Memductance Functions Research Articles

A sampled-data stabilisation analysis on memristive BAM neural networks for memductance function

In this paper, we explore a stabilisation problem for two different class of memductance functions for memristive bidirectional associative memory neural networks (MBAMNNs) with time-varying delay. A new Lyapunov-Krasovskii function (LKF) with double and triple integral terms is derived with some sufficient conditions to guarantee the global asymptotic stability criteria using the Lyapunov stability theory. The controller gain matrix for sampled-data controller is obtained for the state-dependent switched system and state-dependent continuous system in terms of linear matrix inequalities (LMIs). Finally, we stabilise the considered model by visualising the simulation part using standard MATLAB software which demonstrates the validness of the theoretical results.

Design and Analysis of Multiscroll Memristive Hopfield Neural Network With Adjustable Memductance and Application to Image Encryption.

Memristor is an ideal electronic device used as an artificial nerve synapse due to its unique memory function. This article presents a design of a new Hopfield neural network (HNN) that can generate multiscroll attractors by utilizing a new memristor as a synapse in the HNN. Differing from the others, this memristor is constructed with hyperbolic tangent functions. Taking the memristor as a self-feedback synapse of a neuron in the HNN, the memristive HNN can yield multidouble-scroll attractors, and its parameters can be used to effectively control the number of double scrolls contained in an attractor. Interestingly, the generation of multidouble-scroll attractors is independent of the memductance function but depends only on the internal state equation. Thus, the memductance function can be adjusted to yield various complex dynamical behaviors. Moreover, amplitude control effects and quantitatively controllable multistability are revealed by numerical analysis. The accurate reproduction of some dynamical behaviors by a designed circuit verifies the correctness of the numerical analysis. Finally, based on the proposed memristive HNN, a novel image encryption scheme in the 3-D setting is designed and evaluated, demonstrating its good encryption performances.

Initial-condition effects on ReLU-type hyper-jerk system and its application in image encryption

The memristive hyper-jerk system described in this study has a non-smooth Rectified Linear Unit (ReLU) memductance function. A non-smooth memductance function of this kind might cause the system to have three line equilibria within it, bring about the arrival of extreme multistability with infinitely many coexisting attractors. Analyzing the three line equilibria’s stabilities and researching the intricate dynamical effects triggered by memristor and non-memristor initial conditions are related topic. Phase portraits, bifurcation diagrams, Lyapunov exponent spectra, and the basins of attraction all show the dynamical tendencies. According to the results, the novel system’s dynamical behaviors depend on both the memristor initial condition and the non-memristor initial conditions. Moreover, PSIM circuit simulations and MCU-based hardware measurements are run to confirm the coexisting infinitely many attractors. The ReLU-type hyper-jerk system is then utilized to encrypt images. Experimental findings support its exceptional robustness against a range of potential attacks.

A multiplier-free Rulkov neuron under memristive electromagnetic induction: Dynamics analysis, energy calculation, and circuit implementation

Establishing a realistic and multiplier-free implemented biological neuron model is significant for recognizing and understanding natural firing behaviors, as well as advancing the integration of neuromorphic circuits. Importantly, memristors play a crucial role in constructing memristive neuron and network models by simulating synapses or electromagnetic induction. However, existing models lack the consideration of initial-boosted extreme multistability and its associated energy analysis. To this end, we propose a multiplier-free implementation of the Rulkov neuron model and utilize a periodic memristor to represent the electromagnetic induction effect, thereby achieving the biomimetic modeling of the non-autonomous memristive Rulkov (mRulkov) neuron. First, theoretical analysis demonstrates that the stability distribution of the time-varying line equilibrium point is determined by both the parameters and the memristor’s initial condition. Furthermore, numerical simulations show that the mRulkov neuron can exhibit parameter-dependent local spiking, local hidden spiking, and periodic bursting firing behaviors. In addition, based on the periodic characteristics of the memductance function, the topological invariance of the mRulkov neuron is comprehensively proved. Therefore, local basins of attraction, bifurcation diagrams, and attractors related to extreme multistability can be boosted by switching the memristor’s initial condition. Significantly, the novel boosted extreme multistability is discovered in the Rulkov neuron for the first time. More importantly, the energy transition associated with the boosting dynamics is revealed through computing the Hamilton energy distribution. Finally, we develop a simulation circuit for the non-autonomous mRulkov neuron and confirm the effectiveness of the multiplier-free implementation and the accuracy of the numerical results through PSpice simulations.

