As conventional computers based on von Neumann architecture approach their physical and performance limits, unconventional computing paradigms such as Cellular Neural Networks (CellNNs) have emerged as promising platforms for real-time, massively parallel analog computation. However, conventional analog CellNNs suffer from scalability and power constraints due to large cell hardware overhead. This work investigates the integration of memristor-based crossbar arrays into CellNN architectures to address these limitations by exploiting their analog tunability, high density, and low power operation. A 1-Transistor-1-Memristor (1T1R) crossbar is proposed for implementing the coupling weights defining the CellNN gene. Device nonlinearity, asymmetry and stochastic variability are incorporated using the physics-based JART VCM memristor model, enabling accurate mapping of target weights onto memristor conductances through numerical optimization and differential-pair encoding. Simulations of edge detection tasks confirm high functional accuracy and robustness, while Monte Carlo analysis reveals variability’s impact, underscoring the need for variability-aware design of reliable memristor-CNN hardware.

In this work, we introduce a high speed and area efficient Cellular Nonlinear Network (CNN) cell, featuring two circuit variants that utilize threshold switches. The threshold switch (TS) model employed represents a current-controlled nanoscale negative differential resistance (NDR) device which exhibits an S-shaped DC I-V curve as a fingerprint. The proposed cell can be considered as the dual of the standard isolated CNN cell where the bistable cell characteristics, originating from the N-shaped voltage-controlled resistor, is implemented through the S-shaped current-controlled TSs. Similarly, the dynamics induced by the parallel capacitor accompanying the nonlinear resistor in the standard cell version are implemented through the internal inductive dynamics of the TSs, resulting in area and speed efficiency. The proposed CNN cell employs a DC voltage source, two bias resistors and 2 TSs, and essentially, features a differential-mode operation which helps to endow it with a symmetric DC I-V characteristic, as is the case for the standard CNN cell. The differential-mode approach further introduces design flexibility as the cell DC I-V characteristic can be adjusted by tuning circuit parameters. We demonstrate the functionality of the proposed cell by implementing image processing tasks ranging from edge detection and thresholding to logic AND and OR operations.

The symmetry-breaking phenomenon, appearing, under suitable conditions, when identical reaction cells, quiet on their own, are let interact via diffusion processes, is dubbedTuring Instability. Its local form exposes the local destabilization, which allows two multistable cells lose stability at one of its locallyasymptotically-stableoperating points. While the globalTuring Instabilityand its mechanisms have been recently explained in (Ascoli et al., 2022), its local form and an experimental demonstration of these complex effects on a physical memristive medium have not been reported yet. This paper investigates a local form ofTuring Instabilityin a two-cell array, when one of the possiblelocally asymptotically-stableandlocally-activestatic solutions loses stability, when let interact with an identical reaction cell via diffusion processes, resulting in the emergence of two different static solutions after transients fade away. In order to study its mechanisms, this paper first introduces a current-controlled Chua Corsage Memristor (CCM), and demonstrates the operating point destabilization in a single current-controlled CCM-based cell. Adding a dissipative resistor and a capacitor to the current-controlled CCM, preliminarily poised on anedge of chaosoperating point, gives birth to two unstable circuits, inducing a local quiescent bi-stability and a local oscillation, respectively. The mechanisms behind a local form ofTuring Instability, appearing in a current-controlled CCM-based two-cell array, have been elucidated, and the bifurcation, spawning symmetry-breaking effects, locally, across the cellular network, has been identified. Both numerical and experimental results confirm the correctness of the theoretical analysis.

