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.

Equilibrium propagation is a learning technique conceived for training continuous-time recurrent neural networks. It offers some notable advantages when compared to conventional back-propagation-based algorithms and to classical design methods. From an implementation perspective, it demands only a single computational circuit. Theoretically, although it seeks to minimize a cost function, it exhibits similarities to spike-timing-dependent plasticity (STDP), rendering it, to a certain extent, biologically plausible. This paper explores the global dynamic behavior of continuous-time piecewise linear networks trained through equilibrium point propagation. We examine a network in which the target patterns are presented as external inputs rather than as initial conditions. We first show that the learning rules, which extend equilibrium propagation to gradient-like and non-symmetric networks, can be derived as a suitable approximation of Lagrangian optimization. Then, by studying a relatively simple but thoroughly significant case, we demonstrate that a detailed analysis of the equilibrium point distribution yields a profound understanding of the network’s fundamental properties and provides a valuable tool for quantitatively evaluating the network’s accuracy. Compared to classical synthesis techniques, our approach, where patterns are introduced as external inputs, in most cases, circumvents the impractical task of estimating the basins of attraction for sets of multiple equilibrium points. Furthermore, preliminary extensive simulations indicate that the primary dynamic features observed in relatively small networks closely resemble those ensuring the performance and accuracy of large-scale networks.

Second order memristors are two terminal devices that 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, neuromorphic studies and experiments have revealed the existence of different kinds of plasticity and have shown the effect of calcium concentration on synaptic changes. Computational studies have investigated the behavior of spiking networks in the context of supervised, unsupervised, and reinforcement learning. In this paper, we first derive a simplified, almost analytical, model of a second-order memristor, only involving two variables, the mem-conductance, and the temperature, directly attributable to the synaptic efficacy and to the calcium concentration. Then we study in detail the response of a single memristive synapse to the most relevant plasticity models, including cycles of spike pairs, triplets, and quadruplets at different frequencies. Finally, we accurately characterize memristor spiking networks as discrete nonlinear dynamic systems, with mem-conductances as state variables and pre and postsynaptic spikes as inputs and outputs, respectively. The result shows that the model developed in this manuscript can explain and accurately reproduce a significant portion of observed synaptic behaviors, including those not captured by classical spike pair-based STDP models. Furthermore, under such an approach, the global dynamic behavior of memristor networks and the related learning mechanisms can be deeply analyzed by employing advanced nonlinear dynamic techniques.

Recent works have introduced an effective technique to analyze nonlinear dynamics of a class LM of circuits containing ideal flux- or charge-controlled memristors and linear lossless elements (i.e. ideal capacitors and inductors). The technique, named Flux-Charge Analysis Method (FCAM), is based on analyzing the circuits in the flux-charge domain instead of the traditional voltage-current domain. Goal of this paper is to extend the FCAM to a larger class N of circuits containing also nonlinear capacitors and inductors. Nonlinear circuits with memristors and nonlinear lossless elements are widely used to several real nanoscale devices including the well-known Josephson junction. After deriving the constitutive relation in the flux-charge domain of each two-terminal element in N, the work focuses on a relevant subclass of N for which a state equation description can be obtained. State Equations (SE) formulation provides the fundamental basis for studying the chief features of the nonlinear dynamics: presence of invariant manifolds in autonomous circuits; coexistence of infinitely many different reduced-order dynamics on the manifolds; bifurcations due to changing of initial conditions for a fixed set of parameters, a.k.a. bifurcations without parameters.

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.

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.

This paper proposes a simple PSpice implementation of the boundary condition model for memristor nano-structures. The boundary condition model is equivalent to the linear drift model except for the introduction of adaptable boundary conditions, which impose an activation threshold of the state dynamics at the boundaries, i.e. once the state gets clipped at one of the boundaries, it may not be released from it unless the input reverses its sign and gets larger than a certain activation threshold in magnitude. Thanks to the adaptability of the boundary behavior, the boundary condition model is able to describe a variety of physical nano-scale systems, where mem-ristor dynamics arise from distinct physical mechanisms. The proposed PSpice emulator may be used for the investigation of potential applications of memristive systems in integrated circuit design, especially for the development of non-volatile memories and neuromorphic platforms. The accuracy of the PSpice circuit model is validated through comparison with experimental results relative to the Hewlett-Packard memristor.

Chaotic neural networks are able to reproduce chaotic dynamics observable in the brain of various living beings. As a result, study of the dynamical properties of such networks may pave the way towards a better understanding of the memory rules of the brain. In this paper a simple neural circuit employing a theoretical memristive synapse with symmetric charge-flux nonlinearity is found to behave chaotically. After presentation of a novel boundary-condition based model for real memristor nano-structures, conditions under which a suitable arrangement of such nano-structures is dynamically equivalent to the theoretical memristor are derived and validated.

This work presents a novel, simple, accurate and general model capturing the nonlinear dynamics of memristive nano-scale structures including the thin double-layer oxide film manufactured at Hewlett-Packard Labs in 2008. Advantages over other models include ease of analytical integration, existence of closed-form solutions under any input/initial condition combination and opportunity to tune boundary conditions so as to detect either single-valued or multi-valued state-flux characteristics under sign-varying input.

The design of pattern recognition systems based on memristive oscillatory networks need to include a detailed study of the dynamics of the networks and their basic components. A simple two-cell network of this kind, where each cell is made up of a linear circuitry in parallel with a nonlinear memristive element, was found to experience a rich gamut of nonlinear behaviors. In particular, for a synchronization scenario with almost-sinusoidal oscillations, the memristive elements used in the cells exhibited an unusual current-voltage characteristic. This work focuses on the dynamics of the single cell under this synchronization scenario, and, modeling the linear circuitry with a sinusoidal voltage source, analytically derives a rigorous classification of all possible current-voltage characteristics of the periodically-driven memristive element on the basis of amplitude-angular frequency ratio and time hystory of the input source.

Besides being at the basis of next-generation ultra-dense non-volatile memories, a nanoscale memristor also has the potential to reproduce the behavior of a biological synapse. As in a living creature 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 or the charge through it depending on its controlling source. In this manuscript we consider two Hindmarsh-Rose neurons, coupled via a memristive device mimicking a biological synapse. We investigate how the dynamics of the memristive element may influence the syncronization properties of the network.

Networks of coupled nonlinear oscillators are popular mathematical models in many areas of applied sciences. The most successful approach for their analysis is based on phase modeling, founded on the idea to represent each oscillator by a phase variable. Phase models have been analyzed with wealth of details and in a plethora of different variants, but little research has been made in view of the reduction of a physical system to the corresponding phase model. In this paper we propose a technique to obtain the phase model corresponding to a given network. Examples based on Stuart-Landau and van der Pol oscillators are presented.

Oscillatory networks, their archetype being the Turing morphogenesis model, are mathematically represented by large systems of ordinary differential equations and provide an appropriate paradigm for describing many spatial-temporal periodic patterns. The aim of this manuscript is to show that locally connected oscillatory networks (oscillatory CNNs) with linear memoryless and space-invariant interactions may behave as globally connected networks with linear dynamical interactions, if some suitable components of the oscillator state vector are coupled. Space-invariant local connectivity allows to build simple prototype hardware platforms for processing spatial-temporal patterns.

In order to describe genetic regulatory networks several deterministic models based on systems of nonlinear ordinary differential equations (ODEs) have been proposed. The Elowitz repressilator, modeled as a system of three genes that repress each other in a ring, is one of the most outstanding examples. Furthermore, systems that can display a coexistence of different stable attractors are widely exploited in systems biology in order to suitably model the differentiating processes arising in living cells. The aim of the manuscript is to investigate the global periodic oscillations and their bifurcations in networks composed of simple bio-inspired oscillators that have a stable limit cycle and equilibrium point, separated by an unstable limit cycle.

