Research Overview

The sawtooth pattern is that of a "slow" regulatory variable which modulates the fast bursting patterns. Square wave bursting is important since the pancreatic beta cells exhibit this type of electrical behavior. In particular, the rate at which the cells secrete insulin has been strongly correlated to the duration of the active phase. Such periodic oscillations are interesting dynamically since they represent multidimensional limit cycles which can be constructed using singular perturbation techniques (including averaging). The particular model describing the oscillation above has two "fast" variables and one "slow" variable. The periodic orbit of the associated autonomous system is therefore a closed orbit in R³. As with many systems which exhibit bursting, the differential equations contain a small parameter and the bursting solution(s) can be described asymptotically in terms of it's slow and fast subsystem (SS) and (FS), respectively. In the silent phase, the solution tracks a one dimensional  slow manifold (curve) in R³. In the active phase, the solution tracks a different slow manifold (I call it the Averaged Fast Subsystem or AFS) derived using a family of period solutions of (FS). Escape from the active phase typically occurs through a homoclinic bifurcation which can be located using Melnikov theory in some models.

Classification of Bursters:

Below is a figure illustrating how bursters can be classified in terms of the fast parameters (each axis). The lines A-D separate regions describing i) where (FS) equilibria are stable, ii) what their locations are, iii) where Hopf points are, iv) their criticality and v) paremeter values for which homoclinic bifurcations of (FS) exist. Different types of bursters exist in regions bounded by these lines. The shaded region indicates parameter pairs where square-wave bursting (see above) solutions exist. For details, see the 1994 SIAM paper listed in my VITA. What is siginificant about this type of research is that not only are there many different types of bursting oscillations but they are all observed in electrophysiology experiments in nerves, endocrine system cells and muscle. That only two parameters are need to characterize all (relevant) types suggests to me that only a few genetic markers are needed for organisms to develop the correct "burster" with the right dynamics for each organism physiological subsystem. Bursters are thought to control dynamics on time scales of seconds within organisms. They are natural "clocks" and bridge the time barriers of nerve action potentials (msec) and endocrine system physiology (minutes-months).