We study a mathematical model of biological neuronal
networks composed by any finite number N ≥ 2 of non-necessarily
identical cells. The model is a deterministic dynamical system governed by
finite-dimensional impulsive differential equations. The statical structure of
the network is described by a directed and weighted graph whose nodes are certain
subsets of neurons, and whose edges are the groups of synaptical connections among
those subsets. First, we prove that among all the possible networks such as their
respective graphs are mutually isomorphic, there exists a dynamical optimum.
This optimal network exhibits the richest dynamics: namely, it is capable to
show the most diverse set of responses (i.e. orbits in the future) under external stimulus or signals. Second, we prove that
all the neurons of a dynamically optimal neuronal network necessarily satisfy
Dale’s Principle, i.e. each neuron
must be either excitatory or inhibitory, but not mixed. So, Dale’s Principle is
a mathematical necessary consequence of a theoretic optimization process of the
dynamics of the network. Finally, we prove that Dale’s Principle is not
sufficient for the dynamical optimization of the network.