Discrete-time Neural Network Model Research Articles

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).

Global exponential stability of discrete-time Hopfield neural network models with unbounded delays

In this paper, a general setting is presented to study the exponential stability of discrete-time systems with bounded or unbounded delays. Based on the M-matrix theory, we establish sufficient conditions to ensure the global exponential stability of the zero equilibrium of low-order, and high-order, discrete-time Hopfield neural network models with unbounded delays and delay in the leakage terms. A comparison of the literature shows that our results generalize and improve some in recent publications.

Multistability of Delayed Hybrid Impulsive Neural Networks With Application to Associative Memories.

The important topic of multistability of continuous-and discrete-time neural network (NN) models has been investigated rather extensively. Concerning the design of associative memories, multistability of delayed hybrid NNs is studied in this paper with an emphasis on the impulse effects. Arising from the spiking phenomenon in biological networks, impulsive NNs provide an efficient model for synaptic interconnections among neurons. Using state-space decomposition, the coexistence of multiple equilibria of hybrid impulsive NNs is analyzed. Multistability criteria are then established regrading delayed hybrid impulsive neurodynamics, for which both the impulse effects on the convergence rate and the basins of attraction of the equilibria are discussed. Illustrative examples are given to verify the theoretical results and demonstrate an application to the design of associative memories. It is shown by an experimental example that delayed hybrid impulsive NNs have the advantages of high storage capacity and high fault tolerance when used for associative memories.

A continuous-time neural model for sequential action.

Action selection, planning and execution are continuous processes that evolve over time, responding to perceptual feedback as well as evolving top-down constraints. Existing models of routine sequential action (e.g. coffee- or pancake-making) generally fall into one of two classes: hierarchical models that include hand-built task representations, or heterarchical models that must learn to represent hierarchy via temporal context, but thus far lack goal-orientedness. We present a biologically motivated model of the latter class that, because it is situated in the Leabra neural architecture, affords an opportunity to include both unsupervised and goal-directed learning mechanisms. Moreover, we embed this neurocomputational model in the theoretical framework of the theory of event coding (TEC), which posits that actions and perceptions share a common representation with bidirectional associations between the two. Thus, in this view, not only does perception select actions (along with task context), but actions are also used to generate perceptions (i.e. intended effects). We propose a neural model that implements TEC to carry out sequential action control in hierarchically structured tasks such as coffee-making. Unlike traditional feedforward discrete-time neural network models, which use static percepts to generate static outputs, our biological model accepts continuous-time inputs and likewise generates non-stationary outputs, making short-timescale dynamic predictions.

A discrete time neural network model with spiking neurons: II: Dynamics with noise

We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky Integrate-and-Fire neurons, where time is discrete and where neurons are submitted to noise, without restriction on the synaptic weights. We show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics.

A discrete time neural network model with spiking neurons

We derive rigorous results describing the asymptotic dynamics of a discrete time model of spiking neurons introduced in Soula et al. (Neural Comput. 18, 1, 2006). Using symbolic dynamic techniques we show how the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns ("raster plots"). Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of a sharp threshold in the model definition. As a consequence, the model exhibits a dynamical regime indistinguishable from chaos in numerical experiments.

Revisiting time discretisation of spiking network models

A link is built between a biologically plausible generalized integrate and fire (GIF) neuron model with conductance-based dynamics [1] and a discrete time neural network model with spiking neurons [2], for which rigorous results on the spontaneous dynamics has been obtained. More precisely the following has been shown. i) Occurrence of periodic orbits is the generic regime of activity, with a bounded period in the presence of spike-time dependence plasticity, and arbitrary large periods at the edge of chaos (such regime is indistinguishable from chaos in numerical experiments, explaining what is obtained in [2]), ii) the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns (raster plots). This allows a better insight into the possible neural coding in such a network and provides a deep understanding, at the network level, of the system behavior. Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of the sharp spiking threshold [3]. A step further, constructive conditions are derived, allowing to properly implement visual functions on such networks [4]. The time discretisation has been carefully conducted avoiding usual bias induced by e.g. Euler methods and taking into account a rather complex GIF model for which the usual arbitrary discontinuities are discussed in detail. The effects of the discretisation approximation have been analytically and experimentally analyzed, in detail. Figure 1 A view of the numerical experiments software platform raster-plot output, considering either a generic fully connected network or, here, a retinotopic network related to visual functions (top-left: 2D instantaneous spiking activity).

