Dynamics Of Hopfield Neural Networks Research Articles

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.

Memristive Hopfield neural network dynamics with heterogeneous activation functions and its application

Activation functions play a crucial in emulating biological neurons within artificial neural networks. However, the exploration of neural networks composed of various activation functions and their associated dynamics have not been noticed yet. This paper proposes a novel method by introducing heterogeneous activation functions into a memristive Hopfield neural network for the first time. The special feature of the proposed model lies not only in its ability to mimic the diversity of brain neurons, providing a more realistic and adaptable frame for artificial neural networks but also in its rich dynamic properties suitable for engineering applications. Theoretical and experimental investigations into the dynamics of the memristive Hopfield neural network are conducted, employing phase portraits, bifurcation diagrams, Lyapunov exponent spectra, 0–1 tests, and bi-parameter dynamic maps. Complex dynamical behaviors, including periodic bursting, chaotic bursting, and chaotic state jump are revealed by the numerical simulations. Furthermore, a hardware implementation of the proposed neural network is designed and validated through circuit simulation software, which is consistent with the numerical simulation and confirms the validity of the proposed model. Finally, an encryption scheme based on the chaotic bursting is also proposed and evaluated. Results demonstrate that the chaotic bursting attractor exhibits excellent randomness, making it well-suited for image encryption applications. The novel exploration of heterogeneous activation neuronal networks in this paper may pave the way for further research in the field of more bionic networks with complex dynamical behaviors and their applications.

Numerical investigation and deep learning approach for fractal–fractional order dynamics of Hopfield neural network model

This paper investigates the dynamics of Hopfield neural networks involving fractal–fractional derivatives. The incorporation of fractal–fractional derivatives in the neural network framework brings forth novel modeling capabilities, capturing nonlocal dependencies, complex scaling behaviors, and memory effects. The aim of this study is to provide a comprehensive analysis of the dynamics of Hopfield neural networks with fractal–fractional derivatives, including the existence and uniqueness of solutions, stability properties, and numerical analysis techniques. Numerical analysis techniques, including the Adams–Bashforth method, are employed to accurately simulate the fractal–fractional Hopfield neural network system. Moreover, the obtained numerical data serves as validation for developing predictions using Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) neural network methods. The findings contribute to the advancement of both fractional calculus and neural network theory, providing valuable insights for theoretical investigations and practical applications in complex systems analysis.

Five-dimensional memristive Hopfield neural network dynamics analysis and its application in secure communication

PurposeThis study aims to improve the complexity of chaotic systems and the security accuracy of information encrypted transmission. Applying five-dimensional memristive Hopfield neural network (5D-HNN) to secure communication will greatly improve the confidentiality of signal transmission and greatly enhance the anticracking ability of the system.Design/methodology/approachChaos masking: Chaos masking is the process of superimposing a message signal directly into a chaotic signal and masking the signal using the randomness of the chaotic output. Synchronous coupling: The coupled synchronization method first replicates the drive system to get the response system, and then adds the appropriate coupling term between the drive The synchronization error and the coupling term of the system will eventually converge to zero with time. The synchronization error and coupling term of the system will eventually converge to zero over time.FindingsA 5D memristive neural network is obtained based on the original four-dimensional memristive neural network through the feedback control method. The system has five equations and contains infinite balance points. Compared with other systems, the 5D-HNN has rich dynamic behaviors, and the most unique feature is that it has multistable characteristics. First, its dissipation property, equilibrium point stability, bifurcation graph and Lyapunov exponent spectrum are analyzed to verify its chaotic state, and the system characteristics are more complex. Different dynamic characteristics can be obtained by adjusting the parameter k.Originality/valueA new 5D memristive HNN is proposed and used in the secure communication

Dynamics study on the effect of memristive autapse distribution on Hopfield neural network.

As the shortest feedback loop of the nervous system, autapse plays an important role in the mode conversion of neurodynamics. In particular, memristive autapses can not only facilitate the adjustment of the dynamical behavior but also enhance the complexity of the nervous system, in view of the fact that the dynamics of the Hopfield neural network has not been investigated and studied in detail from the perspective of memristive autapse. Based on the traditional Hopfield neural network, this paper uses a locally active memristor to replace the ordinary resistive autapse so as to construct a 2 n-dimensional memristive autaptic Hopfield neural network model. The boundedness of the model is proved by introducing the Lyapunov function and the stability of the equilibrium point is analyzed by deriving the Jacobian matrix. In addition, four scenarios are established on a small Hopfield neural network with three neurons, and the influence of the distribution of memristive autapses on the dynamics of this small Hopfield neural network is described by numerical simulation tools. Finally, the Hopfield neural network model in these four situations is designed and implemented on field-programmable gate array by using the fourth-order Runge-Kutta method, which effectively verifies the numerical simulation results.

