Biological Clocks, Rhythms, and Oscillations: The Theory of Biological Timekeeping

Contents
Preface
Notation
1 Basics
	1.1 Introduction
	1.2 Models
	1.3 Period
	1.4 Phase
	1.5 Frontiers: The Difficulty of Estimating the Phase and Amplitude of a Clock
	1.6 Plotting Circular Data
	1.7 Mathematical Preliminaries, Notations, and Basics
	1.8 Key Problems in the Autonomous Case
	1.9 Perturbations, Phase Response Curves, and Synchrony
	1.10 Key Problems when the External Signal u(t) ≠ 0
	1.11 Frontiers: Probability Distributions with a Focus on Circular Data
	1.12 Frontiers: Useful Statistics for Circular Data
	Exercises
I  MODELS
	2 Biophysical Mechanistic Modeling: Choosing the Right Model Equations
		2.1 Introduction
		2.2 Biochemical Modeling
		2.3 Law of Mass Action: When, Why, and How
		2.4 Frontiers: The Crowded Cellular Environment and Mass Action
		2.5 Three Mathematical Models of Transcription Regulation
		2.6 The Goodwin Model
		2.7 Other Models of Intracellular Processes (e.g., Michaelis-Menten)
		2.8 Frontiers: Bounding Solutions of Biochemical Models
		2.9 On Complex Formation
		2.10 Hodgkin–Huxley and Models of Neuronal Dynamics
		2.11 Frontiers: Rethinking the Ohm’s Law Linear Relationship between Voltage and Current
		2.12 Ten Common Mistakes to Watch for When Constructing Biochemical and Electrophysiological Models
		2.13 Interesting Future Work: Are All Cellular Oscillations Intertwined?
		Code 2.1 Spatial Effects
		Code 2.2 Biochemical Feedback Loops
		Code 2.3 The Hodgkin–Huxley Model
		Exercises
	3 Functioning in the Changing Cellular Environment
		3.1 Introduction
		3.2 Frontiers: Volume Changes
		3.3 Probabilistic Formulation of Deterministic Equations and Delay Equations
		3.4 The Discreteness of Chemical Reactions, Gillespie, and All That
		3.5 Frontiers: Temperature Compensation
		3.6 Frontiers: Crosstalk between Cellular Systems
		3.7 Common Mistakes in Modeling
		Code 3.1 Simulations of the Goodwin Model Using the Gillespie Method
		Code 3.2 Temperature Compensation Counterexample
		Code 3.3 A Black-Widow DNA-Diffusing Transcription Factor Model
		Exercises
	4 When Do Feedback Loops Oscillate?
		4.1 Introduction
		4.2 Introduction to Feedback Loops
		4.3 General Linear Methodology and Analysis of the Goodwin Model
		4.4 Frontiers: Futile Cycles Diminish Oscillations, or Why Clocks Like Efficient Complex Formation
		4.5 Example: Case Study on Familial Advanced Sleep Phase Syndrome
		4.6 Frontiers: An Additional Fast Positive Feedback Loop
		4.7 Example: Increasing Activator Concentrations in Circadian Clocks
		4.8 Bistability and Relaxation Oscillations
		4.9 Frontiers: Calculating the Period of Relaxation Oscillations
		4.10 Theory: The Global Secant Condition
		Code 4.1 Effects of Feedback
		Code 4.2 Effects of the Hill Coefficient on Rhythms in the Goodwin Model
		Exercises
II  BEHAVIORS
	5 System-Level Modeling
		5.1 Introduction
		5.2 General Remarks on Bifurcations
		5.3 SNIC or Type 1 Oscillators
		5.4 Examples of Type 1 Oscillators: Simplifications of the Hodgkin–Huxley Model
		5.5 Hopf or Type 2 Oscillators
		5.6 Examples of Type 2 Oscillators: The Van der Pol Oscillator and the Resonate-and-Fire Model
		5.7 Summary of Oscillator Classification
		5.8 Frontiers: Noise in Type 1 and Type 2 Oscillators
		5.9 Frontiers: Experimentally Testing the Effects of Noise in Squid Giant Axon
		5.10 Example: The Van der Pol Model and Modeling Human Circadian Rhythms
