Applications of the Poincare´ mapping technique
to analysis of neuronal dynamicsPaul Channell Jr.a, Gennady Cymbalyukb, Andrey Shilnikova,
aDepartment of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA
bDepartment of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA
Available online 9 November 2006
Abstract
A single neuron can demonstrate different spiking and bursting patterns which can be elicited naturally depending on modulation
status or artificially due to disturbances caused by distinct recording techniques. For example, when pharmacologically isolated with
bicuculline a leech oscillatory heart interneuron can show an endogenous bursting activity while recorded extracellularly, or the periodic
tonic spiking activity while recorded intracellularly. Transitions between these oscillatory patterns are in general non-local and could not
be understood using only the local analysis of the neuron’s rest states, but the global theory tools such as the Poincare´ return mapping
analysis. The mappings constructed then predict the temporal characteristics of the spiking and bursting patterns and allow one to study
transitions between them. The technique is directly applicable to neuronal models of various types, as well as is aimed to be employed in
neurophysiological experiments.
r 2006 Elsevier B.V. All rights reserved.
Keywords: Tonic spiking; Bursting neuron; Active transitions; Poincare´ mapping
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