Mathematics as a Laboratory Tool - Dynamics, Delays and Noise

Preface
	Laboratory Exercises and Projects
	Acknowledgments
Notation
Tools
Contents
1 Science and the Mathematics of Black Boxes
	1.1 The Scientific Method
	1.2 Dynamical Systems
		1.2.1 Variables
		1.2.2 Measurements
		1.2.3 Units
	1.3 Input–Output Relationships
		1.3.1 Linear Versus Nonlinear Black Boxes
		1.3.2 The Neuron as a Dynamical System
	1.4 Interactions Between System and Surroundings
	1.5 What Have We Learned?
	1.6 Exercises for Practice and Insight
2 The Mathematics of Change
	2.1 Differentiation
	2.2 Differential Equations
		2.2.1 Population Growth
		2.2.2 Time Scale of Change
		2.2.3 Linear ODEs with Constant Coefficients
	2.3 Black Boxes
		2.3.1 Nonlinear Differential Equations
	2.4 Existence and Uniqueness
	2.5 What Have We Learned?
	2.6 Exercises for Practice and Insight
3 Equilibria and Steady States
	3.1 Law of Mass Action
	3.2 Closed Dynamical Systems
		3.2.1 Equilibria: Drug Binding
		3.2.2 Transient Steady States: Enzyme Kinetics
	3.3 Open Dynamical Systems
		3.3.1 Water Fountains
	3.4 The ``Steady-State Approximation''
		3.4.1 Steady State: Enzyme–Substrate Reactions
		3.4.2 Steady State: Consecutive Reactions
	3.5 Existence of Fixed Points
	3.6 What Have We learned?
	3.7 Exercises for Practice and Insight
4 Stability
	4.1 Landscapes in Stability
		4.1.1 Postural Stability
		4.1.2 Perception of Ambiguous Figures
		4.1.3 Stopping Epileptic Seizures
	4.2 Fixed-Point Stability
	4.3 Stability of Second-Order ODEs
		4.3.1 Real Eigenvalues
		4.3.2 Complex Eigenvalues
		4.3.3 Phase-Plane Representation
	4.4 Illustrative Examples
		4.4.1 The Lotka–Volterra Equation
		4.4.2 The van der Pol Oscillator
		4.4.3 Computer: Friend or Foe?
	4.5 Lyapunov's Insight
		4.5.1 Conservative Dynamical Systems
		4.5.2 Lyapunov's Direct Method
	4.6 What Have We Learned?
	4.7 Exercises for Practice and Insight
5 Fixed Points: Creation and Destruction
	5.1 Saddle-Node Bifurcation
		5.1.1 Neuron Bistability
	5.2 Transcritical Bifurcation
		5.2.1 Postponement of Instability
	5.3 Pitchfork Bifurcation
		5.3.1 Finger-Spring Compressions
	5.4 Near the Bifurcation Point
		5.4.1 The Slowing-Down Phenomenon
		5.4.2 Critical Phenomena
	5.5 Bifurcations at the Benchtop
	5.6 What Have We Learned?
	5.7 Exercises for Practice and Insight
6 Transient Dynamics
	6.1 Step Functions
	6.2 Ramp Functions
	6.3 Impulse Responses
		6.3.1 Measuring the Impulse Response
	6.4 The Convolution Integral
		6.4.1 Summing Neuronal Inputs
	6.5 Transients in Nonlinear Dynamical Systems
		6.5.1 Excitability
		6.5.2 Bounded Time-Dependent States
	6.6 What Have We Learned?
	6.7 Exercises for Practice and Insight
7 Frequency Domain I: Bode Plots and Transfer Functions
	7.1 Low-Pass Filters
	7.2 Laplace Transform Toolbox
		7.2.1 Tool 8: Euler's Formula
		7.2.2 Tool 9: The Laplace Transform
	7.3 Transfer Functions
	7.4 Biological Filters
		7.4.1 Crayfish Photoreceptor
		7.4.2 Osmoregulation in Yeast
	7.5 Bode-Plot Cookbook
		7.5.1 Constant Term (Gain)
		7.5.2 Integral or Derivative Term
		7.5.3 Lags and Leads
		7.5.4 Time Delays
		7.5.5 Complex Pair: Amplification
	7.6 Interpreting Biological Bode Plots
		7.6.1 Crayfish Photoreceptor Transfer Function
		7.6.2 Yeast Osmotic Stress Transfer Function
	7.7 What Have We Learned?
	7.8 Exercises for Practice and Insight
8 Frequency Domain II: Fourier Analysis and Power Spectra
	8.1 Laboratory Applications
		8.1.1 Creating Sinusoidal Inputs
		8.1.2 Electrical Interference
		8.1.3 The Electrical World
	8.2 Fourier Analysis Toolbox
		8.2.1 Tool 10: Fourier Series
		8.2.2 Tool 11: The Fourier Transform
	8.3 Applications of Fourier Analysis
		8.3.1 Uncertainties in Fourier Analysis
		8.3.2 Approximating a Square Wave
		8.3.3 Power Spectrum
		8.3.4 Fractal Heartbeats
	8.4 Digital Signals
		8.4.1 Low-Pass Filtering
		8.4.2 Introducing the FFT
		8.4.3 Power Spectrum: Discrete Time Signals
		8.4.4 Using FFTs in the Laboratory
		8.4.5 Power Spectral Density Recipe
	8.5 What Have We Learned?
	8.6 Exercises for Practice and Insight
9 Feedback and Control Systems
