Physica A: Statistical Mechanics and its Applications
Geometrical structure of the neuronal network of Caenorhabditis elegans
Introduction
Complexity of nervous systems is reflected in the complexity of their structural makeup [1]. The purpose of this paper is to characterize the structure of the neuronal networks. We deal with the soil nematode Caenorhabditis elegans (C. elegans), because all the connections of its neuronal network are well known (some data bases are available [2], [3], [4]). C. elegans is a small worm with a relatively simple nervous system. There are only 302 neurons in the adult hermaphrodites. The nervous system is separated into two units as follows. First, the pharyngeal nervous system is composed of 20 cells. The pharyngeal nervous system controls the rhythmical contraction of the pharynx to suck bacteria. The pharyngeal nervous system is nearly completely isolated from the rest. The synaptic connections among the pharyngeal neurons are well documented by Albertson and Thomson [5]. Second, the somatic nervous system consists of the rest neurons. The long processes from the somatic neurons construct bundles; the nerve ring, the ventral cord, etc. The synaptic connections among the 282 somatic neurons are studied by White et al. [6].
In this paper, the real neuronal network is simplified as a graph, which is a set of vertices connected to each other by edges. The internal structure of each cell is ignored. Thus, a vertex stands for a neuron. There are two types of connections; chemical synapse and gap junction. While the former is polarized, the latter is not. Furthermore, each pair of neurons often has more than one connection. For simplicity, type, direction and multiplicity of connection are not taken into account. This simplification is effectual for study of topological feature in diverse networks [7], [8]. Thus, if there exists at least one synaptic connection between a pair of vertices, then they are linked by an edge. We mainly focus on the somatic nervous system. There are 282 vertices and 2268 edges. We use three indicators to characterize the graph; (1) vertex degree, (2) generalized eccentricity (GE), and (3) complete subgraphs [see Fig. 1].
There were some mathematical studies of the neuronal network. Some researchers have studied the neuronal network in terms of vertex degree [2], [11], [12]. However, only considering vertex degree is not sufficient to study the global structure, as we will show. Watts and Strogaz investigated three networks (C. elegans, collaboration of film actors and power grid) with path length and clustering coefficient [7]. The path length corresponds to the total average of GE, and the clustering coefficient relates to complete subgraphs of degree three. However, taking into account the number of complete subgraphs of high degree, we will show that their model is not appropriate for the neuronal network of C. elegans. In this paper, a simple model with geometrical structure is presented.
Section snippets
Characteristic of the neuronal network
First, we use the concept of vertex degree. The degree of a vertex v represents the number of edges meeting at v. There are 13 vertices whose degrees are 0 [17]. The connection of the 13 neurons was not reported in the experimental data by White et al. [6]. We deal with the graph Gce which is obtained by the removal of these 13 isolated vertices. There are 269 vertices and 2268 edges. The solid line in Fig. 2(a) shows the degree sequence, which is the list of the degrees of the vertices which
Simple model
So far, we have seen that the nervous system of C. elegans has the central group of the neurons and the strong clustering structure. Watts and Strogatz [7] proposed a small world model for biological and social networks with the clustering structure. The Watts and Strogatz (WS) model is made by random rewiring from a low-dimensional regular lattice. Thus, the WS model is the mixture of the low-dimensional lattice and the NR graph. Consequently, it is obvious that the WS model has smaller number
Discussions
We have investigated the somatic nervous system of C. elegans, which has the central part and the clustering structure. We proposed the model with the geometrical structure and anticipated its dimension. We can modify the definition of the distance or the distribution in the characteristic space. Such modifications yield quantitative changes, serious qualitative change is hardly seen. It is essential that the characteristic space has a solid boundary. Thus, we conclude that the somatic nervous
Acknowledgements
We would like to thank H. Kagawa, R. Hosono and S. Mitani for their lectures, which motivated this research. We also thank H. Takano, S. Gomi, Y. Iwasaki, K. Omata and E. Akiyama for fruitful discussion and helpful comments. This research is supported by Japan Society for Promotion of Science under the contract number RFTF96I00102.
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