Hi, I'm Applied Math And Computational Sciences Postdoc at the University of Pennsylvania. 

Ion channels embedded in a cell 's membrane (in an axon say) are known to open and close depending upon the voltage potential across the membrane. However, their behavior is also stochastic. Thus, it is their average behavior over some number of neighboring ion channels which can give rise to propagation of an action potential as demonstrated by the one-dimensioanl toy model simulated in the figure to the left.  In the figure we simulate the upstroke of an action potential with a hybrid system.

Sound waves of large wavelength can more easily penetrate walls or other barriers. In much the same way, a compression wave with a long wavelength traveling through a mass-spring lattice (masses connected via springs) can easily overcome the discreteness of the lattice as well as microscopic heterogeneities in its material properties.  If the springs are linear (Hook's law), dispersion eventually leads to the vanishing of the wave; however, if the springs are non-linear, certain waves may propagate unaffected for a much longer time (and distance). The effect of certain random microscopic heterogeneities in the lattice on the wave's propagation over these longer times remains largely a mystery.