	9.1 Feedback Control
	9.2 Pupil Light Reflex (PLR)
	9.3 ``Linear'' Feedback
		9.3.1 Proportional-Integral-Derivative (PID) Controllers
	9.4 Delayed Feedback
		9.4.1 Example: Wright's Equation
	9.5 Production–Destruction
		9.5.1 Negative Feedback: PLR
		9.5.2 Negative Feedback: Gene Regulatory Systems
		9.5.3 Stability: General Forms
		9.5.4 Positive Feedback: Cheyne–Stokes Respiration
	9.6 Electronic Feedback
		9.6.1 The Clamped PLR
	9.7 Intermittent Control
		9.7.1 Virtual Balancing
	9.8 Integrating DDEs
	9.9 What Have We Learned?
	9.10 Exercises for Practice and Insight
10 Oscillations
	10.1 Neural Oscillations
		10.1.1 Hodgkin–Huxley Neuron Model
	10.2 Hopf Bifurcations
		10.2.1 PLR: Supercritical Hopf Bifurcation
		10.2.2 HH Equation: Hopf Bifurcations
	10.3 Analyzing Oscillations
	10.4 Poincaré Sections
		10.4.1 Gait Stride Variability
	10.5 Phase Resetting
		10.5.1 Stumbling
	10.6 Interacting Oscillators
		10.6.1 Periodic Forcing
		10.6.2 Coupled Oscillators
	10.7 What Have We Learned?
	10.8 Exercises for Practice and Insight
11 Beyond Limit Cycles
	11.1 Chaos and Parameters
	11.2 Dynamical Diseases and Parameters
	11.3 Chaos in the Laboratory
		11.3.1 Flour Beetle Cannibalism
		11.3.2 Clamped PLR with ``Mixed Feedback''
	11.4 Complex Dynamics: The Human Brain
		11.4.1 Reduced Hodgkin–Huxley Models
		11.4.2 Delay-Induced Transient Oscillations
	11.5 What Have We Learned?
	11.6 Exercises for Practice and Insight
12 Random Perturbations
	12.1 Noise and Dynamics
		12.1.1 Stick Balancing at the Fingertip
		12.1.2 Noise and Thresholds
		12.1.3 Stochastic Gene Expression
	12.2 Stochastic Processes Toolbox
		12.2.1 Random Variables and Their Properties
		12.2.2 Stochastic Processes
		12.2.3 Statistical Averages
	12.3 Laboratory Evaluations
		12.3.1 Intensity
		12.3.2 Estimation of the Probability Density Function
		12.3.3 Estimation of Moments
		12.3.4 Broad-Shouldered Probability Density Functions
	12.4 Correlation Functions
		12.4.1 Estimating Autocorrelation
	12.5 Power Spectrum of Stochastic Signals
	12.6 Examples of Noise
	12.7 Cross-Correlation Function
		12.7.1 Impulse Response
		12.7.2 Measuring Time Delays
		12.7.3 Coherence
	12.8 What Have We Learned?
	12.9 Problems for Practice and Insight
13 Noisy Dynamical Systems
	13.1 The Langevin Equation: Additive Noise
		13.1.1 The Retina As a Recording Device
		13.1.2 Time-Delayed Langevin Equation
		13.1.3 Skill Acquisition
		13.1.4 Stochastic Gene Expression
		13.1.5 Numerical Integration of Langevin Equations
	13.2 Noise and Thresholds
		13.2.1 Linearization
		13.2.2 Dithering
		13.2.3 Stochastic Resonance
	13.3 Parametric Noise
		13.3.1 On–Off Intermittency: Quadratic Map
		13.3.2 Langevin Equation with Parametric Noise
		13.3.3 Stick Balancing at the Fingertip
	13.4 What Have We Learned?
	13.5 Problems for Practice and Insight
14 Random Walks
	14.1 A Simple Random Walk
		14.1.1 Walking Molecules as Cellular Probes
	14.2 Anomalous Random Walks
	14.3 Correlated Random Walks
		14.3.1 Random Walking While Standing Still
		14.3.2 Gait Stride Variability: DFA
		14.3.3 Walking on the DNA Sequence
	14.4 Delayed Random Walks
	14.5 Portraits in Diffusion
		14.5.1 Finding a Mate
		14.5.2 Facilitated Diffusion on the Double Helix
	14.6 Optimal Search Patterns: Lévy Flights
	14.7 Probability Density Functions
		14.7.1 Master Equation
	14.8 Tool 12: Generating Functions
		14.8.1 Approaches Using the Characteristic Function
	14.9 What Have We Learned?
	14.10 Problems for Practice and Insight
15 Thermodynamic Perspectives
	15.1 Equilibrium
		15.1.1 Mathematical Background
		15.1.2 First and Second Laws of Thermodynamics
		15.1.3 Spontaneity
	15.2 Nonequilibrium
	15.3 Dynamics and Temperature
		15.3.1 Near Equilibrium
		15.3.2 Principle of Detailed Balancing
	15.4 Entropy Production
		15.4.1 Minimum Entropy Production
		15.4.2 Microscopic Entropy Production
	15.5 Far from Equilibrium
		15.5.1 The Sandpile Paradigm
		15.5.2 Power Laws
		15.5.3 Measuring Power Laws
		15.5.4 Epileptic Quakes
	15.6 What Have We Learned?
	15.7 Exercises for Practice and Insight
16 Concluding Remarks
References
Index