We observe network formation in many situations, from blood veines formation to pattern formation of vascular network in plants' leaves. In the latter, we know that Auxin, a phytohormone, is sufficient to induce differenciation of cells into vascular form. Auxin is produced in apical areas of shoots, and is actively pumped to the roots. It is known that the flaw of Auxin is due to polarity and position of transportation proteins in the plasmic membrane of cells.
But we can observe loops in the network of leaves, which is a paradox in polarity of cells. At one point of the loop one cell must be at egal distance in the two directions of the roots. So it can not be polarized. Until now, several models of leaves pattern formation gave patterns similar to primary veines, but no models dealt with secondary or tertiary veines formation, or even with loops.
We investigate a model taking in account the polarity of cells, according to concentration of transportation proteins in cells membran, and the differenciation into vascular cells. We think the position of Auxin production sites or alternatively changing position could generate the same loops as observed in natural leaves and may be sufficient to product secondary and tertiary networks.