Attractor landscapes in Boolean networks with firing memory

TL;DR

This paper investigates the dynamics of Boolean networks with firing memory, demonstrating their computational equivalence to classical networks and analyzing how delays affect attractor landscapes, with applications to biological models.

Contribution

It proves that Boolean networks with firing memory are computationally equivalent to classical networks and characterizes how delays influence network attractors and dynamics.

Findings

Networks with delays > 1 have only fixed points as attractors.

Characterization of delay phase space for two-vertex networks.

Application of delay analysis to biological models of immune control and plant development.

Abstract

In this paper we study the dynamical behavior of Boolean networks with firing memory, namely Boolean networks whose vertices are updated synchronously depending on their proper Boolean local transition functions so that each vertex remains at its firing state a finite number of steps. We prove in particular that these networks have the same computational power than the classical ones, ie any Boolean network with firing memory composed of $m$ vertices can be simulated by a Boolean network by adding vertices. We also prove general results on specific classes of networks. For instance, we show that the existence of at least one delay greater than 1 in disjunctive networks makes such networks have only fixed points as attractors. Moreover, for arbitrary networks composed of two vertices, we characterize the delay phase space, \ie the delay values such that networks admits limit cycles or fixed points. Finally, we analyze two classical biological models by introducing delays: the model of the immune control of the $\lambda$-phage and that of the genetic control of the floral morphogenesis of the plant \emph{Arabidopsis thaliana}.