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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Acknowledgments
The material included in this thesis is related to the research I performed at the University of Groningen (UG). If I made any progress during my education and in particular in the past four years of my PhD, it would not have been accomplished by myself solely. I am greatly indebted to my family, in particular, my parents for their unconditional selfless support, and also my experienced mentors who guided me through obstacles; I sincerely dedicate this thesis to them.
Taking into account the past four years as the last part of my education, I am most indebted to my first advisor, Prof. Ming Cao, who taught me different aspects of scientific research and pointed out my shortcomings and mistakes. I highly appreciate his freedom in choosing research topics, his constructive comments on my drafts, and his career suggestions. I also do like to thank my second advisor, Prof. Jacquelien Scherpen, for all her supports and encouragements and also her management of both the department and the institute.
During my PhD, I mostly discussed and collaborated with† Dr. Hildeberto
Jardón-Kojakhmetov (UG) and Dr. Anton Proskurnikov (UG & TU Delft). I greatly appreciate both of them for all the discussions and also for their time and effort in reading and providing invaluable comments on my drafts. In addition, I am very thankful to Dr. Rose Faghih (MIT & University of Houston) for all our discussions through email, leading to Chapter 3 of this thesis.
I do like to thank Prof. Peter Szmolyan (TU Wien) from whose constructive comments on Chapter 7 of this thesis I greatly benefited during my visit at TU Wien and also from our discussions through email.
I would like to thank my thesis committee members†, Prof. Rick Middleton
(University of Newcastle), Prof. Rodolphe Sepulchre (University of Cambridge), and Prof. Arjan van der Schaft (UG) for their time and effort in assessing this thesis and providing valuable comments.
†alphabetical order
I also want to point out my gratitude to† Prof. Brian D.O. Anderson (Australian
National University), Prof. Henk Broer (UG), Dr. Yu Kawano (UG), Prof. Alexander Medvedev (Uppsala University), and Dr. Andreas Milias Argeitis (UG) for several discussions.
Lastly, I would like to thank all my former and current colleagues and friends at UG, and also those people who indirectly contributed to this thesis. In particular, I am grateful to†Alain Govaert and Dr. Qingkai Yang who will be my paranymphs on
my defense day. Furthermore, thanks to Alain Govaert for translating the summary of this thesis into Dutch.
Hadi Taghvafard Groningen, June 2018
Contents
1 Introduction 1
1.1 Research context . . . 2
1.2 Applications . . . 4
1.2.1 Part I: Application of control theory to endocrinology . . . . 4
1.2.2 Part II: Application of dynamical systems to microbiology . . 6
1.3 Contributions . . . 8
1.3.1 Related publications . . . 11
1.4 Outline of the thesis . . . 11
2 Preliminaries 13 2.1 Basic definitions . . . 13
2.2 Bifurcation theory . . . 16
2.3 Regular versus singular perturbation . . . 17
2.4 Slow-fast systems . . . 18
2.5 Fenichel theory . . . 20
2.6 Blow-up method . . . 22
I
Application of Control Theory to Endocrinology
25
3 Design of intermittent control for cortisol secretion 27 3.1 Introduction . . . 27 3.2 Methods . . . 29 3.3 Results . . . 34 3.3.1 Example 1 . . . 35 3.3.2 Example 2 . . . 37 3.3.3 Example 3 . . . 39 3.4 Discussion . . . 41 ix3.5 Concluding remarks . . . 43
4 Endocrine regulation as a non-cyclic feedback system 45 4.1 Introduction . . . 46
4.2 Goodwin’s model and its extension . . . 48
4.3 Equilibria and local stability properties . . . 50
4.4 Oscillatory properties of solutions . . . 53
4.4.1 Yakubovich-oscillatory solutions . . . 54
4.4.2 The structure of ω-limit set . . . 54
4.5 Numerical simulation . . . 56
4.6 Proof of the results . . . 58
4.7 Concluding remarks . . . 63
5 Impulsive model of endocrine regulation with a local continuous feed-back 65 5.1 Introduction . . . 65
5.2 Extended mathematical model . . . 68
5.3 Main result . . . 70
5.4 Methods . . . 72
5.4.1 Transformation of the system . . . 72
5.4.2 Periodic solutions . . . 74
5.4.3 One-to-one correspondence between 1-cycle solutions . . . 76
5.4.4 Proof of the results . . . 76
5.5 Concluding remarks . . . 81
II
Application of Dynamical Systems to Microbiology
83
6 Parameter-robustness analysis from regular-perturbation perspective 85 6.1 Introduction . . . 856.2 System description . . . 86
6.3 Local analysis . . . 90
6.4 Hopf bifurcation analysis . . . 95
6.5 Global behavior of solutions . . . 98
6.6 On the robustness of bifurcation with respect to parameter changes 103 6.7 Concluding remarks . . . 105
7 Relaxation oscillations in a slow-fast system beyond the standard form107 7.1 Introduction . . . 107
7.2 Preliminary analysis . . . 108
7.2.1 Basic properties and sustained oscillations . . . 108
7.3 Geometric singular perturbation analysis . . . 112
7.3.1 Layer problem and critical manifold . . . 114
7.3.2 Reduced problem, slow manifolds, and slow dynamics . . . 116
7.3.3 Singular cycle . . . 123
7.3.4 Main result . . . 125
7.4 Blow-up analysis . . . 128
7.4.1 Blow-up of the non-hyperbolic line `1× {0} . . . 129
7.4.2 Blow-up of the non-hyperbolic line `2× {0} . . . 146
7.5 Range of parameter γ in Theorem 7.26 . . . 148
7.6 Conclusions . . . 149
8 Conclusions and future research 151 8.1 Conclusions . . . 151
8.1.1 Conclusions of Part I . . . 152
8.1.2 Conclusions of Part II . . . 153
8.2 Directions for future research . . . 154
8.2.1 Potential research directions for Part I . . . 154
8.2.2 Potential research directions for Part II . . . 154
Bibliography 156
Summary 171
