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.

Diffusion-driven instabilities with pattern formation may occur in a network of identical, regularly-spaced, and resistively-coupled cells if and only if the uncoupled cell is poised on a locally-active and stable operating point in the Edge of Chaos domain. This manuscript presents the simplest ever-reported two-cell neural network, combining together only 7 two-terminal components, namely 2 batteries, 3 resistors, and 2 volatile NbOx memristive threshold switches from NaMLab, and subject to diffusion-driven instabilities with the concurrent emergence of Turing patterns. Very remarkably, this is the first time an homogeneous cellular medium, with no other dynamic element than 2 locally-active memristors, hence the attribute all-memristor coined to address it in this paper, is found to support complex phenomena. The destabilization of the homogeneous solution occurs in this second-order two-cell array if and only if the uncoupled cell circuit parameters are chosen from the Edge of Chaos domain. A deep circuit- and system-theoretic investigation, including linearization analysis and phase portrait investigation, provides a comprehensive picture for the local and global dynamics of the bio-inspired network, revealing how a theory-assisted approach may guide circuit design with inherently non-linear memristive devices.

This paper presents a circuit-theoretic analysis of a NbO ₂ -Mott memristor fabricated at Hewlett-Packard Labs. It investigates mechanisms behind the origin of complexity based on local activity, which characterizes the behavior of this outstanding nanodevice. We propose an accurate, particularly simplified version of a recently introduced physical model suitable for large-scale circuit simulations. Following the concept of local activity, we then conduct a small-signal circuit-theoretic derivation of the impedance and associated small-signal equivalent circuit elements to analyze device stability and frequency response. Finally, our analysis reveals locally active operating regions, as well as regions where the device dynamics are positioned on the edge of chaos. The latter regions are crucial for designing bio-inspired computing systems.

As established by the second law of thermodynamics, an isolated system is unable to exhibit complex behaviours. Conversely, a physical system, which interacts with the surrounding environment, may support emergent phenomena, provided some of its constitutive components are capable to amplify infinitesimal fluctuations in energy under suitable polarization, a property which is referred to as Local Activity. Local Activity is in fact a New Universal Physics Principle, which, grounded on solid theoretical foundations, enables to explain emergent phenomena in any open system, e.g. the generation of All-or-None electrical spikes in neurons, and Symmetry-Breaking Effects in biological homogeneous cellular media. The existence of solid-state memristor nano-devices, which, similarly as the sodium and potassium ion channels, may operate in the Local Activity domain under opportune bias conditions, opens up new opportunities to synthesise circuits and systems, which, operating according to biological principles, may outperform traditional computing structures in terms of time and energy efficiency. This tutorial aims to shed light on the precious role that Nonlinear Circuit and System Theory shall assume in the years to come to support the exploration of the full potential of memristor physical realizations, which clearly show signs of Local Activity while admitting a Negative Differential Resistance upon suitable DC excitation, for bio-inspired electronics.

This paper presents an analytical investigation of the transient and steady-state response of non-volatile memristors to high frequency periodic inputs, using as a case study a $\textrm {TaO}_{\textrm {x}}$ -based nano-scale memristor model derived at HP Labs. For the first time, we provide a mathematical proof for the fading memory phenomenon in memristors stimulated by periodic inputs in the high frequency limit. Specifically, we demonstrate that the steady-state response of a non-volatile memristor, exhibiting asymmetric switching kinetics with respect to the polarity of the input, depends only on the amplitude of the testing signal and not on the device initial conditions. Based on the results of our analyses, we provide an alternative method for tuning the memristor state by using high-frequency AC inputs, and introduce a new system-theoretic visualization tool, namely the input-referred High-Frequency Dynamic Route Map (HF-DRM), that allows the reproduction of the memristor time-response to any high-frequency periodic input from each admissible initial condition. The purely theoretical results introduced in this paper could inspire new approaches for modulating the memory states of practical non-volatile memristors.

The Edge of Chaos Principle lies at the origin of emergent phenomena in physical systems. Recurring to powerful concepts from the theory, establishing its rules, it is possible to identify regions of the parameter space of a system, endowing the latter with a high degree of excitability, which can then manifest itself vividly, as complexity emerges across the respective physical medium upon apparently-harmless changes to the environmental conditions. In this manuscript, the first of a two-paper contribution, the Edge of Chaos Principle is invoked to explain the mechanisms, underlying the development of recently-reported yet-unexplained oscillations across a biological cell, consisting of a voltage-driven sodium ion channel, including leakage effects. Our in-depth investigation of the Local Activity and Edge of Chaos of the biological cell, based upon the rigorous mathematical description, first proposed by Hodgkin and Huxley in 1952, identifies the subcritical Hopf, which emerges across its physical medium as it enters one of its two Edge of Chaos domains, giving birth to the aforementioned oscillations, whose unstable nature is thus revealed. The existence of an unstable limit-cycle attractor in the state space of the Hodgkin-Huxley neuron model is crucially important for the All-to-None dynamical phenomenon, which distinctively characterises the evolution of the membrane capacitance voltage under DC synaptic current sweep. Thus the discovery of the capability of the sodium ion channel to generate unstable oscillations on its own, highlights the fundamental role of this biological memristor in the mechanisms behind emergence and extinction of an Action Potential across a neuronal axon. As the cell, studied in this manuscript, is unable to sustain stable oscillations, the second companion paper shall demonstrate how the insertion of a membrane capacitance across the sodium ion channel, reduced to a first-order system by neglecting the dynamics of the fast activation gate variable, is necessary and sufficient to endow the original biological cell with the capability to undergo also a supercritical Hopf bifurcation, besides the subcritical one, which allows the observation of the entire life cycle of a neuronal spike under DC synaptic current sweep, including, especially, the All-to-None phenomenon spawned out of a fold or saddle-node limit cycle bifurcation.