Mathematical modeling with applications in biological systems, physiology, and neuroscience

Doctor of PhilosophyDepartment of MathematicsBacim AlaliDynamical systems modeling is used to describe different biological and physical systems as well as to predict the interactions between multiple components of a system over time. A dynamical system describes the evolution of a given system over time using a set of mathematical laws, typically described by differential equations. There are two main methods to model the dynamical behaviors of a system: continuous time modeling and discrete-time modeling. When the time between two measurements is negligible, the continuous time modeling governs the evolution of the system, however, when there is a gap between any two consecutive measurements, discrete-time system modeling comes into play. Differential equations are used to model continuous systems and iterated maps represent the generations in discrete-time systems. In this dissertation, we study some dynamical systems and present their applications to different problems in biological systems, physiology, and neuroscience. In chapter one, we study the local dynamics of some interesting systems and show the local stable behavior of the system around its critical points. Moreover, we investigate the local dynamical behavior of different systems including the Hénon-Heiles system, the Duffing oscillator, and the Van der Pol equation. Furthermore, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples. In chapter two, we consider some models in computational neuroscience. Due to the complexity of nerve systems, linear modeling methods are not sufficient to understand the various phenomena in neuroscience. We use nonlinear methods and models, which aim at capturing certain properties of the neurons and their complex dynamics. Specifically, we explore the interesting phenomenon of firing spikes and complex dynamics of the Morris-Lecar model. We consider a set of parameters such that the model exhibits a wide range of phenomenon. We investigate the influences of injected current and temperature on the spiking dynamics of Morris-Lecar model. In addition, we study bifurcations, and computational properties of this neuron model. Moreover, we provide a bound for the membrane potential and a certain voltage value or threshold for firing the spikes. Studying the two co-dimension bifurcations demonstrates more complicated behaviors for this single neuron model. Furthermore, we describe the phenomenon of neural bursting and investigate the dynamics of Morris-Lecar model as a square-wave burster, elliptic burster and parabolic burster. Pharmacokinetic models are mathematical models, which provide insights into the interaction of chemicals with certain biological processes. In chapter three, we consider the process of drug and nanoparticle (NPs) distribution throughout the body. We use a tricompartmental model to study the perfusion of NPs in tissues and a six-compartmental model to study drug distribution in different body organs. We perform global sensitivity analysis by LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC). We identify the key parameters that contribute most significantly to the absorption and distribution of drugs and NPs in different organs in the body. In chapter four, we study two infectious disease models and use nonlinear optimization and optimal control theory to help in identifying strategies for transmission control and forecasting the spread of infectious diseases. We analyze the effect of vaccination on the disease transmission in these models. Moreover, we perform global sensitivity analysis to investigate the key parameters in these models. In chapter five, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform local stability analysis for the fixed points of the system and discuss about its persistence for boundary fixed points. This system inherits properties of the dynamics of a one-dimensional Ricker model such as the cascade of period-doubling bifurcation, periodic windows, and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and show the existence of snap-back repeller. In chapter six, we study the problem of chaos synchronization in certain discrete-time dynamical systems. We introduce a drive-response discrete-time dynamical system, which is coupled using convex link function. We investigate a synchronization threshold, after which, the drive-response system uncouples and loses its synchronized behaviors. We apply this method to the synchronized cycles of the Ricker model and show that this model displays a rich cascade of complex dynamics from a stable fixed point and cascade of period-doubling bifurcation to chaos. We numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling affects the synchronization of the system. In chapter seven, we study the synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from a stable fixed point to periodic orbits, quasi periodic orbits and chaos. We introduce a coupling of these two chaotic systems with different initial conditions and show how they synchronize over a short time. We investigate the qualitative behavior of Generalized Nicholson-Bailey model and its synchronized model using time series analysis and its long-time dynamics by using its bifurcation diagram

Aspects of Complex Magnetism: Vortex Phases, Skyrmion Dynamics, and Chaotic Nano-Oscillators

