Title: Boolean canalization in the micro and macro scales
Abstract: Canalization is a property of Boolean automata that characterizes the extent to which subsets of inputs determine (canalize) the output. In this presentation, I describe the role of canalization as a determinant of the dynamical character of Boolean networks (BN). I consider two different measures of canalization introduced by Marques-Pita and Rocha, namely 'effective connectivity' and 'input symmetry,' in a three-pronged approach. First, we show that the mean 'effective connectivity,' a measure of the true mean in-degree of a BN, is a better predictor of the dynamical regime (order or chaos) of random BNs with homogeneous connectivity than the mean in-degree. Next, I combine effective connectivity and input symmetry in a single measure of 'unified canalization' by using a common yardstick of Boolean hypercube “dimension” — a form of fractal dimension. I show that the unified measure is a better predictor of dynamical regime than effective connectivity alone for BNs with large in-degrees. Finally, I introduce 'integrated effective connectivity' as a macro-scale extension of effective connectivity that characterizes the canalization present in BNs coarse-grained in time obtained by iteratively composing a BN with itself. I show that the integrated measure is a better predictor of long-term dynamical regime than just effective connectivity for a small class of BNs known as the elementary cellular automata. The results also help partly explain the chaotic properties of Rule 30 and why it makes sense to use it as a random number generator.