Project Title: Synchronization of
stochastically coupled random 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 finding formulas for the
probability of a node being ON at time t and using these formulas to study the
dynamics of the network. At the same time various authors showed an interest in
studying the synchronization of stochastically coupled cellular automata and
other dynamical systems. We propose to study the synchronization of
stochastically coupled random Boolean networks in the chaotic phase, under
various rules of cellular automata. The phenomenon of synchronization has been
observed in a wide class of natural systems, such as biological populations,
chemical reactions, neural networks, or human social phenomena. For example the
mechanism is useful to investigate the coevolution of
biological species via horizontal genetic exchange.
The student will perform the following tasks:
A.
Become familiar with and write a brief description of
relevant concepts of random Boolean networks, and stochastically coupled
dynamical systems, based on several references such as:
Hung Y-C., Ho
M-C., Lih J-S., Jiang I-M., Chaos synchronization of two stochastically coupled random Boolean
networks, Physics Letters A, 356(2006), 35-43.
Matache M.T., Heidel J., A random Boolean network model exhibiting
deterministic chaos, Phys. Rev. E 69, 056214, 2004.
Matache M.T., Heidel J., Asynchronous
random Boolean network model based on elementary cellular automata rule 126, Physical Review
E 71, 026232 (2005).
Morelli
L.G., Zanette D.H., Synchronization of Kauffman networks, Physical Review E 63, 036204,
2001.
Morelli
L.G., Zanette D.H., Synchronization of coupled extended dynamical systems: a short review,
International Journal of Bifurcation and Chaos, Vol. 13, No. 4, 2003.
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 coupled Boolean networks.
C. Actual study of the dynamics of stochastically
coupled random Boolean networks governed by certain 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 and
stochastic coupling rules for simulation purposes. The student will study the
time evolution of the Hamming distance between the coupled networks and the
synchronization time as a function of the coupling probability. Bifurcation
diagrams, analysis of the sensitivity of the orbits to initial values, and Lyapunov exponent calculations 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 a good understanding of basic probability properties and basic computer
skills. They should become familiar with Matlab
and
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.