Premium
Nonlinear Dynamics of Neuronal Excitability, Oscillations, and Coincidence Detection
Author(s) -
Rinzel John,
Huguet Gemma
Publication year - 2013
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21469
Subject(s) - coincidence , nonlinear system , mathematics , dynamics (music) , coincidence detection in neurobiology , statistical physics , nonlinear oscillations , neuroscience , control theory (sociology) , physics , artificial intelligence , psychology , acoustics , computer science , medicine , quantum mechanics , alternative medicine , pathology , control (management)
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.