Delay-independent stability in bidirectional associative memory networks

Abstract

Abstract-It is shown that if the neuronal gains are small compared with the synaptic connection weights, then a bidirectional associative memory network with axonal signal transmission delays converges to the equilibria associated with exogenous inputs to the network; both discrete and continuously distributed delays are considered; the asymptotic stability is global in the state space of neuronal activations and also is independent of the delays.

Existence and stability of periodic solutions for Cohen–Grossberg neural networks with multiple delays

“…It is also important to incorporate time delay in various neural networks. In recent years, there exist some results on global asymptotical stability, global exponential stability and periodic solutions for the neural networks with constant delays or time-varying delays (see, e.g., [1]- [6], [8], [10]- [17], [19]- [24], [29]- [31]). Although the use of finite delays in models with delayed feedback provides a good approximation to simple circuits consisting of a small number of neurons, neural networks usually should have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.…”

Convergence under dynamical thresholds with delays

“…It is natural to investigate whether these conditions are necessary for the asymptotic stability of the trivial solution of (1); we show now that these conditions are also necessary for the asymptotic stability of the equilibrium of (9).…”

“…The linear variational system corresponding to the trivial solution of (9) is given by (48) The characteristic equation associated with (48) is the transcendental equation (49) The roots of (49) depend continuously on and that for , the only root of (49) satisfies (50) By the continuous dependence of on , it follows from (50) that for small , the roots of (49) have negative real parts, which implies the local asymptotic stability of the equilibrium of (9). If there exists a such that for , (49) has a pair of pure imaginary roots, say then then there exists a for which (49) has roots with zero real parts and the trivial solution of (9) becomes unstable.…”

“…Proposition 3.1: The sufficient conditions required in (14) are also necessary for the existence of a unique equilibrium for (9) and its asymptotic stability.…”

“…The time delays incorporated in the thresholds can lead to wide ranging behavior of the system from relaxation to a stable equilibrium, instability resulting in stable oscillations and "chaos" [15]. There has been recently numerous investigations of Hopfield-type networks with time delays (see, for instance, [1], [2], [9], [10], [16], and the references in these works). Since Hopfield networks with thresholds have not received any attention so far, the system considered in this article is just a beginning in this direction; a multineuron model system with threshold delays will be considered in a future article.…”