One of various families of Neural Networks (NN) that have been used in the study and development of the field of Artificial Intelligence (AI) is the Hodgkin-Huxley (HH) Network. In addition to the computational properties of the HH neuron, it also can be used to reasonably model biological phenomena, and in particular, in modeling neurons which are "synchronized/ desynchronized". The HH Neuron is a nonlinear system with two equilibrium states: A fixed point and a limit cycle. Both of them co-exist and are stable. By using a perturbation method, the behavior of this neuron can be switched between these two equilibria, namely spiking and resting respectively. The process of changing from spiking to resting is referred to as Spike Annihilation. In this paper, we numerically prove the existence of a brief excitation (input) which, when delivered to the HH neuron during its repetitively firing state, annihilates its spikes. We also derive the characteristics of this brief excitation.