  The voltage-conductance kinetic model for the collective behavior of neurons
has been studied by scientists and mathematicians for two decades, but the
rigorous analysis of its solution structure has been only partially obtained in
spite of plenty of numerical evidence in various scenarios. In this work, we
consider a simplified voltage-conductance model in which the velocity field in
the voltage variable is in a separable form. The long time behavior of the
simplified model is fully investigated leading to the following dichotomy:
either the density function converges to the global equilibrium, or the firing
rate diverges as time goes to infinity. Besides, the fast conductance
asymptotic limit is justified and analyzed, where the solution to the limit
model either blows up in finite time, or globally exists leading to time
periodic solutions. An important implication of these results is that the
non-separable velocity field, or physically the leaky mechanism, is a key
element for the emergence of periodic solutions in the original model based on
the available numerical evidence.
