Pi Mu Epsilon - Mathematical Association of America

Thursday MAA Session #5 August 2, 2012
9:10–9:25
Comparison of the Influence of Sub-Networks on the Global Dynamics of
Three-Gene Regulatory Networks in Boolean and Continuous Frameworks
Miranda Henderson
Benedictine University
We discuss the Boolean and continuous dynamics of three-gene regulatory networks. We investigate
how the presence of certain sub-networks dictate the dynamics for the entire network. We
determine the conditions on these sub-networks that lead to the same or different qualitative behavior
of the Boolean and continuous dynamics for the entire network.
9:30–9:45
Pivotal Quantities, Bifurcation Continuation, and Inverse Modeling of Dynamical Systems
Karleigh Cameron and Marissa Saladin
Central Michigan University and Aquinas College
Often when we use a system of equations in mathematical modeling applications, we need to be
able to determine unknown parameters. Conditioning has the potential to improve the identifiability
of the estimation problem. Conditioning likelihoods are typically much simpler to model than
the full joint distribution which may be difficult or impossible to estimate analytically. Bifurcation
continuation can inform us about feature statistics of bi-stable systems. We will be using a likelihood
based estimation procedure and a Monte Carlo simulation to compare bias accuracy among
various estimators. In addition, we will explore how different sources of noise manifest within a
dynamical model and how such noise affects uncertainty in the statistical analysis. Systems analyzed
have applications to neuroscience and cognition.
9:50–10:05
Inverse Modeling of Dynamical Systems: Multi-Dimensional Extensions of a
Stochastic Switching Problem
Erik Bates, Blake Chamberlain, and Rachel Gettinger
Michigan State University, Susquehanna University, and Saint Vincent College
The Buridan’s ass problem is characterized by perpetual indecision between two states, which are
never attained. When this problem is formulated as a dynamical system, indecision is modeled by
a discrete-state Markov process determined by the systems unknown parameters. Interest lies in
estimating these parameters from a limited number of observations. In this talk, we compare estimation
methods and examine how well each can be generalized to multi-dimensional extensions
of this system. We also examine the robustness of these methods when noise is introduced to the
system. Conditioning has the potential to improve the identifiability of these inverse problems.
In addition, conditioning likelihoods are often simpler than the full joint distribution, which may
be difficult or impossible to find analytically. We will quantify various feature statistics such as
amplitude, average value, variance, cumulative power, and frequency. We will compare method of
moments type estimators to likelihood-based Monte Carlo simulations conditioned on these feature
statistics.
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