A theoretical model of the electronic current instability observed experimentally by Bredikhin et al. in the cell Ag(-)/RbAg4I5/C(+) at various constant voltages is proposed. The model is based on the concept of fractal growth of silver dendritic clusters from the Ag(-)/RbAg4I5 interface. It is shown that some of the clusters make connections between opposite electrodes at sufficiently large voltages, which leads to stochastic current oscillations. We show that the number of connecting clusters increases as the voltage is raised and therefore current oscillations become more frequent. All experimental dependences on current instability are well described within the framework of a mean-field approach to the theory of reversible diffusion-limited aggregation without branching.