Hopfield-type Network Research Articles

Study of load distribution in fully connected client server network using feedback neural network architecture

A fully connected client server network reminiscent of a flat group (analogous to a Hopfield type artificial neural network), consisting of a few clients is considered herein. Distinct client nodes are inter-connected in fully connected architecture as in Hopfield’s associative memory model. Weights of connections are symmetric and the states of the clients are bipolar. Stability of such a network can be obtained by using energy function analysis of Hopfield type network. The problem to explore the condition for stability becomes complicated when the server gets connected with the clients, as the connections of the server with the clients are bidirectional but are not symmetric. Hence it does not anymore satisfy the criteria of a Hopfield network to act as an associative memory. In this case the stability is not fixed point stability; rather it is in the form of chaotic or oscillatory states. Under these conditions it is tedious to establish the stability for the entire network in order to determine the optimal load distribution as reflected from connection strengths of the network. This paper explores the stability of clients’ network and subsequently the stability of the entire network after adding the server to investigate correlation. Key words: Artificial neural network, Hopfield energy function, optimized stability, client-server network.

Pattern recall analysis of the Hopfield neural network with a genetic algorithm

This paper describes the implementation of a genetic algorithm to evolve the population of weight matrices for storing and recalling the patterns in a Hopfield type neural network model. In the Hopfield type neural network of associative memory, the appropriate arrangement of synaptic weights provides an associative function in the network. The energy function associated with the stable state of this model represents the appropriate storage of the input patterns. The aim is to obtain the optimal weight matrix for efficient recall of any prototype input pattern. For this, we explore the population generation technique (mutation and elitism), crossover and the fitness evaluation function for generating the new population of the weight matrices. This process continues until the selection of the last weight matrix or matrices has been performed. The experiments incorporate a neural network trained with multiple numbers of patterns using the Hebbian learning rule. In most cases, the recalling of patterns using a genetic algorithm seems to give better results than the conventional recalling with the Hebbian rule. The simulated results suggest that the genetic algorithm is the better searching technique for recalling noisy prototype input patterns.

New LMI-based criteria for global robust stability of delayed neural networks

Some novel, linear matrix inequality based, criteria for the uniqueness and global robust stability of the equilibrium point of Hopfield-type neural networks with delay are presented. A comparison of the present criteria with the previous criteria is made.

Improved global robust stability criterion for interval delayed neural networks

A novel criterion for the global robust stability of Hopfield-type interval neural networks with delay is presented. Some existing criteria turn out to be special cases of the present criterion. An example shows the effectiveness of the present criterion.

Modified criteria for global robust stability of interval delayed neural networks

Two simple criteria for global robust stability of Hopfield-type interval neural networks with delay are presented. The criteria turn out to be modified versions of an earlier criterion due to Cao, Huang, and Qu. Examples show the effectiveness of the modified criteria. Numerical simulations are carried out to confirm the applicability of the modified criteria.

Impulsive Hopfield-type neural network system with piecewise constant argument

In this paper we introduce an impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Sufficient conditions for the existence of the unique equilibrium are obtained. Existence and uniqueness of solutions of such systems are established. Stability criterion based on linear approximation is proposed. Some sufficient conditions for the existence and stability of periodic solutions are derived. An example with numerical simulations is given to illustrate our results.

Performance Analysis of Gradient Neural Network Exploited for Online Time-Varying Matrix Inversion

This technical note presents theoretical analysis and simulation results on the performance of a classic gradient neural network (GNN), which was designed originally for constant matrix inversion but is now exploited for time-varying matrix inversion. Compared to the constant matrix-inversion case, the gradient neural network inverting a time-varying matrix could only approximately approach its time-varying theoretical inverse, instead of converging exactly. In other words, the steady-state error between the GNN solution and the theoretical/exact inverse does not vanish to zero. In this technical note, the upper bound of such an error is estimated firstly. The global exponential convergence rate is then analyzed for such a Hopfield-type neural network when approaching the bound error. Computer-simulation results finally substantiate the performance analysis of this gradient neural network exploited to invert online time-varying matrices.

Delay-Dependent Exponential Stability for Uncertain Stochastic Hopfield Neural Networks With Time-Varying Delays

This paper provides new delay-dependent conditions that guarantee the robust exponential stability of stochastic Hopfield type neural networks with time-varying delays and parameter uncertainties. Both the cases of the time-varying delays which are differentiable and may not be differentiable are considered. The stability conditions are derived by using the recently developed free-weighting matrices technique and expressed in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness of the proposed stability criteria. It is shown that the proposed stability results are less conservative than some previous ones in the literature.

Asymptotic stability of impulsive high-order Hopfield type neural networks

In this paper, we discuss impulsive high-order Hopfield type neural networks. Investigating their global asymptotic stability, by using Lyapunov function method, sufficient conditions that guarantee global asymptotic stability of networks are given. These criteria can be used to analyse the dynamics of biological neural systems or to design globally stable artificial neural networks. Two numerical examples are given to illustrate the effectiveness of the proposed method.

Global exponential stability of impulsive high-order Hopfield type neural networks with delays

In this paper, we investigate the global exponential stability of impulsive high-order Hopfield type neural networks with delays. By establishing the impulsive delay differential inequalities and using the Lyapunov method, two sufficient conditions that guarantee global exponential stability of these networks are given, and the exponential convergence rate is also obtained. A numerical example is given to demonstrate the validity of the results.

