-world. After this striking finding, many more topological principles were found and described which seem to be common principles of many natural networks and led to the definition of complex networks (Barabási and Oltvai, 2004; Barabási, 2009; Bullmore and Sporns, 2009). Natural networks were found to have a modular topology, which means that groups of nodes within a network are more strongly interconnected with each other than with the rest of the network (Ravasz et al., 2002). The importance of individual nodes for the network communication or the network coherence of natural networks can be measured by centrality metrics. Natural networks were shown to have heavy-tailed centrality

distributions, with a small number of nodes connecting the network and coordinating the network function (these nodes are called “hubs”), while the majority of nodes are only of minor importance for the overall network function (van den Heuvel and Sporns, 2013). The first described and simplest measure of centrality is degree centrality, which is defined as the number of connections per node (Barabási et al., 1999; Jeong et al., 2000). A more global aspect of centrality is captured by betweenness centrality, an index of the number of

shortest paths from all single units to all others that pass through that node (Freeman, 1977). In some networks, hubs exhibit a strong tendency to link to each other, forming a so-called rich-club (Colizza et al., 2006). This property can be measured by a rich-club

coefficient that expresses the tendency of highly connected hub nodes to show above-random levels of interconnectivity.

Network analyses of anatomical and functional inter-area brain networks measured with tracers, EEG, MEG, or fMRI also revealed them to be topologically organized, as with complex networks (Bullmore and Sporns, 2009). The regional brain networks of humans and monkeys were found to have a modular and small-world topology (Hilgetag et al., 2000;

Stephan et al., 2000; Bullmore and Sporns, 2009) Further, the centrality distributions of areas were found to be heavy-tailed with hub areas (Achard et al., 2006; Honey and Kötter, 2007; Honey et al., 2007), which were strongly interconnected as a rich-club coordinating global brain communication (Harriger et al., 2012; van den Heuvel et al., 2012).

However, functional network topology analyses of more localized neuronal signals of mammalian brains are lacking in the literature. Three studies analyzing the single neuron functional network of organotypic slices of rat brain showed that the single neuron

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functional connectivity topology was modular, with functional hub neurons organized as a rich-club coordinating the network communication (Bonifazi et al., 2009; Shimono and Beggs, 2014; Schroeter et al., 2015). Nevertheless, the neuronal activity of oranotypic slices is altered compared to the intact brain (Steriade, 2001). Many of the original connections and many neurons in the slice die due to the slicing procedure, no natural sensory inputs are received by the neuronal network, and plasticity effects after the extraction of the slice even further change the neuronal connectivity. Only three studies analyzed the functional

network topology of single neurons recorded in the intact brain. The first study was performed on anesthetized cats passively viewing visual stimuli while many neurons were recorded in parallel in V1, showing a small-world topology of functional connectivity (Yu et al., 2008). The second study was performed on awake monkeys also viewing visual stimuli, while neurons were recorded in parallel in V1. In contrast to the first study, these

investigators suggested that single neuron functional small-world topology is an artifact of distance-dependent functional connectivity (Gerhard et al., 2011). However, the number of recorded neurons was small, and even that small number was most likely due to massive oversorting, questioning the validity of the results from this study. The last and most recent study was performed on awake rats under uncontrolled behavior while neurons were recorded in medial to lateral orbitofrontal cortex. It was reported that the functional single neuron topology could be described as a rich-club (Nigam et al., 2016). Yet, the uncontrolled behavior utilized in that study did not allow for a separation of behaviorally driven common neuronal network activations, such as those triggered by different movements or from synchronization processes reflecting the coordination of network interaction. In summary, it remains unclear how the functional network of local neuronal populations or single neurons is topologically organized within and across areas in order to coordinate information flow.

Since it is so far not feasible to record the majority of neurons in the brain in parallel or of high numbers of areas, an important question is: what is an interesting cortical network from which to record many neurons in parallel? The network should be suitable for analyzing single neuron functional network topology and oscillatory synchronization process in regard to coordination of information flow, as well as the encoding and transformation of

information from perception to behavior.