The electrical engineering community, interested to develop bio-inspired circuits, approaching the efficiency of the neural networks, is searching passionately for accurate yet simple electronic neurons, or neuristors for short. In recent years, the advent of volatile memristor devices, typically referred to as threshold switches, which admit a negative differential resistance under suitable polarization, similarly as the sodium and potassium ion channels across neuronal axon membranes, has opened up new exciting opportunities in neuromorphic circuit design, enabling innovative analogue electronic cells, capable to reproduce closely the intricate dynamical behaviors of biological neurons without requiring a disproportionate use of resources. The study, presented in this manuscript, achieves an important milestone in this area of research, demonstrating, through a circuit design approach based upon concepts and techniques from Dynamical System Theory, how to leverage the rich dynamics of a threshold switch, capable to boost a periodic sine-wave current signal of infinitesimal amplitude, while acting as a source of local energy, when poised on a suitable bias point, lying along the negative differential resistance branch of the respective S-shaped DC current-voltage characteristic, to induce, one after the other, the three fundamental bifurcations, governing the evolution of an electrical voltage spike from birth to extinction via the All-to-None effect across a biological axon membrane under a reverse sweep in the net synaptic current, according to the fourth-order Hodgkin-Huxley neuron model, in a second-order three-element circuit of unprecedented simplicity, as the current, generated by a DC source, appearing in parallel to a linear capacitor as well as to the volatile locally-active memristor, is subject to a monotonic increase.

This manuscript sheds light into the fundamental importance of the Principles of Local Activity and Edge of Chaos for the future design of innovative circuits, which, employing biomimetic memristive devices, are ideally suited for the development of energy-efficient artificially-intelligent technical systems. The focus of the work is the design of a Second-Order Reactance-Less Oscillator, across which oscillations may develop if and only if at least one of its two different volatile thermally-activated memristor physical realizations is biased along a negative differential resistance branch of the respective DC locus, which turns it into a source of local energy. Very importantly, the proposed cell is first found to lock in the oscillatory mode out of a local Hopf Supercritical Bifurcation when its design parameters are chosen from the Edge of Chaos region, providing clear evidence for the high degree of excitability it acquires as a result.

Neuromorphic circuits using emerging memory technologies have recently gained popularity since they facilitate dense integration with reduced design complexity. In this work, we introduce a simplified modeling approach for the analysis of threshold switch (TS) based neuron circuits where, under given constraints, we represent the TS device as a nonlinear resistor in series with a parasitic inductor. As a result, we define the current of the TS as its state variable. In order to demonstrate the feasibility of the proposed approach, we analyze the conventional Leaky Integrate and Fire (LIF) neuron circuit along with two of its modified variants. We validate the accuracy of the provided analysis and key predictions through numerical simulation results. As a significant contribution, we demonstrate the effectiveness of the proposed method in modifying the nullclines of the TS based LIF neuron and qualitatively align them with the nullclines of a 2ndorder biologically plausible neuron model.

Realized memristors exhibit a unique phenomenon called the fading memory effect, where the memristor response to an AC signal is determined by its characteristics (waveform, amplitude, frequency, and DC offset) rather than the memristor initial conditions. Recently, a method for programming Hewlett Packard’s TaOxmemristor to a target state was proposed, involving configuring the DC offset of a high-frequency square-wave AC voltage input. This served as a basic application example that exploits fading memory in non-volatile memristors, but didn’t consider non-ideal effects. Here, we assess the method applicability in a HfOx-based VCM resistive switch from Forschungszentrum Julich incorporating a variability-aware physics-based model.

As demand grows for efficient, localized processing in edge and in-sensor computing, novel architectural approaches are essential to meet low-power, high-density requirements. Memristor Cellular Nonlinear Networks (M-CNNs) offer a promising path forward, leveraging the unique properties of memristors for adaptable and scalable computation. This paper presents a study of novel M-CNN cell configurations designed to enhance computational versatility and address operational challenges in M-CNN-based systems. By leveraging memristor technology within CNN cells, we propose three distinct configurations: (1) incorporating parallel and series resistive elements for refined control over cell dynamics, (2) introducing a fixed bias voltage to expand computational capabilities, and (3) integrating the Full-Range CNN (FR-CNN) model into M-CNNs for the first time. The proposed topologies are evaluated through dynamic route maps (DRM) and vector field analysis to systematically assess stability and performance across varying design parameters.

