K. Gopalsamy

Delay-independent stability in bidirectional associative memory networks

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.

Stability in asymmetric Hopfield nets with transmission delays

Sufficient conditions are derived for the delay independent stability of the equilibria in Hopfields graded response networks of the type dxi(t)/dt = -bixi(t) + Σnj = 1aijfj(μjxj(t - τij)) +Fi(t) (i = 1 , 2, … , n) when the external inputs Fi are held temporally uniform. A generalization to continuously distributed delays is briefly indicated. Several illustrative examples are numerically simulated and the results of simulations are graphically displayed.

Exponential stability of continuous-time and discrete-time cellular neural networks with delays

Convergence characteristics of continuous-time cellular neural networks with discrete delays are studied. By using Lyapunov functionals, we obtain delay independent sufficient conditions for the networks to converge exponentially toward the equilibria associated with the constant input sources. Halanay-type inequalities are employed to obtain sufficient conditions for the networks to be globally exponentially stable. It is shown that the estimates obtained from the Halanay-type inequalities improve the estimates obtained from the Lyapunov methods. Discrete-time analogues of the continuous-time cellular neural networks are formulated and studied. It is shown that the convergence characteristics of the continuous-time systems are preserved by the discrete-time analogues without any restriction imposed on the uniform discretization step size.

Stability of artificial neural networks with impulses

Sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a Hopfield-type neural network with Lipschitzian activation functions without assuming their boundedness, monotonicity or differentiability and subjected to impulsive state displacements at fixed instants of time. Both the continuous-time and their corresponding discrete-time networks are considered. The sufficient conditions of the discrete-time network do not restrict the step-size appearing in the discretization process and these conditions approach as the step-size tends to zero those of the conditions of the continuous-time networks. The sufficient conditions are in terms of the parameters of the network only and are easy to verify; also when the impulsive jumps are absent the results reduce to those of the non-impulsive systems.

On delay differential equations with impulses

Abstract Sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x(t) + ax(t − τ) = ∑j = 1∞ bjx(tj − )δ(t − tj), t ≠ tj and for the existence of a nonoscillatory solution; conditions are also obtained for all solutions to be oscillatory. The asymptotic behaviour of an impulsively perturbed delay-logistic equation is investigated as an extension to a nonlinear equation.

Delay induced periodicity in a neural netlet of excitation and inhibition

Abstract The dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated. It is shown that in the absence of delay, the netlet relaxes to the trivial resting state. If the delay is of sufficient magnitude, the network is excited to a temporally periodic cyclic behaviour. The analytical mechanism for the onset of cyclic behaviour is through a Hopf-type bifurcation. Approximate solusions to the periodic output of the netlet is calculated; stability of the temporally periodic cycle is investigated. It is shown that the bifurcation is supercritical. A related discrete version of the continuous time system is formulated. It is found that the discrete system also displays a cyclic behaviour. Results of a number of computer simulations are displayed graphically; the article concludes with a brief neurobiological discussion.

Convergence under dynamical thresholds with delays

Necessary and sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium of a class of delay differential equations modeling the action of a neuron with dynamical threshold effects.

On a neutral delay logistic equation

The qualitative behaviour of solutions of the neutral delay logistic equation is investigated; sufficient conditions are obtained for the local asymptotic stability of the positive steady state or (*).The osciiiaiory and non-oscinaiory enaracierisucs oi me positive solutions of (*) are also studied.

Global asymptotic stability in Volterra's population systems

Sufficient conditions which can be verified easily are obtained for the global asymptotic stability of the positive steady state in Volterras population system incorporating hereditary effects.

Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons

The dynamical characteristics of a single isolated Hopfield-type neuron with dissipation and time-delayed self-interaction under periodic stimuli are studied. Sufficient conditions for the heteroassociative stable encoding of periodic external stimuli are obtained. Both discrete and continuously distributed delays are included.