Abstract
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: with 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. The state of our network is described as a Markov process. We are interested in the stability of this network, that is, under which conditions the associated Markov process is ergodic. As we will see, when the network is not stable, some of the nodes die, that is, almost surely after a given time, their inhibition never returns to 0 and grows arbitrarily. When these stability 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 kinds of topologies, the fully connected network and related graphs, and the linear network where the nodes are located on a line. A quantity
Citation
Christine Fricker. Philippe Robert. Ellen Saada. Danielle Tibi. "Analysis of some Networks with Interaction." Ann. Appl. Probab. 4 (4) 1112 - 1128, November, 1994. https://doi.org/10.1214/aoap/1177004906
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