This study deals with the evolution of the so called “intelligent” networks (insect society without leader, cells of an organism, brain,…) during their apprenticeship period.
First we summarize briefly the Version 2 (published in French), whose the main characteristics are:

Spacer>   A network connected to its environment is considered as immersed into an information field created by this environment which so dictates to it the apprenticeship constraints.
Spacer>   The used formalism draws one’s inspiration from the one of the Quantum field theory (Principle of stationary action, gauge fields, invariance by symmetry transformations,…).
Spacer>   We obtain Lagrange equations whose solutions describe the network evolution during the whole apprenticeship period.
Spacer>   Then, while proceeding with the same formalism inspiration, we suggest other study ways capable of evolving the knowledge in the considered scope.

In a second part, after a reminder of the points to be improved, we exhibit the Version 3 which brings, we think, relevant improvements. Indeed:

Spacer>   We consider the weighted averages of the variables; this introduces probabilities.
Spacer>   We define two observables (L average of information flux and A activity of the network) which could be measured and so be compared with experimental results.
Spacer>   We find that L , weighted average of information flows, is an invariant.
Spacer>   Finally, we propose two expressions for the conactance, from which we deduce the corresponding Lagrange equations which have to be solved to know the evolution of the considered weighted averages.

But, at the present stage, we think that we can progress only by carrying out experiments (see projects like Human brain project) and discovering invariants, symmetries which would allow us, like in Physics, to classify networks and above all to understand better the connections between them.

Indeed, and that is what we propose among the future research ways, the underlying problem is to understand how, after their apprenticeship period, several networks can connect together to produce, in the brain case for instance, what we call mental states.