Hidden firing patterns and memristor initial condition-offset boosting behavior in a memristive Hindmarsh-Rose neuron model

Electromagnetic induction can effectively induce abundant firing patterns in neurons. In modeling a neuron model with the electromagnetic induction effect, an electromagnetic induction current is frequently added to the state equation of membrane potential. To more properly reflect the non-uniform distribution of the ions inside and outside the neuron membrane, an ideal flux-controlled memristor with sinusoidal memductance function and non-linearly modulated input is raised to depict an electromagnetic induction effect on a Hindmarsh–Rose neuron model, and thereby, a three-dimensional (3D) memristive Hindmarsh–Rose (mHR) neuron model is built in this paper. The proposed mHR neuron model possesses no equilibrium point since the involvement of the ideal flux-controlled memristor, which induces the generation of hidden dynamics. Numerical results declare that the mHR neuron model can generate abundant hidden dynamics, i.e., periodic spiking, chaotic spiking, period-doubling bifurcation route, tangent bifurcation, and chaos crisis. These hidden dynamics are much related to the memristor coupling strength and externally applied stimulus. Afterward, the memristor initial condition-offset boosting behavior is revealed. This can trigger the generation of infinite multiple coexisting firing patterns along the memristor variable coordinate. These coexisting firing patterns have identical attractor topology but different locations in the phase plane. Finally, an analog circuit is designed for implementing the mHR neuron model, and PSIM-based circuit simulation is executed. The circuit-simulated results perfectly verify the generation of hidden infinite multiple coexisting initial condition-offset boosting firing patterns in the proposed mHR neuron model.

New FTFN-Based Tunable Memristor Emulator Circuit and its Mutation to Meminductor and Memcapacitor Emulators

For the first time, a new memristor emulator structure using a single four-terminal floating nullor (FTFN) and a transconductance stage has been presented with tunable circuit configuration. Along with that the circuit requires only a single grounded capacitance and two external MOS transistors to realize both incremental and decremental types of memductance functions. The use of the FTFN block has been demonstrated for the first time to build such a compact memristor emulator, which fully utilizes the employed circuit resources. The wide-band operating frequency range (1 kHz–3 MHz) is another attractive feature of the proposed emulator. Moreover, the mutation of the proposed memristor emulator into meminductor and memcapacitor emulators is also presented by the mutators based on FTFN. All the presented circuits have been tested by performing simulations using PSPICE with 0.18-[Formula: see text]m CMOS technology. The generated simulation results clearly show the ideal nonvolatile nature of the realized memristor, which has also been utilized in an op-amp-based circuit designed to exhibit associative learning phenomena. The proposed FTFN-based memristor has been implemented using commercially available ICs, LM13700, and AD844, and the generated PHL plot is discussed.

Generating Any Number of Initial Offset-Boosted Coexisting Chua’s Double-Scroll Attractors via Piecewise-Nonlinear Memristor

Due to the natural nonlinearity and unique memory characteristics, memristors are promising candidates for the construction of multiscroll attractors having better application potential in the field of information encryption than the traditional double-scroll attractors. This article proposes a novel memristive multidouble-scroll Chua’s system (MMDSCS) via coupling a nonideal flux-controlled memristor with multipiecewise-linear memductance function in Chua’s system directly. Specially, any number of multidouble-scroll chaotic attractors can be generated through adjusting the internal parameters of the memristor conveniently and without changing the original system’s nonlinearity. Moreover, the amount of double scrolls is also closely related to the strength of the memristive coupling. Another striking highlight is that infinite initial offset-boosted coexisting Chua’s double-scroll attractors with the same shape are produced with the variation of the memristor initial conditions, indicating the emergence of an intriguing phenomenon of homogeneous extreme multistability. This unique property and its formation mechanism are investigated in detail using phase portraits, bifurcation diagrams, Lyapunov exponents, time series, and attraction basins. Furthermore, hardware experiments based on the field-programmable gate array are carried out to confirm the numerical simulations. Finally, an image encryption scheme is designed based on the memristor initial offset boosting dynamics from a perspective of engineering application. In comparison with the existing memristive Chua’s systems, the proposed MMDSCS has many merits, such as multidouble-scroll attractors, memristor initial-controlled chaotic sequences with controllability, good robustness, and high security performance, which is more practical in applications involving information confidential communication.

