## Biological Clocks, Rhythms, and Oscillations

by Forger

### Instructor Requests

All areas of biology and medicine contain rhythms, and these behaviors are best understood through mathematical tools and techniques. This book offers a survey of mathematical, computational, and analytical techniques used for modeling biological rhythms, gathering these methods for the first time in one volume. Drawing on material from such disciplines as mathematical biology, nonlinear dynamics, physics, statistics, and engineering, it presents practical advice and techniques for studying biological rhythms, with a common language.

The chapters proceed with increasing mathematical abstraction. Part I, on models, highlights the implicit assumptions and common pitfalls of modeling, and is accessible to readers with basic knowledge of differential equations and linear algebra. Part II, on behaviors, focuses on simpler models, describing common properties of biological rhythms that range from the firing properties of squid giant axon to human circadian rhythms. Part III, on mathematical techniques, guides readers who have specific models or goals in mind. Sections on "frontiers" present the latest research; "theory" sections present interesting mathematical results using more accessible approaches than can be found elsewhere. Each chapter offers exercises. Commented MATLAB code is provided to help readers get practical experience.

The book, by an expert in the field, can be used as a textbook for undergraduate courses in mathematical biology or graduate courses in modeling biological rhythms and as a reference for researchers.

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Contents (pg. vii)
Preface (pg. xiii)
Notation (pg. xvii)
1 Basics (pg. 1)
1.1 Introduction (pg. 1)
1.2 Models (pg. 11)
1.3 Period (pg. 16)
1.4 Phase (pg. 16)
1.5 Frontiers: The Difficulty of Estimating the Phase and Amplitude of a Clock (pg. 17)
1.6 Plotting Circular Data (pg. 19)
1.7 Mathematical Preliminaries, Notations, and Basics (pg. 19)
1.8 Key Problems in the Autonomous Case (pg. 23)
1.9 Perturbations, Phase Response Curves, and Synchrony (pg. 23)
1.10 Key Problems when the External Signal u(t) ≠ 0 (pg. 24)
1.11 Frontiers: Probability Distributions with a Focus on Circular Data (pg. 26)
1.12 Frontiers: Useful Statistics for Circular Data (pg. 30)
Exercises (pg. 32)
I MODELS (pg. 35)
2 Biophysical Mechanistic Modeling: Choosing the Right Model Equations (pg. 37)
2.1 Introduction (pg. 37)
2.2 Biochemical Modeling (pg. 38)
2.3 Law of Mass Action: When, Why, and How (pg. 39)
2.4 Frontiers: The Crowded Cellular Environment and Mass Action (pg. 41)
2.5 Three Mathematical Models of Transcription Regulation (pg. 42)
2.6 The Goodwin Model (pg. 48)
2.7 Other Models of Intracellular Processes (e.g., Michaelis-Menten) (pg. 49)
2.8 Frontiers: Bounding Solutions of Biochemical Models (pg. 53)
2.9 On Complex Formation (pg. 54)
2.10 Hodgkin–Huxley and Models of Neuronal Dynamics (pg. 55)
2.11 Frontiers: Rethinking the Ohm’s Law Linear Relationship between Voltage and Current (pg. 63)
2.12 Ten Common Mistakes to Watch for When Constructing Biochemical and Electrophysiological Models (pg. 64)
2.13 Interesting Future Work: Are All Cellular Oscillations Intertwined? (pg. 65)
