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The brain is one of the most studied and highly complex systems in thebiological world. It is the information center behind all vertebrate and mostinvertebrate life, and thus has become a major focus in current research. Whilemany of these studies have concentrated on studying the brain directly, ourfocus is the structure of the brain itself: at its core an interconnectednetwork of nodes (neurons). A better understanding of the structural aspects ofthe brain should elucidate some of its functional properties. In this paper weanalyze the brain of the nematode Caenorhabditis elegans. Consisting of only302 neurons, it is one of the better-understood neural networks. Using aLaplacian matrix of the 279-neuron "giant component" of the network, we use aneigenvalue counting function to look for fractal-like self similarity. Thismatrix representation is also used to plot (in eigenfunction coordinates) both2 and 3 dimensional visualizations of the neural network. Further analysisexamines the small-world properties of the system, including average pathlength and clustering coefficient. We then test for localization ofeigenfunctions, using graph energy and spacial variance. To better understandthese 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 manystepping-stones in the research of neural networks. While many of thestructures and functions within the brain are known, understanding how the twointeract is also important. A firmer grasp on the structural properties of theneural network is a key step in this process

Analysing properties of the C. Elegans neural network: mathematically modeling a biological system