On Full-Connectivity Properties of Locally Connected Oscillatory Networks

The latest many-core chip technology advances foster highly parallel computing systems. Consequently, it is crucial to conceive hardware oriented architectures and to realize VLSI platforms, with kilo- or mega-processors, that are able to process and recognize spatial-temporal patterns without breaking them into frames. Oscillatory networks, their archetype being the Turing morphogenesis model, represent a suitable paradigm for processing spatial-temporal time-periodic patterns. In this manuscript we aim at pointing out full-connectivity properties of locally connected oscillatory networks (LCONs) with linear memoryless and space-invariant interactions. In particular, it is analytically shown that LCONs can implement any operators of globally connected networks with linear dynamical interactions, if some suitable components of the oscillator state vector are coupled. The key issue of our results is that the inverse of a banded matrix is almost always full, i.e., almost all of its entries are nonzero. Space-invariant local connectivity allows for hardware realizations with straightforward architectures.