Börgers C. An Introduction to Modeling Neuronal Dynamics

Springer International Publishing AG, 2017. — 457 p. — (Texts in Applied Mathematics 66) — ISBN-10 331951170X.This book is intended as a text for a one-semester course on Mathematical and Computational Neuroscience for upper-level undergraduate and beginning graduate students of mathematics, the natural sciences, engineering, or computer science. An undergraduate introduction to differential equations is more than enough mathematical background. Only a slim, high school-level background in physics is assumed, and none in biology.
Topics include models of individual nerve cells and their dynamics, models of networks of neurons coupled by synapses and gap junctions, origins and functions of population rhythms in neuronal networks, and models of synaptic plasticity.
An extensive online collection of MatLAB programs generating the figures accompanies the book.Vocabulary and Notation
The Nernst Equilibrium
The Classical Hodgkin-Huxley ODEs
Numerical Solution of the Hodgkin-Huxley ODEs
Three Simple Models of Neurons in Rodent Brains
The Classical Hodgkin-Huxley PDEs
Linear Integrate-and-Fire (LIF) Neurons
Quadratic Integrate-and-Fire (QIF) and Theta Neurons
Spike Frequency Adaptation
Dynamics of Single Neuron Models
The Slow-Fast Phase Plane
Saddle-Node Collisions
Model Neurons of Bifurcation Type 1
Hopf Bifurcations
Model Neurons of Bifurcation Type 2
Canard Explosions
Model Neurons of Bifurcation Type 3
Frequency-Current Curves
Bistability Resulting from Rebound Firing
Bursting
Modeling Neuronal Communication
Chemical Synapses
Gap Junctions
A Wilson-Cowan Model of an Oscillatory E-I Network
Entrainment, Synchronization, and Oscillations
Entrainment by Excitatory Input Pulses
Synchronization by Fast Recurrent Excitation
Phase Response Curves (PRCs)
Synchronization of Two Pulse-Coupled Oscillators
Oscillators Coupled by Delayed Pulses
Weakly Coupled Oscillators
Approximate Synchronization by a Single Inhibitory Pulse
The PING Model of Gamma Rhythms
ING Rhythms
Weak PING Rhythms
Beta Rhythms
Nested Gamma-Theta Rhythms
Functional Significance of Synchrony and Oscillations
Rhythmic vs. Tonic Inhibition
Rhythmic vs. Tonic Excitation
Gamma Rhythms and Cell Assemblies
Gamma Rhythms and Communication
Synaptic Plasticity
Short-Term Depression and Facilitation
Spike Timing-Dependent Plasticity (STDP)The Bisection Method
Fixed Point Iteration
Elementary Probability Theory
Smooth Approximations of Non-smooth Functions
Solutions to Selected Homework Problems