Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach

For neural networks with constant or time-varying delays, the problems of determining the exponential stability and estimating the exponential convergence rate are studied in this paper. An approach combining the Lyapunov-Krasovskii functionals with the linear matrix inequality is taken to investigate the problems, which provide bounds on the interconnection matrix and the activation functions, so as to guarantee the systems' exponential stability. Some criteria for the exponentially stability, which give information on the delay-dependence property, are derived. The results obtained in this paper provide one more set of easily verified guidelines for determining the exponentially stability of delayed neural networks, which are less conservative and less restrictive than the ones reported so far in the literature.

Dynamic stability conditions for Lotka-Volterra recurrent neural networks with delays

The Lotka-Volterra model of neural networks, derived from the membrane dynamics of competing neurons, have found successful applications in many "winner-take-all" types of problems. This paper studies the dynamic stability properties of general Lotka-Volterra recurrent neural networks with delays. Conditions for nondivergence of the neural networks are derived. These conditions are based on local inhibition of networks, thereby allowing these networks to possess a multistability property. Multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process. Under these nondivergence conditions, a compact set that globally attracts all the trajectories of a network can be computed explicitly. If the connection weight matrix of a network is symmetric in some sense, and the delays of the network are in L2 space, we can prove that the network will have the property of complete stability.

Stability criteria for delayed neural networks

In this paper, delay-independent global asymptotic and exponential stability for a class of delayed neural networks (DNN's) is investigated, and some criteria are established to ensure stability of DNN's by applying the Lyapunov direct method. These criteria are expressed by imposing constraints on weight matrices of the networks, and they are easy to verify and so are applicable in the design of DNN's. Comparisons between our criteria and some earlier results are also made; it is shown that our results generalize some existing criteria in the literature.

Probabilistic computation by neuromine networks

In this paper, we address the question, can biologically feasible neural nets compute more than can be computed by deterministic polynomial time algorithms? Since we want to maintain a claim of plausibility and reasonableness we restrict ourselves to algorithmically easy to construct nets and we rule out infinite precision in parameters and in any analog parts of the computation. Our approach is to consider the recent advances in randomized algorithms and see if such randomized computations can be described by neural nets. We start with a pair of neurons and show that by connecting them with reciprocal inhibition and some tonic input, then the steady-state will be one neuron ON and one neuron OFF, but which neuron will be ON and which neuron will be OFF will be chosen at random (perhaps, it would be better to say that microscopic noise in the analog computation will be turned into a megascale random bit). We then show that we can build a small network that uses this random bit process to generate repeatedly random bits. This random bit generator can then be connected with a neural net representing the deterministic part of randomized algorithm. We, therefore, demonstrate that these neural nets can carry out probabilistic computation and thus be less limited than classical neural nets.

Results concerning the absolute stability of delayed neural networks

We report on results concerning the global asymptotic stability (GAS) and absolute stability (ABST) of delay models of continuous-time neural networks. These results present sufficient conditions for GAS and in case the network has instantaneous signalling as well as delay signalling (for example, a delayed cellular neural network (DCNN)), are milder than previously known criteria; they apply to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. We are therefore able to interpret the results as guarantees of absolute stability of the network with respect to the wide class of admissible activation functions. Furthermore, these results do not assume symmetry of the connection matrices. We also present a sufficient condition for absolute stability in the presence of nonconstant delays.

Periodic solutions and exponential stability in delayed cellular neural networks

Some simple sufficient conditions are given ensuring global exponential stability and the existence of periodic solutions of delayed cellular neural networks (DCNNs) by constructing suitable Lyapunov functionals and some analysis techniques. These conditions are easy to check in terms of system parameters and have important leading significance in the design and applications of globally stable DCNNs and periodic oscillatory DCNNs. In addition, two examples are given to illustrate the theory.

Asymptotic behavior of irreducible excitatory networks of analog graded-response neurons

In irreducible excitatory networks of analog graded-response neurons, the trajectories of most solutions tend to the equilibria. We derive sufficient conditions for such networks to be globally asymptotically stable. When the network possesses several locally stable equilibria, their location in the phase space is discussed and a description of their attraction basin is given. The results hold even when interunit transmission is delayed.

Effect of delay on the boundary of the basin of attraction in a system of two neurons

The behavior of neural networks may be influenced by transmission delays and many studies have derived constraints on parameters such as connection weights and output functions which ensure that the asymptotic dynamics of a network with delay remains similar to that of the corresponding system without delay. However, even when the delay does not affect the asymptotic behavior of the system, it may influence other important features in the system's dynamics such as the boundary of the basin of attraction of the stable equilibria. In order to better understand such effects, we study the dynamics of a system constituted by two neurons interconnected through delayed excitatory connections. We show that the system with delay has exactly the same stable equilibrium points as the associated system without delay, and that, in both the network with delay and the corresponding one without delay, most trajectories converge to these stable equilibria. Thus, the asymptotic behavior of the network with delay and that of the corresponding system without delay are similar. We obtain a theoretical characterization of the boundary separating the basins of attraction of two stable equilibria, which enables us to estimate the boundary. Our numerical investigations show that, even in this simple system, the boundary separting the basins of attraction of two stable equilibrium points depends on the value of the delays. The extension of these results to networks with an arbritrary number of units is discussed.

Effect of delay on the boundary of the basin of attraction in a self-excited single graded-response neuron

Little attention has been paid in the past to the effects of interunit transmission delays (representing axonal and synaptic delays) on the boundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the "location" of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria. The dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.

Optimal solutions for cellular neural networks by paralleled hardware annealing

An engineering annealing method for optimal solutions of cellular neural networks is presented. Cellular neural networks are very promising in solving many scientific problems in image processing, pattern recognition, and optimization by the use of stored program with predetermined templates. Hardware annealing, which is a paralleled version of mean-field annealing in analog networks, is a highly efficient method of finding optimal solutions of cellular neural networks. It does not require any stochastic procedure and henceforth can be very fast. The generalized energy function of the network is first increased by reducing the voltage gain of each neuron. Then, the hardware annealing searches for the globally minimum energy state by continuously increasing the gain of neurons. The process of global optimization by the proposed annealing can be described by the eigenvalue problems in the time-varying dynamic system. In typical nonoptimization problems, it also provides enough stimulation to frozen neurons caused by ill-conditioned initial states.