Better Answers to Real Questions

We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified variables. Implementations of extended quantifier elimination via virtual substitution have been successfully applied to various problems in science and engineering. So far, the answers produced by these implementations included infinitesimal and infinite numbers, which are hard to interpret in practice. We introduce here a post-processing procedure to convert, for fixed parameters, all answers into standard real numbers. The relevance of our procedure is demonstrated by application of our implementation to various examples from the literature, where it significantly improves the quality of the results

Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

International audienceEffective quantifier elimination procedures for first-order theories provide a powerful tool for genericallysolving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifierelimination procedures are based on a fixed set of admissible logical symbolswith an implicitly fixed semantics. Thisadmits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination forthe reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifierelimination we are going to discuss recent results with a subtropical procedure for an existential fragment of thereals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems inchemistry and in the life sciences

Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

Algebraic Aspects of (Bio) Nano-chemical Reaction Networks and Bifurcations in Various Dynamical Systems

The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems

A Generalized Framework for Virtual Substitution

We generalize the framework of virtual substitution for real quantifier elimination to arbitrary but bounded degrees. We make explicit the representation of test points in elimination sets using roots of parametric univariate polynomials described by Thom codes. Our approach follows an early suggestion by Weispfenning, which has never been carried out explicitly. Inspired by virtual substitution for linear formulas, we show how to systematically construct elimination sets containing only test points representing lower bounds

Extension of Generalized Modeling and Application to Problems from Cell Biology

Mathematical modeling is an important tool in improving the understanding of complex biological processes. However, mathematical models are often faced with challenges that arise due to the limited knowledge of the underlying biological processes and the high number of parameters for which exact values are unknown. The method of generalized modeling is an alternative modeling approach that aims to address these challenges by extracting information about stability and bifurcations of classes of models while making only minimal assumptions on the specific functional forms of the model. This is achieved by a direct parameterization of the Jacobian in the steady state, introducing a set of generalized parameters which have a biological interpretation. In this thesis, the method of generalized modeling is extended and applied to different problems from cell biology. In the first part, we extend the method to include also the higher derivatives at the steady state. This allows an analysis of the normal form of bifurcations and thereby a more specific description of the nearby dynamics. In models of gene-regulatory networks, it is shown that the extended method can be applied to better characterize oscillatory systems and to detect bistable dynamics. In the second part, we investigate mathematical models of bone remodeling, a process that renews the human skeleton constantly. We investigate the connection between structural properties of mathematical models and the stability of steady states in different models. We find that the dynamical system operates from a stable steady state that is situated in the vicinity of bifurcations where stability can be lost, potentially leading to diseases of bone. In the third part of this thesis, models of the MAPK signal transduction pathway are analyzed. Since mathematical models for this system include a high number of parameters, statistical methods are employed to analyze stability and bifurcations. Thereby, the parameters with a strong influence on the stability of steady states are identified. By an analysis of the bifurcation structure of the MAPK cascade, it is found that a combination of multiple layers in a cascade-like way allows for additional types of dynamic behavior such as oscillations and chaos. In summary, this thesis shows that generalized modeling is a fruitful alternative modeling approach for various types of systems in cell biology.Mathematische Modelle stellen ein wichtiges Hilfmittel zur Verbesserung des Verständnisses komplexer biologischer Prozesse dar. Sie stehen jedoch vor Schwierigkeiten, wenn wenig über die zugrundeliegende biologischen Vorgänge bekannt ist und es eine große Anzahl von Parametern gibt, deren exakten Werte unbekannt sind. Die Methode des Verallgemeinerten Modellierens ist ein alternativer Modellierungsansatz mit dem Ziel, diese Schwierigkeiten dadurch anzugehen, dass dynamische Informationen über Stabilität und Bifurkationen aus Klassen von Modellen extrahiert werden, wobei nur minimale Annahmen über die spezifischen funktionalen Formen getätigt werden. Dies wird erreicht durch eine direkte Parametrisierung der Jacobimatrix im Gleichgewichtszustand, bei der neue, verallgemeinerte Parameter eingeführt werden, die eine biologische Interpretation besitzen. In dieser Arbeit wird die Methode des Verallgemeinerten Modellierens erweitert und auf verschiedene zellbiologische Probleme angewandt. Im ersten Teil wird eine Erweiterung der Methode vorgestellt, bei der die Analyse höherer Ableitungen im Gleichgewichtszustand integriert wird. Dies erlaubt die Bestimmung der Normalform von Bifurkationen und hierdurch eine spezifischere Beschreibung der Dynamik in deren Umgebung. In Modellen für genregulatorische Netzwerke wird gezeigt, dass die so erweiterte Methode zu einer besseren Charakterisierung oszillierender Systeme sowie zur Erkennung von Bistabilität verwendet werden kann. Im zweiten Teil werden mathematische Modelle zur Knochenremodellierung untersucht, einem Prozess der das menschliche Skelett kontinuierlich erneuert. Wir untersuchen den Zusammenhang zwischen strukturellen Eigenschaften verschiedener Modelle und der Stabilität von Gleichgewichtszuständen. Wir finden, dass das dynamische System von einem stabilen Zustand operiert, in dessen Nähe Bifurkationen existieren, welche das System destabilisieren und so potentiell Knochenkranheiten verursachen können. Im dritten Teil werden Modelle für den MAPK Signaltransduktionsweg analysiert. Da mathematische Modelle für dieses System eine hohe Anzahl von Parametern beinhalten, werden statistische Methoden angewandt zur Analyse von Stabilität und Bifurkationen. Zunächst werden Parameter mit einem starken Einfluss auf die Stabilität von Gleichgewichtszuständen identifizert. Durch eine Analyse der Bifurkationsstruktur wird gezeigt, dass eine kaskadenartige Kombination mehrerer Ebenen zu zusätzliche Typen von Dynamik wie Oszillationen und Chaos führt. Zusammengefasst zeigt diese Arbeit, dass Verallgemeinertes Modellieren ein fruchtbarer alternativer Modellierungsansatz für verschiedene zellbiologische Probleme ist

Complex oscillations with multiple timescales - Application to neuronal dynamics

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations in three-dimensional systems. Two families of such complex oscillations are considered: mixed-mode oscillations (MMOs) in systems with two slow variables, and bursting oscillations in systems with two fast variables. In the last chapter, we present recent results on systems with two slow and two fast variables, where both MMO-type dynamics and bursting-type dynamics can arise and where complex oscillations are also organised by canard solutions. The main application area that we consider here is that of neuroscience, more precisely low-dimensional point models of neurons displaying both sub-threshold and spiking behaviour. We focus on analysing how canard objects allow to control the oscillatory patterns observed in these neuron models, in particular the crossings of excitability thresholds