Periodic solutions of an electromechanical nerve model

The electrical and mechanical models have so far been independently used to inadequately account for the generation and propagation of action potential in nerve fibers. This study seeks

to fill in the lacuna by proposing a hybrid electromechanical model in which phase transitions of lipid biomembrane induces the flow of fluid through the cell membrane, that result in the necessary axoplasmic pressure wave [as highlighted in references: M. M. Rvachev. Rev. Lett. 5(2), 73-88 (2010); A. El Hady and B. B. Machta. Nature Commun. 6, 6697 (2015)]. This triggersmore opening/closing of lipid-ion channels owing to the piezoelectric nature of nerve membrane, hence influencing the propagation of transmembrane voltage. In fact we consider that changes of transmembrane voltage is governed by the modified Hodgkin-Huxley cable equation; where the instantaneous charge stored in the membrane capacitor depends on the difference in densities of the ion-carrying fluid across the nerve membrane. The evolution of axoplasmic pressure wave is modeled by the improved Heimburg-Jackson hydrodynamic equation; in which the membrane capacitance effectively ensures the coupling between the transmembrane voltage and the pressure waves that varies as periodic soliton pulses. We analytically obtained time independent periodic modes that serve as steady state solutions of the transmembrane voltage, with it threshold values vital in determining wavefronts propagation in the nerve axon. An increase in the numerical value of the membrane coupling coefficient generally decrease the amplitude and speed of the electromechanical nerve wavefront, thereby leading to the effective control of neuronal information and other mechanosensory processes. The modified Hodgkin-Huxley equation is equally shown to support static kink modes, as reflected in the spatiotemporal profiles of the transmembrane voltage.