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Memristor devices embedded in pioneering future computing platforms are believed to represent one of the most promising key-enabling technologies for the treatment of massive amount of data. The ideal memristor, a two-terminal circuit element described by a nonlinear relationship between the time integrals of current and voltage (i.e., the current and voltage momenta, a.k.a. charge and flux), was theoretically envisioned by Chua back in 1971 in the seminal paper.

Let us consider a circuit N made of an arbitrary interconnection of (two-terminal) circuit elements. In general, analysis and design of a circuit require to solve a network problem where numerous mathematical (circuit) variables are established and then a set of equations is generated describing the behavior of the composite network. Fundamental principles of Circuit Theory permit to show that circuit equations depend on (a) the structure of the network and (b) the properties of the devices which it contains. If the network contains nonlinear elements, then the equations are usually more difficult to solve.

This chapter is devoted to discuss some basic properties of memristors, memcapacitors, and meminductors, a.k.a. mem-elements, that are both of theoretic and practical interest. In the first part (Sects. 2.1 and 2.2), the main focus is on features of a memristor as a (−1, −1)-element of the periodic table (cf. Chap. 1), hereinafter also named ideal memristor. Such properties include the passivity and the no energy storage property. It is also shown why a memristor is able to store a continuum of stable states and, as such, can implement a nonvolatile memory. In addition, the chapter illustrates some basic signatures of memristors, as the pinched hysteresis loop displayed in the voltage-current (v, i)-domain in response to a zero-mean periodic input (e.g., a sinusoidal signal).

In this brief chapter we discuss some fundamental dynamic phenomena that can be observed in nonlinear circuits containing time-invariant resistors, inductors, capacitors, and dc sources (autonomous RLC circuits). In Chap. 6 we will study analogous dynamic phenomena for nonlinear circuits containing also memristors. While in first-order autonomous circuits any bounded solution converges to an equilibrium point (EP), second-order circuits with locally active nonlinear resistors can display nonvanishing oscillations, as negative resistance oscillators belonging to the class of Van der Pol oscillators. More complex dynamics, as chaotic dynamics, can be observed in third-order autonomous circuits, the most famous example being Chua’s oscillator.

The motion of a nonrelativistic spinless quantum particle moving on a straight line under a sequence of constant potentials is modeled by a cascade of generalized transmission lines. The equivalence between physical quantities of the system and electrical quantities of the model is illustrated. The equivalent circuit can be used for simulation and as an help to understand quantum properties.

Among the recent innovative technologies, memristor (memory-resistor) has attracted 4 researchers’ attention as a fundamental computation element. It has been experimentally shown 5 that memristive elements can emulate synaptic dynamics and are even capable of supporting 6 spike timing dependent plasticity (STDP), an important adaptation rule that is gaining particular 7 interest because of its simplicity and biological plausibility. The overall goal of this work is to 8 provide a novel (theoretical) analogue computing platform based on memristor devices and 9 recurrent neural networks that exploits the memristor device physics to implement two variations 10 of the backpropagation algorithm: recurrent backpropagation and equilibrium propagation. In 11 the first learning technique, the use of memristor–based synaptic weights permits to propagate 12 the error signals in the network by means of the nonlinear dynamics via an analog side network. 13 This makes the processing non-digital and different from the current procedures. However, the 14 necessity of a side analog network for the propagation of error derivatives makes this technique 15 still highly biologically implausible. In order to solve this limitation, it is therefore proposed 16 an alternative solution to the use of a side network by introducing a learning technique used 17 for energy-based models: equilibrium propagation. Experimental results show that both the 18 approaches significantly outperform conventional architectures used for pattern reconstruction. 19 Furthermore, due to the high suitability for VLSI implementation of the equilibrium propagation 20 learning rule, additional results on the classification of the MNIST dataset are here reported.

Second order memristors are two terminal devices which present a conductance depending on two orders of variables, namely the geometric parameters and the internal temperature. They have shown to be able to mimic some specific features of neuron synapses, specifically Spike-Timing-Dependent-Plasticity (STDP), and consequently to be good candidates for neuromorphic computing. In particular memristor crossbar structures appear to be suitable for implementing locally competitive algorithms and for tackling classification problems, by exploiting temporal learning techniques. On the other hand spiking networks have been intensively studied in the context of unsupervised, supervised and reinforcement learning. In this manuscript we briefly present a simplified model of second order memristors, that we have derived in a previous paper. Then we focus on two main results. As a first step we show that the model is capable of accurately reproducing a correct synaptic response to complex inputs, like cycles of spike triplets and quadruplets at different frequencies. Then we show that, exploiting such a simple model, complex spatio-temporal patterns can be almost analytically characterized, in memristor networks connecting an arbitrary number of presynaptic and postsynaptic neurons.

