Nonlinear Dynamics of Neuronal Excitability, Oscillations, and Coincidence Detection
Article first published online: 26 JUN 2013
Copyright © 2013 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Special Issue: Third Special Issue Commemorating the 75th Anniversary of the Courant Institute
Volume 66, Issue 9, pages 1464–1494, September 2013
How to Cite
Rinzel, J. and Huguet, G. (2013), Nonlinear Dynamics of Neuronal Excitability, Oscillations, and Coincidence Detection. Comm. Pure Appl. Math., 66: 1464–1494. doi: 10.1002/cpa.21469
- Issue published online: 26 JUN 2013
- Article first published online: 26 JUN 2013
- Manuscript Received: OCT 2012
- National Institutes of Healthm. Grant Number: DC008543
- Swartz Foundation
- MCyT/FEDER. Grant Numbers: MTM2009–06973, MTM2012–31714, 2009SGR–859
We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability and onset of repetitive firing behavior in single-neuron models and their relation with Hodgkin's classification in 1948 of repetitive firing properties. We discuss the mathematical dissection of bursting oscillations using fast/slow analysis and demonstrate the approach using single-cell and mean-field network models. Finally, we illustrate the properties of Type III excitability in which case repetitive firing for constant or slow inputs is absent. Rather, firing is in response only to rapid enough changes in the stimulus. Our case study involves neuronal computations for sound localization for which neurons in the auditory brain stem perform extraordinarily precise coincidence detection with submillisecond temporal resolution. © 2013 Wiley Periodicals, Inc.