Abstract
Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play important role in the control of various sensory (i.e. stimuli-responsive) organs and homeostasis of living organisms. In this work, the influence of mechanotransduction processes on the generic mechanism of the action potential is investigated analytically, by considering a mathematical model that consists of two coupled nonlinear partial differential equations. One of these two equations is the Korteweg-de Vries equation governing the spatio-temporal evolution of the density difference between intracellular and extracellular fluids across the nerve membrane, and the other is Hodgkin-Huxley cable equation for the transmembrane voltage with a self-regulatory (i.e. diode-type) membrane capacitance. The self-regulatory feature here refers to the assumption that membrane capacitance varies with the difference in density of ion-carrying intracellular and extracellular fluids, thus ensuring an electromechanical feedback mechanism and consequently an effective coupling of the two nonlinear equations. The exact one-soliton solution to the density-difference equation is obtained in terms of a pulse excitation. With the help of this exact pulse solution the Hodgkin-Huxley cable equation is shown to transform, in steady state, to a linear eigenvalue problem some bound states of which can be obtained exactly. Few of such bound-state solutions are found analytically.
Highlights
The mechanism by which the action potential is generated and transmitted along the neuronal network, is a complex phenomenon that has been actively investigated in Mathematical Physics [1,2,3,4,5,6,7,8]
A pioneer model of the nerve-impulse generation was proposed by Hodgkin and Huxley [1]; the model rests on the picture of the nerve impulse as an electric voltage propagating in form of pulse wave along the nerve fiber
This wave is assumed to propagate with a constant shape following a mechanism which can be summarized as follow: During the generation and transmission of the nerve impulse, the leading edge of the depolarization region of the nerve triggers adjacent membranes to depolarize
Summary
The mechanism by which the action potential is generated and transmitted along the neuronal network, is a complex phenomenon that has been actively investigated in Mathematical Physics [1,2,3,4,5,6,7,8]. According to the HH model the nerve impulse is a self-regenerative wave associated with the electrochemical activity of the nerve cells, due to flow of ion currents (Na and K) through specific ion channels [1] This wave is assumed to propagate with a constant shape following a mechanism which can be summarized as follow: During the generation and transmission of the nerve impulse, the leading edge of the depolarization region of the nerve triggers adjacent membranes to depolarize. This causes a selfpropagation of the excitation related to the transmembrane potential down the nerve fiber [1, 9, 10]. In a future work numerical simulations will be carried out on the spatio-temporal model, to check the stability of these exact soliton solutions as they propagate along the nerve
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