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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.

The main focus of this paper is the evolution of complex behavior in a system of coupled nonlinear memristor circuits depending on the applied coupling conditions. Thereby, the parameter space for the local activity and the edge-of-chaos domain will be determined to enable the emergence of the pattern formation in locally coupled cells according to Chua's principle. Each cell includes a Niobium oxide-based memristor, which may feature a locally active behavior once it is suitably biased on the negative differential resistance region of its DC current-voltage characteristic. It will be shown that there exists a domain of parameters under which each uncoupled cell may become locally active around a stable bias state. More specifically, under these conditions, the coupled cells are on the edge-of-chaos, and can support the static and dynamic pattern formation. The emergence of such complex spatio-temporal behavior in homogeneous structures is a prerequisite for information processing. The theoretical results are confirmed by

Neuromorphic circuits will be considered for energy efficient computing based on biological principles in future electronic systems. Thereby, memristors are assumed in neuron models and for synapses in several recent investigations in order to overcome the limits of conventional von Neumann architectures by taking these devices as memory elements and as well as devices for computation also in bioinspired artificial neural networks. Cellular Neural Networks (CNN) are universal high-speed computing systems with stored programmability and are already based on the principle of distributed computing with memory. The dynamical behavior of these spatiotemporal systems will be exploited to solve multidimensional signal processing and classification problems. For example, reaction-diffusion systems can be represented by state equations of socalled reaction-diffusion CNN showing the emergence of complex behavior (e.g. pattern formation) based on local activity and especially on a parameter subset called the edge of chaos. A first approach to the principle of combined sensing and computing will be presented in this contribution. A memristor enhanced control system for a humanoid robot called Myon [1,2] will be shown. For different memristor models new bio-inspired movement control paradigms will be introduced and the performance in achieving a fast energy efficient movement control evaluated for different cases. It will be demonstrated that memristors endow the control system of Myon with the adaptability to changes in its topology. 2010 Mathematics Subject Classification: 92B20, 68U10, 34C28, 35K57.