We present a mathematical analysis, based on numerical explorations, of the bursting patterns that arise in plateau-bursting models of endocrine cells as the position of the equilibrium varies. We consider both square-wave and pseudo-plateau bursting. Within the framework of systems with multiple time scales, it is well known how the underlying fast subsystem organises the behaviour of the model, but such results are valid only in a small enough neighbourhood of the singular limit that defines the fast subsystem. Hence, the slow variable (intracellular calcium concentration) must be very slow, while the physiologically realistic range is moderately slow. Furthermore, the theoretical predictions are also only valid for parameter values such that the equilibrium is close to a homoclinic bifurcation that occurs in the fast subsystem. In this paper, we discuss what happens outside this theoretically known neighbourhood of parameter space. Our results complement our earlier work, in collaboration with Riess and Sherman (Journal of Theoretical Biology 2010, in press), 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.
|Publication status||Unpublished - 2010|
Bibliographical noteSponsorship: HMO was supported by an Advanced Research Fellowship of the Engineering and Physical
Sciences Research Council (EPSRC), UK.
- square wave
- subcritical and supercritical Hopf bifurcation
- homoclinic bifurcation
- plateau bursting