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 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.
Highlights
Based on experimental evidence, Dale’s Principle in Neuroscience postulates that most neurons of a biological neuronal network send the same set of biochemical substances to the other neurons that are connected with them
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
Summary
Dale’s Principle in Neuroscience (see for instance [1,2]) postulates that most neurons of a biological neuronal network send the same set of biochemical substances (called neurotransmitters) to the other neurons that are connected with them. During plastic phases of the nervous systems, the neurotransmitters are released by certain groups of neurons change according to the development of the neuronal network. In this paper we adopt a simplified mathematical model of the neuronal network with a finite number N 2 of neurons, by means of a system of deterministic impulsive differential equations. This model is taken from [11,13], with an adaptation that allows the state variable xi of each cell i to be multidimensional. By means of a rigourous deduction from the abstract mathematical model, we prove that, among all the mathematically theoretic networks of such a model, those exhibiting an optimal dynamics (i.e. the richest or the most versatile dynamics) necessarily satisfy Dale’s Principle (Theorem 16)
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