The papers in this special section were presented at the 2017 IEEE Biomedical Circuits and Systems Conference BioCAS), in Torino, Italy, from October 19–21, 2017.

Synchronization phenomena, frequency shift and phase noise are often limiting key factors in the performances of oscillators. The perturbation projection method allows to characterize how the oscillator's output is modified by these disturbances. In this brief we discuss the appropriate decomposition of perturbations for synchronization and phase noise analysis of planar nonlinear oscillators. We derive analytical formulas for the vectors spanning the directions along which the perturbations have to be projected. We also discuss the implications of this decomposition in control theory and to what extent a simple orthogonal projection is correct.

Goal of the paper is to investigate the analogue computational capabilities of dynamic networks with mem-elements. A relevant class of networks containing memristors and (possibly) nonlinear capacitors and inductors is considered. The paper unfolds the nonlinear dynamics of these networks by highlighting some main features that are potentially useful for real-time signal processing and in-memory computing. These include the presence of bifurcations without parameters and the coexistence of different regimes and complex dynamics. The analysis is conducted by means of a recently developed technique named flux-charge analysis method.

Second order memristors have shown to be able to mimic some specific features of neuron synapses, specifically Spike-Timing-Dependent-Plasticity (STDP), and consequently to be good candidates for neuromorphic computing. In particular, memristor crossbar structures appear to be suitable to implement locally competitive algorithms (LCA) and to tackle classification problems, by exploiting temporal learning techniques. On the other hand neuromorphic studies and experiments have revealed different kinds of plasticity and have shown the effect of calcium concentration on synaptic changes. In this paper we firstly derive a simplified model of a second order memristor, only involving two variables, the mem-conductance and the temperature, directly ascribable to the synaptic efficacy and to the calcium concentration. Then we provide a comparison between second order memristors and neuromorphic models, with reference to some relevant plasticity models, including cycles of spike pairs, triplets and quadruplets at different frequencies. Preliminary results show that, through our almost analytical approach, a significant portion of synaptic behaviors in second order memristors can be easily studied and predicted.

This brief aims at defining a class ${\mathcal{ D}}$ of extended memristors by using a combination of fundamental algebraic circuit elements corresponding to ideal memristors and nonlinear resistors. By massaging the characteristics of the constitutive elements, such class ${\mathcal{ D}}$ of extended memristors is able to approximate rectifying effects and asymmetric pinched hysteresis loops of real non-volatile switching memristor devices. Finally, it is shown that if the nonlinear resistors are described in terms of piecewise linear characteristics, then the flux-charge analysis method permits to analyze the dynamics of nonlinear circuits with extended memristors in ${\mathcal{ D}}$ .

Phase-change memory (PCM) is one of the most promising non-volatile memory technologies and is finding applications in areas such as storage-class memory and emerging non-von Neumann computing systems. Even though powerful physics-based models have been developed for these devices, there is a lack of simple and accurate circuit models to describe these elements. In this brief, we exploit memristor theory to obtain a simple and reliable circuit model based on electrical variables such as charge and flux . This model is based on experimental measurements of PCM devices fabricated in the 90 nm technology node.

The physical constraints underlying the concept of quantum circuit are considered. In particular it is shown that the point of departure for their modeling starts from the interconnection of the components into a classical network, followed by quantization of the latter, and not by the interconnection of already quantized components. The procedure is straightforward for lossless networks but cannot be worked out in presence of resistors for the impossibility of constructing a Lagrangian function. However the difficulty is circumvented by distinguishing thermal from radiative resistors, the former being the usual ones, the latter being realized by semi-infinite LC transmission lines, for which the Lagrangian exists. In the complex plane s = σ + jω the impedance of a thermal resistor is Z(s) = R in the entire plane while that of a radiative resistor is Rsign(σ), so that the latter is lossless and does not dissipate energy but conveys it to the infinity. Comparison with Lindblad approach shows that the resistor fits into it in the RHP. Radiative resistors make evolution reversible, which is not true for a physical system including thermal.