Code 2.1 Spatial Effects (pg. 65)
Code 2.2 Biochemical Feedback Loops (pg. 67)
Code 2.3 The Hodgkin–Huxley Model (pg. 68)
Exercises (pg. 69)
3 Functioning in the Changing Cellular Environment (pg. 73)
3.1 Introduction (pg. 73)
3.2 Frontiers: Volume Changes (pg. 73)
3.3 Probabilistic Formulation of Deterministic Equations and Delay Equations (pg. 75)
3.4 The Discreteness of Chemical Reactions, Gillespie, and All That (pg. 78)
3.5 Frontiers: Temperature Compensation (pg. 84)
3.6 Frontiers: Crosstalk between Cellular Systems (pg. 92)
3.7 Common Mistakes in Modeling (pg. 93)
Code 3.1 Simulations of the Goodwin Model Using the Gillespie Method (pg. 94)
Code 3.2 Temperature Compensation Counterexample (pg. 96)
Code 3.3 A Black-Widow DNA-Diffusing Transcription Factor Model (pg. 97)
Exercises (pg. 98)
4 When Do Feedback Loops Oscillate? (pg. 101)
4.1 Introduction (pg. 101)
4.2 Introduction to Feedback Loops (pg. 102)
4.3 General Linear Methodology and Analysis of the Goodwin Model (pg. 104)
4.4 Frontiers: Futile Cycles Diminish Oscillations, or Why Clocks Like Efficient Complex Formation (pg. 109)
4.5 Example: Case Study on Familial Advanced Sleep Phase Syndrome (pg. 112)
4.6 Frontiers: An Additional Fast Positive Feedback Loop (pg. 115)
4.7 Example: Increasing Activator Concentrations in Circadian Clocks (pg. 118)
4.8 Bistability and Relaxation Oscillations (pg. 119)
4.9 Frontiers: Calculating the Period of Relaxation Oscillations (pg. 122)
4.10 Theory: The Global Secant Condition (pg. 124)
Code 4.1 Effects of Feedback (pg. 126)
Code 4.2 Effects of the Hill Coefficient on Rhythms in the Goodwin Model (pg. 127)
Exercises (pg. 128)
II BEHAVIORS (pg. 131)
5 System-Level Modeling (pg. 133)
5.1 Introduction (pg. 133)
5.2 General Remarks on Bifurcations (pg. 134)
5.3 SNIC or Type 1 Oscillators (pg. 136)
5.4 Examples of Type 1 Oscillators: Simplifications of the Hodgkin–Huxley Model (pg. 139)
5.5 Hopf or Type 2 Oscillators (pg. 142)
5.6 Examples of Type 2 Oscillators: The Van der Pol Oscillator and the Resonate-and-Fire Model (pg. 145)
5.7 Summary of Oscillator Classification (pg. 150)
5.8 Frontiers: Noise in Type 1 and Type 2 Oscillators (pg. 150)
5.9 Frontiers: Experimentally Testing the Effects of Noise in Squid Giant Axon (pg. 156)
5.10 Example: The Van der Pol Model and Modeling Human Circadian Rhythms (pg. 157)
5.11 Example: Refining the Human Circadian Model (pg. 159)
5.12 Example: A Simple Model of Sleep, Alertness, and Performance (pg. 163)
5.13 Frontiers: Equivalence of Neuronal and Biochemical Models (pg. 165)
Code 5.1 Simulation of Type 1 and Type 2 Behavior in the Morris-Lecar Model (pg. 167)
Exercises (pg. 168)
6 Phase Response Curves (pg. 171)
6.1 Introduction and General Properties of Phase Response Curves (pg. 171)
6.2 Type 1 Response to Brief Stimuli in Phase-Only Oscillators (pg. 173)
6.3 Perturbations to Type 2 Oscillators (pg. 175)
6.4 Instantaneous Perturbations to the Radial Isochron Clock (pg. 177)
6.5 Frontiers: Phase Resetting with Pathological Isochrons (pg. 182)
6.6 Phase Shifts for Weak Stimuli (pg. 183)
6.7 Frontiers: Phase Shifting in Models with More Than Two Dimensions (pg. 185)
6.8 Winfree’s Theory of Phase Resetting (pg. 186)
6.9 Experimental PRCs (pg. 187)
