Mechanisms for nonuniform propagation along excitable cables.

We have discussed two classes of mechanisms that can lead to propagation with nonconstant velocity, and to disruption of temporal patterning of action potentials. Inhomogenieties along the cable due to geometrical change or to altered cell coupling can result in conduction delays, with the possibility of block or reflection. Such conduction irregularities have been considered relevant to cardiac reentry phenomena. Our simulations with a discrete number of excitable cells, coupled by gap junctions, showed that the underlying mathematical structure of a saddle point threshold in an ionic model also contributes in an important way to creating a long delay. Such threshold behavior, although not yet demonstrated for some of the most well-studied models of excitability, should not be viewed as extraordinary; we have seen it in models other than those of references 6 and 7, and have produced it in the Hodgkin-Huxley model with reasonable parameter variations (but have not yet checked for reflections with these modifications). We are unaware of any computations with theoretical models of cardiac membrane that yield robust reflection behavior. Perhaps modifications of these models will be necessary in order to obtain adequate delays for reflection. The mechanism we have described here may serve as a guideline for additional features to seek in such parametric tuning. A different class of factors that contribute to interferring with action potential timing include the effects of previous activity on propagation speed. These influences may be described in terms of the dispersion relation, c(T), the dependence of speed on time between action potentials. The form of this function, for large T, reflects the exponential behavior of the action potential's return to rest. Supernormal conduction reveals itself in the dispersion relation when there is an overshoot of excitability in the return to rest, either a single overshoot or an alternating sequence of over- and undershoots (as seen in some nerve membrane models). A simple kinematic recipe was described for quantitatively predicting timing changes during propagation without having to solve the full cable equations. To apply these concepts to cardiac models it will be necessary to compute the dispersion relation for these models. By studying the dependence of c(T) and the waveform trajectory (including conductances as well as membrane potential) on various parameters one may gain insight into the ionic basis for experimentally observed supernormal conduction.