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

Multiscroll hidden attractor in memristive autapse neuron model and its memristor-based scroll control and application in image encryption

In current neurodynamic studies, memristor models using polynomial or multiple nested composite functions are primarily employed to generate multiscroll attractors, but their complex mathematical form restricts both research and application. To address this issue, without relying on polynomial and multiple nested composite functions, this study devises a unique memristor model and a memristive autapse HR (MAHR) neuron model featuring multiscroll hidden attractor. Specially, the quantity of scrolls within the multiscroll hidden attractors is regulated by simulation time. Besides, a simple control factor is incorporated into the memristor to improve the MAHR neuron model. Numerical analysis further finds that the quantity of scrolls within the multiscroll hidden attractor from the improved MAHR neuron model can be conveniently adjusted by only changing a single parameter or initial condition of the memristor. Moreover, a microcontroller-based hardware experiment is conducted to confirm that the improved MAHR neuron model is physically feasible. Finally, an elegant image encryption scheme is proposed to explore the real-world applicability of the improved MAHR neuron model.

Dynamic Neural Network Structure: A Review for its Theories and Applications

The dynamic neural network (DNN), in contrast to the static counterpart, offers numerous advantages, such as improved accuracy, efficiency, and interpretability. These benefits stem from the network's flexible structures and parameters, making it highly attractive and applicable across various domains. As the broad learning system (BLS) continues to evolve, DNNs have expanded beyond deep learning (DL), orienting a more comprehensive range of domains. Therefore, this comprehensive review article focuses on two prominent areas where DNN structures have rapidly developed: 1) DL and 2) broad learning. This article provides an in-depth exploration of the techniques related to dynamic construction and inference. Furthermore, it discusses the applications of DNNs in diverse domains while also addressing open issues and highlighting promising research directions. By offering a comprehensive understanding of DNNs, this article serves as a valuable resource for researchers, guiding them toward future investigations.

Coexistence of Cyclic Sequential Pattern Recognition and Associative Memory in Neural Networks by Attractor Mechanisms

Neural networks are developed to model the behavior of the brain. One crucial question in this field pertains to when and how a neural network can memorize a given set of patterns. There are two mechanisms to store information: associative memory and sequential pattern recognition. In the case of associative memory, the neural network operates with dynamical attractors that are point attractors, each corresponding to one of the patterns to be stored within the network. In contrast, sequential pattern recognition involves the network memorizing a set of patterns and subsequently retrieving them in a specific order over time. From a dynamical perspective, this corresponds to the presence of a continuous attractor or a cyclic attractor composed of the sequence of patterns stored within the network in a given order. Evidence suggests that the brain is capable of simultaneously performing both associative memory and sequential pattern recognition. Therefore, these types of attractors coexist within the neural network, signifying that some patterns are stored as point attractors, while others are stored as continuous or cyclic attractors. This article investigates the coexistence of cyclic attractors and continuous or point attractors in certain nonlinear neural networks, enabling the simultaneous emergence of various memory mechanisms. By selectively grouping neurons, conditions are established for the existence of cyclic attractors, continuous attractors, and point attractors, respectively. Furthermore, each attractor is explicitly represented, and a competitive dynamic emerges among these coexisting attractors, primarily regulated by adjustments to external inputs.

Plane coexistence behaviors for Hopfield neural network with two-memristor-interconnected neurons

Memristors are commonly used as the connecting parts of neurons in brain-like neural networks. The memristors, unlike the existing literature, possess the capability to function as both self-connected synaptic weights and interconnected synaptic weights, thereby enabling the generation of intricate initials-regulated plane coexistence behaviors. To demonstrate this dynamical effect, a Hopfield neural network with two-memristor-interconnected neurons (TMIN-HNN) is proposed. On this basis, the stability distribution of the equilibrium points is analyzed, the related bifurcation behaviors are studied by utilizing some numerical simulation methods, and the plane coexistence behaviors are proved theoretically and revealed numerically. The results clarify that TMIN-HNN not only exhibits complex bifurcation behaviors, but also has initials-regulated plane coexistence behaviors. In particular, the coexistence attractors can be switched to different plane locations by the initial states of the two memristors. Finally, a digital experiment device is developed based on STM32 hardware board to verify the initials-regulated plane coexistence attractors.

