Multiple geometric viewpoints of mixed mode dynamics associated with pseudo-plateau bursting (Q2871348)

Pseudo-plateau bursting, commonly found in neural bursting models, is a particular type of bursting which features small amplitude oscillations or spikes in the active phase superimposed on relaxation type oscillations. In this regard, an essential feature of mixed mode oscillations (MMOs), understood as oscillatory trajectories in which there is an alternation between large amplitude and small amplitude spiking, is the multiple timescale structure of the governing system, often featuring 3 distinct timescales. Such 3-timescale systems have been typically treated in the framework of geometric singular perturbation theory, but as 2-timescale problems, fixing one of the two perturbation parameters.NEWLINENEWLINEThe authors investigate a recent model for the electrical activity and calcium signaling in a 4-dimensional pituitary lactotroph, showing by means of dimensional analysis that this model belongs a certain large class of 3-timescale systems of general concern in the paper. By finding the bifurcation structure of the model, several regions in the parameter space where MMOs exist are first identified.NEWLINENEWLINEDriven by a geometric viewpoint, the authors then perform a singular perturbation analysis of the full 3-timescale problem, formulating their analysis in a general way to emphasize that their approach can easily be adapted to the study of other 3-timescale problems. Further, they observe that the usual 2-timescale analysis techniques (one which treats the calcium concentration as a slowly varying parameter and considers a parametrized family of fast subsystems and one which divides the system so that there is only one fast variable and shows that the bursting arises from canard dynamics) are different unfoldings of the 3-timescale analysis.NEWLINENEWLINEIt is observed that the 3-timescale decomposition retains the predictive power of the 2-timescale methodologies, used by the authors to explain the transient and long-term dynamics of the pituitary lactotroph model under consideration. The paper is remarkably comprehensive and reasonably self-contained (it contains, among other things, brief but concentrated itineraries through generic geometric singular perturbation analysis, singular orbits, canards and various MMOs), benefiting also from a intuitive presentational style.