Image encryption is a robust method to secure information transmission over public, unprotected networks. Thanks to their complex dynamics, chaotic systems are gaining interest for encryption scheme development. This paper presents a novel method to generate pseudorandom sequences designing a compact and cost-effective circuit that mimics the dynamics of the logistic map. This hardware platform employs a standard microcontroller to turn the raw chaotic time series into balanced binary sequences, made mutually-orthogonal one to the other through cross-correlation calculations. The resulting binary codes passed all statistical tests in the Institute of Standards and Technology (NIST) SP 800-22 suite, with success rates up to 99.6%. Here, we discuss the integration of the proposed hardware platform into an image encryption system aimed at securing clinical communications. We exploited the unique properties of the chaotic codes to implement a DNA-inspired image encryption algorithm. Robustness was evaluated against four clinical images of skin ulcers, one for each severity class (Wound Bed Preparation standard). An automated data classification procedure confirmed that the encryption and decryption processes do not degrade the diagnostic content of the images. Six security and robustness tests were also passed successfully. We thus present an economic hardware solution, amenable to integration into standard communication platforms, delivering security and enabling novel data protection methods in clinical environments.

We investigate multistability, noise-induced transitions, and stochastic resonance in a second-order nonlinear oscillator incorporating a nonvolatile memristive device. The memristor provides a programmable nonlinear conductance, enabling bistable dynamics with two asymptotically stable equilibrium points separated by a saddle. Under periodic excitation, the system exhibits coexisting limit cycles, period-doubling cascades, boundary crises, and transitions to chaos. Lyapunov exponent analysis reveals repeated crossings of the edge-of-chaos regime, where the largest nonzero exponent approaches zero, marking a balance between stability and sensitivity to perturbations. The effects of additive Gaussian white noise are analyzed by reformulating the dynamics in terms of an effective potential landscape, where noise induces random transitions between coexisting attractors. Transition rates are accurately described in the weak-noise regime by the Eyring–Kramers formula. When periodic forcing and noise act jointly, the system exhibits stochastic resonance, with optimal synchronization occurring when the forcing period matches the mean noise-induced transition time. These results demonstrate that memristor-based nonlinear circuits naturally operate near critical dynamical regimes and provide a compact hardware platform for studying noise-assisted computation and edge-of-chaos dynamics in neuromorphic systems.

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.

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.

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.

Nonvolatile memristive devices display nonlinear characteristics suitable for implementing circuits exhibiting oscillations or more complex dynamic behaviors, including chaos. However, the results presented in related works are mostly limited to simulations and employing ideal memristor models whose resistance is governed by a charge-flux relation that is not connected to real devices, thus hindering the realization of such nonlinear oscillators. In this work, we present the framework for the physical implementation of a tunable memristor Chua’s circuit, which is based on a nonvolatile memristive device that provides the nonlinear conductance required by the circuit and the possibility to tune it for the purpose of selecting among different oscillation patterns. We first establish design guidelines to guarantee complex oscillations in the tunable memristor Chua’s circuit. Further, we physically implement the circuit after characterizing and modeling the tunable current-voltage characteristic of a real device. Our circuit successfully generates different oscillation patterns just by programming the nonvolatile memristive device to different states. The devised design guidelines and device modeling were used to extend the experimental work and draw further requirements for device properties for a successful circuit implementation.

Analog computing platforms have been a popular and promising research area that suggest efficient ways of computation compared to its digital counterparts. Memristor based crossbars drew attention by computing the vector-matrix calculation intensive tasks such as Artificial Intelligence (AI) and Machine Learning (ML) in one time step. Although they provide an energy efficient way of computing these tasks, analog computation in general suffers from non-idealities and systematic errors in the circuitry, which could degrade the performance and accuracy significantly. One of the issues is the random offset associated with the op-amps in the system resulting from the process and mismatch variations. In this paper, a novel technique is offered to reduce the negative effects of the random offset and increase the output accuracy. This newly proposed system uses minimum extra circuitry and additional power consumption and only requires the crossbar to be enlarged by two extra rows. The intrinsic issue of the analog crossbars, interconnect parasitics, must be incorporated into the problem, and a way to separate the offset and wire resistance issues from each other is offered. The functionality of the system has been shown with a case study in the results section where the op-amps have$\sigma_{offset}=3mV$. The effectiveness of the offered technique demonstrates a 6× better accuracy with the mitigation of the offset problem. The proposed method can be used in memristor and other analog crossbars to achieve a greater performance and thus improve their competitiveness.

