Boolean Threshold Networks as Models of Genotype-Phenotype Maps

Boolean threshold networks (BTNs) are a class of mathematical models used to describe complex dynamics on networks. They have been used to study gene regulation, but also to model the brain, and are similar to artificial neural networks used in machine learning applications. In this paper we study BTNs from the perspective of genotype-phenotype maps, by treating the network’s set of nodes and connections as its genotype, and dynamic behaviour of the model as its phenotype. We show that these systems exhibit (1) Redundancy, that is many genotypes map to the same phenotypes; (2) Bias, the number of genotypes per phenotypes varies over many orders of magnitude; (3) Simplicity bias, simpler phenotypes are exponentially more likely to occur than complex ones; (4) Large robustness, many phenotypes are surprisingly robust to random perturbations in the parameters, and (5) this robustness correlates positively with the evolvability, the ability of the system to find other phenotypes by point mutations of the parameters. These properties should be relevant for the wide range of systems that can be modelled by BTNs.