Project Title: Robustness to stochastic noise of certain Boolean networks

Adviser: Dora Matache

Description: Random Boolean Networks are networks of nodes that can be in one of two possible states ON or OFF, and whose evolution from one time point to another is governed by given Boolean rules. Each node’s evolution is influenced by the state of other nodes called its parents. If all the nodes are updated at the same time then the network is called synchronous, otherwise it is called asynchronous. Recent research has focused on the study of sensitivity of Boolean functions to certain types of noise procedures. The goal of this project is to extend those results to the study of robustness of Boolean networks to similar types of noise procedures.

The student will perform the following tasks:

A. Become familiar with and write a brief description of relevant concepts of noisy Boolean networks, based on several references such as:

Matache M.T., Matache V., On the Sensitivity to Noise of a Boolean Function. To appear in Journal of Mathematical Physics 2009.

Beck G., Matache M.T., Dynamical Behavior and Influence of Stochastic Noise on Certain Generalized Boolean Networks. Physica A: Statistical Mechanics and its Applications, Volume 387, Issues 19-20, August 2008, Pages 4947-4958,

Alligood K.T., Sauer T.D., Yorke J.A., Chaos: and introduction to dynamical systems, Springer-Verlag, 1996. (short topics from the book)

This will help the student become familiar with the topic and will serve as an introductory section of the final research report.

B. Get familiar with the Matlab software in order to be able to write and run several programs related to the study of the dynamics of the noisy Boolean networks.

C. Actual study of the dynamics of noisy Boolean networks governed by certain generalized totalistic rules of elementary cellular automata. The student will work with the adviser to develop a mathematical model and to determine a selection of Boolean networks/functions and noise rules for simulation purposes. Bifurcation diagrams, analysis of the sensitivity of the orbits to initial values, Lyapunov exponent calculations, and other graphical tools will be employed as necessary. The findings of this study will become the core of the research report.

D. Create a final research report to be presented at the MAM Symposium.

OTHER REQUIREMENTS: The students interested in the project above are expected to have mathematical maturity, a good understanding of basic probability properties and basic computer skills. They should become familiar with Matlab and TeX, (a popular markup text-editor for mathematical papers), as the project is under way.

NOTE: The results of this research will represent the core of a research paper that later will be sent for publication to a suitable journal.