An analytical approach to neuronal connectivity

This paper describes how to analytically characterize the connectivity of neuromorphic networks taking into account the morphology of their elements. By assuming that all neurons have the same shape and are regularly distributed along a two-dimensional orthogonal lattice with parameter , we obtain the exact number of connections and cycles of any length by applying convolutions and the respective spectral density derived from the adjacency matrix. It is shown that neuronal shape plays an important role in defining the spatial distribution of synapses in neuronal networks. In addition, we observe that neuromorphic networks typically present an interesting property where the pattern of connections is progressively shifted along the spatial domain for increasing connection lengths. This arises from the fact that the axon reference point usually does not coincide with the cell center of mass of neurons. Morphological measurements for characterization of the spatial distribution of connections, including the adjacency matrix spectral density and the lacunarity of the connections, are suggested and illustrated. We also show that Hopfield networks with connectivity defined by different neuronal morphologies, which are quantified by the analytical approach proposed herein, lead to distinct performances for associative recall, as measured by the overlap index. The potential of our approach is illustrated for digital images of real neuronal cells.