Classification of bursting patterns: A tale of two ducks
Fig 5
Small homoclinic/big homoclinic bursting, corresponding to Fig 88 of [7]; shown is a new simulation with the same parameter values (available in [7]).
Panel (a) shows the slow-fast dissection in the (u, V) phase plane; the inset shows a zoom corresponding to the dashed rectangle, which better reveals the shape of the bursting cycle while the main plot better highlights the fast subsystem bifurcation structure. Labels HB, LP, and Ho refer to Hopf bifurcation, saddle-node bifurcation (fold or “limit point”), and homoclinic bifurcation of the fast subsystem, respectively. Panel (b) shows the V-time series of this bursting solution. Izhikevich’s classification allowed to characterize bursting patterns beyond square-wave, elliptic and parabolic, and already opened the door toward explaining more complex biological data. In particular, one can mention pathological brain activity related to, e.g., epileptic seizure [52] or spreading depolarization [16,53]. According to Izhikevich’s classification, bursting oscillations where the burst initiates via a fold bifurcation of the fast subsystem are termed fold-initiated bursting. In the present work, we will propose an extended classification based upon fold-initiated bursting cases.