Amongst the important phenomena in neurophysiology, nerve pulse generation and propagation is fundamental. Scientists have studied this phenomena using mathematical models based on experimental observations on the phy...Amongst the important phenomena in neurophysiology, nerve pulse generation and propagation is fundamental. Scientists have studied this phenomena using mathematical models based on experimental observations on the physiological processes in the nerve cell. Widely used models include: the Hodgkin-Huxley (H-H) model, which is based entirely on the electrical activity of the nerve cell;and the Heimburg and Jackson (H-J), model based on the thermodynamic activity of the nerve cell. These classes of models do not, individually, give a complete picture of the processes that lead to nerve pulse generation and propagation. Recently, a hybrid model proposed by Mengnjo, Dikandé and Ngwa (M-D-N), takes into consideration both the electrical and thermodynamic activities of the nerve cell. In their work, the first three bound states of the model are analytically computed and they showed great resemblance to some of the experimentally observed pulse profiles. With these bound states, the M-D-N model reduces to an initial value problem of a linear parabolic partial differential equation with variable coefficients. In this work we consider the resulting initial value problem and, using the theory of function spaces, propose and prove conditions under which such equations will admit unique solutions. We then verify that the resulting initial value problem from the M-D-N model satisfies these conditions and so has a unique solution. Given that the derived initial value problem is complex and there are no known analytic techniques that can be deployed to obtain its solution, we designed a numerical experiment to estimate the solutions. The simulations revealed that the unique solution is a stable pulse that propagates in the x-t plane with constant velocity and maintains the shape of the initial profile.展开更多
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 impo...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.展开更多
文摘Amongst the important phenomena in neurophysiology, nerve pulse generation and propagation is fundamental. Scientists have studied this phenomena using mathematical models based on experimental observations on the physiological processes in the nerve cell. Widely used models include: the Hodgkin-Huxley (H-H) model, which is based entirely on the electrical activity of the nerve cell;and the Heimburg and Jackson (H-J), model based on the thermodynamic activity of the nerve cell. These classes of models do not, individually, give a complete picture of the processes that lead to nerve pulse generation and propagation. Recently, a hybrid model proposed by Mengnjo, Dikandé and Ngwa (M-D-N), takes into consideration both the electrical and thermodynamic activities of the nerve cell. In their work, the first three bound states of the model are analytically computed and they showed great resemblance to some of the experimentally observed pulse profiles. With these bound states, the M-D-N model reduces to an initial value problem of a linear parabolic partial differential equation with variable coefficients. In this work we consider the resulting initial value problem and, using the theory of function spaces, propose and prove conditions under which such equations will admit unique solutions. We then verify that the resulting initial value problem from the M-D-N model satisfies these conditions and so has a unique solution. Given that the derived initial value problem is complex and there are no known analytic techniques that can be deployed to obtain its solution, we designed a numerical experiment to estimate the solutions. The simulations revealed that the unique solution is a stable pulse that propagates in the x-t plane with constant velocity and maintains the shape of the initial profile.
文摘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.
