TRANSIENT OSCILLATIONS IN CONTINUOUS-TIME EXCITATORY RING NEURAL NETWORKS WITH DELAY

A ring neural network is a closed chain in which each unit is connecte d unidirectionally to the next one. Numerical investigations indicate that continuous-time excitatory ring networks composed of graded-respo nse units can generate oscillations when interunit transmission is del ayed. These oscillations appear for a wide range of initial conditions . The mechanisms underlying the generation of such patterns of activit y are studied. The analysis of the asymptotic behavior of the system s hows that (i) trajectories of most initial conditions tend to stable e quilibria, (ii) undamped oscillations are unstable, and can only exist in a narrow region forming the boundary between the basins of attract ion of the stable equilibria. Therefore the analysis of the asymptotic behavior of the system is not sufficient to explain the oscillations observed numerically when interunit transmission is delayed. This anal ysis corroborates the hypothesis that the oscillations are transient. In fact, it is shown that the transient behavior of the system with de lay follows that of the corresponding discrete-time excitatory ring ne twork. The latter displays infinitely many nonconstant periodic oscill ations that transiently attract the trajectories of the network with d elay, leading to long-lasting transient oscillations. The duration of these oscillations increases exponentially with the inverse of the cha racteristic charge-discharge time of the neurons, indicating that they can outlast observation windows in numerical investigations. Therefor e, for practical applications, these transients cannot be distinguishe d from stationary oscillations. It is argued that understanding the tr ansient behavior of neural network models is an important. complement to the analysis of their asymptotic behavior, since both living nervou s systems and artificial neural networks may operate in changing envir onments where long-lasting transients are functionally indistinguishab le from asymptotic regimes.