Google matrix analysis of C.elegans neural network

We study the structural properties of the neural network of the C.elegans (worm) from a directed graph point of view. The Google matrix analysis is used to characterize the neuron connectivity structure and node classifications are discussed and compared with physiological properties of the cells. Our results are obtained by a proper definition of neural directed network and subsequent eigenvector analysis which recovers some results of previous studies. Our analysis highlights particular sets of important neurons constituting the core of the neural system. The applications of PageRank, CheiRank and ImpactRank to characterization of interdependency of neurons are discussed. Introduction. – The human brain neural network has an enormous complexity containing about 10 neurons and 10 synapses linking various neurons [1]. Such a complex network can only be compared with the World Wide Web (WWWW) which indexed size is estimated to be of about 10 pages [2]. This comparison gives an idea that the methods of computer science, developed for WWW analysis, can be suitable for the investigations of neural networks. Among these methods the PageRank algorithm of the Google matrix of WWW [3] clearly demonstrated its efficiency being at the heart of Google search engine [4]. Thus we can expect that the Google matrix analysis can find useful applications for the neural networks. This approach has been tested in [5] on a reduced brain model of mammalian thalamocortical systems studied in [6]. However, it is more interesting to perform the Google matrix analysis for real neural networks. In this Letter we apply this analysis to characterize the properties of neural network of C.elegans (worm). The full connectivity of this directed network is known and documented at [7]. The number of linked neurons (nodes) is N = 279 with the number of synaptic connections and gap junctions (links) between them being Nl = 2990. Recently, there is a growing interest to the complex network approach for investigation of brain neural networks [8, 9], [10, 11], [12]. Generally these networks are directional but it is difficult to determine directionality of links by physical and physiological measurements. Thus, at present, the worm network is practically the only example of neural network where the directionality of all links is established [7]. The analysis of certain properties this directed network has been reported recently in [11, 12], however, the approach based on the Google matrix has not been used yet. Thus we think that this study will allow to highlight the features of worm network using recent advancements of computer science. Google matrix construction. – The Google matrix G of C.elegans is constructed using the connectivity matrix elements Sij = Ssyn,ij + Sgap,ij , where Ssyn is an asymmetric matrix of synaptic links whose elements are 1 if neuron j connects to neuron i through a chemical synaptic connection and 0 otherwise. The matrix part Sgap is a symmetric matrix describing gap junctions between pairs of cells, Sgap,ij = Sgap,ji = 1 if neurons i and j are connected through a gap junction and 0 otherwise. Following the standard rule [3,4], the matrix elements Sij are renormalized (Sij = Sij/ ∑ i Sij) for each column with non-zero elements; the columns with all zero elements are replaced by columns with all elements 1/N . Thus the sum of elements in each column is equal to unity and the Google matrix takes the form Gij = αSij + (1 − α)/N . (1) Here α is the damping factor introduced in [3]. In the context of the WWW, the last term of the equation describes a probability for a random surfer to jump on any node of the network [4]. We use the usual value α = 0.85 [4]. All matrix elements Ssyn,ij , Sgap,ij , Sij are given at [13] The eigenspectrum λi and right eigenvectors ψi(j) of G