Spectra of the spike flow graphs of recurrent neural networks

Abstract

Recently the notion of power law networks in the context of neural networks has gathered considerable attention. Some empirical results show that functional correlation networks in human subjects solving certain tasks form power law graphs with exponent approaching ≈∈2. The mechanisms leading to such a connectivity are still obscure, nevertheless there are sizable efforts to provide theoretical models that would include neural specific properties. One such model is the so called spike flow model in which every unit may contain arbitrary amount of charge, which can later be exchanged under stochastic dynamics. It has been shown that under certain natural assumptions about the Hamiltonian the large-scale behavior of the spike flow model admits an accurate description in terms of a winner-take-all type dynamics. This can be used to show that the resulting graph of charge transfers, referred to as the spike flow graph in the sequel, has scale-free properties with power law exponent γ=∈2. In this paper we analyze the spectra of the spike flow graphs with respect to previous theoretical results based on the simplified winner-take-all model. We have found numerical support for certain theoretical predictions and also discovered other spectral properties which require further theoretical investigation. © 2009 Springer Berlin Heidelberg.