We study the quasi-periodicity phenomena occurring at the transition between tonic spiking and bursting activities in exemplary biologically plausible Hodgkin-Huxley type models of individual cells and reduced phenomenological models with slow and fast dynamics. Using the geometric slow-fast dissection and the parameter continuation approach, we show that the transition is due to either the torus bifurcation or the period-doubling bifurcation of a stable periodic orbit on the 2D slow-motion manifold near a characteristic fold. Various torus bifurcations including stable and saddle torus-canards, resonant tori, the co-existence of nested tori, and the torus breakdown leading to the onset of complex and bistable dynamics in such systems are examined too.