GWN-induced bursting, spiking, and random subthreshold impulsing oscillation before hopf bifurcations in the chay model

Gaussian white noise (GWN), as an intrinsic noise source, can give rise to various firing activities at the rest. state before a supercritical or subcritical Hopf bifurcation (supH or subH) in the Chay system without or with external current input, when V-K, V-C, lambda(n) and I are considered as changeable control parameters. These firing activities are closely related to the global bifurcation mechanism of the whole system and the fast/slow dynamical subsystems, and can be tackled by means of bifurcation analysis. GWN can induce some typical bursting phenomena in the stochastic Chay system. Firstly, integer multiple fold/homoclinic or circle/homoclinic bursting due to GWN, with only one spike per burst, can arise from rest states before both subH and supH (with respect to the parameter V-K), and their respective trajectories have the same shape and property. However, less spikes appear and their peaks are lower before supH, comparing with those before subH. Secondly, a fold/fold point-point hysteresis loop bursting due to GWN is generated before supH (with respect to the parameter V-C) on the upper branch of a Z-shaped bifurcation curve between two fold bifurcations of the fast system. Thirdly, at a rest state before subH (with respect to the additional current I) situated on the lower branch of a S-shaped bifurcation curve between two fold bifurcations of the fast system, a GWN-induced firing pattern appears and is classified as Hopf/homoclinic bursting via fold/homoclinic point-point hysteresis loop. GWN-induced firing activities other than bursting can also be observed in the stochastic Chay system. For example, sometimes GWN-induced continuous spiking without any particular shape may arise at a rest state before supH (with respect to the parameter V-K) for certain values of parameters. Moreover, under the situation that a stable node and a stable focus coexist before subH (with respect to the parameter I) and the attractive region of the stable node is larger than that of the stable focus, GWN only provoke random subthreshold impulsing oscillation near the stable node.