Dale’s Principle Is Necessary for an Optimal Neuronal Network’s Dynamics

Abstract

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 connection s 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.