A Compact and Continuous Reformulation of the Strachan TaOx Memristor Model With Improved Numerical Stability

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

Spatiotemporal pattern in a neural network with non-smooth memristor

<abstract> <p>Considering complicated dynamics of non-smooth memductance function, an improved Hindmarsh-Rose neuron model is introduced by coupling with non-smooth memristor and dynamics of the improved model are discussed. Simulation results suggest that dynamics of the proposed neuron model depends on the external stimuli but not on the initial value for the magnetic flux. Furthermore, a network composed of the improved Hindmarsh-Rose neuron is addressed via single channel coupling method and spatiotemporal patterns of the network are investigated via numerical simulations with no-flux boundary condition. Firstly, development of spiral wave are discussed for different coupling strengths, different external stimuli and various initial value for the magnetic flux. Results suggest that spiral wave can be developed for coupling strength $ 0 < D < 1 $ when the nodes are provided with period-1 dynamics, especially, double-arm spiral wave appear for $ D = 0.4 $.External stimuli changing can make spiral wave collapse and the network demonstrates chaotic state. Alternation of initial value for the magnetic flux hardly has effect on the developed spiral wave. Secondly, formation of target wave are studied for different coupling strengths, different sizes of center area with parameter diversity and various initial value for the magnetic flux. It can be obtained that, for certain size of center area with parameter diversity, target wave can be formed for coupling strength $ 0 < D < 1 $, while for too small size of center area with parameter diversity, target wave can hardly be formed. Change of initial value for the magnetic flux has no effect on the formation of target wave. Research results reveal the spatiotemporal patterns of neuron network to some extent and may provide some suggestions for exploring some disease of neural system.</p> </abstract>

A Novel Nonideal Flux-Controlled Memristor Model for Generating Arbitrary Multi-Double-Scroll and Multi-Double-Wing Attractors

This paper proposes a novel nonideal flux-controlled memristor model with a multipiecewise linear memductance function, which can be used to construct a memristive multi-scroll or multi-wing chaotic system. Importantly, arbitrary multi-double-scroll and multi-double-wing attractors can be generated depending on this memristor model directly and without the need to change the original nonlinear terms of the system. Another highlight is that the odd or even number of the double-scroll and double-wing attractors can also be freely controlled by the memristor model. To further illustrate these unique features, by introducing the memristor model into two classical chaotic systems, i.e. Jerk system and Lorenz system, multi-double-scroll and multi-double-wing chaotic attractors are obtained respectively. The formation mechanism of the multi-double-wing and multi-double-scroll attractors is also discussed. Moreover, the randomness of the chaotic binary sequences generated by the proposed memristor model is tested by the National Institute of Standards and Technology test suite. The tested results are better than those of the well-known Lorenz system. Furthermore, the corresponding circuits are constructed. The experimental results and the numerical simulations coincide well with each other, showing the effectiveness and feasibility of the proposed memristor model.

Persistence and coexistence of infinite attractors in a fractal Josephson junction resonator with unharmonic current phase relation considering feedback flux effect

