EFFECT OF DELAY ON THE BOUNDARY OF THE BASIN OF ATTRACTION IN A SYSTEM OF 2 NEURONS

The behavior of neural networks may be influenced by transmission dela ys and many studies have derived constraints on parameters such as con nection weights and output functions which ensure that the asymptotic dynamics of a network with delay remains similar to that of the corres ponding system without delay. However, even when the delay does not af fect the asymptotic behavior of the system, it may influence other imp ortant features in the system's dynamics such as the boundary of the b asin of attraction of the stable equilibria. In order to better unders tand such effects, we study the dynamics of a system constituted by tw o neurons interconnected through delayed excitatory connections. We sh ow that the system with delay has exactly the same stable equilibrium points as the associated system without delay, and that, in both the n etwork with delay and the corresponding one without delay, most trajec tories converge to these stable equilibria. Thus, the asymptotic behav ior of the network with delay and that of the corresponding system wit hout delay are similar. We obtain a theoretical characterization of th e boundary separating the basins of attraction of two stable equilibri a, which enables us to estimate the boundary. Our numerical investigat ions 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 a n arbritrary number of units is discussed. (C) 1998 Elsevier Science L td. All rights reserved.