Structural principles for dynamics of glass networks
| dc.contributor.author | Lu, Linghong | |
| dc.contributor.supervisor | Edwards, Roderick | |
| dc.date.accessioned | 2008-04-26T00:46:30Z | |
| dc.date.available | 2008-04-26T00:46:30Z | |
| dc.date.copyright | 2008 | en_US |
| dc.date.issued | 2008-04-26T00:46:30Z | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Master of Science M.Sc. | en_US |
| dc.description.abstract | Gene networks can be modeled by piecewise-linear (PL) switching systems of differential equations, called Glass networks after their originator. Networks of interacting genes that regulate each other may have complicated interactions. From a `systems biology' point of view, it would be useful to know what types of dynamical behavior are possible for certain classes of network interaction structure. A useful way to describe the activity of this network symbolically is to represent it as a directed graph on a hypercube of dimension $n$ where $n$ is the number of elements in the network. Our work here is considering this problem backwards, i.e. we consider different types of cycles on the $n$-cube and show that there exist parameters, consistent with the directed graph on the hypercube, such that a periodic orbit exists. For any simple cycle on the $n$-cube with a non-branching vertex, we prove by construction that it is possible to have a stable periodic orbit passing through the corresponding orthants for some sets of focal points $F$ in Glass networks. When the simple cycle on the $n$-cube doesn't have a non-branching vertex, a structural principle is given to determine whether it is possible to have a periodic orbit for some focal points. Using a similar construction idea, we prove that for self-intersecting cycles where the vertices revisited on the cycle are not adjacent, there exist Glass networks which have a periodic orbit passing through the corresponding orthants of the cycle. For figure-8 patterns with more than one common vertex, we obtain results on the form of the return map (Poincar{\'e} map) with respect to how the images of the returning cones of the 2 component cycle intersect the returning cone themselves. Some of these allow complex behaviors. | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/900 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights | Available to the World Wide Web | en_US |
| dc.subject | Switching network | en_US |
| dc.subject | Structural principles | en_US |
| dc.subject | Periodic orbit | en_US |
| dc.subject.lcsh | UVic Subject Index::Sciences and Engineering::Mathematics::Pure mathematics | en_US |
| dc.title | Structural principles for dynamics of glass networks | en_US |
| dc.type | Thesis | en_US |