Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach, we study the relationship between the network connectivity structure and its low dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and post-synaptic neurons through a sufficiently smooth function g of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of g. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low-dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.

18The structure in cortical micro-circuits deviates from what would be expected in a purely random 19 network, which has been seen as evidence of clustering. To address this issue we sought to reproduce 20 the non-random features of cortical circuits by considering several distinct classes of network topology, 21including clustered networks, networks with distance-dependent connectivity and those with broad 22 degree distributions. To our surprise we found that all these qualitatively distinct topologies could 23 account equally well for all reported non-random features, despite being easily distinguishable from 24 one another at the network level. This apparent paradox was a consequence of estimating network 25 properties given only small sample sizes. In other words, networks which differ markedly in their global 26 structure can look quite similar locally. This makes inferring network structure from small sample sizes, 27 a necessity given the technical difficulty inherent in simultaneous intracellular recordings, problematic. 28We found that a network statistic called the sample degree correlation (SDC) overcomes this difficulty. 29The SDC depends only on parameters which can be reliably estimated given small sample sizes, and 30 is an accurate fingerprint of every topological family. We applied the SDC criterion to data from rat 31 visual and somatosensory cortex and discovered that the connectivity was not consistent with any of 32 these main topological classes. However, we were able to fit the experimental data with a more general 33 network class, of which all previous topologies were special cases. The resulting network topology could 34 be interpreted as a combination of physical spatial dependence and non-spatial, hierarchical clustering. 35Significance Statement 36The connectivity of cortical micro-circuits exhibits features which are inconsistent with a simple random 37 network. Here we show that several classes of network models can account for this non-random structure 38 despite qualitative differences in their global properties. This apparent paradox is a consequence of the small 39 numbers of simultaneously recorded neurons in experiment: when inferred via small sample sizes many 40 networks may be indistinguishable, despite being globally distinct. We develop a connectivity measure 41 which successfully classifies networks even when estimated locally, with a few neurons at a time. We show 42 that data from rat cortex is consistent with a network in which the likelihood of a connection between 43 neurons depends on spatial distance and on non-spatial, asymmetric clustering.

Meanfield theory for networks of spiking neurons based on the so-called diffusion approximation has been used to calculate certain measures of neuronal activity which can be compared with experimental data. This includes the distribution of firing rates across the network. However, the theory in its current form applies only to networks in which there is relatively little heterogeneity in the number of incoming and outgoing connections per neuron. Here we extend this theory to include networks with arbitrary degree distributions. Furthermore, the theory takes into account correlations in the in-degree and out-degree of neurons, which would arise e.g. in the case of networks with hub-like neurons. Finally, we show that networks with broad and postively correlated degrees can generate a large-amplitude sustained response to transient stimuli which does not occur in more homogeneous networks.