Adaptive Intermittent Pinning Control for Synchronization of Delayed Nonlinear Memristive Neural Networks With Reaction-Diffusion Items

In this article, the global exponential synchronization problem is investigated for a class of delayed nonlinear memristive neural networks (MNNs) with reaction-diffusion items. First, using the Green formula, Lyapunov theory, and proposing a new fuzzy adaptive pinning control scheme, some novel algebraic criteria are obtained to ensure the exponential synchronization of the concerned networks. Furthermore, the corresponding control gains can be promptly adjusted based on the current states of partial nodes of the networks. Besides, a fuzzy adaptive aperiodically intermittent pinning control law is also designed to synchronize the fuzzy MNNs (FMNNs). The controller with intermittent mechanism can obtain appropriate rest time and save energy consumption. Finally, some numerical examples are provided to confirm the effectiveness of the results in this article.

An FHN-HR Neuron Network Coupled With a Novel Locally Active Memristor and Its DSP Implementation

In this article, a novel locally active memristor (LAM) model is designed and its characteristics are studied in detail. Then, the LAM model is applied to couple FitzHugh-Nagumo (FHN) and Hindmarsh-Rose (HR) neuron. The simple neuron network is built to emulate connection of separate neurons and transmission of information from FHN neuron to HR neuron. The equilibrium point about this FHN-HR model is analyzed. Under the influence of varied parameters, dynamical characteristics for the model are explored with various analysis methods, including phase diagram, time series, bifurcation diagram, and Lyapunov exponent spectrum (LEs). The spectral entropy (SE) complexity and sequence randomness of the model are studied. In addition to observing chaotic and periodic attractors, multiple types of attractor coexistence and particular state transition phenomena are also found in the coupled FHN-HR model. Furthermore, geometric control is used for modulating the amplitude and offset of attractor and neuron firing signals, involving amplitude control and offset control. Finally, DSP implementation is finished, proving digital circuit feasibility of the FHN-HR model. The research imitates the coupling and information transmission between different neurons and has potential applications to secrecy or encryption.

The dynamical behavior effects of different numbers of discrete memristive synaptic coupled neurons

Two types of neuron models are constructed in this paper, namely the single discrete memristive synaptic neuron model and the dual discrete memristive synaptic neuron model. Firstly, it is proved that both models have only one unstable equilibrium point. Then, the influence of the coupling strength parameters and neural membrane amplification coefficient of the corresponding system of the two models on the rich dynamical behavior of the systems is analyzed. Research has shown that when the number of discrete local active memristor used as simulation synapses in the system increases from one to two, the coupling strength parameter of the same memristor has significantly different effects on the dynamical behavior of the system within the same range, that is, from a state with periodicity, chaos, and periodicity window to a state with only chaos. In addition, under the influence of coupling strength parameters and neural membrane amplification coefficients, the complexity of the system weakens to varying degrees. Moreover, under the effect of two memristors, the system exhibits a rare and interesting phenomenon where the coupling strength parameter and the neural membrane amplification coefficient can mutually serve as control parameter, resulting in the generation of a remerging Feigenbaum tree. Finally, the pseudo-randomness of the chaotic systems corresponding to the two models are detected by NIST SP800-22, and relevant simulation results are verified on the DSP hardware experimental platform. The discrete memristive synaptic neuron models established in this article provide assistance in studying the relevant working principles of real neurons.

ANN-based chaotic PRNG in the novel jerk chaotic system and its application for the image encryption via 2-D Hilbert curve

In this paper, we introduce a category of Novel Jerk Chaotic (NJC) oscillators featuring symmetrical attractors. The proposed jerk chaotic system has three equilibrium points. We show that these equilibrium points are saddle-foci points and unstable. We have used traditional methods such as bifurcation diagrams, phase portraits, and Lyapunov exponents to analyze the dynamic properties of the proposed novel jerk chaotic system. Moreover, simulation results using Multisim, based on an appropriate electronic implementation, align with the theoretical investigations. Additionally, the NJC system is solved numerically using the Dormand Prince algorithm. Subsequently, the Jerk Chaotic System is modeled using a multilayer Feed-Forward Neural Network (FFNN), leveraging its nonlinear mapping capability. This involved utilizing 20,000 values of x1, x2, and x3 for training (70%), validation (15%), and testing (15%) processes, with the target values being their iterative values. Various network structures were experimented with, and the most suitable structure was identified. Lastly, a chaos-based image encryption algorithm is introduced, incorporating scrambling technique derived from a dynamic DNA coding and an improved Hilbert curve. Experimental simulations confirm the algorithm's efficacy in enduring numerous attacks, guaranteeing strong resiliency and robustness.

