Homoclinic bifurcations of both equilibria and periodic orbits are argued to be critical for understanding the dynamics of the Hindmarsh-Rose model in particular, as well as of some squarewave bursting models of neurons of the Hodgkin-Huxley type. They explain very well various transitions between the tonic spiking and bursting oscillations in the model. We present the approach that allows for constructing Poincaré return mapping via the averaging technique. We show that a modified model can exhibit the blue sky bifurcation, as well as, a bistability of the coexisting tonic spiking and bursting activities. A new technique for localizing a slow motion manifold and periodic orbits on it is also presented.

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Understanding of the onset and generic mechanisms of transitions between distinct patterns of activity in realistic models of individual neurons and neural networks presents a fundamental challenge for the theory of applied dynamical systems. We use three examples of slow-fast neural systems to demonstrate a suite of new computational tools to study diverse neuronal systems.