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

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Dynamical 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.