Heterogeneous coexisting attractors, large-scale amplitude control and finite-time synchronization of central cyclic memristive neural networks

Memristors are of great theoretical and practical significance for chaotic dynamics research of brain-like neural networks due to their excellent physical properties such as brain synapse-like memorability and nonlinearity, especially crucial for the promotion of AI big models, cloud computing, and intelligent systems in the artificial intelligence field. In this paper, we introduce memristors as self-connecting synapses into a four-dimensional Hopfield neural network, constructing a central cyclic memristive neural network (CCMNN), and achieving its effective control. The model adopts a central loop topology and exhibits a variety of complex dynamic behaviors such as chaos, bifurcation, and homogeneous and heterogeneous coexisting attractors. The complex dynamic behaviors of the CCMNN are investigated in depth numerically by equilibrium point stability analysis as well as phase trajectory maps, bifurcation maps, time-domain maps, and LEs. It is found that with the variation of the internal parameters of the memristor, asymmetric heterogeneous attractor coexistence phenomena appear under different initial conditions, including the multi-stable coexistence behaviors of periodic-periodic, periodic-stable point, periodic-chaotic, and stable point-chaotic. In addition, by adjusting the structural parameters, a wide range of amplitude control can be realized without changing the chaotic state of the system. Finally, based on the CCMNN model, an adaptive synchronization controller is designed to achieve finite-time synchronization control, and its application prospect in simple secure communication is discussed. A microcontroller-based hardware circuit and NIST test are conducted to verify the correctness of the numerical results and theoretical analysis.

Dynamic analysis of FN-HR neural network coupled of bistable memristor and encryption application based on Fibonacci Q-Matrix

In this paper, a cosine hyperbolic memristor model is proposed with bistable asymmetric hysteresis loops. A neural network of coupled hyperbolic memristor is constructed by using the Fitzhugh-Nagumo model and the Hindmarsh-Rose model. The coupled neural network with a large number of equilibrium points is obtained by numerical analysis. In addition, the coexisting discharge behavior of the coupled neural network is revealed using local attractor basins. The complex dynamic properties of the memristor-coupled neural network are verified by analyzing the two-parameter Lyapunov exponential map and spectral entropy map, and the equivalent circuit of the coupled neural network is designed to prove the accuracy of the numerical analysis. Finally, an image encryption algorithm is proposed, which combines coupled neural network and Fibonacci Q-Matrix. The numerical analysis demonstrates that the algorithm exhibits strong security and resistance against cracking attempts.

Dynamics of heterogeneous Hopfield neural network with adaptive activation function based on memristor

Memristor and activation function are two important nonlinear factors of the memristive Hopfield neural network. The effects of different memristors on the dynamics of Hopfield neural networks have been studied by many researchers. However, less attention has been paid to the activation function. In this paper, we present a heterogeneous memristive Hopfield neural network with neurons using different activation functions. The activation functions include fixed activation functions and an adaptive activation function, where the adaptive activation function is based on a memristor. The theoretical and experimental study of the neural network's dynamics has been conducted using phase portraits, bifurcation diagrams, and Lyapunov exponents spectras. Numerical results show that complex dynamical behaviors such as multi-scroll chaos, transient chaos, state jumps and multi-type coexisting attractors can be observed in the heterogeneous memristive Hopfield neural network. In addition, the hardware implementation of memristive Hopfield neural network with adaptive activation function is designed and verified. The experimental results are in good agreement with those obtained using numerical simulations.

