### Abstract

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.

Original language | English (US) |
---|---|

Pages (from-to) | 16-34 |

Number of pages | 19 |

Journal | BioSystems |

Volume | 88 |

Issue number | 1-2 |

DOIs | |

State | Published - Mar 1 2007 |

### Fingerprint

### Keywords

- Asynchrony
- Boolean network
- Cellular automata rule 126
- Chaos
- Stochastic process

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Biochemistry, Genetics and Molecular Biology(all)
- Applied Mathematics

### Cite this

*BioSystems*,

*88*(1-2), 16-34. https://doi.org/10.1016/j.biosystems.2006.04.002

**Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes.** / Deng, Xutao; Geng, Huimin; Matache, Mihaela T.

Research output: Contribution to journal › Article

*BioSystems*, vol. 88, no. 1-2, pp. 16-34. https://doi.org/10.1016/j.biosystems.2006.04.002

}

TY - JOUR

T1 - Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes

AU - Deng, Xutao

AU - Geng, Huimin

AU - Matache, Mihaela T

PY - 2007/3/1

Y1 - 2007/3/1

N2 - An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.

AB - An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.

KW - Asynchrony

KW - Boolean network

KW - Cellular automata rule 126

KW - Chaos

KW - Stochastic process

UR - http://www.scopus.com/inward/record.url?scp=33846847652&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846847652&partnerID=8YFLogxK

U2 - 10.1016/j.biosystems.2006.04.002

DO - 10.1016/j.biosystems.2006.04.002

M3 - Article

VL - 88

SP - 16

EP - 34

JO - BioSystems

JF - BioSystems

SN - 0303-2647

IS - 1-2

ER -