A discrete-time neural network for solving nonlinear convex problems with hybrid constraints

This paper investigates a discrete-time neural network model for solving nonlinear convex programming problems with hybrid constraints. The neural network finds the solution of both primal and dual problems and converges to the corresponding exact solution globally. We prove here that the proposed neural network is globally exponentially stable. Furthermore, we extend the proposed neural network for solving a class of monotone variational inequality problems with hybrid constraints. Compared with other existing neural networks for solving such problems, the proposed neural network has a low complexity for implementation without a penalty parameter and converge an exact solution to convex problem with hybrid constraints. Some numerical simulations for justifying the theoretical analysis are also given. The numerical simulations are shown that in the new model note only the cost of the hardware implementation is not relatively expensive, but also accuracy of the solution is greatly good.

Behavior of solutions for a class of difference systems

In this paper, the discrete-time neural network model of two neurons with piecewise constant argument is considered. Some sufficient conditions under which every solution is either periodic or convergent are obtained.

Periodic oscillation for discrete-time Hopfield neural networks

By using the continuation theorem of coincidence degree theory, we obtain some sufficient conditions for the existence of periodic solutions of discrete-time Hopfield neural network model. Our results can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Moreover, these conclusions are presented in terms of system parameters and can be easily verified.

Recurrent neural networks for solving linear inequalities and equations

This paper presents two types of recurrent neural networks, continuous-time and discrete-time ones, for solving linear inequality and equality systems. In addition to the basic continuous-time and discrete-time neural-network models, two improved discrete-time neural networks with faster convergence rate are proposed by use of scaling techniques. The proposed neural networks can solve a linear inequality and equality system, can solve a linear program and its dual simultaneously, and thus extend and modify existing neural networks for solving linear equations or inequalities. Rigorous proofs on the global convergence of the proposed neural networks are given. Digital realization of the proposed recurrent neural networks are also discussed.

Robustness analysis of a class of discrete-time recurrent neural networks under perturbations

A robustness analysis is conducted for a large class of discrete-time recurrent neural networks for associative memories under perturbations of system parameters. The present paper aims to give an answer to the following question. Given a discrete-time neural network with specified stable memories (specified asymptotically stable equilibria), under what conditions will a perturbed model of the discrete-time neural network possess stable memories that are close (in distance) to the stable memories of the unperturbed discrete-time neural network model? Robustness stability results for perturbed discrete-time neural network models are established and conditions are obtained for the existence of asymptotically stable equilibria of the perturbed discrete-time neural network models which are near the asymptotically stable equilibria of the original unperturbed neural networks. In the present results, quantitative estimates (explicit estimates of bounds) are given for the distance between the corresponding equilibrium points of the unperturbed and perturbed discrete-time neural network models considered herein.

Chaos control in multistable delay-differential equations and their singular limit maps

The multistability exhibited by first-order delay-differential equations (DDE's) at large delay-to-response ratios $R$ is useful for the design of dynamical memory devices. This paper first characterizes multistability in the Mackey-Glass DDE at large $R.$ The extended control of its unstable periodic orbits (UPO's), based on additional feedback terms evaluated at many times in the past, is then presented. The method enhances the control of UPO's and of their harmonics. Further, the discrete-time map obtained in the singular perturbation limit of the controlled DDE is useful to characterize the range of parameters where this extended control occurs. Our paper then shows how this singular limit map leads to an improved method of controlling UPO's in continuous-time difference equations and in discrete-time maps. The performance of the method in the general contexts of extended additive and parametric control is evaluated using the logistic map and the Mackey-Glass map. The applicability of the method is finally illustrated on the Nagumo-Sato discrete-time neural network model.

A study of network dynamics

The dynamics of a discrete-time neural network model are investigated. First, a numerical survey of network power spectra is reported for networks of varying size with random weight matrices and initial states. The steepness of the logistic function and a symmetry measure of the weight matrix are taken as control parameters. Summary statistics are presented to give gross measures of the model's temporal activity in parameter space. Second, a detailed study of the dynamics of a particular network is described. Complex dynamical behavior is observed, including Hopf bifurcations, the Ruelle-Takens-Newhouse route to chaos (showing mode-locking at rational winding numbers and the destruction of an invariant torus), and the period-doubling route to chaos.