Towards Google matrix of brain

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 G is constructed on the basis of neuronal network of a brain model discussed in PNAS 105, 3593 (2008). We show that the spectrum of eigenvalues of 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 α. The properties of other eigenstates are also analyzed. We discuss further parallels and similarities between the World Wide Web and neuronal networks. More than 50 years ago John von Neumann traced first parallels between architecture of the computer and the brain [1]. Since that time computers became an unavoidable element of the modern society forming a computer network connected by the World Wide Web (WWW). The WWW demonstrates a continuous growth approaching to 10 11 web pages spread all over the world (see e.g. starts to become even larger than 10 10 neurons in the brain. Each neuron can be viewed as an independent processing unit connected with about 10 4 other neurons by synaptic links (see e.g. [2–4]). About 20% of these links are unidirectional [5] and hence the brain can be viewed as a directed network of neuron links. At present, more and more experimental information about neurons and their links becomes available and the investigation of properties of neuronal networks attracts an active interest of many groups (see e.g. [6–13]. The WWW is also a directed network where a site j points to a site i but no necessary vice versa. The classification of web sites and information retrieval from such an enormous data base as the WWW becomes a formidable challenge of modern society where the search engines like Google are used by internet users in everyday life. An efficient way to classify and extract the information from WWW is based on the PageRank Algorithm (PRA), proposed by Brin and Page in 1998 [14], which forms the basis of the Google search engine. The PRA is based on the construction of the Google matrix which can be written as (see e.g. [15] for details): G = αS + (1 − α)E/N. (1) Here the matrix S is constructed from the adjacency matrix A of directed network links between N nodes so that S ij = A ij …