Over the last decade, Gaussian processes (GPs) have become popular in the area of machine learning and data analysis for their flexibility and robustness. Despite their attractive formulation, practical use in large-scale problems remains out of reach due to computational complexity. Existing direct computational methods for manipulations involving large-scale$n\times n$covariance matrices require$O(n^{3})$calculations. In this work, we present the design and evaluation of a simulated computing platform for exact GP inference, that achieves true model parallelism using memristive crossbars. To achieve a one-shot solution, a linear equation solver and a vector-matrix multiplication solver crossbar configurations are used together, reducing the number of operations from$O(n^{3})$to$O(n)$. The transistor level op-amps, ADC models for quantization, circuit and interconnect parasitics, together with the finite memristor precision are incorporated into the system simulation. The analog system resulted in %1.51 mean error and %2.93 average variance error in solving a nonlinear regression problem. The proposed method achieved 9× to 144× better energy efficiency compared to TPU and 7× compared to a custom analog linear regression solver.

Determining the dominant eigenvector of matrices and graphs is one of the most fundamental tasks in many machine learning problems, including spectral clustering, Hyperlink Induced Topic Search (HITS), Markov Chains, PageRank and eigenvector centralities. Among the several algorithms used, the Power Method is one of the simplest iterative approaches. It relies on multiple vector-matrix multiplications (VMMs) and a normalization step to prevent divergent behaviours. Recently, efficiency of the memristor crossbars in solving VMMs have been demonstrated using fundamental laws of the circuit theory. In this work, we propose a circuit to accelerate the Power iteration algorithm including current-mode termination for the memristor crossbars and a normalization circuit. The normalization step together with the feedback loop of the complete circuit ensure stability and convergence of the dominant eigenvector. The system allows the observation of the evolution of the outputs. We implement a transistor level peripheral circuitry around the memristor crossbar and take non-idealities such as wire parasitics, source driver resistance and finite memristor precision into account. We compute the eigenvector centrality to demonstrate the performance of the proposed system. We compare our results to the ones coming from the conventional digital computers and observe significant energy savings while maintaining a competitive accuracy.

Weakly Connected Oscillatory Networks (WCONs) are bio-inspired models which exhibit associative memory properties and can be exploited for information processing. It has been shown that the nonlinear dynamics of WCONs can be reduced to equations for the phase variable if oscillators admit stable limit cycles with nearly identical periods. Moreover, if connections are symmetric, the phase deviation equation admits a gradient formulation establishing a one-to-one correspondence between phase equilibria, limit cycle of the WCON and minima of the system’s potential function. The overall objective of this work is to provide a simulated WCON based on memristive connections and Van der Pol oscillators that exploits the device mem-conductance programmability to implement a novel local supervised learning algorithm for gradient models: Equilibrium Propagation (EP). Simulations of the phase dynamics of the WCON system trained with EP show that the retrieval accuracy of the proposed novel design outperforms the current state-of-the-art performance obtained with the Hebbian learning.

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.

Resistive devices such as memristors have attracted the researchers' attention as fundamental computing elements. The prime objective of this work is to provide a simulated analogue computing platform based on memristor devices and recurrent neural networks that exploits the memristor device conductance programmability to implement local learning algorithms. We present the application of two simple two-phase learning procedures for Dynamic Neural Networks used to solve a pattern reconstruction task. The first learning scheme is related to Energy-based models and the second generalizes this method to generic vector field dynamics, relaxing the requirement of an energy function. Experimental results show that both the two approach significantly outperforms conventional learning rules used for pattern reconstruction.

