Oscillations in networks of networks stem from adaptive nodes with memory

Optimal responsiveness and information flow in networks of heterogeneous neurons

Cerebral cortex is characterized by a strong neuron-to-neuron heterogeneity, but it is unclear what consequences this may have for cortical computations, while most computational models consider networks of identical units. Here, we study network models of spiking neurons endowed with heterogeneity, that we treat independently for excitatory and inhibitory neurons. We find that heterogeneous networks are generally more responsive, with an optimal responsiveness occurring for levels of heterogeneity found experimentally in different published datasets, for both excitatory and inhibitory neurons. To investigate the underlying mechanisms, we introduce a mean-field model of heterogeneous networks. This mean-field model captures optimal responsiveness and suggests that it is related to the stability of the spontaneous asynchronous state. The mean-field model also predicts that new dynamical states can emerge from heterogeneity, a prediction which is confirmed by network simulations. Finally we show that heterogeneous networks maximise the information flow in large-scale networks, through recurrent connections. We conclude that neuronal heterogeneity confers different responsiveness to neural networks, which should be taken into account to investigate their information processing capabilities.

Distribution of shortest path lengths in subcritical Erdős-Rényi networks

Networks that are fragmented into small disconnected components are prevalent in a large variety of systems. These include the secure communication networks of commercial enterprises, government agencies, and illicit organizations, as well as networks that suffered multiple failures, attacks, or epidemics. The structural and statistical properties of such networks resemble those of subcritical random networks, which consist of finite components, whose sizes are nonextensive. Surprisingly, such networks do not exhibit the small-world property that is typical in supercritical random networks, where the mean distance between pairs of nodes scales logarithmically with the network size. Unlike supercritical networks whose structure has been studied extensively, subcritical networks have attracted relatively little attention. A special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies. The overall statistics of the network can be obtained by a summation over all the components with suitable weights. We use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths (DSPL) on components of given sizes and topologies in subcritical Erdős-Rényi (ER) networks. From this expansion we obtain an exact analytical expression for the DSPL of the entire subcritical network, in the asymptotic limit. The DSPL, which accounts for all the pairs of nodes that reside on the same finite component (FC), is found to follow a geometric distribution of the form P_{FC}(L=ℓ|L<∞)=(1-c)c^{ℓ-1}, where c<1 is the mean degree. Using computer simulations we calculate the DSPL in subcritical ER networks of increasing sizes and confirm the convergence to this asymptotic result. We also obtain exact asymptotic results for the mean distance, 〈L〉_{FC}, and for the standard deviation of the DSPL, σ_{L,FC}, and show that the simulation results converge to these asymptotic results. Using the duality relations between subcritical and supercritical ER networks, we obtain the DSPL on the nongiant components of ER networks above the percolation transition.

Distribution of shortest cycle lengths in random networks

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdős-Rényi network), degenerate distribution (random regular graph), and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.