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 probability theory 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.