The paper considers a relevant class of networks containing memristors and (possibly) nonlinear capacitors and inductors. The goal is to unfold the nonlinear dynamics of these networks by highlighting some main features that are potentially useful for real-time signal processing and in-memory computing. In particular, an analytic treatment is provided for dynamic phenomena as the presence of invariant manifolds, the coexistence of different regimes, complex dynamics and attractors and the phenomenon of bifurcations without parameters, i.e., bifurcations due to changing the initial conditions of the state variables for a fixed set of circuit parameters. The paper also addresses the issue of how to design pulse independent voltage or current sources to steer the network dynamics through different manifolds and attractors. Two relevant examples are worked out in details, namely, a variant of Chua's circuit with a memristor and a nonlinear capacitor and a relaxation oscillator with a memristor and a nonlinear inductor. In the latter example, the paper also studies the effect on manifolds and coexisting dynamics when real memristive devices are accounted for using a class of extended memristor models. The analysis is conducted by means of a recently developed technique named flux-charge analysis method (FCAM). Numerical simulations are presented to confirm the theoretic findings.

Problems involving discrete Markov Chains are solved mathematically using matrix methods. Recently, several research groups have demonstrated that matrix-vector multiplication can be performed analytically in a single time step with an electronic circuit that incorporates an open-loop memristor crossbar that is effectively a resistive random-access memory. Ielmini and co-workers have taken this a step further by demonstrating that linear algebraic systems can also be solved in a single time step using similar hardware with feedback. These two approaches can both be applied to Markov chains, in the first case using matrix-vector multiplication to compute successive updates to a discrete Markov process and in the second directly calculating the stationary distribution by solving a constrained eigenvector problem. We present circuit models for open-loop and feedback configurations, and perform detailed analyses that include memristor programming errors, thermal noise sources and element nonidealities in realistic circuit simulations to determine both the precision and accuracy of the analog solutions. We provide mathematical tools to formally describe the trade-offs in the circuit model between power consumption and the magnitude of errors. We compare the two approaches by analyzing Markov chains that lead to two different types of matrices, essentially random and ill-conditioned, and observe that ill-conditioned matrices suffer from significantly larger errors. We compare our analog results to those from digital computations and find a significant power efficiency advantage for the crossbar approach for similar precision results.

Circuits containing memelements (memory elements) are suitable for the design of new unconventional computational schemes. The coexistence of a rich variety of different attractors is one of the appealing property of these circuits, which has stimulated the so-called “multistability control” problem. This paper considers the multistability control problem for a circuit with a charge-controlled memristor. It is shown how pulse control inputs can be generated via an external current generator in order to drive the system dynamics from an attractor to another one in a given finite time interval.

Nonlinear dynamic memory elements, as memristors, memcapacitors, and meminductors (also known as mem-elements), are of paramount importance in conceiving the neural networks, mem-computing machines, and reservoir computing systems with advanced computational primitives. This paper aims to develop a systematic methodology for analyzing complex dynamics in nonlinear networks with such emerging nanoscale mem-elements. The technique extends the flux-charge analysis method (FCAM) for nonlinear circuits with memristors to a broader class of nonlinear networks N containing also memcapacitors and meminductors. After deriving the constitutive relation and equivalent circuit in the flux-charge domain of each two-terminal element in N , this paper focuses on relevant subclasses of N for which a state equation description can be obtained. On this basis, salient features of the dynamics are highlighted and studied analytically: 1) the presence of invariant manifolds in the autonomous networks; 2) the coexistence of infinitely many different reduced-order dynamics on manifolds; and 3) the presence of bifurcations due to changing the initial conditions for a fixed set of parameters (also known as bifurcations without parameters). Analytic formulas are also given to design nonautonomous networks subject to pulses that drive trajectories through different manifolds and nonlinear reduced-order dynamics. The results, in this paper, provide a method for a comprehensive understanding of complex dynamical features and computational capabilities in nonlinear networks with mem-elements, which is fundamental for a holistic approach in neuromorphic systems with such emerging nanoscale devices.