Oscillations in networks composed of Cyclic Negative Feedback systems (CNF systems) are widely used to mimic many periodic phenomena occurring in systems biology. In particular, the possible coexistence of different attractors permits to suitably describe the differentiating processes arising in living cells. The aim of the manuscript is to characterize, through a spectral based techniques, the complex global dynamical behaviors emerging in arrays of diffusively coupled CNF systems.

An equivalent circuit is presented for a quantum system composed of two spin 1/2 particles. Such a circuit shows that the entire dynamics of the system, including single-particle and two-particle annihilation and creation, as well as the single particle transitions between ground and excited states, can be described as the superposition of the variables of two uncoupled resonant circuits.

The analysis and the detection of spatial-temporal patterns are extremely important to unfold the main features of numerous biological phenomena. It is also essential to conceive hardware oriented architectures in order to realize VLSI platforms that are able to process and recognize spatial-temporal patterns without breaking them into frames. Oscillatory networks, whose dynamical behavior is described by large system of ordinary differential equation, represent a suitable paradigm to describe many spatial-temporal time-periodic patterns. The aim of the manuscript is to show that locally connected oscillatory networks (oscillatory CNNs) with linear memoryless and space-invariant interactions act as globally connected oscillatory networks with linear dynamical interactions under the constraint that the couplings involve at least two components of the oscillator state vector. The space-invariant local connectivity permits to realize simple prototype hardware platforms for processing spatial-temporal patterns.

Negative feedback control loops are widely used in numerous models of periodic phenomena occurring in systems biology, since they give rise to sustained oscillations. In particular, the possible coexistence of different attractors permits to suitably describe the differentiating processes arising in the cell. The aim of this work is to study coupled systems of such kind through spectral techniques, in order to characterize the various complex dynamical behaviors that can emerge due to the couplings.

The existence and the stability of waves and phase locked oscillations in lattices composed by oscillators with delayed interactions is investigated. In the neighborhood of a multiple Hopf bifurcation, the equations governing the dynamics of the whole network reduce to an amplitude-phase model, reducing the research of phase locked oscillations to the prospecting of equilibrium points. The stability of the solutions is determined analytically and the possible coexistence of waves and phase locked oscillations is shown.

Spiral waves are the most universal form of patterns arising in dissipative media of oscillatory and excitable nature. By focusing on oscillatory networks, whose cells admit of a Lur'e description and are linearly connected through weak couplings, the occurrence of spiral waves has been studied. In particular, the global dynamic behavior of such networks is investigated through the phase deviation equation obtained by the joint application of the harmonic balance method and Malkin's theorem. Furthermore, a simple condition for verifying the occurrence of spiral waves is provided.

The aim of this work is to study periodic oscillations and bifurcations in cellular nonlinear networks composed by oscillatory cells and connected through arbitrary couplings. In order to characterize each oscillator by using amplitude and phase variables, a method based on a generalized version of the describing function technique is proposed. Furthermore, by exploiting the method of multiple scales a set of ordinary differential equations governing the amplitude and phase dynamics is derived. The results also permit to study accurately weakly connected oscillatory networks. Finally, the method is compared to a spectral technique, based on the harmonic balance approach, by considering a chain of Chua's circuits.

The global dynamics of an artificial neural network composed by oscillators with delays is investigated. Using center manifold reduction and normal form theory, the equation governing the whole network dynamics is reduced to an amplitude-phase model (i.e. a set of coupled differential equations describing the evolution of both the amplitudes and the phases of the oscillators). The analysis of a network with a simple architecture reveals that different kind of phase locked oscillations is admissible, and the possible coexistence of in-phase and anti-phase locked solutions.

