Mutual Information between Boolean Net Regions

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Boolean nets are sequential, dynamical systems that generalize cellular automata. An -Boolean net can be represented by a directed graph with nodes, in which each node is associated with a (different) Boolean function of arguments. All functions are evaluated synchronously, their arguments being identified by the edges incident to each node. This Demonstration supports the creation of Boolean nets with an even number of nodes (), partitioned into regions and of equal sizes; additionally, you can specify the number of edges connecting and (coupling factor). The main output is an estimate of the asymptotic mutual information between the and components of the global state as the net evolves.

Contributed by: Tommaso Bolognesi (May 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

This Demonstration shows the creation of a random Boolean net (RBN) with a user-defined even number of nodes (), in which each node is associated with a randomly chosen Boolean function of arguments (). The node set of the net is split into parts and of equal size, and you can assign the - coupling factor "coup", defined as the number of edges that bridge between and . These edges enable a flow of information between the two net regions. The slider "- unravel" affects only the layout of the net, providing better visual separation between the nodes in and .

This Demonstration is concerned with the mutual information between the states of the and net components as the net evolves; these states are -tuples of bits and are seen as random variables, also denoted by and . The mutual information between two random variables measures the amount of information that one variable provides about the other and is defined in terms of their joint and marginal distributions. When two variables are independent, their mutual information is 0.

For computational efficiency, this Demonstration adopts an algorithm that provides an approximate but sufficiently accurate estimate of the asymptotic value of or of its average in the case of periodic fluctuation. The approximation is obtained by analyzing the global state graph of the net and its attractors, depicted in the right subpanel. Each global state is an -tuple of bits (only tuple is actually shown in the graph) that we conceive as split into two tuples of equal length, colored in blue and green and corresponding to random variables and . We assume an initial uniform distribution of all global states. Under this distribution, variables and are initially independent: any value of is equally likely to be paired with any value of . If coup, and remain independent as the net evolves, the theoretical value of the corresponding asymptotic mutual information is . Interestingly, the two variables remain independent also when coup assumes its maximum possible value , yielding again . (Occasionally, for these two extreme coup values the shown in the lower-right slot of the panel may slightly depart from 0, due to the approximation procedure.) Conversely, nonzero values of coup typically yield nonzero asymptotic values of . On average, would describe a reversed parabola as coup moves from 0 to .

References

[1] S. Kauffman, "Homeostasis and Differentiation in Random Genetic Control Networks," Nature, 224, 1969 pp. 177–178. doi:10.1038/224177a0.

[2] S. Kauffman, At Home in the Universe: The Search for Laws of Self-Organization and Complexity, New York: Oxford University Press, 1995.

[3] D. Balduzzi and G. Tononi, "Integrated Information in Discrete Dynamical Systems: Motivation and Theoretical Framework," PLOS Computational Biology, 4(6), 2008 e1000091. doi:10.1371/journal.pcbi.1000091.



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