Applications of the Poincaré mapping technique to analysis of neuronal dynamics

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 Poincaré 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.