# ABSTRACT

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:

**>** 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**.

**>** 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,…).

**>** We obtain **Lagrange equations** whose solutions describe the **network evolution during the whole apprenticeship period**.

**>** 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:

**>** We consider the **weighted averages of the variables**; this introduces **probabilities**.

**>** 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**.

**>** We find that **L** , weighted average of information flows, is an **invariant**.

**>** 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.