Experiments on vein regeneration (Jost 1942; Jacobs 1952) suggest that a signal of some kind, which can cause the differentiation of veins, flows from a source in the growing tissues of leaves towards the root. Sachs (1969, 1978) has given evidence that the capacity of a given pathway to transport this signal increases with the flux it carries. He suggests that this progress could cause the canalization of signal flow into a pattern of discrete strands, which subsequently differentiate into veins. I formulate here a mathematical model based on these assumptions, and show that it can generate well defined strands. Given plausible estimates for diffusion constants and polar transport rates, it appears that vein formation could occur by this mechanism over an appropriate distance within an acceptably short time. I show that this model can simulate Sachs's experiments on vein formation. I also show that, with suitable assumptions about the distribution of source activity, the model can generate elaborate networks, with branches and loops of the kind seen in the leaves of higher plants.