An Introduction to Modeling Neuronal Dynamics

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.- Modeling a Single Neuron.- 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 Nuronal Communication.- Chemical Synapses.- Gap Junctions.- A Wilson-Cowan Model of an Oscillatory E-I Network.- Entertainment, Synchronization, and Oscillations.- Entertainment 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).- Appendices.- A. The Bisection Method.- Fixed Point Iteration.- Elementary Probability Theory.- Smooth Approximations of Non-smooth Functions.- Solutions to Selected Homework Problems.