Derivation of reduced scalar equations for synchronous boolean networks

Abstract EN

This paper studies reduced scalar equations that are used to model synchronous Boolean networks.

We distinguish between individual minimal equations, which are typically different for the various scalar variables, and a unified minimal equation, which is common for all the scalar variables.

We review and enhance the existing ad hoc method for the derivation of these equations by seeking and utilizing certain orthogonality relations among certain successive instances of the same scalar variable.

We also present a novel general algorithm for deriving the reduced scalar equations via powers of either the function matrix or the transition matrix of the network.

The algorithm simplifies considerably in the case of affine equations.

We present three classical examples to illustrate our techniques, present corrections of previously published results, and demonstrate how the reduced scalar equations can be supplemented by techniques of number theory, Diophantine equations and Boolean equations in making subtle inferences about Boolean networks.