Dale's Principle is necessary for an optimal neuronal network's dynamics

TL;DR

This paper demonstrates that Dale's Principle is a necessary condition for the optimal dynamics of neuronal networks, which are modeled as directed graphs governed by impulsive differential equations, but it is not sufficient for optimality.

Contribution

It proves that Dale's Principle is a necessary condition for dynamical optimality in neuronal networks modeled mathematically, linking biological constraints to network dynamics.

Findings

Optimal networks exhibit the richest dynamics.

Dale's Principle is necessary for optimality.

Dale's Principle is not sufficient for optimality.

Abstract

We study a mathematical model of biological neuronal networks composed by any finite number $N \geq 2$ 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 that 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.