Energy-Performance Assessment of Oscillatory Neural Networks Based on VO2 Devices for Future Edge AI Computing

Oscillatory neural network (ONN) is an emerging neuromorphic architecture composed of oscillators that implement neurons and are coupled by synapses. ONNs exhibit rich dynamics and associative properties, which can be used to solve problems in the analog domain according to the paradigm let physics compute. For example, compact oscillators made of VO2 material are good candidates for building low-power ONN architectures dedicated to AI applications at the edge, like pattern recognition. However, little is known about the ONN scalability and its performance when implemented in hardware. Before deploying ONN, it is necessary to assess its computation time, energy consumption, performance, and accuracy for a given application. Here, we consider a VO2-oscillator as an ONN building block and perform circuit-level simulations to evaluate the ONN performances at the architecture level. Notably, we investigate how the ONN computation time, energy, and memory capacity scale with the number of oscillators. It appears that the ONN energy grows linearly when scaling up the network, making it suitable for large-scale integration at the edge. Furthermore, we investigate the design knobs for minimizing the ONN energy. Assisted by technology computer-aided design (TCAD) simulations, we report on scaling down the dimensions of VO2 devices in crossbar (CB) geometry to decrease the oscillator voltage and energy. We benchmark ONN versus state-of-the-art architectures and observe that the ONN paradigm is a competitive energy-efficient solution for scaled VO2 devices oscillating above 100 MHz. Finally, we present how ONN can efficiently detect edges in images captured on low-power edge devices and compare the results with Sobel and Canny edge detectors.

Assessing sigmoidal function on memristive maps

Memristors offer a crucial element for constructing discrete maps that have garnered significant attention in complex dynamics and various potential applications. In this study, we have integrated memristive and sigmoidal function to propose innovative mapping techniques. Our research confirms that the amalgamation of memristor and sigmoidal functions represents a promising approach for creating both 2D and 3D maps. Particularly noteworthy are the chaotic maps featuring multiple sigmoidal functions and multiple memristors, as highlighted in our findings. Specifically focusing on the novel STMM1 map, we delve into its dynamics and assess its feasibility. Intriguingly, the introduction of sigmoidal functions leads to alterations in the quantity of fixed points and the symmetry of the map.

Synchronization in scale-free neural networks under electromagnetic radiation

The functional networks of the human brain exhibit the structural characteristics of a scale-free topology, and these neural networks are exposed to the electromagnetic environment. In this paper, we consider the effects of magnetic induction on synchronous activity in biological neural networks, and the magnetic effect is evaluated by the four-stable discrete memristor. Based on Rulkov neurons, a scale-free neural network model is established. Using the initial value and the strength of magnetic induction as control variables, numerical simulations are carried out. The research reveals that the scale-free neural network exhibits multiple coexisting behaviors, including resting state, period-1 bursting synchronization, asynchrony, and chimera states, which are dependent on the different initial values of the multi-stable discrete memristor. In addition, we observe that the strength of magnetic induction can either enhance or weaken the synchronization in the scale-free neural network when the parameters of Rulkov neurons in the network vary. This investigation is of significant importance in understanding the adaptability of organisms to their environment.

Offset boosting in a discrete system

Offset boosting plays an important role in chaos application in electronic engineering. A direct variable substitution typically will destroy the dynamics of a discrete map even though the initial condition is well considered. The internal fundamental reason is that the left-hand side of a discrete system does not have the dimension of variable differentiation (DVD) like the one of a continuous system. When the key property of DVD is completely preserved, the offset boosting based on a parameter or the initial condition can be reasonably achieved like in a differential system. Consequently, by the initial condition-oriented offset boosting, flexible multistability like attractor self-reproducing or attractor doubling can be further realized. A circuit experiment is completed for the verification of reliable offset boosting. The systematic exploration of offset boosting in a map will cast a new light on chaos regulation and attractor transportation in a discrete map. As a simple case, a two-dimensional Hénon map is taken as the example demonstrating the achievement of offset boosting via the parameter or initial condition.

Hyperchaos of neuron under local active discrete memristor simulating electromagnetic radiation

Memristor enables the coupling of magnetic flux to membrane voltage and is widely used to investigate the response characteristics of neurons to electromagnetic radiation. In this paper, a local active discrete memristor is constructed and used to study the effect of electromagnetic radiation on the dynamics of neurons. The research results demonstrate that increasing electromagnetic radiation intensity could induce hyperchaotic attractors. Furthermore, this neuron model generates hyperchaotic and three points coexistence attractors with the introduction of the memristor. A digital circuit is designed to implement the model and evaluate the randomness of its output sequence. Neuronal models exhibit a rich dynamic behavior with electrical radiation stimulation, which can provide new directions for exploring the production mechanisms of certain neurological diseases.