A recent work [1] introduced a flux-charge analysis method (FCAM) to study the nonlinear dynamics and bifurcations of a large class of memristor circuits. FCAM relies on the use of Kirchhoff Flux and Charge Laws and constitutive relations of circuits elements in the flux-charge domain. In [1], the saddle-node bifurcations of equilibrium points in the simplest memristor circuit composed of an ideal flux-controlled memristor and a capacitor, were studied. This paper is devoted to analyze via FCAM more complex bifurcations, such as Hopf bifurcations and period-doubling bifurcations originating complex attractors, in higher-order memristor circuits. It is shown analytically and quantitatively how these bifurcations can be induced by varying the initial conditions of dynamic circuit elements in the voltage-current domain while assuming that circuits parameters are held fixed. Such bifurcations are known in the literature as bifurcations without parameters.

We investigate noise effects in second-order oscillators described in terms of amplitude and phase. We show that phase noise in an oscillator is a convection-diffusion problem, i.e., noise is responsible for both phase diffusion and frequency shift. The magnitude of a noise-induced frequency shift depends on both the noise intensity and on the curvature of iscohrons, i.e., manifolds describing the phase of a nonlinear oscillator. We discuss this dependence and to what extent amplitude fluctuations can be neglected. In the case of additive noise, ignoring amplitude fluctuations gives a trivial phase macromodel that can be integrated directly, and the resulting phase variable is a Gaussian-distributed random process.

Recent articles introduced a new method, named flux-charge analysis method (FCAM), for studying nonlinear dynamics and bifurcations of a large class of memristor circuits. FCAM is based on Kirchhoff flux and charge Laws, and constitutive relations of basic circuits elements, expressed in the flux-charge domain. As such, FCAM is in contrast with other traditional methods for studying the dynamics of memristor circuits, that are instead based on the analysis in the standard voltage-current domain. So far, FCAM has been used to study saddle-node bifurcations of equilibrium points in the simplest memristor circuit composed of an ideal flux-controlled memristor and a capacitor, and more complex Hopf and period doubling bifurcations in certain classes of second- and third-order oscillatory memristor circuits. These bifurcations may be induced by varying initial conditions for a fixed set of circuit parameters (bifurcations without parameters). A peculiar property proved via FCAM is that the state space of a memristor circuit can be decomposed in infinitely many invariant manifolds, where each manifold is characterized by a different reduced-order dynamics and attractors. In this paper, FCAM is used for studying synchronization phenomena that can be observed in resistively-coupled arrays of chaotic memristor circuits. In particular, the paper considers two coupled memristor circuits such that each uncoupled circuit displays a double-scroll chaotic attractor on a certain invariant manifold. It is demonstrated via simulations how phase-synchronization of the two coupled attractors can be achieved depending on the choice of the coupling strength.

The present manuscript relies on the companion paper entitled “Memristor Circuits: Flux–Charge Analysis Method,” which has introduced a comprehensive analysis method to study the nonlinear dynamics of memristor circuits in the flux–charge $(\varphi ,q)$ –domain. The Flux–Charge Analysis Method is based on Kirchhoff Flux and Charge Laws and constitutive relations of circuit elements in terms of incremental fluxes and incremental charges. The straightforward application of the method has previously provided a full portrait of the nonlinear dynamics and bifurcations of the simplest memristor circuit composed by a capacitor and a flux-controlled memristor. This paper aims to show that the method is effective to analyze nonlinear dynamics and bifurcations in memristor circuits with more complex dynamics including Hopf bifurcations (originating persistent oscillations) and period–doubling cascades (leading to chaotic behavior). One key feature of the method is that it makes clear how initial conditions give rise to bifurcations for an otherwise fixed set of circuit parameters. To the best of the authors’ knowledge, these represent the first results that relate such bifurcations, which are referred to in the paper as Bifurcations without Parameters , with physical circuit variables as the initial conditions of dynamic circuit elements.

In this chapter, theory, circuit design methodologies and possible applications of Cellular Nanoscale Networks (CNNs) exploiting memristor technology are reviewed. Memristor-based CNNs platforms (MCNNs) make use of memristors to realize analog multiplication circuits that are essential to perform CNN calculation with low power and small area.

