MATHEMATICS AS A LABORATORY TOOL dynamics, delays and noise.

Preface to the Second Edition
Preface to the First Edition
Acknowledgements for the Second Edition
Contents
Notation
Tools
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 Computer: Friend or Foe?
	4.5 Cubic nonlinearity: excitable cells
		4.5.1 The van der Pol Oscillator
		4.5.2 Fitzhugh–Nagumo equation
	4.6 Lyapunov's Insight
		4.6.1 Conservative Dynamical Systems
		4.6.2 Lyapunov's Direct Method
	4.7 What Have We Learned?
	4.8 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
	6.6 Neuron spiking thresholds
		6.6.1 Bounded Time-Dependent States
	6.7 What Have We Learned?
	6.8 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 Convolution using the FFT
		8.4.4 Power Spectrum: Discrete Time Signals
		8.4.5 Using FFTs in the Laboratory
		8.4.6 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: frequency domain approaches
	9.2 ``Linear'' Feedback
		9.2.1 Proportional–Integral–Derivative (PID) Controllers
	9.3 Feedback control: time-domain approaches
		9.3.1 Gene regulatory systems
	9.4 Production–Destruction
	9.5 Time-delayed feedback: Pupil Light Reflex (PLR)
		9.5.1 PLR: frequency-domain analysis
		9.5.2 PRL: time-domain analysis
	9.6 Clamping the PLR
	9.7 The future
	9.8 What Have We Learned?
	9.9 Exercises for Practice and Insight
10 Time delays
	10.1 Classification of DDEs
	10.2 Estimating time delays
	10.3 Delays versus lags
	10.4 Jump discontinuities
	10.5 Stability analysis: First-order DDEs
		10.5.1 Wright's Equation
		10.5.2 Hayes equation
		10.5.3 Positive Feedback: Cheyne–Stokes Respiration
	10.6 Stability analysis: Second-order DDEs
		10.6.1 Planar pendulum
		10.6.2 Stability analysis
	10.7 Human balancing
		10.7.1 The vibration paradox
	10.8 Intermittent control
		10.8.1 Event-driven intermittent control
		10.8.2 Switching feedback: clock driven
	10.9 Numerical Integration
		10.9.1 XPPAUT
		10.9.2 Method of steps (dde23, desolve, NDSolve)
		10.9.3 dde23 (matlab)
