University of Groningen
Modeling, analysis, and control of biological oscillators
Taghvafard, Hadi
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Publication date: 2018
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Taghvafard, H. (2018). Modeling, analysis, and control of biological oscillators. University of Groningen.
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Summary
This book is devoted to the study of rhythms, so-called “oscillators”. In particular, it is concerned with modeling, analysis, and control of biological oscillators. It is divided into two parts, where Part I is devoted to the application of control theory to endocrinology, and Part II is allocated to the application of dynamical systems to microbiology.
Part I develops three mathematical models of endocrine regulation. The first model, which is a second-order impulsive differential equation, describes the cortisol’s diurnal patterns. Through an analytical approach, we design an impulsive controller to identify the timing and amplitude of secretory events, while the blood cortisol levels are restricted to a specific circadian range. By proposing an algorithm and employing it into various examples, we show that the achieved cortisol levels lead to the circadian and ultradian rhythms, which are in line with the known physiology of cortisol secretion.
The second model, which is a third-order ordinary differential equation, gener-ally describes the control mechanisms in the hypothalamic-pituitary axes, controlled by the brain. For this model, which is an extension of the conventional Goodwin’s oscillator with an additional nonlinear feedback, we establish the relationship between its local behavior at the equilibrium point and its global behavior, i.e., the convergence of solutions to periodic orbits.
The last model, which is a third-order impulsive differential equation, describes the pulsatile secretion of the hypothalamic-pituitary axes. This model, obtained from an impulsive version of the Goodwin’s oscillator, has an additional affine feedback. For this model, we present conditions for the existence, uniqueness, and positivity of a type of periodic solution.
Part II studies a biochemical oscillator model (known as “Frzilator”), which describes the social-behavior transition phase of myxobacteria, a kind of soil bacteria. This part studies the Frzilator from two different perspectives, namely, regular perturbation, and geometric singular perturbation, respectively. Using regular
172 Summary
perturbation theory, we investigate parameter-robustness analysis of the Frzilator. In particular, after identifying and unifying some small parameters of the system, we establish the relation between its local and global behavior at the equilibrium point. Moreover, we explicitly give certain parameter regimes in which solutions of the system converge to a finite number of periodic orbits.
Using geometric singular perturbation theory and the blow-up method, we analyze the dynamics of the Frzilator in the limit of small parameters of the system. We prove that, within certain parameter regimes, there exists a strongly attracting periodic orbit for the system. Moreover, we give the detailed description of the structure of such an orbit as well as the timescales along it. The existence of multiple time scales along the orbit demonstrates that the Frzilator is a relaxation oscillator.
