In this paper we study a stochastic network model introduced recently in the analysis of neural networks. In this model the interaction between the nodes of the network is local : to each node is associated some real number (the inhibition in the language of neural networks) which is decreasing linearly with time. When this number reaches 0, it sends out some random input to its neighbors (a spike) and restarts with some random value. We are interested in the asymptotic behavior of the network, that is under which conditions the nodes do not die (i.e. their inhibition is not converging to infinity). When these conditions are not satisfied, we analyze the set of nodes which are likely to die. We consider networks with a finite number of nodes and two kind of topologies, the fully connected network and related graphs and the linear network where the nodes are located on a line. A quantity p is associated to this network and the stability properties of the network depend only on it. For the fully connected network, we give the necessary and sufficient condition for the stability of the network (p < 1) as well as the explicit expression for the invariant measure of the Markov process associated to this model. For the stability of the linear network of size N, we prove that the critical value for p is 1/2 if N is odd, otherwise it is the constant.F(1;2cos p/(N+1)). When the network is not stable, the set of possible asymptotic states is analyzed.