On simulation modelling of nerve impulse

V. S. Novoselov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 4 (2011), pp. 73-83

Résumé

It is shown that when an ionic excitable membrane is incorporated into a nonlinear diffusion process, it can give rise to travelling nonlinear waves of electrical excitation. The fundamental equation describing the propagation of nonlinear waves in a one-dimensional case in medium weakly dispersion is the Korteweg–de Vries equation (KdV), with solution as stable solitary wave structures, i. e. solitons. A travelling pulse (often called a solitary pulse) is a travelling wave solution that starts and ends at the same steady state of the governing equations. We derive the KdV equation from the full set of neurodynamic equations (for the propagation of action potential along the axon of a nerve). The showed KdV equation indicated that this action potential should propagate along an axon with fixed velocity, which could be calculated. To study travelling waves define the travelling wave coordinate $\xi=x-vt$, where $v=\text{const}>0$ is the wave velocity. Because of application of the travelling coordinate the partial differential equations become the ordinary differential equations. To understand the structure of a travelling pulse it is helpful to study travelling pulse solution in the partial piecewise linear equation mit three or four regions. In each region the differential equation is linear and can be analytically solved. The regional solutions are then joined at $u=u_{\text{п}}$, and $u=u_{{\rm max}}$ by stipulating. The variable $u$ is a membrane potential. To understand the structure of a travelling pulse for the cardiac muscle it is helpful first to study travelling pulse solution piecewise linear equation in the FitzHugh–Nagumo model. To understand the structure of a travelling pulse for the nonmyelinated nerve fibers it is helpful to study travelling pulse solution of a piecewise linear equation in the Hodgkin–Huxley model. In this article three piecewise linear models are examined. The solutions of piecewise linear equations give initial approximations. Here upon the travelling pulses of the FitzHugh–Nagumo and Hodgkin–Huxley equations can be numerically computed.

Détail

Mots-clés : soliton
Keywords: wave equation of long waves, travelling nerve pulse, action potential, piecewise linear model.

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