Binary threshold networks as a natural null model for biological networks.

Spin models of neural networks and genetic networks are considered elegant as they are accessible to statistical mechanics tools for spin glasses and magnetic systems. However, the conventional choice of variables in spin systems may cause problems in some models when parameter choices are unrealistic from a biological perspective. Obviously, this may limit the role of a model as a template model for biological systems. Perhaps less obviously, also ensembles of random networks are affected and may exhibit different critical properties. We consider here a prototypical network model that is biologically plausible in its local mechanisms. We study a discrete dynamical network with two characteristic properties: Nodes with binary states 0 and 1, and a modified threshold function with Θ(0)(0)=0. We explore the critical properties of random networks of such nodes and find a critical connectivity K(c)=2.0 with activity vanishing at the critical point. Finally, we observe that the present model allows a more natural implementation of recent models of budding yeast and fission yeast cell-cycle control networks.

[1]  S. Bornholdt,et al.  Boolean Network Model Predicts Cell Cycle Sequence of Fission Yeast , 2007, PloS one.

[2]  I. Nakamura.  Dynamics of threshold network on non-trivial distribution degree , 2004 .

[3]  S. Bornholdt,et al.  Self-organized critical neural networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Stuart A. Kauffman,et al.  Maximal Information Transfer and Behavior Diversity in Random Threshold Networks , 2009, J. Comput. Biol..

[5]  Thilo Gross,et al.  Adaptive coevolutionary networks: a review , 2007, Journal of The Royal Society Interface.

[6]  S. Bornholdt,et al.  The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model. , 2008, Journal of theoretical biology.

[7]  G. Parisi,et al.  Relevant elements, magnetization and dynamical properties in Kauffman networks: a numerical study , 1998 .

[8]  Stefan Bornholdt,et al.  Boolean network models of cellular regulation: prospects and limitations , 2008, Journal of The Royal Society Interface.

[9]  K. E. Kurten.  Correspondence between neural threshold networks and Kauffman Boolean cellular automata , 1988 .

[10]  S. Bornholdt,et al.  Topological evolution of dynamical networks: global criticality from local dynamics. , 2000, Physical review letters.

[11]  Anders Krogh,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[12]  Barbara Drossel,et al.  The phase diagram of random threshold networks , 2008, 0807.0429.

[13]  S. Bornholdt,et al.  Criticality in random threshold networks: annealed approximation and beyond , 2002, cond-mat/0201079.

[14]  Carla Perrone-Capano,et al.  Activity-dependent neural network model on scale-free networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  J. M. Herrmann,et al.  Dynamical synapses causing self-organized criticality in neural networks , 2007, 0712.1003.

[16]  J. M. Herrmann,et al.  Phase transitions towards criticality in a neural system with adaptive interactions. , 2009, Physical review letters.

[17]  K. E. Kürten.  Critical phenomena in model neural networks , 1988 .

[18]  Stefan Bornholdt,et al.  Less Is More in Modeling Large Genetic Networks , 2005, Science.

[19]  J. J. Torres,et al.  Complex behavior in a network with time-dependent connections and silent nodes , 2007, 0710.3320.

[20]  B. Derrida,et al.  Phase Transitions in Two-Dimensional Kauffman Cellular Automata , 1986 .

[21]  Thilo Gross,et al.  Adaptive self-organization in a realistic neural network model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  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.

[23]  G. Parisi,et al.  Closing probabilities in the Kauffman model: an annealed computation , 1995, cond-mat/9510137.

[24]  L. de Arcangelis,et al.  Self-organized criticality model for brain plasticity. , 2006, Physical review letters.

[25]  Q. Ouyang,et al.  The yeast cell-cycle network is robustly designed. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[26]  K. Sneppen,et al.  Specificity and Stability in Topology of Protein Networks , 2002, Science.

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

[28]  B. Derrida,et al.  Random networks of automata: a simple annealed approximation , 1986 .

[29]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[30]  K. E. Kürten,et al.  Critical dynamics of randomly assembled and diluted threshold networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Sompolinsky,et al.  Spin-glass models of neural networks. , 1985, Physical review. A, General physics.