Since the 2008-dated discovery of memristor behavior at the nano-scale, Hewlett Packard is credited for, a large deal of efforts have been spent in the research community to derive a suitable model able to capture the nonlinear dynamics of the nano-scale structures. Despite a considerable number of models of different complexity have been proposed in the literature, there is an ongoing debate over which model should be universally adopted for the investigation of the unique opportunities memristors may offer in integrated circuit design. In order to shed some light into this passionate discussion, this paper compares some of the most noteworthy memristor models present in the literature. The strength of the Pickett?s model stands in its experiment-based development and in its ability to describe some physical mechanism at the origin of memristor dynamics. Since its parameter values depend on the excitation of the memristor and/or on the circuit employing the memristor, it may be assumed as a reference for comparison only in those scenarios for which its parameters were reported in the literature. In this work various noteworthy memristor models are fitted to the Pickett's model under one of such scenarios. This study shows how three models, Biolek's model, the Boundary Condition Memristor model and the Threshold Adaptive Memristor model, outperform the others in the replica of the dynamics observed in the Pickett's model. In the second part of this work the models are used in a couple of basic circuits to study the variance between the dynamical behaviors they give rise to. This analysis intends to make the circuit designers aware of the different behaviors which may occur in memristor-based circuits according to the memristor model under use.

In this brief, the design and the experimental behavior of a memristor emulator is presented. An npn and a pnp transistor control the switching transition to the low resistance state in forward and reverse input bias, while a capacitor is responsible for the hysteresis loop. The circuit does not need external bias and in this sense it operates in a passive mode. The measured i-v characteristic exhibits a memristive hysteretic loop of Type II. Consequently, this circuit is a non-ideal memristor emulator.

The development of neuromorphic systems has increased in pace in the past years since the birth of industrial samples of memristors. In addition to developing the general technology, a strong interest in generating new applications for memristive systems is emerging. Cellular automata (CA) can be utilized for biomedical applications, primarily for image processing. Here, we have developed a CA that can be used to improve patient care, through the follow-up and monitoring of patients affected by topic wounds such as cutaneous ulcers.

In this article we exploit memristor-based chaotic circuits for pseudo-random generators. In particular, the simplest memristor-based chaotic circuit is considered. The circuit consists of only three series elements: inductor, capacitor and memristor. The statistical properties of the generated pseudo-random sequences have been verified by means of the NIST test. The results show that the developed generator can be used in cryptographic applications.

Purpose In the past few years there has been growing interest in memristive devices. These devices rely on thin metal oxide films with a peculiar structure and composition, making precise control of oxide features vital. To this end, anodic oxidation allows a very large range of oxides to be formed on the surface of valve metals, whose thickness, structure and functional properties depend on the process parameters introduced. This work reports how memristive anodic oxides were obtained on titanium and other valve metals, such as niobium and tantalum. Methods Anodic oxidation was performed on valve metals by immersion in H2SO4 or H3PO4 electrolytes and application of voltages ranging from 10 to 90 V. The memristive behavior was evaluated by cyclic voltammetry. Results The behavior of differently grown oxides was compared to identify the best conditions to achieve good memristive performances. High voltages were identified as not suitable due to the excessive oxide thickness, while below 20 V the film was not thick and uniform enough to give a good response. Surface preparation also played a major role in the observation of memristive properties. Conclusions Optimal surface preparation and anodizing conditions were seen to give high memristive perfomances on both titanium and niobium oxides, while on tantalum oxides no reproducibility was achieved.

The peculiar features of the memristor, a fundamental passive two-terminal element characterized by nonlinear relationship between charge and flux, promise to revolutionize integrated circuit design in the next few decades. Besides its most popular potential applications, ultra-dense non-volatile memories and brain-simulating systems, much research has been lately devoted to their use in chaotic circuits. Although the physical memristor is inherently-asymmetric, complementary behaviors arise in devices with opposite polarity. In this work we demonstrate that this makes it feasible to devise a number of practical realizations of a monotone-increasing odd-symmetric charge-flux nonlinearity suitable for chaos-based applications through the sole use of physical memristors of the kind recently fabricated at Hewlett-Packard Labs. Confirmation for such claim is obtained through comparison of chaotic behavior of two modified Chua's oscillators, in which the nonlinear diode is replaced in one case with an artificial memristor with symmetric nonlinearity and in the other case with one of the proposed symmetric memristor combinations

