The reliability of firing of excitable-oscillating systems is studied through the response of the active rotator, a neuronal model evolving on the unit circle, to white gaussian noise. A stochastic return map is introduced that captures the behavior of the model. This map has two fixed points: one stable and the other unstable. Iterates of all initial conditions except the unstable point tend to the stable fixed point for almost all input realizations. This means that to a given input realization, there corresponds a unique asymptotic response. In this way, repetitive stimulation with the same segment of noise realization evokes, after possibly a transient time, the same response in the active rotator. In other words, this model responds reliably to such inputs. It is argued that this results from the nonuniform motion of the active rotator around the unit circle and that similar results hold for other neuronal models whose dynamics can be approximated by phase dynamics similar to the active rotator.

Little attention has been paid in the past to the effects of interunit transmission delays (representing a xonal and synaptic delays) ontheboundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the “location” of the boundary of the basin of attraction 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 within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.