Phase transitions in stochastic non-linear threshold Boolean automata networks on Z2: The boundary impact

Abstract

This paper addresses the question of the impact of the boundary on the dynamical behaviour of finite Boolean automata networks on Z 2. The evolution over discrete time of such networks is governed by a specific stochastic threshold non-linear transition rule derived from the classical rule of formal neural networks. More precisely, the networks considered in this paper are finite but the study is done for arbitrarily large sizes. Moreover, the boundary impact is viewed as a classical definition of a phase transition in probability theory, characterising in our context the fact that a network admits distinct asymptotic behaviours when different boundary instances are assumed. The main contribution of this paper is the highlight of a formula for a necessary condition for boundary sensitivity, whose sufficiency and necessity are entirely proven with natural constraints on interaction potentials.

Entropy as a Robustness Marker in Genetic Regulatory Networks

“…A previous work [ 40 ] gives an algebraic formula allowing for the calculation of the number of attractors of a Boolean network; in their Jacobian interaction graph were two tangential circuits ( Table 1 ): one positive of length right, (involved in the richness in attractors, as predicted by [ 32 , 33 , 34 ], and one negative of length four (responsible of the trajectory stability, as predicted by [ 36 , 89 ]). This predicted number (11) is the same as that calculated from the simulation of all trajectories from all possible initial conditions summarized in Figure 8 , and more results both theoretical and applied to real Boolean networks can be found in more recent literature [ 46 , 47 , 48 , 49 , 90 ] showing a large spectrum of possible applications in genetic, metabolic or social Boolean networks.…”

“…In the present paper, we will focus on the study of robustness of genetic regulatory networks driven by Hopfield’ stochastic rule [ 35 ], by using the Kolmogorov-Sinaï entropy of the Markov process underlying the state transition dynamics [ 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 ]. In Section 2 , we define the concepts underlying the relationships between complexity, stability and robustness in the Markov framework of genetic threshold Boolean random regulatory networks.…”