  Point neuron models with a Heaviside firing rate function can be ill-posed.
That is, the initial-condition-to-solution map might become discontinuous in
finite time. If a Lipschitz continuous, but steep, firing rate function is
employed, then standard ODE theory implies that such models are well-posed and
can thus, approximately, be solved with finite precision arithmetic. We
investigate whether the solution of this well-posed model converges to a
solution of the ill-posed limit problem as the steepness parameter, of the
firing rate function, tends to infinity. Our argument employs the
Arzel\`{a}-Ascoli theorem and also yields the existence of a solution of the
limit problem. However, we only obtain convergence of a subsequence of the
regularized solutions. This is consistent with the fact that we show that
models with a Heaviside firing rate function can have several solutions. Our
analysis assumes that the Lebesgue measure of the time the limit function,
provided by the Arzel\`{a}-Ascoli theorem, equals the threshold value for
firing, is zero. If this assumption does not hold, we argue that the
regularized solutions may not converge to a solution of the limit problem with
a Heaviside firing function.