A thorough investigation of the nonlinear dynamics of networks of memristor oscillators is a key step towards the design of systems based upon them, such as neuromorphic circuits and dense nonvolatile memories. A wide gamut of complex dynamic behaviors, including chaos, is observed even in a simple network of memristor oscillators, proposed here as a good candidate for the realization of oscillatory associative and dynamic memories. A detailed study of number and stability of all periodic and nonperiodic oscillations appearing in the network may not leave aside a preliminary deep understanding of the local and global behavior of the basic oscillator. Depending on two bifurcation parameters, controlling memristor nonlinearity, the oscillator exhibits different dynamic behaviors, analyzed here through application of state-of-the-art techniques from the theory of nonlinear dynamics to the oscillator model.

Phase models represent the ideal framework to investigate the synchronization of a nonlinear oscillator with an external forcing. While many researches focused the attention to their analysis, little work has been done about the reduction of a physical system to the corresponding phase model. In this paper we show how, resorting to averaging techniques, it is possible to obtain the phase model corresponding to a given set of state equations. As examples, we derive the phase equations and investigate the synchronization properties of two popular nonlinear oscillators.

Oscillatory nonlinear networks represent a circuit architecture for image and information processing. In particular they have associative properties and can be exploited for dynamic pattern recognition. In this manuscript the global dynamic behavior of weakly connected cellular networks of oscillators is investigated. It is assumed that each cell admits of a Lur'e description. In case of weak coupling the main dynamic features of the network are revealed by the phase deviation equation (i.e., the equation that describes the phase deviation due to the weak coupling). Firstly a very accurate analytic expression of the phase deviation equation is derived via the joint application of the describing function technique and of Malkin's Theorem. Then a complete analysis of the phase-deviation equation is carried out for 1-D arrays of oscillators. It is shown that the total number of periodic limit cycles with their stability properties can be estimated. Finally, in order to show the accuracy of the proposed approach, two networks containing second-order and third-order oscillators, respectively, are studied in detail.

International audience; Synchronization of coupled oscillators is a paradigm for complexity in many areas of science and engineering. Any realistic network model should include noise effects. We present a description in terms of phase and amplitude deviation for nonlinear oscillators coupled together through noisy interactions. In particular, the coupling is assumed to be modulated by white Gaussian noise. The equations derived for the amplitude deviation and the phase are rigorous, and their validity is not limited to the weak noise limit. We show that using Floquet theory, a partial decoupling between the amplitude and the phase is obtained. The decoupling can be exploited to describe the oscillator’s dynamics solely by the phase variable. We discuss to what extent the reduced model is appropriate and some implications on the role of noise on the frequency of the oscillators.

In the past few years many significant innovations in the fabrication of solid-state nanopores have triggered a large number of new biophysics applications, due to the small dimension (5-20 nm) reached using a large scale of membrane materials. These significant improvements catalyzed a deeper understanding in the mechanism of charge diffusion inside nanopores (concerning the ionic flux). To correlate the physical phenomena (under dynamic excitation) inside the nanopore result highly important to easily predict and control the ionic transport throughout the nanopore. Here, we show an innovative study, based on Poisson-Nernst-Planck (PNP) equations, to derive a circuit model which is useful in designing nanopores with specific properties. We obtained analytical simulations which highlighted pinched hysteresis voltage-current loops (i.e. a memristor) related to the local ionic concentration response to AC external sources. Through finite difference space discretization, we are able to show how the memristor and the conductance parameters are derived. Finally, a space-varying transmission line with memristor defines the transport phenomena in the nanopore. We believe the combination of the electrical circuit presented here, correlated by the physics in the nanopores, can provide an effective tool to more accurately predict the diffusion in case of networks of nanopores in a membrane.

International audience; This paper aims to provide some insight into the mechanisms underlying the emergence of synchronization between two memristor-coupled Hindmarsh–Rose oscillatory neural cells. Extensive numerical investigations show that in some cases the history of the nonlinear dynamics of the memristor plays a key role in the development of synchronous oscillations in the network. In fact, an equivalent diffusive network, lacking the plasticity properties featured by the memristor-based synaptic coupling, fails to lock into synchronization under the same simulation settings. These results are then confirmed by a deep theoretical analysis based on the contraction mapping theory. This work sheds light on some aspects of the nonlinear behavior of the still largely unexplored memristor, which is doomed to make a revolutionary impact in integrated circuit design in the years to come.

