  We investigated the effects of synaptic depression on the macroscopic
behavior of stochastic neural networks. Dynamical mean field equations were
derived for such networks by taking the average of two stochastic variables: a
firing state variable and a synaptic variable. In these equations, their
average product is decoupled as the product of averaged them because the two
stochastic variables are independent. We proved the independence of these two
stochastic variables assuming that the synaptic weight is of the order of 1/N
with respect to the number of neurons N. Using these equations, we derived
macroscopic steady state equations for a network with uniform connections and a
ring attractor network with Mexican hat type connectivity and investigated the
stability of the steady state solutions. An oscillatory uniform state was
observed in the network with uniform connections due to a Hopf instability.
With the ring network, high-frequency perturbations were shown not to affect
system stability. Two mechanisms destabilize the inhomogeneous steady state,
leading two oscillatory states. A Turing instability leads to a rotating bump
state, while a Hopf instability leads to an oscillatory bump state, which was
previous unreported. Various oscillatory states take place in a network with
synaptic depression depending on the strength of the interneuron connections.
