Algebraic Representation of Asynchronous Multiple-Valued Networks and Its Dynamics

In this paper, dynamics of asynchronous multiple-valued networks (AMVNs) are investigated based on linear representation. By semitensor product of matrices, we convert AMVNs into the discrete-time linear representation. A general formula to calculate all of network transition matrices of a specific AMVN is achieved. A necessary and sufficient algebraic criterion to determine whether a given state belongs to loose attractors of length s is proposed. Formulas for the numbers of attractors in AMVNs are provided. Finally, algorithms are presented to detect all of the attractors and basins. Examples are shown to demonstrate the feasibility of the proposed scheme.

[1]  Richard Banks,et al.  University of Newcastle upon Tyne Computing Science a High-level Petri Net Framework for Multi-valued Genetic Regulatory Networks a High-level Petri Net Framework for Multi-valued Genetic Regulatory Networks Bibliographical Details a High-level Petri Net Framework for Multi-valued Genetic Regulatory , 2022 .

[2]  Fangfei Li,et al.  Controllability of probabilistic Boolean control networks , 2011, Autom..

[3]  Diederik Aerts,et al.  Contextual Random Boolean Networks , 2003, ECAL.

[4]  Konstantin Klemm,et al.  Finding attractors in Asynchronous Boolean Dynamics , 2010, Adv. Complex Syst..

[5]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Frank Dellaert,et al.  Toward an evolvable model of development for autonomous agent synthesis , 1994 .

[7]  Robert K. Brayton,et al.  Simplification of non-deterministic multi-valued networks , 2002, IWLS.

[8]  A. Adamatzky.  On dynamically non-trivial three-valued logics: oscillatory and bifurcatory species , 2003 .

[9]  D. Cheng,et al.  Analysis and control of Boolean networks: A semi-tensor product approach , 2010, 2009 7th Asian Control Conference.

[10]  B. Drossel,et al.  Number and length of attractors in a critical Kauffman model with connectivity one. , 2004, Physical review letters.

[11]  Tamer Kahveci,et al.  Large-Scale Signaling Network Reconstruction , 2012, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[12]  Janet Wiles,et al.  Asynchronous dynamics of an artificial genetic regulatory network , 2004 .

[13]  Fangfei Li,et al.  Observability of Boolean Control Networks With State Time Delays , 2011, IEEE Transactions on Neural Networks.

[14]  L. Kadanoff,et al.  Boolean Dynamics with Random Couplings , 2002, nlin/0204062.

[15]  John Maloney,et al.  Finding Cycles in Synchronous Boolean Networks with Applications to Biochemical Systems , 2003, Int. J. Bifurc. Chaos.

[16]  Elena Dubrova,et al.  Random Multiple-Valued Networks: Theory and Applications , 2006, 36th International Symposium on Multiple-Valued Logic (ISMVL'06).

[17]  Jerrold E. Marsden,et al.  Perspectives and Problems in Nonlinear Science , 2003 .

[18]  Xutao Deng,et al.  Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes , 2007, Biosyst..

[19]  L. Jason Steggles,et al.  Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation , 2011, ArXiv.

[20]  E. D. Paolo,et al.  Rhythmic and non-rhythmic attractors in asynchronous random Boolean networks , 2001 .

[21]  Zhong Mai,et al.  Boolean network-based analysis of the apoptosis network: irreversible apoptosis and stable surviving. , 2009, Journal of theoretical biology.

[22]  Carlos Gershenson,et al.  Classification of Random Boolean Networks , 2002, ArXiv.

[23]  Giovanni De Micheli,et al.  An Efficient Method for Dynamic Analysis of Gene Regulatory Networks and in silico Gene Perturbation Experiments , 2007, RECOMB.

[24]  Ming Liu,et al.  Finding Attractors in Synchronous Multiple-Valued Networks Using SAT-Based Bounded Model Checking , 2010, 2010 40th IEEE International Symposium on Multiple-Valued Logic.

[25]  Florian Greil,et al.  Attractor and basin entropies of random Boolean networks under asynchronous stochastic update. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Aurélien Naldi,et al.  Decision Diagrams for the Representation and Analysis of Logical Models of Genetic Networks , 2007, CMSB.

[27]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[28]  Maxim Teslenko,et al.  A SAT-Based Algorithm for Finding Attractors in Synchronous Boolean Networks , 2011, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[29]  Florian Greil,et al.  Critical Kauffman networks under deterministic asynchronous update , 2007, 0707.1450.

[30]  Christof Teuscher,et al.  Critical Values in Asynchronous Random Boolean Networks , 2003, ECAL.

[31]  John Maloney,et al.  Scalar equations for synchronous Boolean networks with biological applications , 2004, IEEE Transactions on Neural Networks.

[32]  A. Garg,et al.  Modeling of Multiple Valued Gene Regulatory Networks , 2007, 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[33]  Florian Greil,et al.  Dynamics of critical Kauffman networks under asynchronous stochastic update. , 2005, Physical review letters.

[34]  Daizhan Cheng,et al.  Controllability and observability of Boolean control networks , 2009, Autom..