  We study collective dynamics of complex networks of stochastic excitable
elements, active rotators. In the thermodynamic limit of infinite number of
elements, we apply a mean-field theory for the network and then use a Gaussian
approximation to obtain a closed set of deterministic differential equations.
These equations govern the order parameters of the network. We find that a
uniform decrease in the number of connections per element in a homogeneous
network merely shifts the bifurcation thresholds without producing qualitative
changes in the network dynamics. In contrast, heterogeneity in the number of
connections leads to bifurcations in the excitable regime. In particular we
show that a critical value of noise intensity for the saddle-node bifurcation
decreases with growing connectivity variance. The corresponding critical values
for the onset of global oscillations (Hopf bifurcation) show a non-monotone
dependency on the structural heterogeneity, displaying a minimum at moderate
connectivity variances.
