Nonlinear fractional waves at elastic interfaces

Abstract

We derive the nonlinear fractional surface wave equation that governs compression waves at an interface that is coupled to a viscous bulk medium. The fractional character of the differential equation comes from the fact that the effective thickness of the bulk layer that is coupled to the interface is frequency dependent. The nonlinearity arises from the nonlinear dependence of the interface compressibility on the local compression, which is obtained from experimental measurements and reflects a phase transition at the interface. Numerical solutions of our nonlinear fractional theory reproduce several experimental key features of surface waves in phospholipid monolayers at the airwater interface without freely adjustable fitting parameters. In particular, the propagation length of the surface wave abruptly increases at a threshold excitation amplitude. The wave velocity is found to be of the order of 40 cm/s both in experiments and theory and slightly increases as a function of the excitation amplitude. Nonlinear acoustic switching effects in membranes are thus shown to arise purely based on intrinsic membrane properties, namely the presence of compressibility nonlinearities that accompany phase transitions at the interface.

Similarities between action potentials and acoustic pulses in a van der Waals fluid

“…4a,b). At low amplitudes of excitation the pulse response was qualitatively similar to previous theoretical results obtained in a small-amplitude analysis [36]. However, at larger amplitudes of excitation the amplitude of the density pulse saturated at ρ=3ρ c .…”

“…Here, ρ b is the bulk density, η b the bulk viscosity, t p the pulse duration, and ρ i the density of the interface. For a typical lipid interface coupled to bulk water, this results in attenuation by 2-3 orders of magnitudes, and agrees with experimental observation [30,36]. Furthermore, these scales also agree with experimental measurements of APs in living cells (∼ 10 −3 − 10 1 s and ∼ 10 −3 − 10 2 m/s, respectively [6,7,11]).…”

“…Two main criticisms of the model have been an overestimation of the propagation velocity and the lack of annihilation of pulses. Kappler et al later demonstrated that the viscous coupling between the interface and a bulk fluid attenuates the propagation velocity of interfacial longitudonal waves, and results in a corrected velocity that agrees with experimental evidence [36]. Still, some observations are not captured by these previous works, including the saturation of pulse amplitude at large stimulation amplitudes, annihilation of pulses upon collision, as well as a thorough analysis of different aspects of the pulses under adiabatic conditions (e.g., density, pressure, temperature and electrical properties).…”

“…The discrepancy in velocity and length scales results from our attempt to keep the model simple, and would disappear by extending the model to include the non-negligible coupling to the bulk. It was previously demonstrated that the viscous coupling between a compressible interface and the bulk attenuates the velocity and length of acoustic pulses by a factor of ∼ 1 + √ ρ b η b t p /ρ i [36]. Here, ρ b is the bulk density, η b the bulk viscosity, t p the pulse duration, and ρ i the density of the interface.…”

Collision and annihilation of nonlinear sound waves and action potentials in interfaces

“…For example, we observe an increase in velocity with an increase in amplitude, while Heimburg's soliton model predicts that velocity decreases with an increase in amplitude [30]. We believe that the qualitative disagreement in the observed nonlinearity indicates a fundamental difference in the dispersion relation assumed in the soliton model and its applicability to a lipid monolayer [31]. With respect to collisions in particular, the predictions of the theoretical analysis of colliding pulses reveals no annihilation (less than 4%) in soliton models [32,33], which is in clear contrast to our results that demonstrate significant (up to 80%) decrease in amplitude upon collision.…”