Thanks, CrossRoads! 
When I'll arrive home tonight, I'll take some current measurements on the other anode layers. I plan to do this by inserting a multimeter between these other anode layers and the drain of the MOSFET which drives them. I hope that's right...
Honestly the thought that some layers turn on in addition to the layer which is really supposed to be on, has planted the feeling in me too that there's something wrong with the pullup resistors that I have today between the gate and the source of the MOSFETS (I have 5.6 KOhm there today). I'll try replacing them with some stronger resistors. I think I have 220 KOhms lying around, I'll give them a try. I hope that's not too much. I wonder if this can also explain why the two middle layers turn on when no layer is supposed to be on.
You mentioned that if I don't manage to solve the problem with bigger pulllups, I might need some pulldowns. From what I've read on Wikipedia, I understood that a pulldown is a resistor which you connect between the gate of the MOSFET and the ground. Is this correct? Also, is it good practice to use pullups and pulldowns combined, or you need to decide which one and use just one?
Thank you!