Non-deterministic updates of Boolean networks

TL;DR

This paper explores non-deterministic update mechanisms in Boolean networks, unifying various complex and ad-hoc dynamics under a common framework to better understand their behavior and potential transitions.

Contribution

It introduces a framework for extending updating modes to non-deterministic configurations, encompassing previously ad-hoc dynamics as part of Boolean network updates.

Findings

Non-deterministic updates can be expressed as composition of set-mappings.

Complex dynamics like Memory Boolean networks are integrable into this framework.

The framework enables transitions beyond deterministic update capabilities.

Abstract

Boolean networks are discrete dynamical systems where each automaton has its own Boolean function for computing its state according to the configuration of the network. The updating mode then determines how the configuration of the network evolves over time. Many of updating modes from the literature, including synchronous and asynchronous modes, can be defined as the composition of elementary deterministic configuration updates, i.e., by functions mapping configurations of the network. Nevertheless, alternative dynamics have been introduced using ad-hoc auxiliary objects, such as that resulting from binary projections of Memory Boolean networks, or that resulting from additional pseudo-states for Most Permissive Boolean networks. One may wonder whether these latter dynamics can still be classified as updating modes of finite Boolean networks, or belong to a different class of dynamical systems. In this paper, we study the extension of updating modes to the composition of non-deterministic updates, i.e., mapping sets of finite configurations. We show that the above dynamics can be expressed in this framework, enabling a better understanding of them as updating modes of Boolean networks. More generally, we argue that non-deterministic updates pave the way to a unifying framework for expressing complex updating modes, some of them enabling transitions that cannot be computed with elementary and non-elementary deterministic updates.