		10.9.4 Semi-discretization
	10.10 What have we learned?
	10.11 Exercises for Practice and Insight
11 Oscillations
	11.1 Hodgkin–Huxley Neuron Model
	11.2 Ion channels
	11.3 Excitability
		11.3.1 Type 1 excitability
		11.3.2 Type 2 excitability
		11.3.3 Type 3 excitability
	11.4 Creating and destroying limit cycles
		11.4.1 Oscillation onset bifurcations
		11.4.2 Oscillation destroying bifurcations
	11.5 AUTO
	11.6 Reduced Hodgkin–Huxley Models
		11.6.1 Wilson-type reduced neurons
		11.6.2 Izhikevich-type reduced neurons
	11.7 What Have We Learned?
	11.8 Exercises for Practice and Insight
12 Characterizing and Manipulating  Oscillations
	12.1 Poincaré Sections
		12.1.1 Gait Stride Variability
	12.2 Phase Resetting
		12.2.1 Stumbling
	12.3 Interacting Oscillators
		12.3.1 Periodic Forcing: Continuous stimulation
	12.4 Periodic forcing: Pulsatile stimulation
	12.5 An illustrative biological example
	12.6 Coupled Oscillators
	12.7 Dynamic clamping
	12.8 Switching bistable states
	12.9 What Have We Learned?
	12.10 Exercises for Practice and Insight
13 Beyond Limit Cycles
	13.1 Chaos and Parameters
	13.2 Dynamical Diseases and Parameters
	13.3 Chaos in the Laboratory
		13.3.1 Flour Beetle Cannibalism
		13.3.2 Clamped PLR with ``Mixed Feedback''
	13.4 Microchaos
		13.4.1 The quail map
		13.4.2 Microchaotic dynamics
		13.4.3 Microchaotic map
	13.5 Multistability
		13.5.1 Multistability: Integrate-and-fire models
		13.5.2 Multistability: Hodgkin–Huxley models with delayed recurrent loops
		13.5.3 Multistability: 2-neuron coupled networks with delay
	13.6 Delay-induced transient oscillations
	13.7 What Have We Learned?
	13.8 Exercises for Practice and Insight
14 Random Perturbations
	14.1 Noise and Dynamics
		14.1.1 Stick Balancing at the Fingertip
		14.1.2 Noise and Thresholds
		14.1.3 Stochastic Gene Expression
	14.2 Stochastic Processes Toolbox
		14.2.1 Random Variables and Their Properties
		14.2.2 Stochastic Processes
		14.2.3 Statistical Averages
	14.3 Bayesian Inferences
		14.3.1 Extensions to general Bayes' theorem
		14.3.2 Priori and Posteriori probabilities
		14.3.3 Bayesian Updates
		14.3.4 Monty Hall Problem
		14.3.5 Applications to neuroscience
	14.4 Laboratory Evaluations
		14.4.1 Intensity
		14.4.2 Estimation of the Probability Density Function
		14.4.3 Estimation of Moments
		14.4.4 Signal-averaging
		14.4.5 Broad-Shouldered Probability Density Functions
		14.4.6 Survival functions
	14.5 Correlation Functions
		14.5.1 Estimating Autocorrelation
	14.6 Power Spectrum of Stochastic Signals
	14.7 Examples of Noise
	14.8 Cross-Correlation Function
		14.8.1 Impulse Response
		14.8.2 Measuring Time Delays
		14.8.3 Coherence
	14.9 What Have We Learned?
	14.10 Problems for Practice and Insight
15 Noisy Dynamical Systems
	15.1 The Langevin Equation: Additive Noise
		15.1.1 The Retina As a Recording Device
		15.1.2 Time-Delayed Langevin Equation
		15.1.3 Skill Acquisition
		15.1.4 Stochastic Gene Expression
		15.1.5 Numerical Integration of Langevin Equations
	15.2 Noise and Thresholds
		15.2.1 Linearization
		15.2.2 Dithering
		15.2.3 Stochastic Resonance
	15.3 Parametric Noise
		15.3.1 On–Off Intermittency: Quadratic Map
		15.3.2 Langevin Equation with Parametric Noise
		15.3.3 Pole Balancing at the Fingertip
	15.4 What Have We Learned?
	15.5 Problems for Practice and Insight
16 Random Walks
	16.1 A Simple Random Walk
		16.1.1 Random walked diffusion
		16.1.2 Sampling
		16.1.3 Walking Molecules as Cellular Probes
	16.2 Random walk: Probability distribution function
	16.3 Random walk: Power spectrum
	16.4 Anomalous Random Walks
	16.5 Correlated Random Walks
		16.5.1 Random Walking While Standing Still
		16.5.2 Gait Stride Variability: DFA
		16.5.3 Walking on the DNA Sequence
	16.6 Portraits in Diffusion
		16.6.1 Finding a Mate
		16.6.2 Facilitated Diffusion on the Double Helix
	16.7 Optimal Search Patterns: Lévy Flights
	16.8 Fokker–Planck equation
		16.8.1 Master Equation
	16.9 Tool 12: Generating Functions
		16.9.1 Approaches Using the Characteristic Function
	16.10 Stable random walks
		16.10.1 Ehrenfest urn model
		16.10.2 Auto–Correlation function: Ehrenfest random walk
		16.10.3 Delayed Random Walks
	16.11 Postural sway
		16.11.1 Transient auto–correlation function
	16.12 Delayed Fokker–Planck equation
	16.13 What Have We Learned?
	16.14 Problems for Practice and Insight
17 Thermodynamic Perspectives
	17.1 Equilibrium
		17.1.1 Mathematical Background
		17.1.2 First and Second Laws of Thermodynamics
		17.1.3 Spontaneity
	17.2 Nonequilibrium
	17.3 Dynamics and Temperature
		17.3.1 Near Equilibrium
		17.3.2 Principle of Detailed Balancing
	17.4 Entropy Production
		17.4.1 Minimum Entropy Production
		17.4.2 Microscopic Entropy Production
	17.5 Far from Equilibrium
		17.5.1 The Sandpile Paradigm
		17.5.2 Power Laws
		17.5.3 Measuring Power Laws
		17.5.4 Epileptic Quakes
	17.6 What Have We Learned?
	17.7 Exercises for Practice and Insight
18 Concluding Remarks
References
Index