Hysteresis-loop bursting in the DTB bursting model.

A: This is one portion of the unfolding in which SN/SH bursting can be placed, together with other classes [45]. Saddle-Node (SN) and supercritical Hopf (SupH) curves meet at the Takens-Bogdanov point TB. Bifurcation curves partition the map in five regions with different state space configurations (Roman Numerals). When this map is used for the fast subsystem of a fast-slow bursters with a hysteresis-loop mechanism for the slow variable, possible classes in the map are: SN/SH, SN/SupH, SupH/SH and SupH/SupH plus SN/SN where the system alternates between the two stable fixed points [45]. When more than one fixed point exist, the resting (or inter ictal) state is the one on the right, the other one we call ‘active rest’. The resting state corresponds to the upper branch of fixed points in panel B. B: Typical bifurcation diagram for the SN/SH class. When the system is at rest, z increases until the fixed point destabilizes through the SN bifurcation and the system jumps into the stable limit cycle. Now that the system is far from rest, z decreases until the limit cycle destabilizes through a SH bifurcation and the system jumps back to rest. If the destabilization of the fixed point/limit cycle is obtained through a different bifurcation, we will have a different onset/offset class and a different appearance of the burster’s timeseries.