Analysing properties of the C. Elegans neural network: mathematically   modeling a biological system
  The brain is one of the most studied and highly complex systems in the biological world. It is the information center behind all vertebrate and most invertebrate life, and thus has become a major focus in current research. While many of these studies have concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural aspects of the brain should elucidate some of its functional properties. In this paper we analyze the brain of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian matrix of the 279-neuron "giant component" of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot (in eigenfunction coordinates) both 2 and 3 dimensional visualizations of the neural network. Further analysis examines the small-world properties of the system, including average path length and clustering coefficient. We then test for localization of eigenfunctions, using graph energy and spacial variance. To better understand these results, all of these calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been "rewired" to have small-world properties. This analysis is one of many stepping-stones in the research of neural networks. While many of the structures and functions within the brain are known, understanding how the two interact is also important. A firmer grasp on the structural properties of the neural network is a key step in this process 
Daniel J. Kelleher, Tyler M. Reese, Dylan T. Yott, Antoni Brzoska
q-bio.NC physics.bio-ph q-bio.QM