Different bifurcations and slow dynamics underlying different stochastic dynamics of slow, medium, and fast bursting of β-cell

The bursting with long burst duration of the pancreatic beta-cells related to the diabetes is helpful to maintain the normal blood glucose concentration. Then, nonlinear dynamics underlying burst duration is an important issue, which is investigated in a theoretical model containing two slow variables (s and z) in the present paper. For the deterministic case, different shapes and sizes of the bursting trajectory show that the slow bursting is modulated by s, z, and s-nullcline in wide ranges, the medium bursting by s and z in medium ranges, and the fast bursting mainly by s in narrow ranges. Through two-parameter bifurcation analysis, the burst of the three bursting patterns terminates at saddle homoclinic orbit (SH) bifurcation curve. For the slow and medium bursting patterns, the latter part of the burst mainly runs along z-direction and closely to the saddle surface. Then, noise can induce the burst to terminate earlier than the SH via passage through the saddle surface, resulting in decreased burst duration. Compared with the drastic decrease for the medium bursting, slow bursting exhibits small reduction, since the large size induces two phases insensitive to noise. One is the latter part of the quiescent state, which runs along the s-nullcline to exhibit extra-stable dynamics, and the other is the former part of the burst, which runs far from the saddle surface. However, the fast bursting mainly runs along s-direction and far from the saddle surface exclusive the SH, then, noise can only induce small changes. Then, the different changing trends of the stochastic slow, medium, and fast bursting patterns are well explained with the different stochastic responses around different bifurcation points or critical phases. The results present dynamical mechanism for the long burst duration which is favor for the maintenance of the normal blood glucose concentration.