Chua corsage memristor (CCM) is characterized by its local activity and can be used to construct neuron circuits. Edge of chaos is a subset of the locally active domain, which is responsible for the emergence of complexity and neuromorphic behaviors. When two identical resting “dead” CCM neurons poised on the edge of chaos are coupled through a linear passive resistor, these two neurons can be activated and a couple of oscillations appear. This phenomenon is referred to as the Smale paradox, which has not been observed from hardware circuits. The present paper addresses this issue by proposing the stability criterion of the two-port coupled system using the small-signal analysis method and then derives an emergence condition of the Smale paradox based on two coupled “dead” CCM neurons in terms of the parameter value ranges. Simulation results demonstrate the correctness of the theoretical analysis. Interestingly, anti-phase synchronization is observed after two identical neurons are coupled with a linear resistor, which is different from the traditional in-phase synchronization between resistively coupled oscillators. The resistively coupled memristive neurons are implemented by hardware based on the poor man’s circuit. The experimental results confirm the reproduction of the Smale paradox and reveal the effect of the coupling resistance on the dynamics of the system.

Inspired by recent research reported in [1], this paper investigates the bio-inspired bifurcation patterns of a simple memristive neuron on edge of chaos. The adopted memristive neuron, comprising a DC current source, a current-controlled locally active memristor, and a capacitor, successfully reproduces the bifurcation cascade patterns observed in the Hodgkin-Huxley (H-H) neuron model, including fold limit cycle bifurcation (FLCB), subcritical Hopf bifurcation (SUB-HB), and supercritical Hopf bifurcation (SUP-HB). Through attraction basin analysis and pulse-based initial state regulation, we verify the coexistence phenomenon of stable and unstable limit cycles induced by FLCB, as well as the bistable behaviors triggered by SUB-HB. Furthermore, taking resistively coupled memristive neurons as an example, we explore the influence of the dynamics of individual neurons on the bifurcation patterns of coupled networks, where two neurons have identical parameters but different initial states. The results demonstrate that the three bifurcation modes also emerge in memristive coupled networks, and their evolutionary patterns are closely related to the dynamic behaviors of individual neurons. Finally, hardware experiments successfully reproduce the bifurcation cascade phenomenon thereby validating the correctness of theoretical analysis and simulation results.

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.

In this paper, the design of an isolated Memristor Cellular Nonlinear Network (CNN) cell with discrete electronic elements is presented. The proposed versatile circuit allows for adjustable cell dynamical characteristics, controlled by design parameters, while the discrete element approach enables simple on-board implementation without the need for large-scale integration, which is necessary for testing hardware with individual fabricated memristors. A voltage-mode approach, that makes use of the diversity of operational amplifiers, is preferred here over a current-mode one that necessitates a large number of individual transistors. The dynamical properties of the system are initially investigated through the calculation of equilibrium points and further illustrated applying the concept of State Dynamic Routes (SDRs) for the cell assuming that the memristor dynamics are much slower than the capacitor voltage dynamics. Moreover, the effect of design parameters on the cell dynamics is being investigated, showing how the scaling of the operating voltage, as well as a plethora of CNN variants -i.e., the Chua-Yang and Full Range models-, can be implemented within the same design. Finally, the nonlinear conductance properties of real memristor devices are incorporated into the study, demonstrating interesting bifurcation phenomena between the cell monostability and bistability for specific parameter values.

Biological synapses behave as dynamically-rich nonlinear elements, participating in complicated computing tasks through their adaptation due to external stimuli. Such adaptivity constitutes an intrinsic property of non-volatile memristor devices, which are also able to maintain their internal state, under zero input, enabling novel bio-inspired learning operations. In this work, a synaptic element based on a memristive bridge, containing two resistors and two memristors, is studied, aiming to investigate complex memristor-based topologies that may result in rich synaptic dynamics. The proposed memristive bridge allows the realization of both positive and negative synaptic weights, while an asymmetric tuning of a weight, stemming from memristor's features and bridge topology, is demonstrated. In particular, by properly selecting the memristor's position and polarity within the bridge, different tuning behaviors have been observed, showcasing versatile learning properties of the topology. Along with the synaptic weight tuning, the read overall process of the synaptic weight, necessary for inference operations, is also investigated. We explore the dynamics of the bridge via numerical simulations.