Nonlinear Dynamics of Three-Neurons-Based Hopfield Neural Networks (HNNs): Remerging Feigenbaum Trees, Coexisting Bifurcations and Multiple Attractors

In this work, the dynamics of a simplified model of three-neurons-based Hopfield neural networks (HNNs) is investigated. The simplified model is obtained by removing the synaptic weight connection of the third and second neuron in the original Hopfield networks introduced in Ref. 11 . The investigations have shown that the simplified model possesses three equilibrium points among which origin of the systems coordinates. It is found that the origin is always unstable while the symmetric pair of fixed points with conditional stability has values depending on synaptic weight between the second and the first neuron that is used as bifurcation control parameter. Numerical simulations, carried out in terms of bifurcation diagrams, graph of Lyapunov exponents, phase portraits, Poincaré section, time series and frequency spectra are employed to highlight the complex dynamical behaviors exhibited by the model. The results indicate that the modified model of HNNs exhibits rich nonlinear dynamical behaviors including symmetry breaking, chaos, periodic window, antimonotonicity (i.e., concurrent creation and annihilation of periodic orbits) and coexisting self-excited attractors (e.g., coexistence of two, four and six disconnected periodic and chaotic attractors) which have not been reported in previous works focused on the dynamics of HNNs. Finally, PSpice simulations verify the results of theoretical analyses of the simplified model of three-neurons-based HNNs.

On the Dynamics of Hopfield Neural Networks on Unit Quaternions.

In this paper, we first address the dynamics of the elegant multivalued quaternionic Hopfield neural network (MV-QHNN) proposed by Minemoto et al. Contrary to what was expected, we show that the MV-QHNN, as well as one of its variation, does not always come to rest at an equilibrium state under the usual conditions. In fact, we provide simple examples in which the network yields a periodic sequence of quaternionic state vectors. Afterward, we turn our attention to the continuous-valued quaternionic Hopfield neural network (CV-QHNN), which can be derived from the MV-QHNN by means of a limit process. The CV-QHNN can be implemented more easily than the MV-QHNN model. Furthermore, the asynchronous CV-QHNN always settles down into an equilibrium state under the usual conditions. Theoretical issues are all illustrated by examples in this paper.

Parallel and distributed computations of maximum independent set by a Hopfield neural net embedded into a wireless sensor network

This paper, as the first one in a three-paper sequence, presents a proposed framework to employ a wireless sensor network as a hardware computation platform for fully parallel and distributed computation of maximum independent set of a given graph through a Hopfield neural network. Theoretical and mathematical foundations of the proposed framework will be discussed. Mapping the maximum independent set problem to Hopfield neural network dynamics is presented. This is followed by the demonstration of embedding the Hopfield neural network as a static optimizer into the wireless sensor network in fully parallel and distributed mode. The outcome is a wireless sensor network operating as a parallel and distributed computing hardware platform for a Hopfield neural network configured to solve a static optimization problem. The nesC-TinyOS model of the proposed computational framework and the corresponding simulation study are deferred to the second and third papers, respectively, in the three-paper sequence.

On stability of disturbed Hopfield neural networks with time delays

In real application, the dynamics of Hopfield neural network is often affected by disturbing signals and time delays, so it is worthwhile to study dynamical properties of this type of neural network. Firstly, the ideal solution is defined as the solution of the network without disturbing signals. In order to ensure uniqueness, L 2-gain stability, global stability or global exponential stability of the ideal solution, corresponding sufficient conditions are presented, respectively, using homotopic method, inequality techniques, M-matrix properties or one time-delay inequality. All the obtained results are illustrated by several simulations.

Global dynamics of Hopfield neural networks involving variable delays

In this article, a model describing dynamics of Hopfield neural networks involving variable delays is considered. Existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established. Further, we derive sufficient criteria of global asymptotic stability (GAS) of the equilibrium point.

Dynamics of neural networks relevant to properties of proteins

We studied how the dynamics of Hopfield neural networks depend on computational and physical properties of the network. The dynamics of the network was characterized by the distribution of first passage times (FPT) between the states. The FPT distributions depended on the updating scheme, temperature, connectivity range, and number of stored memories. The FTP distributions were different for synchronous and asynchronous updating, and were more physically consistent for the synchronous than for the asynchronous updating scheme. Neural networks and proteins share common features such as many degrees of freedom, conflicting constraints on energy minimization, and energy functions with many local minima. Thus the general lessons learned here on how the dynamics of neural networks depends on their physical properties may be relevant in understanding how the dynamics of proteins is influenced by similar physical properties.