  We follow up an earlier work (briefly reviewed below) to investigate the
temporal stability of an exact travelling front solution, constructed in the
form of an integral expression, for a one-dimensional discrete Nagumo-like
model without recovery. Since the model is a piecewise linear one with an
on-site reaction function involving a Heaviside step function, a
straightforward linearisation around the front solution presents problems, and
we follow an alternative approach in estimating a `stability multiplier' by
looking at the variational problem as a succession of linear evolution of the
perturbations, punctuated with `kicks' of small but finite duration. The
perturbations get damped during the linear evolution, while the kicks amplify
only the perturbations located at specific sites (the `significant
perturbations', see below) with reference to the propagating front. Comparison
is made with results of numerical integration of the reaction-diffusion system
whereby it appears likely that the travelling front is temporally stable for
all parameter values characterising the model for which it exists. We modify
the system by introducing a slow variation of a relevant recovery parameter and
perform a leading order singular perturbation analysis to construct a pulse
solution in the resulting model. In addition, we obtain (in the leading order)
a 1-parameter family of periodic pulse trains for the system, modelling
re-entrant pulses in a one-dimensional ring of excitable cells.