		5.11 Example: Refining the Human Circadian Model
		5.12 Example: A Simple Model of Sleep, Alertness, and Performance
		5.13 Frontiers: Equivalence of Neuronal and Biochemical Models
		Code 5.1 Simulation of Type 1 and Type 2 Behavior in the Morris-Lecar Model
		Exercises
	6 Phase Response Curves
		6.1 Introduction and General Properties of Phase Response Curves
		6.2 Type 1 Response to Brief Stimuli in Phase-Only Oscillators
		6.3 Perturbations to Type 2 Oscillators
		6.4 Instantaneous Perturbations to the Radial Isochron Clock
		6.5 Frontiers: Phase Resetting with Pathological Isochrons
		6.6 Phase Shifts for Weak Stimuli
		6.7 Frontiers: Phase Shifting in Models with More Than Two Dimensions
		6.8 Winfree’s Theory of Phase Resetting
		6.9 Experimental PRCs
		6.10 Entrainment
		Code 6.1 Calculating a Predicted Human PRC
		Code 6.2 Iterating PRCs
		Exercises
	7 Eighteen Principles of Synchrony
		7.1 Basics and Definitions of Synchrony
		7.2 Synchrony in Pulse-Coupled Oscillators
		7.3 Heterogeneous Oscillators
		7.4 Subharmonic and Superharmonic Synchrony
		7.5 Frontiers: The Counterintuitive Interplay between Noise and Coupling
		7.6 Nearest-Neighbor Coupling
		7.7 Frontiers: What Do We Gain by Looking at Limit-Cycle Oscillators?
		7.8 Coupling Damped Oscillators
		7.9 Amplitude Death and Beyond
		7.10 Theory: Proof of Synchrony in Homogeneous Oscillators
		Code 7.1 Two Coupled Biochemical Feedback Loops
		Code 7.2 Pulse-Coupled Oscillators
		Code 7.3 Inhibitory Pulse-Coupled Oscillators
		Code 7.4 Noisy Coupled Oscillators
		Code 7.5 Coupled Chain of Oscillators
		Code 7.6 Amplitude Death
		Exercises
III  ANALYSIS AND COMPUTATION
	8 Statistical and Computational Tools for Model Building: How to Extract Information from Timeseries Data
		8.1 How to Find Parameters of a Model
		8.2 Frontiers: Theoretical Limits on Fitting Timecourse Data
		8.3 Discrete Models, Noise, and Correlated Error
		8.4 Maximum Likelihood and Least-Squares
		8.5 The Kalman Filter
		8.6 Calculating Least-Squares
		8.7 Frontiers: Using the Kalman Filter for Problems with Correlated Errors
		8.8 Examples
		8.9 Theory: The Akaike Information Criterion
		8.10 A Final Word of Caution about Stationarity
		Code 8.1 Fitting Protein Data
		Exercises
	9 How to Shift an Oscillator Optimally
		9.1 Asking the Right Biological Questions
		9.2 Asking the Right Mathematical Questions
		9.3 Frontiers: A Geometric Interpretation of Optimality
		9.4 Influence Functions
		9.5 Frontiers: Two Additional Derivations of the Influence Functions
		9.6 Adding the Cost to the Hamiltonian
		9.7 Numerical Methods for Finding Optimal Stimuli
		9.8 Frontiers: Optimal Stimuli for the Hodgkin–Huxley Equation
		9.9 Examples: Analysis of Minimal Time Problems
		9.10 Example: Shifting the Human Circadian Clock
		Code 9.1 Optimal Stimulus for the Hodgkin–Huxley Equations
		Code 9.2 An Alternate Method to Calculate Optimal Stimuli for the Hodgkin–Huxley Model
		Exercises
	10 Mathematical and Computational Techniques for Multiscale Problems
		10.1 Simplifying Multiscale Systems
		10.2 Frontiers: Averaging in Systems with More Than Two Variables
		10.3 Frontiers: Piecewise Linear Approximations to Nonlinear Equations
		10.4 Frontiers: Poincaré Maps and Model Reduction
		10.5 Ruling out Limit Cycles
		Code 10.1 Five Simulations of the Goodwin Model
		Code 10.2 Poincaré Maps of a Detailed Mammalian Model
		Code 10.3 Chaotic Motions
		Exercises
Glossary
Bibliography
Index