An Analytical Approach to Neuronal Connectivity

TL;DR

This paper presents an analytical method to characterize neuronal connectivity in neuromorphic networks, emphasizing the role of neuron shape and spatial distribution, with applications to real neuronal images.

Contribution

It introduces an analytical framework for fully characterizing neuronal network connectivity based on neuron shape and spatial arrangement, including spectral and morphological measures.

Findings

Connectivity properties can be derived analytically from neuron shape and placement.

Connections tend to shift progressively along the spatial domain.

Number of cycles follows a power law with respect to their length.

Abstract

This paper describes how realistic neuromorphic networks can have their connectivity properties fully characterized in analytical fashion. By assuming that all neurons have the same shape and are regularly distributed along the two-dimensional orthogonal lattice with parameter $\Delta$, it is possible to obtain the accurate number of connections and cycles of any length from the autoconvolution function as well as from the respective spectral density derived from the adjacency matrix. It is shown that neuronal shape plays an important role in defining the spatial spread of network connections. In addition, most such networks are characterized by the interesting phenomenon where the connections are progressively shifted along the spatial domain where the network is embedded. It is also shown that the number of cycles follows a power law with their respective length. Morphological measurements for characterization of the spatial distribution of connections, including the adjacency matrix spectral density and the lacunarity of the connections, are suggested. The potential of the proposed approach is illustrated with respect to digital images of real neuronal cells.