Neural networks are tackled through probabilities for neurons to be activated by other neurons. They are represented by doubly stochastic matrices, named brain matrices, the polytope of which is the convex hull of the permutation matrices which are vertices of this Birkhoff polytope. Each permutation matrix enables to identify loops of neurons associated with a given neurotransmitter. The entropy of evolution of one network is defined and a short study of the optimal information transport in this network leads to consider two thresholds that give rise to questioning about the foundations of classical psychoanalysis within the construction of an extended and more realistic matrix of the neural network. A parallel is emphasized between the expansions in permutation matrices of the brain matrix and the quantum measurement theory through the collapse of the wave function. At a higher scale all the neural networks can be integrated in a global model that can be studied on the same ground as individual brain matrices or through specific thresholds in order to define the origins of ethical behaviors as well as what can lead to mental disability.
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