Following the experimental evidences of filament forming in organic thin film memories, we developed a semiclassical drift-diffusion model of electrical conductivity in the filament. We show that the global behavior of a memory device and the total current can be accounted for by fully-formed and well-connected filaments. We investigated and ruled out the eventual influence of coherent quantum tunneling in disconnected filaments. It is also shown how a heating model of the filament can be used to check if assumptions on the number of filaments and their radii are physically plausible.