Low Frequency Sound Propagation in Lipid Membranes

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.