Quantifying the functional importance of neuronal assemblies in the brain using Laplacian Hückel graph Energy

Determining the functional relationships between nodes in complex networks such as the neuronal networks is important. In recent years, graph theory has been employed to characterize the functional network structure of the brain from neurophysiological data such as the electroencephalogram (EEG). Current work on graph theoretic analysis of brain networks focuses on global characteristics of the network such as small world network measures. However, it is as important to be able to extract local features of the graph and quantify the vulnerability and robustness of different brain regions. In this paper, we explore how a well-known measure in signal processing, energy, can be extended toward understanding the functional role of neural assemblies in the brain network as represented by a graph. For this purpose, we introduce the Laplacian-Hückel Energy to quantify the local contribution of the nodes to the organization of any scale-free graph and determine anomalies in the graph. The proposed measure is evaluated for both the well-known Zachery karate network and a brain network constructed from an electroencephalogram study.