Information about Boolean Network
A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. They are a particular case of discrete dynamical networks, where time and states are discrete, i.e. they have a bijection onto an integer series. Boolean and elementary cellular automata are particular cases of Boolean networks, where the state of a variable is determined by its spatial neighbors.
A Random Boolean network (RBN) is a system of N binary-state nodes (representing genes) with K inputs to each node representing regulatory mechanisms. The two states (on/off) represent respectively, the status of a gene being active or inactive. The variable K is typically held constant, but it can also be varied across all genes, making it a set of integers instead of a single integer. In the simplest case each gene is assigned, at random, K regulatory inputs from among the N genes, and one of the possible Boolean functions of K inputs. This gives a random sample of the possible ensembles of NK networks. The state of a network at any point in time is given by the current states of all N genes. Thus the state space of any such network is 2N.
Simulation of RBNs is done in discrete time steps. The state of a node at time t+1 is a function of the state of its input nodes and the boolean function associated with it. The behavior of specific RBNs and generalized classes of them has been the subject of much of Kauffman's (and others) research.
Such models are also known as NK models, or Kauffman networks.
Classical model
The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks (Kauffman 1969, 1993).A Random Boolean network (RBN) is a system of N binary-state nodes (representing genes) with K inputs to each node representing regulatory mechanisms. The two states (on/off) represent respectively, the status of a gene being active or inactive. The variable K is typically held constant, but it can also be varied across all genes, making it a set of integers instead of a single integer. In the simplest case each gene is assigned, at random, K regulatory inputs from among the N genes, and one of the possible Boolean functions of K inputs. This gives a random sample of the possible ensembles of NK networks. The state of a network at any point in time is given by the current states of all N genes. Thus the state space of any such network is 2N.
Simulation of RBNs is done in discrete time steps. The state of a node at time t+1 is a function of the state of its input nodes and the boolean function associated with it. The behavior of specific RBNs and generalized classes of them has been the subject of much of Kauffman's (and others) research.
Such models are also known as NK models, or Kauffman networks.
Attractors
A Boolean network has 2N possible states. Since the dynamics are deterministic, sooner or later it will reach a previously visited state, thus falling into an attractor.Dynamics
Order, chaos, and the edgeTopologies
- homogeneous
- normal
- scale-free (Aldana, 2003)
Updating Schemes
- synchronous
- asynchronous (Harvey and Bossomaier, 1997)
- semi-synchronous (Gershenson, 2002)
- deterministic asynchronous
- deterministic semi-synchronous
Applications
- genetic regulatory networks
References
- Aldana, M. (2003). *Boolean dynamics of networks with scale-free topology. Physica D 185:45–66
- Aldana , M., Coppersmith, S., and Kadanoff, L. P. (2003). Boolean dynamics with random couplings. In Kaplan, E., Marsden, J. E., and Sreenivasan, K. R., editors, Perspectives and Problems in Nonlinear Science. A Celebratory Volume in Honor of Lawrence Sirovich. Springer Applied Mathematical Sciences Series.
- Kauffman, S. A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22:437-467.
- Kauffman, S. A. (1993). Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press. Technical monograph. ISBN 0-19-507951-5
- Gershenson, C. (2002). *Classification of random Boolean networks. In Standish, R. K., Bedau, M. A., and Abbass, H. A., editors, Artificial Life VIII:Proceedings of the Eight International Conference on Artificial Life, pages 1-8. MIT Press.
- Harvey, I. and Bossomaier, T. (1997). Time out of joint: Attractors in asynchronous random Boolean networks. In Husbands, P. and Harvey, I., editors, Proceedings of the Fourth European Conference on Artificial Life (ECAL97), pages 67-75. MIT Press.
- Wuensche, A. (1998). *Discrete dynamical networks and their attractor basins. In Standish, R., Henry, B., Watt, S., Marks, R., Stocker, R., Green, D., Keen, S., and Bossomaier, T., editors, Complex Systems'98, University of New South Wales, Sydney, Australia.
External links
In Graph theory, a network is a digraph with weighted edges. These networks have become an especially useful concept in analysing the interaction between biology and mathematics.
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In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.
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f(x) = y.
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A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory, mathematics, and theoretical biology. It consists of a regular grid of cells, each in one of a finite number of states.
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Stuart Alan Kauffman (born September 28, 1939) is a theoretical biologist and complex systems researcher, who has given much thought to the origin of life on Earth. He is best known for arguing that the complexity of biological systems and organisms might result as much from
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random is used to express lack of order, purpose, cause, or predictability in non-scientific parlance. A random process is a repeating process whose outcomes follow no describable deterministic pattern, but follow a probability distribution.
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A gene regulatory network (also called a GRN or genetic regulatory network) is a collection of DNA segments in a cell which interact with each other (indirectly through their RNA and protein expression products) and with other substances in the cell, thereby
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SET may stand for:
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- Sanlih Entertainment Television, a television channel in Taiwan
- Secure electronic transaction, a protocol used for credit card processing,
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A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. These play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers.
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An attractor is a set to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed.
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