We study the dynamics of randomly connected excitatory networks of excitable spike-response neuron models. Large networks can exhibit a nonmonotonic collective response to a stimulation when the coupling strength between neurons lies within an appropriate range. With such a coupling, noise imposed upon neurons induces synchronization of the units and oscillations of the network activity, consisting of a succession of bursts. Furthermore, the regularity of this rhythmic activity goes through a maximum as the noise amplitude is increased. This nonmonotonic dependence on the noise amplitude relies on the fact that noise acts in two antagonistic ways. Noise of low amplitude shortens the interval between two successive bursts, leading to an increase of the dynamics regularity, whereas noise of strong amplitude deteriorates the regularity of the dynamics during a burst. In order to study the influence of the noise amplitude and the coupling on the generation of collective oscillations quantitatively, we consider a simpler network model of excitable units. We derive a discrete map, including the noise amplitude and the coupling strength as parameters, which describes the network dynamics in the limit of a large number of neurons. This map reproduces all characteristic features of the activity dynamics obtained with simulated networks. From the analysis of the bifurcation structure of this map, we obtain parameter regions where noise-induced oscillations occur. Using this map we also study the effect of the network connectivity on the generation of oscillations. We show that such noise-induced coherent oscillations in fully connected networks are related to special initial conditions, and are sensitive to perturbations, whereas they can be the only asymptotically stable regime in sparsely connected networks.

• Received 11 February 1998

DOI:https://doi.org/10.1103/PhysRevE.58.3610