Securing image-based document transmission in logistics and supply chain management through cheating-resistant visual cryptographic protocols

In today's digital landscape, securing multimedia visual information-specifically color images-is of critical importance across a range of sectors, including the burgeoning fields of logistics and supply chain management. Traditional Visual Cryptography (VC) schemes lay the groundwork for encrypting visual data by fragmenting a secret image into multiple shares, thereby ensuring no single share divulges the secret. Nevertheless, VC faces challenges in ascertaining the integrity of reconstructed images, especially when shares are manipulated maliciously. Existing solutions often necessitate additional shares or a trusted third party for integrity verification, thereby adding complexity and potential security risks. This paper introduces a novel Cheating-Resistant Visual Cryptographic Protocol (CRVC) for Color Images that aims to address these limitations. Utilizing self-computational models, this enhanced protocol simplifies the integrated integrity verification process, eliminating the need for extra shares. A standout feature is its capability to securely transmit meaningful shares for color images without compromising the quality of the reconstructed image as the PSNR maintains to be ∞. Experimental findings substantiate the protocol's resilience against quality degradation and its effectiveness in verifying the authenticity of the reconstructed image. This innovative approach holds promise for a wide array of applications, notably in sectors requiring secure document transmission, such as Logistics and Supply Chain Management, E-Governance, Medical and Military Applications.

Theoretical study and circuit implementation of three chain-coupled self-driven Duffing oscillators

In this paper, we describe the scenario from the birth of oscillations to multi-spiral chaos in a novel system composed of three chain-coupled self-driven Duffing oscillators. Eight of the equilibrium points develop (multiple) Hopf bifurcation when varying a parameter (e.g., coupling coefficient). Considering the computer integration of the state equations, the combined exploitation of Lyapunov exponent plots, bifurcation diagrams, basins of attraction, and phase portraits, unusual and attractive features were highlighted including the coexistence of eight bifurcation branches, Hopf bifurcations, a multitude of coexisting types of oscillations and a six-spiral chaotic attractor, just to cite a few. Using basic electronic components, the electronic circuit of the three chain-coupled Duffing oscillator system is performed. Orcad-PSpice simulated dynamics of the proposed chain-coupled analog circuit confirm the theoretically disclosed features. Moreover, the practical feasibility of the coupled system is demonstrated by considering microcontroller-based hardware realization.

Fractal-Based Hybrid Cryptosystem: Enhancing Image Encryption with RSA, Homomorphic Encryption, and Chaotic Maps

Protecting digital data, especially digital images, from unauthorized access and malicious activities is crucial in today's digital era. This paper introduces a novel approach to enhance image encryption by combining the strengths of the RSA algorithm, homomorphic encryption, and chaotic maps, specifically the sine and logistic map, alongside the self-similar properties of the fractal Sierpinski triangle. The proposed fractal-based hybrid cryptosystem leverages Paillier encryption for maintaining security and privacy, while the chaotic maps introduce randomness, periodicity, and robustness. Simultaneously, the fractal Sierpinski triangle generates intricate shapes at different scales, resulting in a substantially expanded key space and heightened sensitivity through randomly selected initial points. The secret keys derived from the chaotic maps and Sierpinski triangle are employed for image encryption. The proposed scheme offers simplicity, efficiency, and robust security, effectively safeguarding against statistical, differential, and brute-force attacks. Through comprehensive experimental evaluations, we demonstrate the superior performance of the proposed scheme compared to existing methods in terms of both security and efficiency. This paper makes a significant contribution to the field of digital image encryption, paving the way for further exploration and optimization in the future.

Star Memristive Neural Network: Dynamics Analysis, Circuit Implementation, and Application in a Color Cryptosystem

At present, memristive neural networks with various topological structures have been widely studied. However, the memristive neural network with a star structure has not been investigated yet. In order to investigate the dynamic characteristics of neural networks with a star structure, a star memristive neural network (SMNN) model is proposed in this paper. Firstly, an SMNN model is proposed based on a Hopfield neural network and a flux-controlled memristor. Then, its chaotic dynamics are analyzed by using numerical analysis methods including bifurcation diagrams, Lyapunov exponents, phase plots, Poincaré maps, and basins of attraction. The results show that the SMNN can generate complex dynamical behaviors such as chaos, multi-scroll attractors, and initial boosting behavior. The number of multi-scroll attractors can be changed by adjusting the memristor's control parameters. And the position of the coexisting chaotic attractors can be changed by switching the memristor's initial values. Meanwhile, the analog circuit of the SMNN is designed and implemented. The theoretical and numerical results are verified through MULTISIM simulation results. Finally, a color image encryption scheme is designed based on the SMNN. Security performance analysis shows that the designed cryptosystem has good security.