Since memristors are widely studied and the range of applications developed for these devices are becoming increasingly broad, circuital implementations exhibiting fingerprints of memristive behavior have become highly relevant. A class of memristor circuits was recently obtained by cascading a static nonlinear two-port with a dynamical one-port. In general, these circuits are classifiable as extended memristors and may be controlled either in current or in voltage. This paper has the aim of presenting a novel element from this class which experiences various complex behaviors, including periodic oscillations and chaos. This thorough investigation of the rich nonlinear dynamics emerging in the proposed circuit may shed light on interesting engineering applications that memristors may be suited for.

Several physical and mathematical models have been proposed to describe all 2-terminal non-volatile memory devices based on resistance switching (i.e. memristors), regardless of the device material and physical operating mechanisms. A complete and precise classification of memristor, including memcapacitor and meminductors as well, is instrumental in matching model parameters to physical phenomena and in boosting applications of memristors in unconventional computing systems. The aim of the paper is to provide a theoretical approach to the various classes of mem-devices (i.e. memristors, memcapacitors and meminductors) as nonlinear dynamical systems whose characteristic curves (i.e. dynamic characteristics) are pinched at the origin when driven by bipolar excitations. This theory provides a practical tool to describe mem-devices developed for non-volatile memory applications and neuromorphic systems.

Certain two-terminal devices exhibiting the finger-prints of memristive behavior offer the possibility to mimic the dynamics of biological synapses with a higher level of accuracy as compared to their current electronic realizations. It has been recently shown that neural networks with memristive synapses may exhibit distinct synchronization properties over equivalent diffusively-coupled systems. Applying concepts from the contraction mapping theory, this paper derives analytical conditions for the emergence of synchronization in a memristive neural network of Hindmarsh-Rose neurons. The results reveal the crucial impact the initial memristor state has on the development of synchronous oscillations in the network.

We present a novel phase-amplitude model for noisy oscillators described by Ito stochastic differential equations. The model is completely rigorous and it holds for any value of the noise intensity. The phase and amplitude equations depend on the choice of an appropriate set of basis vectors. We show that using Floquet's basis, a phase-amplitude description is obtained analogous to others, previously proposed. We also show how, using moment closure techniques, information on the expected angular frequency, oscillation amplitude and amplitude variance can be obtained from the phase-amplitude model without solving the equations explicitly.

In this paper, the theory, circuit design, and possible applications of Cellular Nanoscale Networks (CNNs) which are based on memristor technology are reviewed. In the memristor-based CNNs, memristors can be used to realize the analog multiplication circuit that is essential in performing the computation functions of CNNs with low-power consumption and small area. Compared to the memristor-based crossbar architecture that can be used to mimic the fundamental neuron-cell-level operation such as Spike Time Dependent Plasticity (STDP), the memristor-based CNN circuit is more suitable in mimicking the advanced sensory systems such as image processing of human's retina. In this paper, we explain the basics of CNN computation at first and we discuss the previous memristor-based CNN circuits that are very useful in performing analog multiplication. And, also, we think of some practical issues of CNN circuits and discuss the possible solutions. For the CNN applications using memristors, we show the simulation results of CNN circuit with Laplacian template that can be used in the edge detection of various images.

The development of neuromorphic systems has increased in pace in the past years since the birth of industrial samples of memristors. In addition to developing the general technology, a strong interest in generating new applications for memristive systems is emerging. In this paper we present a Cellular Automata (CA) that works exploiting the Belief Propagation Inspired (BPI) algorithm. To foster and to prove the the equivalence among the two systems a image processing example is introduced.

Recently, scientific communities in electrical engineering, material science, biophysics and nano-technologies have paid special attention to memristor devices. Several physical and mathematical memristor models have been proposed to describe devices developed for nonvolatile memory applications and neuromorphic systems. The aim of the paper is to provide a theoretical approach to the various classes of memristive devices as nonlinear dynamical systems whose voltage-current curves (i.e., dynamic characteristics) are pinched at the origin when driven by bipolar excitations. Off-origin dynamic characteristics are discussed and mathematical criteria to model such devices are provided as well. Finally, passivity and losslessness properties of memristor are briefly analized.

