Can the Hodgkin-Huxley neuron incorporate desensitization?
The Hodgkin-Huxley model does have the ability to accurately reflect adaptation processes in a neuron (e.g., Awiszus, (1989));
A perfect integrator model is also suitable (Goldman et al., 2007); 
A leaky integrator model is often used to model spiking behavior as well (Mihalas &amp; Niebur, 2009).


References
- Awiszus, Biol Cybern (1988); 59(4-5): 295-302
- Goldman et al., New Encycopedia of Neuroscience (2007)
- Mihalas &amp; Niebur, Neural Comput (2009); 21(3): 704–18