Neuronal cells in isolation or as an assembly exhibit bursting behavior on two different time scales. We introduce a simple one-dimensional model which requires only one phase variable to describe the phenomenon of parabolic bursting. The analysis in the continuum limit reveals that for any unimodal distribution of frequencies, the qualitative properties of the full and the reduced model are identical. Further, we derive analytically an exact low-dimensional description of a globally coupled network of bursting oscillators for our model. Study of the stability for this low-dimensional model reveals different dynamical signatures in the parameter space. We demonstrate that the structure of the parameter space remains independent of the number of spikes per burst.

• Received 22 May 2009

DOI:https://doi.org/10.1103/PhysRevE.80.041930