On a Model for Nerve Impulse Generation Mediated by Electromechanical Processes

Abstract

The generation of action potential involves specific mechanosensory stimuli that are manifest in the variation of membrane capacitance, related to the selective membrane permeability to ion exchanges, showing the central role of electromechanical processes in the buildup mechanism of the nerve impulse. It has been established by Gross et al. (Cellular and Molecular Neurobiology 3:89, 27) that in these electromechanical processes, the net instantaneous charge stored in the membrane capacitor is regulated by the rate of change of the net fluid density through the membrane, corresponding to the difference in densities of extracellular and intracellular fluids. In the present work, an electromechanical model for the nerve is considered, in which mechanical forces are assumed to be generated by fluid flow through the nerve membrane. These mechanical forces induce pressure waves that stimulate the membrane, and hence control the net charge stored in the membrane capacitor. The mathematical model features two coupled nonlinear partial differential equations, namely the familiar cable equation for the transmembrane voltage in which the membrane capacitor now acts like a capacitive diode, and the Heimburg-Jackson’s nonlinear hydrodynamic equation for the pressure wave assumed to control the instantaneous charge in the membrane capacitor. In the stationary regime, the variable-capacitance cable equation reduces to a linear eigenvalue problem with a null spectral parameter, the exact bound states of which are Legendre polynomials. In the dynamical regime, numerical simulations of the modified cable equation lead to a variety of wave profiles for the transmembrane voltage.