@mthiboust “Did I get it well ?”
You did very well.
These are excellent references and I can see that you grasp the concepts to the detail necessary to start to see how it in embodied in the biology.
I am hoping that as we start to see funcional models expressed in code there will be more light shed into the large scale structure of the functioning of the brain. Many disparate models and papers should have a basic framework to tie them all together, much the same way that tectonic plate theory greatly enhanced our understanding of geology.
The interaction with the thalamus and L6 is also very important and I suggest that an excellent starting place is this paper. I found it to be a hard read but well worth the effort: