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Abstract
Introduction: Boolean networks (BN) are used for modeling networks whose node activity can be described as a binary value. We focus on partially nested canalizing functions (PNCF) which are a combination of two types of Boolean functions: totalistic and canalizing. A totalistic function only depends on the summation of the inputs. A function with a canalizing input at its so-called canalizing value automatically determines the output with no regard to the remaining inputs.
Background: Past studies have looked at the stability of fully and partially nested canalizing functions, but never in conjunction with a totalistic function. Conflicting results were found with regard to the presence of chaotic behavior. Moreover, these studies consider an ergodic network in which each state is equally likely. We relax this assumption and provide a model for the sensitivity of the network to perturbations.
Objective: Use the mathematical model to observe the dynamics of a BN governed by the ensemble of PNCFs and determine for what parameter values phase transitions occur.
Methods: We utilize bifurcation diagrams to determine the behavior of a twodimensional map for the density of ones of the network and the probability the output of a function is 1 if none of the canalizing inputs are at their canalizing value. We use Derrida plots to match the mathematical model to the actual network dynamics. Finally, we generate phase transition diagrams to indicate parameter values that cause a transition from order to chaos.
Conclusions: The bifurcation diagrams show the two-dimensional map exhibits stability with pitch-fork bifurcations. Moreover, once a certain threshold of canalizing depth is reached, there are no major changes in the graphs. The Derrida plots indicate our model matches the actual network dynamics for a wide range of parameters. Similar results were found in the phase diagrams: the depth of canalization does not cause major changes once a certain threshold is reached. However, both chaos and order exist for various parameter combinations. The edge-of-chaos curve is identified in the simulations.