6.10 Entrainment (pg. 188)
Code 6.1 Calculating a Predicted Human PRC (pg. 190)
Code 6.2 Iterating PRCs (pg. 191)
Exercises (pg. 192)
7 Eighteen Principles of Synchrony (pg. 195)
7.1 Basics and Definitions of Synchrony (pg. 195)
7.2 Synchrony in Pulse-Coupled Oscillators (pg. 199)
7.3 Heterogeneous Oscillators (pg. 199)
7.4 Subharmonic and Superharmonic Synchrony (pg. 205)
7.5 Frontiers: The Counterintuitive Interplay between Noise and Coupling (pg. 206)
7.6 Nearest-Neighbor Coupling (pg. 209)
7.7 Frontiers: What Do We Gain by Looking at Limit-Cycle Oscillators? (pg. 211)
7.8 Coupling Damped Oscillators (pg. 213)
7.9 Amplitude Death and Beyond (pg. 214)
7.10 Theory: Proof of Synchrony in Homogeneous Oscillators (pg. 214)
Code 7.1 Two Coupled Biochemical Feedback Loops (pg. 218)
Code 7.2 Pulse-Coupled Oscillators (pg. 218)
Code 7.3 Inhibitory Pulse-Coupled Oscillators (pg. 219)
Code 7.4 Noisy Coupled Oscillators (pg. 220)
Code 7.5 Coupled Chain of Oscillators (pg. 221)
Code 7.6 Amplitude Death (pg. 222)
Exercises (pg. 223)
III ANALYSIS AND COMPUTATION (pg. 225)
8 Statistical and Computational Tools for Model Building: How to Extract Information from Timeseries Data (pg. 227)
8.1 How to Find Parameters of a Model (pg. 227)
8.2 Frontiers: Theoretical Limits on Fitting Timecourse Data (pg. 236)
8.3 Discrete Models, Noise, and Correlated Error (pg. 240)
8.4 Maximum Likelihood and Least-Squares (pg. 243)
8.5 The Kalman Filter (pg. 244)
8.6 Calculating Least-Squares (pg. 246)
8.7 Frontiers: Using the Kalman Filter for Problems with Correlated Errors (pg. 247)
8.8 Examples (pg. 249)
8.9 Theory: The Akaike Information Criterion (pg. 252)
8.10 A Final Word of Caution about Stationarity (pg. 254)
Code 8.1 Fitting Protein Data (pg. 255)
Exercises (pg. 258)
9 How to Shift an Oscillator Optimally (pg. 261)
9.1 Asking the Right Biological Questions (pg. 261)
9.2 Asking the Right Mathematical Questions (pg. 265)
9.3 Frontiers: A Geometric Interpretation of Optimality (pg. 267)
9.4 Influence Functions (pg. 271)
9.5 Frontiers: Two Additional Derivations of the Influence Functions (pg. 273)
9.6 Adding the Cost to the Hamiltonian (pg. 275)
9.7 Numerical Methods for Finding Optimal Stimuli (pg. 276)
9.8 Frontiers: Optimal Stimuli for the Hodgkin–Huxley Equation (pg. 277)
9.9 Examples: Analysis of Minimal Time Problems (pg. 279)
9.10 Example: Shifting the Human Circadian Clock (pg. 281)
Code 9.1 Optimal Stimulus for the Hodgkin–Huxley Equations (pg. 282)
Code 9.2 An Alternate Method to Calculate Optimal Stimuli for the Hodgkin–Huxley Model (pg. 286)
Exercises (pg. 288)
10 Mathematical and Computational Techniques for Multiscale Problems (pg. 291)
10.1 Simplifying Multiscale Systems (pg. 291)
10.2 Frontiers: Averaging in Systems with More Than Two Variables (pg. 303)
10.3 Frontiers: Piecewise Linear Approximations to Nonlinear Equations (pg. 307)
10.4 Frontiers: Poincaré Maps and Model Reduction (pg. 310)
10.5 Ruling out Limit Cycles (pg. 311)
Code 10.1 Five Simulations of the Goodwin Model (pg. 315)
Code 10.2 Poincaré Maps of a Detailed Mammalian Model (pg. 317)
Code 10.3 Chaotic Motions (pg. 319)
Exercises (pg. 320)
Glossary (pg. 323)
Bibliography (pg. 329)
Index (pg. 341)

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