A Mathematical Model for Evaluating the Functional Connectivity Strongness in Healthy People

The human brain is a really complex organization of connectivity whose principal elements are neurons, synapses and brain regions. Up to now this connectivity is not fully understood, and recent impulse in investigating its structure has been given by Graph Theory. However, some points remain unclear, mainly due to possible mismatching between the Mathematical and the Neuroscientific approach. It is known that neural connectivity is classified into three categories: structural (or anatomical) connectivity, functional connectivity and effective connectivity. The point is that these categories demand different kinds of graphs, except in the case of the resting state, and sometimes topological and metrical parameters are involved simultaneously, without a specific distinction of their roles. In this paper we propose a mathematical model for treating the functional connectivity, based on directed graphs with weighted edges. The function W(i,j,t), representing the weight of the edge connecting nodes i,j at time t, is obtained by splitting the model in two parts, where different parameters have been introduced step by step and rigorously motivated. In particular, there is a double role played by the notion of distance, which, according to the different parts of the model, assumes  a discrete  or an Euclidean meaning. Analogously, the time t appears both from a local and from a global perspective. The local aspect relates to a specific task submitted to an health volunteer (in view of possible future applications also to subjects affected by neurological diseases), while the global one concerns the different periods in the human life that characterize the main changes in the neural brain network. In the particular case of the resting state, we have shown that W reduces to the usually employed probabilistic growth laws for the edge formation. We tested the correctness of our model by means of synthetic data, where the selection of all involved parameters has been motivated according to what is known from the available literature. It turns out that simulated outputs fit well with the expected results, which encourages further analysis on real data, and possible future applications to neurological pathologies.