Intro
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