Network topology shapes the ability of neural networks to store and transmit information. In neuronal network models the topology is represented by a directed graph, which is a set of vertices (the neurons) and a set of directed edges that connect them in a precise manner (the synapses). One of the challenges of network modeling is, therefore, the construction of random graphs that reflect both the variability and the main structural properties of real neural networks. Erdös-Rényi (ER) graphs, defined by a single parameter p which determines the probability of any directed edge, are among the simplest random models and have been used in a broad range of theoretical studies. Some experimental studies suggest, however, that cortical microcircuits are not well represented by ER models , .
Here we discuss several alternative classes of network models for fitting the available data: 1 - networks with distinct neuronal clusters, 2 - networks with spatially decaying connectivity, and 3 - networks with broad, correlated in-degree and out-degree distributions. We find that all three classes of networks fit the available data well, including doublet and triple motifs  and the increase in the likelihood of connectivity between neurons as a function of common neighbors . Interestingly, networks of the third class, namely, with broad, correlated degrees, do not include any clustering, indicating that one should use caution in drawing inferences about such higher order structure from lower order statistics.
Finally, we study the dynamics of networks of spiking neurons for each of the three classes of network connectivity. Specifically, we consider the networks in the so-called fluctuation-driven in which recurrent excitatory and inhibitory currents are large and balanced, leading to spiking statistics similar to that seen in-vivo: e.g. CV of ISI near 1, low mean firing rate, broad firing rate distributions. In this regime we look for dynamic signatures of each network class which could be used to aid in distinguishing the underlying network structure. We find that the network response to transient stimuli can temporarily break the balance of currents in the stationary regime, thereby generating dynamics which is strongly shaped by the underlying network structure.
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