We review the state space decomposition techniques for the assessment of the noise properties of autonomous oscillators, a topic of great practical and theoretical importance for many applications in many different fields, from electronics, to optics, to biology. After presenting a rigorous definition of phase, given in terms of the autonomous system isochrons, we provide a generalized projection technique that allows to decompose the oscillator fluctuations in terms of phase and amplitude noise, pointing out that the very definition of phase (and orbital) deviations depends of the base chosen to define the aforementioned projection. After reviewing the most advanced theories for phase noise, based on the use of the Floquet basis and of the reduction of the projected model by neglecting the orbital fluctuations, we discuss the intricacies of the phase reduction process pointing out the presence of possible variations of the noisy oscillator frequency due to amplitude-related effects.

We present a theoretical approach to the various classes of memristive systems as nonlinear dynamical systems whose dynamic characteristics, in presence of zero mean periodic excitations, are pinched at the origin. We also discuss their passivity and losslessness properties.

International audience; Besides being at the core of novel ultra-high density low-power non-volatile memories and innovative pattern recognition systems based upon oscillatory associative and dynamic memories, the nano-scale memristor also has the potential to reproduce the behavior of a biological synapse more efficiently and accurately than any conventional electronic emulator. As in a living being the weight of a synapse is adapted by the ionic flow through it, so the conductance of a memristor is adjusted by the flux across it. This article is organized according to the regulations of the ISCAS2014 special session on the state-of-the-art in memristor-based nonlinear circuits and architectures. In this work we focus on arrays of oscillatory cells locally coupled through memristors. Our investigations shows how the nonlinear dynamics of the memristor plays a key role in the mechanisms underlying the synchronization properties of the networks. This work provides new insights into the nonlinear behavior of the still largely unexplored memristor element, which promises to revolutionize integrated circuit design in the incoming years.

Cellular Nonlinear/Nanoscale Networks (CNNs) that can provide parallel processing in massive scale are known to be suitable for neuromorphic applications such as vision systems. In CNNs, synaptic weights can be calculated by digital or analog multiplications. Though conventional CMOS digital circuits can be used in calculating these multiplications for CNN applications, they occupy very large area and consume a large amount of power, especially when multiplications should be calculated in parallel in massive scale. On the other hand, analog circuits seem to be very attractive for calculating multiplications for CNN applications. One possible approach is to multiply the input current by the programmable resistance of a memristor before applying the resulting voltage to a differential pair for the final voltage-to-current conversion. In this chapter we introduce some analog circuits for CNN applications that use the resistance of a memristor in calculating multiplications. In addition we discuss memristor models and some practical problems in CNN circuits that should be resolved using analog memristor-based implementations.

Since the first experimental evidence for the existence of the memristor in nature, a large number of memristor mathematical models have been proposed in the literature. Among them the generalized Boundary Condition Memristor model sticks out for the adaptability of the dynamics at the boundaries and for the tunability of the nonvolatile behavior. The first part of the paper describes in some detail the PSpice implementation of the generalized Boundary Condition Memristor model. Such PSpice emulator constitutes a reliable tool for computer-aided design of memristor-based circuits. As a conclusion to the first part, the use of the emulator demonstrates the ability of the memristor to capture the Hebbian learning rule, which governs the rate of change of synaptic strength in biological neural networks. The second part of the paper is devoted to the presentation of a novel class of purely passive memristor circuits. Each element from the class is composed of the cascade between a static nonlinear two-port and a linear dynamic one-port and employs solely standard electrical components from Circuit Theory. The state equations of the proposed circuits fall into the class of memristor systems.

Spin torque oscillator (STO) nanodevices have been brought into focus of engineering hoping they could provide for a platform of computation beyond Moore's law. In this paper we propose hybrid-architectures (i.e. combining CMOS units and STO nanodevices) useful to realize Oscillatory Cellular Nonlinear Network (O-CNN) arrays that can be used for associative memory (AM) problem-solving. The fundamental components of the AM O-CNN are (1) a CMOS preprocessing unit generating input feature vectors from picture inputs, (2) an AM cluster generating signature outputs composed of spin torque oscillator (STO) cells and local spin-wave interactions, as an oscillatory CNN (O-CNN) array unit, applied several times arranged in space, and (3) a classification unit (CMOS). In this manuscript we focus on the AM cluster composed of several STO and we aim at showing how local spin-wave interactions lead to global indirect interactions. In addition, a mathematical methodology is proposed in order to design the fully-connected AM cluster of STO exploiting the local spin-wave interactions due to physical limits of the implementation.