The steady state performance of a two-state quantum system interacting with a thermal bath at a fixed temperature and with a classical electromagnetic wave is analyzed by the harmonic balance technique. Thus the time-variant equations can be solved to any chosen approximation and the corrections to be brought to the classical RWA approximation to take into account the effect of the counter-rotating field can be calculated.

The aim of this work is to study periodic oscillations and bifurcations in oscillatory networks with arbitrary couplings. In order to characterize each oscillator by using amplitude and phase variables, a method based on a generalized version of the describing function technique is proposed. It allows us to derive a set of ordinary differential equations governing the amplitude and phase dynamics. The results also permit to study accurately weakly connected oscillatory networks. Finally, the method is compared to a spectral technique, based on the harmonic balance approach, by considering a chain of Chua's circuits.

Many studies in neuroscience have shown that nonlinear dynamic networks represent a bio-inspired model for information and image processing. Recent studies on the thalamo-cortical system have shown that weakly connected oscillatory networks, forced by an external input, have the capability of modelling the architecture of a neurocomputer. In particular they have associative properties and can be exploited for dynamic pattern recognition. In this manuscript the global dynamic behavior of such networks is investigated. In case of weak coupling, their main dynamic features 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. Furthermore, a complete analysis of the phase-deviation equation shows that the proposed technique can be effectively exploited for designing dynamic associative memories

Cellular neural networks (CNN) are described by large systems of locally coupled nonlinear differential equations. In most applications the connectivity are specified through space-invariant templates. As far as the dynamic behavior is concerned, CNNs can be divided in two main classes: stable CNNs, with the property that each trajectory (with exception of a set of measure zero) converges towards an equilibrium point; unstable CNNs, that exhibit at least one attractor, that is not a stable equilibrium point. Due to their complex dynamics, only a few methods for template design have been so far proposed. We propose a rigorous design algorithm for stable CNNs and we identify the class of templates to which such an algorithm can be applied.

Cellular neural/nonlinear networks (CNNs) are analog dynamic processor arrays, that present local interconnections. CNN models with polynomial interactions among the cells (Polynomial type CNNs) have been recently introduced. They are useful for solving some complex computational problems and for real-time implementation of PDE-based algorithms. This manuscript provides some simple and rigorous sufficient conditions for stability of polynomial type CNNs. A particular emphasis is given to conditions that can be expressed in terms of template elements, since they can be exploited for design purposes.

It is known that a cellular neural network (CNN) can be analyzed as a system that depends on one or two discrete space and one continuous time coordinates. In this paper the state of the network is represented as a linear combination of a suitable space mode basis. The nonlinear differential equations, that originally describe the CNN, are transformed into an equivalent set of equations that involve the space mode coefficients. Such a system of equations is able to describe the network in the whole state space and not only in the CNN linear region. It is shown that the study of the time evolution of the most significant space modes allows one to understand the behavior of a CNN as a nonlinear space filter and to develop useful design strategies.

A spectral technique is proposed for studying and predicting chaos in a one-dimensional array of Chua's circuits. By use of a double Fourier transform the network is reduced to a scalar Lur'e system to which the describing function technique is applied for discovering the existence of periodic wave. Finally, by the computation of the distortion index an approximate tool is given for detecting the occurrence of chaos.

The authors present a method to prove equivalence of a pair of FSMs, described at the gate level with D-type flip-flops and a reset signal available to bring them into the all-zero initial state. This method restricts investigation to that minimum subset of states that can be reached from the reset condition and are necessary to reach the goal. The equivalence condition is expressed in theoretical terms within the framework of the product machine. Without any loss of information, it is possible to reduce the product machine to a deterministic finite automaton (DFA). considerably reducing the number of states. The DFA is dynamically built by an explicit enumeration algorithm and, in general, only a very small part of the automaton is actually considered. The equivalence condition becomes a proof of the reachability of the DFA's final state. Search is performed in breadth-first. Experimental results on some pairs of ISCAS'89 circuits are reported.>