Modeling, analysis, and control of biological oscillators

This thesis 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 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.

The second model generally 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.

The last model 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.