  We study complex networks of stochastic two-state units. Our aim is to model
discrete stochastic excitable dynamics with a rest and an excited state. Both
states are assumed to possess different waiting time distributions. The rest
state is treated as an activation process with an exponentially distributed
life time, whereas the latter in the excited state shall have a constant mean
which may originate from any distribution. The activation rate of any single
unit is determined by its neighbors according to a random complex network
structure. In order to treat this problem in an analytical way, we use a
heterogeneous mean-field approximation yielding a set of equations general
valid for uncorrelated random networks. Based on this derivation we focus on
random binary networks where the network is solely comprised of nodes with
either of two degrees. The ratio between the two degrees is shown to be a
crucial parameter. Dependent on the composition of the network the steady
states show the usual transition from disorder to homogeneous ordered
bistability as well as new scenarios that include inhomogeneous ordered and
disordered bistability as well as tristability. The various steady states
differ in their spiking activity expressed by a state dependent spiking rate.
Numerical simulations agree with analytic results of the heterogeneous
mean-field approximation.
