  We establish the existence and nonlinear stability of travelling wave
solutions for a class of lattice differential equations (LDEs) that includes
the discrete FitzHugh-Nagumo system with alternating scale-separated diffusion
coefficients. In particular, we view such systems as singular perturbations of
spatially homogeneous LDEs, for which stable travelling wave solutions are
known to exist in various settings.
  The two-periodic waves considered in this paper are described by singularly
perturbed multi-component functional differential equations of mixed type
(MFDEs). In order to analyze these equations, we generalize the spectral
convergence technique that was developed by Bates, Chen and Chmaj to analyze
the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm
properties from the spatially homogeneous to the spatially periodic setting.
Our results hence do not require the use of comparison principles or
exponential dichotomies.
