Origin of chaos in a simple two-dimensional map model replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: stable fixed point (replicating silence) and superstable limit cycle (replicating spikes). Coupling this subsystem with the slow subsystem leads to the generation of periodic or chaotic spiking-bursting behavior. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases.

Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltage-gated calcium channels and calcium-sensitive potassium channels, they can be very different. For example, in insulin-secreting pancreatic β-cells, plateau bursting is characterized by well-defined spikes during the depolarized phase whereas in pituitary cells, bursting features fast, irregular, small amplitude spikes. The latter has been termed “pseudo-plateau bursting ” because the spikes are transients around a depolarized steady state rather than stable oscillations in the fast subsystem. In this study we systematically investigate the bursting patterns found in endocrine cell models. We show that this class of voltage and calcium gated conductance based models can be reduced to the polynomial model of Hindmarsh and Rose (25). This reduction preserves the main properties of the biophysical class of models that we consider and allows for detailed bifurcation analysis of the full fast-slow system. Our analysis does not require decomposition of the full system into fast and slow subsystems and reveals properties of endocrine bursting that are not captured by the standard fast-slow analysis.