On the equivalence between locally and fully connected oscillatory networks

The analysis and the detection of spatial-temporal patterns are extremely important to unfold the main features of numerous biological phenomena. It is also essential to conceive hardware oriented architectures in order to realize VLSI platforms that are able to process and recognize spatial-temporal patterns without breaking them into frames. Oscillatory networks, whose dynamic behavior is described by large systems of ordinary differential equations, represent a suitable paradigm to describe many spatial-temporal periodic. The aim of this manuscript is to show that locally connected oscillatory networks (oscillatory CNNs) with linear memoryless and space-invariant interactions act as globally connected networks with linear dynamical interactions, if some suitable components of the oscillator state vector are coupled. The space-invariant local connectivity permits to realize simple prototype hardware platforms for processing spatial-temporal patterns.