New neural network algorithm for image reconstruction from fan-beam projections

Neural networks have some applications in computerized tomography, in particular to reconstruct an image from projections. The presented paper describes a new practical approach to the reconstruction problem using a Hopfield-type neural network. The methodology of this reconstruction algorithm resembles a transformation formula—the so-called ρ-filtered layergram method. The method proposed in this work is adapted for discrete fan beam projections, already used in practice. Performed computer simulations show that the neural network reconstruction algorithm designed to work in this way outperforms conventional methods in obtained image quality, and in perspective of hardware implementation in the speed of the reconstruction process.

Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture

This paper is devoted to the analysis of a discrete-time-delayed Hopfield-type neural network of p neurons with ring architecture. The stability domain of the null solution is found, the values of the characteristic parameter for which bifurcations occur at the origin are identified and the existence of Fold/Cusp, Neimark-Sacker and Flip bifurcations is proved. These bifurcations are analyzed by applying the center manifold theorem and the normal form theory. It is proved that resonant 1:3 and 1:4 bifurcations may also be present. It is shown that the dynamics in a neighborhood of the null solution become more and more complex as the characteristic parameter grows in magnitude and passes through the bifurcation values. A theoretical proof is given for the occurrence of Marotto's chaotic behavior, if the magnitudes of the interconnection coefficients are large enough and at least one of the activation functions has two simple real roots.

Formula omitted]-matrix structure and harmless delays in a Hopfield-type neural network

Motivated by work in [W. Ma, T. Hara, Y. Takeuchi, Stability of a 2-dimensional neural network with time delays, J. Biol. Syst. 8 (2000) 177–193; S.A. Campbell, Delay independent stability for additive neural networks, in: New Millennium Special Issue on Neural Networks and Neurocomputing—Theory, Models and Applications Part I, Differential Equations Dynam. Syst. 9 (2001) 115–138; S. Zhang, W. Ma, Y. Kuang, Necessary and sufficient conditions for global attractivity of Hopfield-type neural networks with time delays, Rocky Mountain J. Math. 38 (2008) 1829–1840], this work gives some necessary and sufficient conditions for global attractivity of an n -dimensional Hopfield-type neural network model with time delays based on the M -matrix and a class of nonlinear algebra equations with n variables.

Limit set trichotomy and dichotomy for skew-product semiflows with applications

The limit set trichotomy and dichotomy for skew-product semiflows are established under suitable monotone and concave assumptions. As an application, a delayed Hopfield-type neural network model is considered.

A new approach for estimating the attraction domain for Hopfield-type neural networks.

In this letter, using methods proposed by E. Kaslik, St. Balint, and their colleagues, we develop a new method, expansion approach, for estimating the attraction domain of asymptotically stable equilibrium points of Hopfield-type neural networks. We prove theoretically and demonstrate numerically that the proposed approach is feasible and efficient. The numerical results that obtained in the application examples, including the network system considered by E. Kaslik, L. Brăescu, and St. Balint, indicate that the proposed approach is able to achieve better attraction domain estimation.

A New Approach for Estimating the Attraction Domain for Hopfield-Type Neural Networks

In this letter, using methods proposed by E. Kaslik, St. Balint, and their colleagues, we develop a new method, expansion approach, for estimating the attraction domain of asymptotically stable equilibrium points of Hopfield-type neural networks. We prove theoretically and demonstrate numerically that the proposed approach is feasible and efficient. The numerical results that obtained in the application examples, including the network system considered by E. Kaslik, L. Brăescu, and St. Balint, indicate that the proposed approach is able to achieve better attraction domain estimation.

Necessary and Sufficient Conditions for Global Attractivity of Hopfield-Type Neural Networks with Time Delays

Necessary and Sufficient Conditions for Global Attractivity of Hopfield-Type Neural Networks with Time Delays

Convergent dynamics for multistable delayed neural networks

This investigation aims at developing a methodology to establish convergence of dynamics for delayed neural network systems with multiple stable equilibria. The present approach is general and can be applied to several network models. We take the Hopfield-type neural networks with both instantaneous and delayed feedbacks to illustrate the idea. We shall construct the complete dynamical scenario which comprises exactly 2n stable equilibria and exactly (3n − 2n) unstable equilibria for the n-neuron network. In addition, it is shown that every solution of the system converges to one of the equilibria as time tends to infinity. The approach is based on employing the geometrical structure of the network system. Positively invariant sets and componentwise dynamical properties are derived under the geometrical configuration. An iteration scheme is subsequently designed to confirm the convergence of dynamics for the system. Two examples with numerical simulations are arranged to illustrate the present theory.

A 2D approach to tomographic image reconstruction using a Hopfield-type neural network

In this paper a new approach to tomographic image reconstruction from projections is developed and investigated. To solve the reconstruction problem a special neural network which resembles a Hopfield net is proposed. The reconstruction process is performed during the minimizing of the energy function in this network. To improve the performance of the reconstruction process an entropy term is incorporated into energy expression. The approach presented in this paper significantly decreases the complexity of the reconstruction problem.

Comment on “Existence and global exponential stability of periodic solution for delayed high-order Hopfield-type neural networks” [Phys. Lett. A 352 (2006) 341

Comment on “Existence and global exponential stability of periodic solution for delayed high-order Hopfield-type neural networks” [Phys. Lett. A 352 (2006) 341