Dynamics of Boolean Networks

TL;DR

This paper presents a method to analyze the long-term behavior of Boolean networks by translating their polynomial functions into matrices, enabling the determination of fixed points and limit cycles.

Contribution

It introduces a novel matrix-based algorithm to derive dynamical properties of Boolean networks from their polynomial representations.

Findings

Algorithm effectively determines fixed points from polynomial expressions.

Method allows construction of Boolean networks with specific limit cycles.

Provides practical examples illustrating the analysis process.

Abstract

Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in studying these dynamical systems is to link their long term behaviors to the structures of the functions that define them. In this paper, a method for deriving a Boolean network's dynamical information via its disjunctive normal form is explained. For a given Boolean network, a matrix with entries 0 and 1 is associated with the polynomial function that represents the network, then the information on the fixed points and the limit cycles is derived by analyzing the matrix. The described method provides an algorithm for the determination of the fixed points from the polynomial expression of a Boolean network. The method can also be used to construct Boolean networks with prescribed limit cycles and fixed points. Examples are provided to explain the algorithm.