Spatial graphs and Convolutive Models

In the last two decades, many complex systems have benefited from the use of graph theory, and these approaches have shown robust applicability in the field of finance, computer circuits and in biological systems. Large scale models of brain systems make also a great use of random graph models. Graph theory can be instrumental in modeling the connectivity and spatial distribution of neurons, through a characterization of the relative topological properties. However, all approaches in studying brain function have been so far limited to use experimental constraints obtained at a macroscopic level (e.g. fMRI, EEG, MEG, DTI, DSI). In this contribution, we present a microscopic use (i.e. at the single neuron level) of graph theory to introduce a new model, which we call spatial convolutive model (SCM). Such a model is able to merge random graphs and Power Law models in such a way to quantitatively reproduce the topological and spatial connection distributions observed in real systems.