Simple cyclic memristive neural networks with coexisting attractors and large-scale amplitude control

The memristor's unique memory function and non-volatile nature make it an ideal electronic bionic device for artificial neural synapses. This paper aims to construct a class of memristive neural networks (MNNs) with a simple circular connection relationship and complex dynamics by introducing a generic memristor as synapse. For placing the memristive synapse in different coupling positions, three MNNs with the same coupling cyclic connection are yielded. One remarkable feature of the proposed MNNs is that they can yield complex dynamics, in particular, abundant coexisting attractors and large-scale parameter-relied amplitude control, by comparing with some existing MNNs. Taking one of the MNNs as an example, the complex dynamics (including chaos, period-doubling bifurcation, symmetric coexisting attractors, large-scale amplitude control) and circuit implementation are studied . The number of equilibria and their stabilities are discussed. The parameter-relied dynamic evolution and the coexisting attractors are numerically shown by using bifurcations and phase portraits. A microcontroller-based hardware circuit is given to realize the network, which verifies the correctness of the numerical results and experimental results.

Robust Multi-Mode Synchronization of Chaotic Fractional Order Systems in the Presence of Disturbance, Time Delay and Uncertainty with Application in Secure Communications

This paper investigates the robust adaptive synchronization of multi-mode fractional-order chaotic systems (MMFOCS). To that end, synchronization was performed with unknown parameters, unknown time delays, the presence of disturbance, and uncertainty with the unknown boundary. The convergence of the synchronization error to zero was guaranteed using the Lyapunov function. Additionally, the control rules were extracted as explicit continuous functions. An image encryption approach was proposed based on maps with time-dependent coding for secure communication. The simulations indicated the effectiveness of the proposed design regarding the suitability of the parameters, the convergence of errors, and robustness. Subsequently, the presented method was applied to fractional-order Chen systems and was encrypted using the chaotic masking of different benchmark images. The results indicated the desirable performance of the proposed method in encrypting the benchmark images.

Efficient Colour Image Encryption Algorithm Using a New Fractional-Order Memcapacitive Hyperchaotic System

In comparison with integer-order chaotic systems, fractional-order chaotic systems exhibit more complex dynamics. In recent years, research into fractional chaotic systems for the utilization of image cryptosystems has become increasingly highlighted. This paper describes the development, testing, numerical analysis, and electronic realization of a fractional-order memcapacitor. Then, a new four-dimensional (4D) fractional-order memcapacitive hyperchaotic system is suggested based on this memcapacitor. Analytically and numerically, the nonlinear dynamic properties of the hyperchaotic system have been explored, where various methods, including equilibrium points, phase portraits of chaotic attractors, bifurcation diagrams, and the Lyapunov exponent, are considered to demonstrate the chaos behaviour of this new hyperchaotic system. Consequently, an encryption cryptosystem algorithm is used for colour image encryption based on the chaotic behaviour of the memcapacitive model, where every pixel value of the original image is incorporated in the secret key to strengthen the encryption algorithm pirate anti-attack robustness. For generating the keyspace of that employed cryptosystem, the initial condition values, parameters, and fractional-order derivative value(s) (q) of the memcapacitive chaotic system are utilized. The common cryptanalysis metrics are verified in detail by histogram, keyspace, key sensitivity, correlation coefficient values, entropy, time efficiency, and comparisons with other recent related fieldwork in order to demonstrate the security level of the proposed cryptosystem approach. Finally, images of various sizes were encrypted and recovered to ensure that the utilized cryptosystem approach is capable of encrypting/decrypting images of various sizes. The obtained experimental results and security metrics analyses illustrate the excellent accuracy, high security, and perfect time efficiency of the utilized cryptosystem, which is highly resistant to various forms of pirate attacks.