Have you thought of why you get tired or why you get hungry? Something in your body keeps track of time. It is almost like you have a clock that tells you all those things. And indeed, in the suparachiasmatic region of our hypothalamus reside cells which each act like an oscillator, and together form a coherent circadian rhythm to help our body keep track of time. In fact, such circadian clocks are not limited to mammals but can be found in many organisms including single-cell, reptiles and birds. The study of such rhythms constitutes a field of biology, chronobiology, and forms the background for my research and this thesis. Pioneers of **chronobiology**, Pittendrigh and Aschoff, studied biological clocks from an input-output view, across a range of organisms by observing and analyzing their overt activity in response to stimulus such as light. Their study was made without recourse to knowledge of the **biological **underpinnings of the circadian pacemaker. The advent of the new biology has now made it possible to “break open the box” and identify biological feedback systems comprised of gene transcription and protein translation as the core mechanism of a biological clock. My research has focused on a simple transcription-translation clock model which nevertheless possesses many of the features of a circadian pacemaker including its entrainability by light. This model consists of two nonlinear coupled and delayed differential equations…

*Contents*

I Introduction

1 Introduction

2 Model of the Transcriptional-Translational Process

3 Analysis Ideas

4 Motivation

5 Previous Work

6 Future Research

7 Explanation of Frequently Used Expressions

References

II Papers

A Describing Function Analysis

1 Introduction

2 Background

3 A Dual-Input Describing Function Analysis – Considering the Bias and the First Harmonic

3.1 Discussion

4 Multiple-Input Describing Function Analysis – Considering the Bias,

the First and the Second Harmonic

4.1 Discussion

A Derivation of the Dual-Input Describing Function for a General

Case – Considering the Bias and the First Harmonic

B Derivation of the Multiple-Input Describing Function for a General

Case – Considering the Bias, First and the Second Harmonic

References

B Multiple Scales Analysis

1 Introduction

2 Model of the Transcriptional-Translational Process

3 Dimensional Analysis and Scaling

4 The Equilibrium Solution

5 Simpliﬁcations

ix6 Stability of the Equilibrium Solution

7 Periodic Orbits

8 Numerical Analysis

8.1 Analysis Using DDE-biftool

9 Two-Scales Expansion

10 Discussion

References

C Multiple Scales’s Dependency of Linear Coordinate Transforma-tions

1 Introduction

2 The Method of Multiple Scales Applied to the van Der Pol Oscillator

3 A System With a Hopf Bifurcation

4 Conclusion

References

Author: Ohlsson, Henrik

Source: Linköping University

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