Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

The Journal of Mathematical Neuroscience

Table 3 Bursting classes in the unfolding of the deg. TB singularity, focus case

Class	Label	Bifurcations sequence	Point A	Point B
LCs region
SN/SH	c2s	SH_l, SN_r	SH_l	SN_r ∈ [P₂, SNLs_r]
SN/supH	c3s	SN_l, H, SN_r	SN_l ∈ [TB_l, C_s]	SN_r ∈ [P₂, SNLs_r]
supH/SH	c10s	SH_l, H, SN_r	SH_l	SN_r ∈ [C_s, P₂]
supH/supH	c11s	SN_l, H, H, SN_r	SN_l ∈ [TB_l, C_s]	SN_r ∈ [C_s, P₂]
LCb region
SN/SH	c2b	SHb, SN_r ^a	SHb ∈ [DL, SNLb_r]	SN_r ∈ [SNLb_r,TBr]
SN/FLC	c4b	FLC, SN_r ^a	FLC	SN_r ∈ [SNLb_r, TBr]
subH/SH	c14b	SHb, H^a	SHb ∈ [DL, SNLb_r]	H ∈ [B, TB_r]
subH/FLC	c16b	FLC, H^a	FLC	H ∈ [B, TB_r]

For each of the possible classes in the LCs or LCb region we state the requirements in terms of sequence of bifurcations the system has to go through and location of the initial and final points of the path.
^aThese classes can cross SHb ∈ [SNLb_l, DL], SH_r, SN_l and SNIC ∈ [SNLb_l, SNLs_l] between the offset and the onset point, without destroying the bursting trajectory.