Mathematical Sciences
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Mathematical research at MSU is focused primarily on related topics in pure and applied mathematics. Research programs complement each other and are often applied to problems in science and engineering. Research in statistics encompasses a broad range of theoretical and applied topics. Because the statisticians are actively engaged in interdisciplinary work, much of the statistical research is directed toward practical problems. Mathematics education faculty are active in both qualitative and quantitative experimental research areas. These include teacher preparation, coaching and mentoring for in-service teachers, online learning and curriculum development.
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Item Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models(2019-04) Crawford-Kahrl, Peter; Cummins, Bree; Gedeon, TomasModeling the dynamics of biological networks introduces many challenges, among them the lack of first principle models, the size of the networks, and difficulties with parameterization. Discrete time Boolean networks and related continuous time switching systems provide a computationally accessible way to translate the structure of the network to predictions about the dynamics. Recent work has shown that the parameterized dynamics of switching systems can be captured by a combinatorial object, called a Dynamic Signatures Generated by Regulatory Networks (DSGRN) database, that consists of a parameter graph characterizing a finite parameter space decomposition, whose nodes are assigned a Morse graph that captures global dynamics for all corresponding parameters. We show that for a given network there is a way to associate the same type of object by considering a continuous time ODE system with a continuous right-hand side, which we call an L-system. The main goal of this paper is to compare the two DSGRN databases for the same network. Since the L-systems can be thought of as perturbations (not necessarily small) of the switching systems, our results address the correspondence between global parameterized dynamics of switching systems and their perturbations. We show that, at corresponding parameters, there is an order preserving map from the Morse graph of the switching system to that of the L-system that is surjective on the set of attractors and bijective on the set of fixed-point attractors. We provide important examples showing why this correspondence cannot be strengthened.Item DSGRN: Examining the Dynamics of Families of Logical Models(2018-06) Cummins, Bree; Gedeon, Tomas; Harker, Shaun; Mischaikow, KonstantinWe present a computational tool DSGRN for exploring the dynamics of a network by computing summaries of the dynamics of switching models compatible with the network across all parameters. The network can arise directly from a biological problem, or indirectly as the interaction graph of a Boolean model. This tool computes a finite decomposition of parameter space such that for each region, the state transition graph that describes the coarse dynamical behavior of a network is the same. Each of these parameter regions corresponds to a different logical description of the network dynamics. The comparison of dynamics across parameters with experimental data allows the rejection of parameter regimes or entire networks as viable models for representing the underlying regulatory mechanisms. This in turn allows a search through the space of perturbations of a given network for networks that robustly fit the data. These are the first steps toward discovering a network that optimally matches the observed dynamics by searching through the space of networks.Item Global dynamics for switching systems and their extensions by linear differential equations(2018-11) Huttinga, Zane; Cummins, Bree; Gedeon, Tomas; Mischaikow, KonstantinSwitching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.Item Combinatorial Representation of Parameter Space for Switching Networks(2016-11) Cummins, Bree; Harker, Shaun; Mischaikow, Konstantin; Mok, Kafung; Gedeon, TomasWe describe the theoretical and computational framework for the Dynamic Signatures Generated by Regulatory Networks (DSGRN) database. The motivation stems from an urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and have decades of parameters. The input to the database computations is a regulatory network, i.e., a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate) and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piecewise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type of information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5 nodes and a 29-dimensional parameter space.Item Convergence Properties of Posttranslationally Modified Protein-Protein Switching Networks with Fast Decay Rates(2016-06) Fan, G.; Cummins, Bree; Gedeon, TomasA significant conceptual difficulty in the use of switching systems to model regulatory networks is the presence of so-called "black walls" co-dimension 1 regions of phase space with a vector field pointing inward on both sides of the hyperplane. Black walls result from the existence of direct negative self-regulation in the system. One biologically inspired way of removing black walls is the introduction of intermediate variables that mediate the negative self-regulation. In this paper, we study such a perturbation. We replace a switching system with a higher-dimensional switching system with rapidly decaying intermediate proteins, and compare the dynamics between the two systems. We find that the while the individual solutions of the original system can be approximated for a finite time by solutions of a sufficiently close perturbed system, there are always solutions that are not well approximated for any fixed perturbation. We also study a particular example, where global basins of attraction of the perturbed system have a strikingly different form than those of the original system. We perform this analysis using techniques that are adapted to dealing with non-smooth systems.Item On the Efficacy of State Space Reconstruction Methods in Determining Causality(2015-03) Cummins, Bree; Gedeon, Tomas; Spendlove, Kelly T.We present a theoretical framework for inferring dynamical interactions between weakly or moderately coupled variables in systems where deterministic dynamics plays a dominating role. The variables in such a system can be arranged into an interaction graph, which is a set of nodes connected by directed edges wherever one variable directly drives another. In a system of ordinary differential equations, a variable $x$ directly drives $y$ if it appears nontrivially on the right-hand side of the equation for the derivative of $y$. Ideally, given time series measurements of the variables in a system, we would like to recover the interaction graph. We introduce a comprehensive theory showing that the transitive closure of the interaction graph is the best outcome that can be obtained from state space reconstructions in a purely deterministic system. Our work depends on extensions of Takens\' theorem and the results of Sauer et al. [J. Stat. Phys., 65 (1991), pp. 579--616] that characterize the properties of time-delay reconstructions of invariant manifolds and attractors. Along with the theory, we discuss practical implementations of our results. One method for empirical recovery of the interaction graph is presented by Sugihara et al. [Science, 338 (2012), pp. 496--500], called convergent cross-mapping. We show that the continuity detection algorithm of Pecora et al. [Phys. Rev. E, 52 (1995), pp. 3420--3439] is a viable alternative to convergent cross-mapping that is more consistent with the underlying theory. We examine two examples of dynamical systems for which we can recover the transitive closure of the interaction graph using the continuity detection technique. The strongly connected components of the recovered graph represent distinct dynamical subsystems coupled through one-way driving relationships that may correspond to causal relationships in the underlying physical scenario.