In this paper, we present a new compact model of threshold switching devices which is suitable for efficient circuit-level simulations. First, a macro model, based on a compact transistor based circuit, was implemented in LTSPICE. Then, a descriptive model was extracted and implemented in MATLAB, which is based on the macro model. This macro model was extended to develop a physical model that describes the processes that occur during the threshold switching. The physical model derived comprises a delay structure with few electrical components adjacent to the second junction. The delay model incorporates an internal state variable, which is crucial to transform the descriptive model into a compact model and to parameterize it in terms of electrical parameters that represent the component’s behavior. Finally, we applied our model by fitting measured$i-v$data of an OTS device manufactured by Western Digital Research.

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.

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.

We present a compact, continuous, and numerically stable version of a tantalum oxide (TaOx) memristor model which can be employed for robust and reliable simulations of large scale memristor based circuits. The original model contains a piecewise differentiable function in the memductance expression and discontinuous step functions in the state equation. Additionally, the original model does not set a proper upper bound for the state variable and may admit blowing up solutions due to an exponential power term, preventing the use of it for numerically reliable simulations. Considering these drawbacks, we modify the original model so as to i) simplify the memductance function while removing its piecewise differentiable nonlinearity, ii) include a proper window function for the ON state dynamics, which is missing in the original model, iii) modify and bound the exponential power term to prevent an uncontrollable blow-up of the solutions, and iv) apply a process called unification, allowing us to remove the step functions inherent in the model, which is a novelty in state-limited memristor models. We validate the accuracy of the proposed model via DC and transient simulations, dynamic route map analysis and a Spice implementation of an anti-series configuration, showing the applicability of the model.

This paper presents the analytical investigation of complex pattern formation in a Memristor Cellular Nonlinear Network (M-CNN) by applying the theory of local activity. The proposed M-CNN has the conventional two dimensional (2D) planar structure, where all the memristive cells are identical and resistively coupled to each other. The single cell is composed of a suitable combination of a DC voltage source, a bias resistor, a locally active NbOx memristor, and a capacitor. The locally active memristor has a simplified generic form, enhancing the simulation speed, and a functional AC equivalent circuit, facilitating further inspections. The stability analysis of the single cell is followed by the extraction of the parameters of the local activity, edge-of-chaos, and sharp-edge-of-chaos domains. Simulation results demonstrate that pattern formation can emerge in a dissipatively coupled M-CNN with locally active memristors.

We present and experimentally validate two minimal compact memristive models for spiking neuronal signal generation using commercially available low-cost components. The first neuron model is called the Memristive Integrate-and-Fire (MIF) model, for neuronal signaling with two voltage levels: the spike-peak, and the rest-potential. The second model MIF2 is also presented, which promotes local adaptation by accounting for a third refractory voltage level during hyperpolarization. We show both compact models are minimal in terms of the number of circuit elements and integration area. Using the MIF and MIF2 models, we postulate the design of a memristive solid-state brain with an estimation of its surface area and power consumption. Analytical projections show that a memristive solid-state brain could be realized within (i) the surface area of the median human brain, 2,400cm2, (ii) the same volume of the median human brain, and (iii) a total power budget of approximately 20 W using a 3.5 nm technology. Distinct from the past decade of memristive neuron literature, our benchmarks are attained using generic commercially available memristors that are reproducible using off-the-shelf components. We expect this work can promote more experimental demonstrations of memristive circuits that do not rely on prohibitively expensive fabrication processes.

This paper presents a circuit-theoretic analysis of a NbO2-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.

A new chaotic oscillator which can be implemented on standard CMOS process is presented. The circuit generates double-scroll chaotic attractor and is suitable for high-frequency operation. As a possible application, the use of the circuit as the core of a random number generator (RNG) is illustrated. The quality of the RNG is investigated using standard statistical tests (according to FIPS-140-1 and NIST 800-22) and it is shown that the generated binary sequences have good statistical properties.