Bifurcation, Bursting and Spike Generation in a Neural Model

Bursting is an important behavior pattern of nerve cells and in mathematical models it is often related to bifurcations. However, the onset of bursting and the birth of new spikes are also possible without any bifurcation. We present a numerical study of a model of Plant which exhibits a periodically bursting behavior in a relevant range of two parameters. In this two-dimensional range the behavior changes continuously, except for a number of tiny regions where spikes are born or die. These islands are extensions of bifurcation islands. It appears that the generation of spikes can be related to local extrema of the periods of the bursting periodic orbits. This allows to detect the birth of spikes also in parameter regions where no bifurcations occur. From a biological point of view the average instantaneous spike frequency is probably the most important output variable of a neuron. We show that in the Plant model (in contrast to some other models) this frequency is essentially determined by the number of spikes in a burst.