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

Abstract

A boolean network is a map F : {0, 1} n → {0, 1} n that defines a discrete dynamical system by the subsequent iterations of F . Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.

Intrinsic universality in automata networks I: Families and simulations

“…On the other hand, majority networks under partially asynchronous updates (precisely block-sequential update modes) were shown to have super-polynomial periodic orbits and a PSPACE-complete prediction problem [30,5]. A similar result was obtained recently for conjunctive networks under a more general update mode called 'firing memory' [31].…”

“…• use of this proof technique to show how some non-universal concrete families can recover universality by changing the update schedule of the system, thus extending previous results like [31,17,16] (done in the third paper of the series [53]).…”