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. We examine 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.Neurons exhibit a multiplicity of oscillatory patterns such as periodic, quasiperiodic and chaotic tonic-spiking and bursting oscillations including their mixed modes. The corresponding mathematical models fall into the class of slowfast systems, which are characterized by the existence and interaction of diverse characteristic timescales. Study of typical transitions between these oscillatory regimes, described in terms of the bifurcation theory, is one of the trends in mathematical neuroscience. Quasiperiodic oscillations, associated with frequency locking and synchronization two close or commensurable frequencies, typically occur in coupled or periodically forced nonlinear dynamical systems. The phase space of the system with quasiperiodic oscillations contains a resonant torus, whose dimension is determined by the number of the interacting characteristic frequencies. In slow-fast systems quasiperiodic oscillations emerge through nonlinear reciprocal interactions when the fast subsystem experiences a bottle-neck effect equating its time scale to that of the slow subsystem. The torus breakdown, being one of the routes to onset of complex dynamics, turns out to also underlie a typical mechanism of transition from tonic-spiking to regular or chaotic bursting in one class of slow-fast neuronal models. The study of quasiperiodicity and torus bifurcation is a challenging task that requires the aggregated use of computational approaches based on non-local bifurcation techniques. In this paper we follow the so-called bottom-up approach, starting with the highly detailed Hodgkin-Huxley type models, and ending with formal reduced models. Our goal is to demonstrate how various techniques: geometrical slow-fast dissection, parameter continuation and averaging, Poincaré return maps, when combined, allow us to elaborate on all fine details of the theory of torus bifurcations and breakdown in the illustrative applications.We dedicate this paper to our dear colleague and friend, Valentin Afraimovich, who passed away suddenly in February, 2018. He made many fundamental contributions to the theory of dynamical systems and bifurcations, including his co-works on quasiperodicity that set the stage for the current understanding of this phenomenon in dissipative nonlinear systems.
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