Modularity of the <i>C. elegans</i> neuronal network.

The modularity of the network is measured using the parameter , which requires a knowledge of the partitions or communities which divide the network. We obtain the modularity measure, , on assuming the communities to correspond to the ganglia. Its positive values indicate that neurons in the same ganglion have high density of inter-connections. We have also obtained by determining the modules of the network using a spectral method, the corresponding values being indicated by . The relatively high values of compared to , indicates that the ganglia do not match with this optimal partitioning of the network. The measures, and , as well as the number of modules, , have been obtained for both unweighted and weighted networks consisting of either gap junctions or synapses or both. We calculate the overlap between the ganglionic and the optimal partition of the network using the normalized mutual information index, . For the case of perfect match between the two, the index, , whereas if they are independent of each other, . The measured values of indicate that the overlap between the different modules and the anatomically defined ganglia is not high. The modular nature of the somatic nervous system is emphasised by comparing the empirical network with networks obtained by randomizing the connections, keeping the degree of each neuron fixed. The mean and standard deviation of the modularity and the corresponding number of partitions are shown for both weighted and unweighted networks, and for the different types of connections. For all cases, the randomized networks show a significantly lower modularity than the empirical network.