%0 Journal Article
%T A showcase of torus canards in neuronal bursters
%A John Burke
%A Mathieu Desroches
%A Anna M Barry
%A Tasso J Kaper
%A Mark A Kramer
%J The Journal of Mathematical Neuroscience
%D 2012
%I Springer
%R 10.1186/2190-8567-2-3
%X The primary unit of brain electrical activity - the neuron - generates a characteristic dynamic behavior: when excited sufficiently, a rapid (on the order of milliseconds) increase then decrease in the neuronal voltage occurs, see for example [1]. This action potential (or ¡®spike¡¯) mediates communication between neurons, and therefore is fundamental to understanding brain activity [2-4]. Neurons exhibit many different types of spiking behavior including regular periodic spiking and bursting, which consists of a periodic alternation between intervals of rapid spiking and quiescence, or active and inactive phases, respectively, [5-7]. Bursting activity may serve important roles in neuronal communication, including robust transmission of signals and support for synaptic plasticity [8,9].Computational models of spiking and bursting allow a detailed understanding of neuronal activity. Perhaps the most famous computational model in neuroscience - developed by Hodgkin and Huxley [1] - provided new insights into the biophysical mechanisms of spike generation. Subsequently, the dynamical processes that support spiking and bursting have been explored, see for example [10-12]. Recent research has led to a number of classification schemes of bursting, including a scheme by Izhikevich [7] based on the bifurcations that support the onset and termination of the burst¡¯s active phase. This classification requires identifying the separate time scales of the bursting activity: a fast time scale supporting rapid spike generation, and a slow time scale determining the duration of the active and inactive burst phases. This separation of time scales naturally decomposes the full model into a fast system and a slow system. Understanding the bifurcation structure of the isolated fast system is the principal element of the classification scheme. Within this scheme, the onset of the burst¡¯s active phase typically corresponds to a loss of fixed point stability in the fast system, and the termi
%K Bursting
%K torus canards
%K saddle-node of periodic orbits
%K torus bifurcation
%K transition to bursting
%K mixed-mode oscillations
%K Hindmarsh-Rose model
%K Morris-Lecar equations
%K Wilson-Cowan model
%U http://www.mathematical-neuroscience.com/content/2/1/3