Christoph Börgers is Professor of Mathematics at Tufts University, USA. His expertise is in mathematical neuroscience, applied dynamical systems, numerical analysis, and scientific computing.

Includes access to Matlab codes generating the figures

Exercises at widely varying levels are included at the end of every chapter

Detailed writing allows for self-study opportunities

Includes supplementary material: [...]

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.