MATHEMATICS AS A LABORATORY TOOL dynamics, delays and noise.

Description

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