Abstract
A novel algebraic approach is proposed to study dynamics of asynchronous random Boolean networks where a random number of nodes can be updated at each time step (ARBNs). In this article, the logical equations of ARBNs are converted into the discrete-time linear representation and dynamical behaviors of systems are investigated. We provide a general formula of network transition matrices of ARBNs as well as a necessary and sufficient algebraic criterion to determine whether a group of given states compose an attractor of length in ARBNs. Consequently, algorithms are achieved to find all of the attractors and basins in ARBNs. Examples are showed to demonstrate the feasibility of the proposed scheme.
Highlights
The gene regulatory networks (GRNs) as complex systems present diverse dynamical behaviors
The nodes in ARBNs take values from the set f0,1g corresponding to two levels of gene expressions. fi is a Boolean logic from f0,1gp?f0,1g, which is assigned to node i
We have presented a new approach to study the dynamics of random Boolean networks under asynchronous stochastic update
Summary
The gene regulatory networks (GRNs) as complex systems present diverse dynamical behaviors. Each gene is represented by a node in RBNs with two possible states, i.e. the logical 0 and 1. Since dynamics is deterministic and the state space is finite, the states of a particular RBN eventually converge into a series of periodically recurring states, which are called attractors including the fixed points and cycles, starting from an arbitrary initial condition. Attractors and their transient states compose the basins of attraction, which present the dynamical behaviors of systems. Along with the research of networks in molecular biology, chemistry, neurobiology, and economy etc [4,5,6,7,8,9,10,11,12,13,14,15,16], RBNs have been extensively investigated and a wealth of results have been achieved [17,18,19,20,21,22,23,24,25,26,27,28]
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