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 ne ural networks. As a first step toward a better understanding of the in fluence of delay, we study the dynamics of a single graded-response ne uron with a delayed excitatory self-connection. The behavior of this s ystem is representative of that of a family of networks composed of gr aded-response neurons in which most trajectories converge to stable eq uilibrium points for any delay value. It is shown that changing the de lay modifies the ''location'' of the boundary of the basin of attracti on 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 wi thin the adjacent basins. Our results suggest that when dealing with n etworks with delay, it is important to study not only the effect of th e delay on the asymptotic convergence of the system but also on the bo undary of the basins of attraction of the equilibria.