Dynamics of Plateau Bursting Depending on the Location of its Equilibrium

We present a mathematical analysis of the dynamics that underlies plateau bursting in models of endocrine cells under variation of the location of the (unstable) equilibrium around which these bursting patterns are organised. We focus primarily on the less well-studied case of pseudo-plateau bursting, but also consider the square-wave case. The behaviour of such models is explained using the theory for systems with multiple time scales and it is well known that the underlying so-called fast subsystem organises their dynamics. However, such results are valid only in a sufficiently small neighbourhood of the singular limit that defines the fast subsystem. Hence, the slow variable (intracellular calcium concentration) must be very slow, which is actually not the case for pseudo-plateau bursting. Furthermore, the theoretical predictions are also only valid for parameter values such that the equilibrium is close to a homoclinic bifurcation occuring in the fast subsystem. In the present study, we use numerical explorations to discuss what happens outside this theoretically known neighbourhood of parameter space. In particular, we consider what happens as the equilibrium moves outside a small neighbourhood of the homoclinic bifurcation that occurs in the fast subsystem, and relatively fast speeds are allowed for the slow variable which is controlled by a relatively large value of a parameter epsilon. The results obtained complement our earlier work [Tsaneva-Atanasova et al. (2010) J Theor Biol 264, 1133-1146], which focussed on how the bursting patterns vary with the rate of change epsilon of the slow variable: we fix epsilon and move the equilibrium over the full range of the bursting regime. Our findings show that the transitions between different bursting patterns are rather similar for square-wave and pseudo-plateau bursting, provided that the value of epsilon for the pseudo-plateau-bursting model is chosen so that it is much larger than for the square-wave bursting model. Furthermore, the two families of tonic spiking and plateau bursting, which are generally viewed as two separately generated families, are actually connected into a single family in the two-parameter plane through branches of unstable periodic orbits.