  In the recent years we have shown that cylindrical biological membranes such
as nerve axons under physiological conditions are able to support stable
electromechanical pulses called solitons. These pulses share many similarities
with the nervous impulse, e.g., the propagation velocity as well as the
measured reversible heat production and changes in thickness and length that
cannot be explained with traditional nerve models. A necessary condition for
solitary pulse propagation is the simultaneous existence of nonlinearity and
dispersion, i.e., the dependence of the speed of sound on density and
frequency. A prerequisite for the nonlinearity is the presence of a chain
melting transition close to physiological temperatures. The transition causes a
density dependence of the elastic constants which can easily be determined by
experiment. The frequency dependence is more difficult to determine. The
typical time scale of a nerve pulse is 1 ms, corresponding to a characteristic
frequency in the range up to one kHz. Dispersion in the sub-kHz regime is
difficult to measure due to the very long wave lengths involved. In this
contribution we address theoretically the dispersion of the speed of sound in
lipid membranes and relate it to experimentally accessible relaxation times by
using linear response theory. This ultimately leads to an extension of the
differential equation for soliton propagation.
