Robustness of power laws in degree distributions for spiking neural networks

Power law graphs are an actively studied branch of random graph theory, motivated by a number of recent empirical discoveries which revealed power law degree distributions in a variety of networks. Power laws often coexist with some degree of self-organization either based on growth and preferential attachment (which seems to be the case in sociological/technological networks) or duplication (which seems to be the case for biological/methabolic networks). Quite recently a power law graph with exponent gamma ap 2 has been observed in fMRI brain studies of correlations of functional centers of activity. We study the model we introduced previously to explore possible mechanisms existing in large neural networks that might lead to power law connectivity. The model (referred to as the spike flow model) resembles a kind of spiking neural network and yields a power law graph with exactly gamma = 2 as a byproduct of its dynamical behavior. In this paper we investigate whether the power law is robust under certain changes to the model´s dynamics. In particular we study the effect of merging the model with a random Erdos-Renyi graph which can be interpreted as an addition of long range myelinated connections. Our numerical results indicate that as long as the density of Erdos-Renyi fraction is bounded by a constant, the power law is preserved in systems of appropriate size.