Journal of Applied Mathematics
Volume 2003 (2003), Issue 8, Pages 397-407
Identifying generalized Fitzhugh-Nagumo equation from a numerical solution of Hodgkin-Huxley model
Section of Biodynamics and Biorheology, Institute of Mechanics, Acad. G. Bonchev St., bl. 4, Sofia 1113, Bulgaria
Received 25 November 2002; Revised 16 January 2003
Copyright © 2003 Nikola V. Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An analytic time series in the form of numerical solution (in an appropriate finite time interval) of the Hodgkin-Huxley current clamped (HHCC) system of four differential equations, well known in the neurophysiology as an exact empirical model of excitation of a giant axon of Loligo, is presented. Then we search for a second-order differential equation of generalized Fitzhugh-Nagumo (GFN) type, having as a solution the given single component (action potential) of the numerical solution. The given time series is used as a basis for reconstructing orders, powers, and coefficients of the polynomial right-hand sides of GFN equation approximately governing the process of action potential. For this purpose, a new geometrical method for determining phase space dimension of the unknown dynamical system (GFN equation) and a specific modification of least squares method for identifying unknown coefficients are developed and applied.