  We rigorously prove the existence of traveling fronts in neural field models
with lateral inhibition coupling types and smooth sigmoidal firing rates. With
Heaviside firing rates as our base point (where unique traveling fronts exist),
we repeatedly apply the implicit function theorem in Banach spaces to provide a
non-monotone version of the homotopy approach originally proposed by Ermentrout
and McLeod (1993) in their seminal study of monotone fronts in purely
excitatory models. By comparing smooth and Heaviside firing rates, we develop
global wave speed and profile comparisons that guide our analysis, leading to
uniqueness (modulo translation) in the perturbative case. Moreover, we
establish a meaningful a priori existence result; we prove existence holds for
a range of firing rates, independent of continuation path.
