Classification of 3-Node Restricted Excitatory-Inhibitory Networks

TL;DR

This paper classifies all connected 3-node excitatory-inhibitory networks based on their structure, considering equivalence and minimality, to facilitate understanding their dynamics and bifurcations in biological systems.

Contribution

It extends previous 2-node classifications to 3-node networks, providing a comprehensive structural categorization under specific constraints.

Findings

Complete classification of 3-node networks based on ODE-equivalence and minimality.

Identification of network structures with valence ≤ 2.

Foundation for analyzing dynamics and bifurcations in biological networks.

Abstract

We classify connected 3-node restricted excitatory-inhibitory networks, extending our previous paper (`Classification of 2-node Excitatory-Inhibitory Networks', Mathematical Biosciences 373 (2024) 109205). We assume that there are two node-types and two arrow-types, excitatory and inhibitory; all excitatory arrows are identical and all inhibitory arrows are identical; and excitatory (resp. inhibitory) nodes can only output excitatory (resp. inhibitory) arrows. The classification is performed under the following two network perspectives: ODE-equivalence and minimality; and valence less or equal to 2. The results of this and the previous work constitute a first step towards analysing dynamics and bifurcations of excitatory-inhibitory networks and have potential applications to biological network models.