Structural Stability Conditions for Boolean Delay Equations
Boolean delay equations (BDEs) are a potential tool for modeling physical systems with threshold behavior and nonlinear feedbacks. BDEs are evolution equations for discrete vector variables x. The value of each component x_i(t), 0 or 1, depends on previous values of all components after some delays. Basically, the delays model the effects of diffusion or cellular transport. These take some time until a threshold is reached. Depending on the state of the cell and the environment, sometimes it may take a little longer, sometimes it goes a little faster. These effects can be modeled by giving a perturbation to the delay matrix. Models whose dynamics is very sensitive to the exact values of the delays are appear to be useless for modeling biological processes. Hence realistic models of biological processes should exhibit some form of structural stability. Structural stability is the stability of solutions with respect to the parameters; in our case, to the delays. In the context of Boolean delay systems, structural stability means that solutions should stay close when the delay matrix is slightly perturbed.
In this thesis we use Matlab to implement the algorithm in  to give a visual intuition of the solution for Boolean delay system. In order to find the distance between solutions, we first modify the problematic definition of norm in . Finally, we provide various possible formal definitions of structural stability and give some examples to show the relationship between these definitions.
School Location:USA - Ohio
Source Type:Master's Thesis
Keywords:boolean delay equations matlab bdes
Date of Publication:01/01/2008