Project I Vortex Phase in Spiral Antiferromagnets Spiral antiferromagnets are characterized by a Dzyaloshinskii-Moriya interaction that stabilizes spatially modulated phases of the staggered order parameter. In the framework of a Ginzburg-Landau theory, it is shown that a magnetic field leads to the formation of a topological phase constituting a square lattice of vortices and antivortices. An orthogonal alignment of the antiferromagnetic staggered order parameter with an external magnetic field is energetically favorable since both sublattices of a spiral antiferromagnet cannot minimize their Zeeman energy simultaneously, and energy can be gained from spin canting. This spin-flop mechanism has the same effect as easy-plane anisotropy, which leads the vortices to form topological defects with vanishing core. Thus, the vortex phase is only stable close to the Néel temperature. At lower temperatures, the square-lattice vortex phase undergoes spontaneous symme- try breaking into a rectangular phase. We investigate the stability of this rectangular phase with respect to mixed DMI and in-plane magnetic fields. Since any modulated magnetic texture induces a ferroelectric polarization, the vortices of both the vortex and the rectangular phase carry an electrical charge which makes them amenable to the ma- nipulation with in-plane electric fields. Finally, the relevance of these results for the chiral antiferromagnet Ba2CuGe2O7 is discussed. Project II High-Energy Magnons of a Skyrmion Lattice The energy bands of magnons in the skyrmion lattice phase of a chiral magnet, which were recently measured experimentally, show a peculiar, parabola-shaped superstructure when plotted in an extended zone scheme. They are described theoretically in the con- tinuum approximation by a bosonic Bogoliubov-de Gennes equation. In this project, a high-energy approximation is developed, which takes the form of a Schrödinger equa- tion, describing these magnons as charged particles in the emergent magnetic field of the skyrmion lattice. It is known that charged particles in a periodically modulated magnetic field can form runaway orbits, skipping between regions of positive and regions of negative magnetic field values and effectively behaving as free particles. A semiclassical analysis shows the magnon eigenfunctions corresponding to the parabola-shaped superstructures focus on the runaway orbits in the periodically modulated emergent magnetic field experienced by the magnons in this high-energy description. Hence, they can be explained by classical runaway orbits, skipping along the high-symmetry directions of the skyrmion lattice phase, which may be used as magnon waveguides. Project III Chaotic Spin-Torque Nano-Oscillator A spin current transversing a magnetic material exerts a spin-transfer torque onto the magnetic textures, which may lead to oscillations of the magnetization, which constitutes a so-called spin-torque nano-oscillator. Understanding the dynamics of spin-torque nano- oscillators is a prerequisite for applications in reservoir and stochastic computing and designing hardware that emulates artificial neural networks with low power consumption. This project analyses a specific setup for an antiferromagnetic spin-torque nano-oscillator, where a spin current drives a collinear easy-axis antiferromagnet, including damping, and with an external magnetic field applied perpendicular to it. First, it characterizes the static, uniform states and their excitations, yielding the eigenfrequencies of the nano- oscillator. Next, it analyzes the regular dynamics, investigating the stability of a limit cycle at the spin-flop field. Finally, it is shown by calculating the Lyapunov spectrum that this model features chaotic dynamics intrinsically. The transition to chaos is analyzed us- ing bifurcation diagrams, and it is shown that for large damping, chaos is controlled by period-halving bifurcations

Discrete Geometric Singular Perturbation Theory

We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold $S$. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of $S$. The persistence of the critical manifold $S$, local stable/unstable manifolds $W^{s/u}_{loc}(S)$ and foliations of $W^{s/u}_{loc}(S)$ by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincar\'e maps, the geometry and dynamics of which can be described in detail using DGSPT.Comment: Updated to include minor corrections made during the review process (no major changes

Regelungstheorie

The workshop “Regelungstheorie” (control theory) covered a broad variety of topics that were either concerned with fundamental mathematical aspects of control or with its strong impact in various fields of engineering

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand

Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed