Towards Google matrix of brain

TL;DR

This paper applies the Google matrix approach to neuronal networks, revealing spectral properties and delocalization phenomena, and draws parallels between brain networks and the World Wide Web.

Contribution

It introduces a novel application of the Google matrix to brain networks, analyzing eigenvalue spectra and PageRank behavior in neuronal models.

Findings

Eigenvalue spectrum of the Google matrix has a gapless structure.

PageRank becomes delocalized at certain damping factors.

Neuronal networks exhibit spectral properties similar to the Web.

Abstract

We apply the approach of the Google matrix, used in computer science and World Wide Web, to description of properties of neuronal networks. The Google matrix ${\bf G}$ is constructed on the basis of neuronal network of a brain model discussed in PNAS {\bf 105}, 3593 (2008). We show that the spectrum of eigenvalues of ${\bf G}$ has a gapless structure with long living relaxation modes. The PageRank of the network becomes delocalized for certain values of the Google damping factor $\alpha$. The properties of other eigenstates are also analyzed. We discuss further parallels and similarities between the World Wide Web and neuronal networks.