  We establish the existence and nonlinear stability of travelling pulse
solutions for the discrete FitzHugh-Nagumo equation with infinite-range
interactions close to the continuum limit. For the verification of the spectral
properties, we need to study a functional differential equation of mixed type
(MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and
phase spaces, by building on a technique developed by Bates, Chen and Chmaj for
the discrete Nagumo equation. This allows us to transfer several crucial
Fredholm properties from the PDE setting to our discrete setting.
