Abstract
The conditions when solutions of Huxley equation can be expressed in special form and the procedure of finding exact solutions are presented in this paper. Huxley equation is an evolution equation that describes the nerve propagation in biology. It is often useful to obtain a generalized solitary solution for fully understanding its physical meanings. It is shown that the solution produced by the Exp-function method may not hold for all initial conditions. It is proven that the analytical condition describing the existence of the produced solution in the space of initial conditions (or even in the space of the system's parameters) can not be derived by the Exp-function method because the question about the existence of that solution is omitted. The proposed operator method, on the contrary, brings the load of symbolic computations before the structure of the solution is identified. The method for the derivation of the solution is based on the concept of the rank of the Hankel matrix constructed from the sequence of coefficients representing formal solution in the series form. Moreover, the structure of the algebraic-analytic solution is generated automatically together with all conditions of the solution's existence. Computational experiments are used to illustrate the properties of derived analytical solutions.
Highlights
Huxley equation is a core mathematical framework for modern biophysically based neural modelling
Definitions and theorems associated with stating conditions, under which the power series solution of a differential equation can be reduced to a finite sum of standard functions, are presented below
The solution produced by the Exp-function method may not hold for all initial conditions
Summary
Huxley equation is a core mathematical framework for modern biophysically based neural modelling. It is often useful to obtain a generalized analytical solitary solution for fully understanding physical meanings of nonlinear processes taking place in Hodgkin-Huxley models. An analytical criterion determining if a solution of a differential equation can be expressed in an analytical form comprising exponential functions is developed in [12]. Huxley equation is a nonlinear partial differential equation of second order of the form ut = uxx + u (k − u) (u − 1) This equation is an evolution equation that describes the nerve propagation in biology. The Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons It proposes that signals travel along the cell’s membrane in the form of certain kinds of sound (or density) pulses known as solitons.
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