  We analyse small parameters in selected models of biological excitability,
including Hodgkin-Huxley (1952) model of nerve axon, Noble (1962) model of
heart Purkinje fibres, and Courtemanche et al. (1998) model of human atrial
cells. Some of the small parameters are responsible for differences in the
characteristic timescales of dynamic variables, as in the traditional singular
perturbation approaches. Others appear in a way which makes the standard
approaches inapplicable. We apply this analysis to study the behaviour of
fronts of excitation waves in spatially-extended cardiac models. Suppressing
the excitability of the tissue leads to a decrease in the propagation speed,
but only to a certain limit; further suppression blocks active propagation and
leads to a passive diffusive spread of voltage. Such a dissipation may happen
if a front propagates into a tissue recovering after a previous wave, e.g.
re-entry. A dissipated front does not recover even when the excitability
restores. This has no analogy in FitzHugh-Nagumo model and its variants, where
fronts can stop and then start again. In two spatial dimensions, dissipation
accounts for break-ups and self-termination of re-entrant waves in excitable
media with Courtemanche et al. (1998) kinetics.
