Propagation failure is an important factor in the initiation of re-entrant arryhthmias, and can play a significant role in their break-up or self-termination (Biktasheva et al 2003). One mechanism for propagation failure is accumulation of Na inactivation, which leads to inward current insufficient to maintain propagation. Phenomenologically, this mechanism is seen as a loss of the sharp gradient of the action potential profile, or ‘dissipation’ of the propagation front (DPF) (Biktashev 2002). DPF cannot be obtained in the FitzHugh-Nagumo caricatures of excitation, in which there is only one fast variable, but is seen in ionic models for excitation and its propagation. We have analysed the structure of Hodgkin and Huxley (1952), Noble (1962) and Courtemanche et al (1998) ionic models, identified small parameters that appear in those models in non-standard ways, and developed an asymptotic approach based on these small parameters. Further simplifications have been achieved by appropriate approximation of nonlinear functions in the models. Contrary to common belief, the fast Na current inactivation gating variable h is not necessarily slow compared to the transmembrane voltage V during the upstroke of the propagating action potential. Interplay between V and h is responsible for the DPF. We suggest a simplified model, which emerges from the asymptotic analysis, and considers V and h as equally fast variables. This model reproduces DPF and admits analytical study. In particular, it yields conditions for the DPF. This interpretation can be applied to explain the breakup and self-termination of re-entrant waves in detailed realistic ionic models. FitzHugh-Nagumo type caricatures, although successfully describing successfull propagation, fail to correctly describe propagation failure. Thus using such models to describe processes involving initiation, block of propagation or re-entrant waves in cardiac tissue may misrepresent most important features. The new simplified model or its analogue should be used instead.