# On the effects of firing memory in the dynamics of conjunctive networks

## TL;DR

This paper investigates how firing memory delays influence the dynamics of conjunctive Boolean networks, revealing complex attractor behaviors and proving the PSPACE-completeness of a related prediction problem.

## Contribution

It introduces the concept of firing memory delay in Boolean networks, analyzes its effects on dynamics, and establishes computational complexity results for state prediction.

## Key findings

- Existence of attractors with non-polynomial period due to firing memory
- Firing memory significantly alters network dynamics
- Prediction of vertex state changes is PSPACE-complete

## Abstract

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several \emph{update schemes} that range from parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called \emph{firing memory} in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is {\bf PSPACE}-complete.