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Logical models offer a simple but powerful means to understand the complex dynamics of biochemical regulation, without the need to estimate kinetic parameters. However, even simple automata components can lead to collective dynamics that…

Using Boolean networks as prototypical examples, the role of symmetry in the dynamics of heterogeneous complex systems is explored. We show that symmetry of the dynamics, especially in critical states, is a controlling feature that can be…

We prove that nested canalizing functions are the minimum-sensitivity Boolean functions for any activity ratio and we determine the functional form of this boundary which has a nontrivial fractal structure. We further observe that the…

Gene regulatory networks exhibit remarkable stability, maintaining functional phenotypes despite genetic and environmental perturbations. Discrete dynamical models, such as Boolean networks, provide systems biologists with a tractable…

Boolean networks constitute relevant mathematical models to study the behaviours of genetic and signalling networks. These networks define regulatory influences between molecular nodes, each being associated to a Boolean variable and a…

This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory…

Canalization is a classic concept in Developmental Biology that is thought to be an important feature of evolving systems. In a Boolean network it is a form of network robustness in which a subset of the input signals control the behavior…

Boolean network models of molecular regulatory networks have been used successfully in computational systems biology. The Boolean functions that appear in published models tend to have special properties, in particular the property of being…

Randomness is an unavoidable feature of the intracellular environment due to chemical reactants being present in low copy number. That phenomenon, predicted by Delbr\"uck long ago \cite{delbruck40}, has been detected in both prokaryotic…

Gene regulatory networks can be successfully modeled as Boolean networks. A much discussed hypothesis says that such model networks reproduce empirical findings the best if they are tuned to operate at criticality, i.e. at the borderline…

The past decade has seen great advances in our understanding of the role of noise in gene regulation and the physical limits to signaling in biological networks. Here we introduce the spectral method for computation of the joint probability…

We present schema redescription as a methodology to characterize canalization in automata networks used to model biochemical regulation and signalling. In our formulation, canalization becomes synonymous with redundancy present in the logic…

Boolean networks are used to model biological networks such as gene regulatory networks. Often Boolean networks show very chaotic behaviour which is sensitive to any small perturbations. In order to reduce the chaotic behaviour and to…

Living systems operate in a critical dynamical regime -- between order and chaos -- where they are both resilient to perturbation, and flexible enough to evolve. To characterize such critical dynamics, the established 'structural theory' of…

For years, we have been building models of gene regulatory networks, where recent advances in molecular biology shed some light on new structural and dynamical properties of such highly complex systems. In this work, we propose a novel…

This paper studies the mathematical properties of collectively canalizing Boolean functions, a class of functions that has arisen from applications in systems biology. Boolean networks are an increasingly popular modeling framework for…

We present a method to eliminate redundancy in the transition tables of Boolean automata: schema redescription with two symbols. One symbol is used to capture redundancy of individual input variables, and another to capture permutability in…

The alignment of biological networks has the potential to teach us as much about biology and disease as has sequence alignment. Sequence alignment can be optimally solved in polynomial time. In contrast, network alignment is $NP$-hard,…

Boolean networks with canalizing functions are used to model gene regulatory networks. In order to learn how such networks may behave under evolutionary forces, we simulate the evolution of a single Boolean network by means of an adaptive…

It has been shown that uniform as well as non-uniform cellular automata (CA) can be evolved to perform certain computational tasks. Random Boolean networks are a generalization of two-state cellular automata, where the interconnection…