On Wavelet based Modeling of Neural Networks using Graph-theoretic Approach

A graph is an abstract representation of complex networks. Many practical problems can be represented by graphs. With graphs, it is possible to model many types of relations and process dynamics in physical, biological, social and information systems.. More specifically, the relationships among data in several areas of science and engineering, e.g. computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs. For example, graph analysis has been used in the study of models of neural networks, anatomical connectivity, and functional connectivity based upon functional magnetic resonance imaging (fMRI), electroencephalography (EEG) and magnetoencephalography (MEG). Of late, many important properties of complex networks have been delineated and a significant progress has been made in understanding the relationship between the structural properties of networks and the nature of dynamics taking place on these networks with the help of graph theoretical approach. These developments in the theory of complex networks have inspired new applications in the upcoming field of neuroscience. In this work, a novel wavelet based neural network stochastic model that extends existing neural network methods for processing the data represented in a graph domain is proposed. Here, the approach is based on defining random walks on arbitrary infinite graphs representing neural networks. The random walk itself is a stochastic process characterized by some probability distribution. More so, random walks exhibit fractal-like patterns that, in turn, legitimately allow the use of wavelet methods for visualizing and processing. The wavelet transform is constructed for the random walk represented by Gaussian function with the vanishing momenta engendered by Gaussian function as an analyzing function. The wavelet transform so devised is well defined since both the testing function and analyzing function employed in the construction belonged to a class of Gaussian functions. Theory is further extended to discrete format for numerical implementation of the transform. The robustness of the proposed model as against the existing ones has been justified by highlighting the potential applications of neuroscience. In this work, an attempt is made to combine wavelet transform and graph theory. The work is not example specific but it provides a theoretical framework for analyzing the complex structures of neural networks representing various neuroscientific phenomena.