Josephson junction resonators are the devices which exhibit complex behaviours as a consequence of their inductive properties. Even though the insulating medium between Josephson junctions (JJs) is normally considered homogeneous, the fact that lithography is used to form the layer, it has fractal substrates. Such JJs are identified as fractal Josephson junctions (FJJs). In this paper, a new chaotic oscillator based on memristor and FJJ has been investigated. Superconductor properties can dramatically change its operating points especially voltage and heat that are related to Josephson tunnelling. Some changes in the operating points can cause the Josephson tunnelling junctions to oscillate in different oscillation modes in very high frequencies. This can be achieved by considering the potential across the junction with its flux feedback. In order to model the magnetic flux effect, we use a memristor whose memductance function is considered as an exponential function. By varying the type of the bias current, we could observe the property of infinitely coexisting attractors in the memristor-fractal Josephson junction oscillator, which is considered as a rare phenomenon in physical circuits. The proposed Josephson-Memristor circuit model is developed, and its equilibrium points, bifurcation and Lyapunov exponents are computed. As an engineering application, modelling the trajectories of the moving object has been realized. First, the SURF algorithm, which is not affected by the scale and rotations of the object, is used in the images to identify an object that tracks the states of the proposed Josephson-Memristor circuit. In this way, the coordinates of the orbits on which the object moves were determined on the image. In order to reproduce the orbits of the specified object, the coordinate information of the object has been trained to the artificial neural network model and the orbits of the object have been reproduced.

Spiral waves in externally excited neuronal network: Solvable model with a monotonically differentiable magnetic flux.

Spiral waves are particular spatiotemporal patterns connected to specific phase singularities representing topological wave dislocations or nodes of zero amplitude, witnessed in a wide range of complex systems such as neuronal networks. The appearance of these waves is linked to the network structure as well as the diffusion dynamics of its blocks. We report a novel form of the Hindmarsh-Rose neuron model utilized as a square neuronal network, showing the remarkable multistructure of dynamical patterns ranging from characteristic spiral wave domains of spatiotemporal phase coherence to regions of hyperchaos. The proposed model comprises a hyperbolic memductance function as the monotone differentiable magnetic flux. Hindmarsh-Rose neurons with an external electromagnetic excitation are considered in three different cases: no excitation, periodic excitation, and quasiperiodic excitation. We performed an extensive study of the neuronal dynamics including calculation of equilibrium points, bifurcation analysis, and Lyapunov spectrum. We have found the property of antimonotonicity in bifurcation scenarios with no excitation or periodic excitation and identified wide regions of hyperchaos in the case of quasiperiodic excitation. Furthermore, the formation and elimination of the spiral waves in each case of external excitation with respect to stimuli parameters are investigated. We have identified novel forms of Hindmarsh-Rose bursting dynamics. Our findings reveal multipartite spiral wave formations and symmetry breaking spatiotemporal dynamics of the neuronal model that may find broad practical applications.

On Designing Feedback Controllers for Master-Slave Synchronization of Memristor-Based Chua’s Circuits

This paper is concerned with designing feedback controllers for master-slave synchronization of two chaotic memristor-based Chua’s circuits. The memductance function of memristor-based Chua’s circuits is a bounded function with a bounded derivative which is more generalized than those piecewise constant-valued functions or quadratic functions in some existing papers. The main contributions are that one master-slave synchronization criterion is established for two chaotic memristor-based Chua’s circuits, and the feedback controller gain is easily obtained by solving a set of linear matrix inequalities. One numerical example is given to illustrate the effectiveness of the design method.

A hyperchaotic memristor system with exponential and discontinuous memductance function

Most of the memristor chaotic and hyperchaotic oscillators discussed in the literatures use the cubic flux controlled memristor model. The drawback of this model is that in spite of knowing the terminal voltage polarity, one cannot easily determine whether the memductance increase or decrease. Hence we propose new memristor hyperchaotic oscillators derived using exponential memductance and discontinuous memductance functions. Dynamical analysis of the proposed oscillators are conducted using equilibrium points, stability of equilibrium, Eigen values and Lyapunov exponents. Bifurcation plots are derived to understand the parameter dependence of the proposed oscillators. Multistability and coexisting attractors are exhibited by these memristor oscillators.

Complex dynamics of a neuron model with discontinuous magnetic induction and exposed to external radiation.

