Neurons can demonstrate various types of activity; tonically spiking, bursting as well as silent neurons are frequently observed in electrophysiological experiments. The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. Here we demonstrate that a bifurcation of a codimension one can explain a transition between tonic spiking behavior and bursting behavior. Namely, we argue that the Lukyanov-Shilnikov bifurcation of a saddle-node periodic orbit with noncentral homoclinics may initiate a bistability observed in a model of a leech heart interneuron under defined pharmacological conditions. This model can exhibit two coexisting types of oscillations: tonic spiking and bursting, depending on the initial state of the neuron model. Moreover, the neuron model also generates weakly chaotic bursts when a control parameter is close to the bifurcation values that correspond to homoclinic bifurcations of a saddle or a saddle-node periodic orbit.