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.

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.

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.

A class of memristor circuits is obtained by cascading a static nonlinear two-port with a dynamical one-port. The terminals of the input port of the static nonlinearity represent the access nodes for each memristor in the class. The class may be splitted into two sub-classes, namely the current- and voltagecontrolled memristors. Two further sets of memristors may be identied within each of such sub-classes, particularly the current- and voltage state memristors. The simplest memristor circuits from the proposed class employ solely passive two-terminal elements from circuit theory. This represents an absolute novelty in the panorama of memristor emulators. The passive elements from the class are volatile memories. However, non-volatile memory behaviour may arise in case the dynamical one-port contains active elements. The versatile nature of the circuit topologies of the proposed memristors allows the emergence of a wide variety of complex dynamical behaviours, which may enable the accomplishment of novel signal processing tasks or lead to improvements in the performance of conventional circuits.

We present a novel phase-amplitude model for noisy oscillators described by Itô 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.

Neural Networks trained with the Belief Propagation Inspired (BPI) algorithm are able to learn a number of associations close to the theoretical limit in time that is sublinear in the number of input. Using binary synapses, implemented by a memristor, a single layer perceptron with BPI has been proposed. It well know that perceptrons with step function type nonlinearity can be implemented by a suitable class of Cellular Neural/Nonlinear Networks. This paper aims to present a statistical analysis on the learning efficiency of Memristor-based Cellular Nonlinear Networks (M-CNNs) with Belief Propagation Inspired (BPI) algorithm. Monte Carlo simulations permit to assess that the learning efficiency of M-CNNs with BPI is not regardless of the input signals given to train the perceptron.

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.

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.

Biological neural systems use self- reconfigurable and self-learning primitive elements (synapses) to extract relevant information from complex and noisy environments, to detect specific spatio-temporal patterns in the data of interest and to compute and simultaneously store some significant features. All these desirable attributes may be realized by using two-terminal elements, memristors (memory resistors), which most closely resemble biological synapses. This article is organized according to the rule of the ISCAS2013 special session having the same title. We present a short summary of the state-of-the-art of memristor theory and Hodgkin-Huxley neural model. In addition, we briefly introduce a comprehensive nonlinear circuit-theoretic foundation for a novel circuit implementation of the Hodgkin-Huxley neural model with memristors.

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.

In this paper we analyze synchronization in networks of identical and nearly identical Chua's oscillators. Using time-domain simulations and the Master Stability Function (MSF) approach in time and frequency domain, we show that losses can lower down the coupling bounds for which a given network of oscillators synchronizes. We also show that by using the extended MSF losses reduce the synchronization error when the oscillators are nearly identical, i.e. there is a bounded mismatch of the parameters of the oscillators.

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.

Locally Connected Oscillatory Networks (LCONs) are a special class of Cellular Neural Networks (CNNs) where each cell (neuron) exhibits time periodic behavior. In this paper we investigate the dynamics of LCONs whose neurons exhibit the coexistence of a stable equilibrium point and a stable limit cycle. We consider a constant external stimulus applied to each neuron, which influences the neuron's own natural frequency. We show that new interesting dynamics, namely synchronous oscillations of various amplitudes, may arise due to the interaction between different kind of attractors. We also show that neurons subjected to different external stimuli are able to synchronize if their local couplings are strong enough.

An efficient tool to study synchronization of periodic oscillations in networks of coupled nonlinear oscillators, given by the joint application of the Master Stability Function and the Harmonic Balance, is applied to the to study of synchronization in networks of Chua's circuits coupled with resistors and capacitors.

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.

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 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

Drusen characterize the age-related macular degeneration. Automatic procedures for their identification have been recently developed. In this paper a cellular neural network based algorithm for drusen identification in fundus photograph is proposed. The algorithm is composed by different image processing steps: noise reduction, histogram normalization and a novel procedure of adaptive segmentation. The algorithm has been validated by using images provided from an ophthalmic medical center

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.