Bursting activity is crucial for neuronal control of rhythmic behavior of animals. The co-existence of bursting activity and silence is common in the neuron models. We described a new mechanism for the co-existence of bursting activity and silence and the transition between these two regimes. This mechanism is based on Homoclinic and Andronov-Hopf bifurcations. The area of the co-existence is limited my boarders which are defined by Homoclinic and Andronov-Hopf bifurcation. The unstable hyperpolarized sub-threshold oscillations (periodic orbit) occur through an Andronov-Hopf bifurcation on equilibrium and disappear on the Homoclinic bifurcation. The boarder is defined by the homoclinic bifurcation exihibit to the phenomenon of intermittent transition from bursting into silence. This mechanism is shown first in the simplified four dimensional model and then confirmed in the full fourteen dimensional model of leech heart interneuron.