The last two decades have seen many literatures on the mathematical and computational analysis of neuronal activities resulting in many mathematical models to describe neuron. Many of those models have described the membrane potential of a neuron in terms of the leakage current and the synaptic inputs. Only recently researchers have proposed a new neuron model based on the electromagnetic induction theorem, which considers inner magnetic fluctuation and external electromagnetic radiation as a significant missing part that can participate in neural activity. While the flux coupling of the membrane is considered equivalent to a memductance function of a memristor, standard memductance model of has been used in the literatures, but in this paper we propose a new memductance function based on discontinuous flux coupling. Various dynamical properties of the neuron model with discontinuous flux coupling are studied and interestingly the proposed model shows hyperchaotic behavior which was not identified in the literatures. Furthermore, we consider a ring network of the proposed model and investigate whether the chimera state can emerge. The chimera state relates to the state with simultaneously coherence and incoherence in oscillatory networks and has received much attention in recent years.

Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function

In this paper, a simplified memristor-based fractional-order neural network (MFNN) with discontinuous memductance function is proposed. It is essentially a switched system with irregular switching laws and consists of eight fractional-order neural network (FNN) subsystems. The nonlinear dynamics of the simplified MFNN including equilibrium points and their stability, bifurcation and chaos is investigated analytically and numerically. In particular, the effect of the switching jump on the dynamics of the simplified MFNN is explored for the first time. Taking the fractional order, the memristive connection weight or the switching jump as the bifurcation parameter, the dynamics such as chaotic motion, tangent bifurcation and intermittent chaos is identified over a wide range of some specified parameter. It can be seen that the incorporation of the memristors greatly improves and enriches the dynamics of the corresponding FNN. Different from the period-doubling route to chaos, this paper reveals that the mechanism behind the emergence of chaos for the simplified MFNN is the intermittency route to chaos. In particular, for some typical parameter, the existence of chaotic attractors is verified with the phase portraits, bifurcation diagrams, Poincare sections and maximum Lyapunov exponents, respectively. This paper not only provides a way of designing chaotic MFNN with discontinuous memductance function but also suggests a possible method of generating more complicated chaotic attractors, such as multi-scroll or multi-wing attractors.

Global dissipativity analysis for memristor-based uncertain neural networks with time delay in the leakage term

This paper is concerned with the problem of global dissipativity analysis for a class of memristor-based uncertain neural networks with leakage delay and state-dependent switched memductance functions. By combining differential inclusions with set-valued maps and constructing a proper Lyapunov-Krasovskii functional, delaydependent criteria in terms of linear matrix inequalities are obtained for the global dissipativity of the memristive uncertain neural networks. Finally, a numerical example is given to illustrate the feasibility of the theoretical results.

Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system.

In this paper, two kinds of novel ideal active flux-controlled smooth multi-piecewise quadratic nonlinearity memristors with multi-piecewise continuous memductance function are presented. The pinched hysteresis loop characteristics of the two memristor models are verified by building a memristor emulator circuit. Using the two memristor models establish a new memristive multi-scroll Chua's circuit, which can generate 2N-scroll and 2N+1-scroll chaotic attractors without any other ordinary nonlinear function. Furthermore, coexisting multi-scroll chaotic attractors are found in the proposed memristive multi-scroll Chua's circuit. Phase portraits, Lyapunov exponents, bifurcation diagrams, and equilibrium point analysis have been used to research the basic dynamics of the memristive multi-scroll Chua's circuit. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design.

Delay-Dependent Passivity and Stability Analysis for a Class of Memristor-Based Neural Networks with Time Delay in the Leakage Term

This paper is concerned with the problem of passivity and stability analysis for a class of memristor-based neural networks with leakage delay and state-dependent switched memductance functions. By combining differential inclusions with set-valued maps and constructing a proper Lyapunov–Krasovskii functional, delay-dependent criteria in terms of linear matrix inequalities are obtained for the passivity of the memristive neural networks. Meanwhile, based on the derived criteria, stability criteria are obtained for the networks via Barbalat’s lemma. Finally, a numerical example is given to illustrate the feasibility of the theoretical results.

Passivity analysis of memristive neural networks with mixed time-varying delays and different state-dependent memductance functions

This paper is concerned with the problem of passivity analysis for a class of memristive neural networks with mixed time-varying delays and different state-dependent memductance functions. By employing the theories of differential inclusions and set-valued maps, delay-dependent criteria in terms of linear matrix inequalities are obtained for the passivity of the memristive neural networks. Finally, numerical examples are given to illustrate the feasibility of the theoretical results.