Period adding bifurcation scenarios with chaotic and stochastic bursting in an experimental neural pacemaker

The similarity and distinction between period adding bifurcation scenarios from period 1 bursting to period 4 bursting with chaotic and stochastic bursting discovered in the experiments on neural pacemaker are identified. The period 1 bursting is changed into period 2 bursting directly in bifurcation scenarios with chaos and indirectly via a stochastic bursting in the bifurcation scenarios with stochastic bursting. The stochastic and chaotic bursting lying between period k and period k+1 bursting (k=2,3) exhibit similarities in chaos-like characteristics such as positive Lyapunov exponent, non-periodic spike trains and deterministic structures including periodic orbit and distinction in the following aspects. The spike trains and first return map of ISI series of the chaotic bursting are more complex than those of the stochastic bursting. The evolutions of the periodic orbit with respect to bifurcation parameter are different. Multi-peaked histogram can be seen in stochastic bursting while not in the chaotic bursting. The results not only reveal the dynamics of two kinds of bifurcation scenarios, but also give practical indicators to distinguish two kinds of bifurcation scenarios, and the chaotic bursting and stochastic bursting.