Theoretical Foundations of Memristor Cellular Nonlinear Networks: Stability Analysis With Dynamic Memristors

If the memristor, used in each cell of a memristive variant of the standard space-invariant Cellular Nonlinear Network (CNN), undergoes analogue memductance changes, the processing element operates as a second-order system. The Dynamic Route Map (DRM) technique, applicable to investigate first-order systems only, is no longer relevant. In this manuscript, a recently introduced methodology, generalizing the DRM technique to second-order systems, is applied to the models of Memristor CNN (M-CNN) cells, accomodating dynamic memristors. This allows to gain insights into the operating principles of these cellular structures, which make computations through the evolution of their states toward prescribed equilibria. Our analysis uncovers all possible local and global phenomena, which may emerge in the cell phase space under zero offset current for any self-feedback synaptic weight. Under these hypotheses, the dynamics of the M-CNN cell may significantly differ from those of a standard space-invariant CNN counterpart. The insertion of an offset current into each cell endows it with further properties, including monostability. The analysis method is used to demonstrate how a non-autonomous memristive array exploits the capability of its cells to feature monostability or bistability, depending upon the respective offset currents, to compute the element-wise logical AND between two binary images.