On the Distribution of Successor States in Boolean Threshold Networks

We study the distribution of successor states in Boolean networks (BNs). The state vector ${\mathbf{y}}$ is called a successor of ${\mathbf{x}}$ if ${\mathbf{y}}= \textbf {F}({\mathbf{x}})$ holds, where ${\mathbf{x}}, {\mathbf{y}}\in \{0,1\}^{n}$ are state vectors and $\textbf {F}$ is an ordered set of Boolean functions describing the state transitions. This problem is motivated by analyzing how information propagates via hidden layers in Boolean threshold networks (discrete model of neural networks) and is kept or lost during time evolution in BNs. In this article, we measure the distribution via entropy and study how entropy changes via the transition from ${\mathbf{x}}$ to ${\mathbf{y}}$ , assuming that ${\mathbf{x}}$ is given uniformly at random. We focus on BNs consisting of exclusive OR (XOR) functions, canalyzing functions, and threshold functions. As a main result, we show that there exists a BN consisting of $d$ -ary XOR functions, which preserves the entropy if $d$ is odd and $n > d$ , whereas there does not exist such a BN if $d$ is even. We also show that there exists a specific BN consisting of $d$ -ary threshold functions, which preserves the entropy if $n \mod d = 0$. Furthermore, we theoretically analyze the upper and lower bounds of the entropy for BNs consisting of canalyzing functions and perform computational experiments using BN models of real biological networks.