Local activity is the capability of a system to amplify infinitesimal fluctuations in energy. Complex phenomena, including the generation of action potentials in neuronal axon membranes, may never emerge in an open system unless some of its constitutive elements operate in a locally active regime. As a result, the recent discovery of solid-state volatile memory devices, which, biased through appropriate DC sources, may enter a local activity domain, and, most importantly, the associated stable yet excitable sub-domain, referred to as edge of chaos, which is where the seed of complexity is actually planted, is of great appeal to the neuromorphic engineering community. This paper applies fundamentals from the theory of local activity to an accurate model of a niobium oxide volatile resistance switching memory to derive the conditions necessary to bias the device in the local activity regime. This allows to partition the entire design parameter space into three domains, where the threshold switch is locally passive (LP), locally active but unstable, and both locally active and stable, respectively. The final part of the article is devoted to point out the extent by which the response of the volatile memristor to quasi-static excitations may differ from its dynamics under DC stress. Reporting experimental measurements, which validate the theoretical predictions, this work clearly demonstrates how invaluable is non-linear system theory for the acquirement of a comprehensive picture of the dynamics of highly non-linear devices, which is an essential prerequisite for a conscious and systematic approach to the design of robust neuromorphic electronics. Given that, as recently proved, the potassium and sodium ion channels in biological axon membranes are locally active memristors, the physical realization of novel artificial neural networks, capable to reproduce the functionalities of the human brain more closely than state-of-the-art purely CMOS hardware architectures, should not leave aside the adoption of resistance switching memories, which, under the appropriate provision of energy, are capable to amplify the small signal, such as the niobium dioxide micro-scale device from NaMLab, chosen as object of theoretical and experimental study in this work.

No isolated system may ever support complexity. Emergent phenomena may however appear in an open system, if, as established by the Edge of Chaos theory, some of its constitutive elements feature the capability to amplify infinitesimal fluctuations in energy, provided an external source supplies them with a sufficient amount of DC power, which is known to be a signature for locally-active behaviour. In particular, complex behaviours, including static and dynamic pattern formation, may emerge in arrays of identical diffusively-coupled cells, if and only if the basic unit is poised on a particular sub-domain of the Local Activity regime, referred to as Edge of Chaos, within which a quiet state hides in fact a high degree of excitability. Here we show, for the first time, that these counterintuitive phenomena may emerge in a basic memristor cellular neural network, consisting of two identical diffusively-coupled second-order cells. The proposed bio-inspired array represents the simplest ever-reported open system, which reproduces the shocking phenomenon, reported by Smale in 1974, when, while studying a model from cellular biology, he observed two identical reaction cells, “mathematically dead” on their own, pulsating together upon diffusive coupling. Impressively, the bio-inspired two-cell reaction-diffusion network contains only nine circuit elements, specifically two DC voltage sources, three linear resistors, two linear capacitors, and two functional niobium oxide (NbO) memristors from NaMLab. Applying the theory of Local Activity to an accurate model of the memristor oscillator, a comprehensive picture for its local and global dynamics may be drawn, providing a systematic method to tune the design parameters of the two-cell array to enable diffusion-driven instabilities therein.

In the memristive version of a standard space-invariant Cellular Nonlinear Network, each cell accommodates one first-order non-volatile memristor in parallel with a capacitor. In case, the resistance switching memory may only undergo almost-instantaneous switching transitions between two possible resistive states, acting at any time, as either the on or the off resistor, the processing elements effectively operate as first-order dynamical systems, and the classical Dynamic Route Map technique may be applied to investigate their operating principles. On the contrary, in case the memristors experience smooth conductance changes, as the bioinspired array implements memcomputing paradigms, each cell truly behaves as a second-order dynamical system. The recent extension of the Dynamic Route Map analysis tool to systems with two degrees of freedom constitutes a powerful technique to investigate the nonlinear dynamics of memristive cellular networks in these scenarios. This paper exploits this system-theoretic technique, called Second-Order Dynamic Route Map, to introduce a novel systematic procedure to design memristive arrays, in which a given memcomputing task is executed by ensuring that, depending upon the network inputs and initial conditions, the analogue dynamic routes of the states of the processing elements, namely capacitor voltages and memristor states, asymptotically converge toward pre-defined stable equilibria.

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.