Exp-Function Method for Solving Huxley Equation

Abstract

Huxley equation is a core mathematical framework for modern biophysically based neural modeling. It is often useful to obtain a generalized solitary solution for fully understanding its physical meanings. There are many methods to solve the equation, but each method can only lead to a special solution. This paper suggests a relatively new method called the Exp-function method for this purpose. The obtained result includes all solutions in open literature as special cases, and the generalized solution with some free parameters might imply some fascinating meanings hidden in Huxley equation.

1. Introduction

The core mathematical framework for modern biophysically based neural modeling was developed half a century ago by Hodgkin and Huxley [1]. In a series of papers published in 1952, they presented the results of an elegant series of electrophysiological experiments in which they investigated the flow of electric current through the surface membrane of the giant nerve fiber of a squid. Huxley equation [2–6] is a nonlinear partial differential equation of second order of the form 𝑢_𝑡=𝑢_𝑥𝑥+𝑢(𝑘−𝑢)(𝑢−1). (1.1)

This equation is an evolution equation that describes the nerve propagation in biology from which molecular CB properties can be calculated. It also gives a phenomenological description of the behavior of the myosin heads II. This equation has many fascinating phenomena such as bursting oscillation [7], interspike [8], bifurcation, and chaos [9]. A generalized exact solution can gain an insight into these phenomena. There is not a universal method for nonlinear equations. The traditional approaches to this task are the variational iteration method [10–12], the homotopy perturbation method [13, 14], Adomain’s decomposition method [15], and the tanh method [16–18]; however, many methods may sometimes fail or the solution procedure is complex as the degree of nonlinearity increases, for example, calculation of Adomian polynomials in Adomian’s method is terribly tedious. Recently, the study showed that the homotopy perturbation method [19–21] and the variational iteration method [22, 23] can completely overcome the difficulty. If we are really determined to extract physical meanings from analytic formulations of biological processes, we must resort to amelioration of the classical methods using modern mathematical tools. Exp-function method [24–29] is at this moment the most promising candidate theory for this purpose.

2. Basic Idea of Exp-Function Method

Rational approximation for soliton and soliton-like solutions was first proposed by Hirota [30] and further developed by many authors [31, 32]. In this paper, we will apply the Exp-function method to the discussed problem. The basic idea of the Exp-function was proposed in He’s monograph [30]. Some illustrative examples in [24–27] showed that this method is very effective to search for various solitary and periodic solutions of nonlinear equations. Zhu applied the method to some difference-differential equations [28, 29].

Consider the following general partial differential equation: 𝑃(𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,𝑢_𝑥𝑥𝑥,…)=0. (2.1) We first unite the independent variables 𝑥 and 𝑡 into one wave variable 𝜂=𝜔𝑥+𝜗𝑡, leading (2.1) to an ordinary differential equation, 𝑄(𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…)=0. (2.2) The Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form [30]: ∑𝑢(𝜂)=^{𝑑𝑛=−𝑐}𝑎_𝑛exp(𝑛𝜂)∑^𝑞_𝑛=−𝑝𝑏_𝑚=𝑎exp(𝑚𝜂)^𝑐exp(𝑐𝜂)+⋯+𝑎_−𝑑exp(−𝑑𝜂)𝑏_𝑝exp(𝑝𝜂)+⋯+𝑎_−𝑞exp(−𝑞𝜂), (2.3) where 𝑐, 𝑑, 𝑝, and 𝑞 are positive integers which are unknown to be further determined, and 𝑎_𝑛 and 𝑏_𝑚 are unknown constants. To determine the values of 𝑐 and 𝑝, we balance the linear term of highest order in (2.2) with the highest-order nonlinear term. Similarly to determine the values of 𝑑 and 𝑞, we balance the linear term of lowest order in (2.2) with the lowest-order nonlinear term.

3. Application to Huxley Equation

Using the wave variable 𝜂=𝜔𝑥+𝜗𝑡, we have −𝜗𝑢^′+𝜔²𝑢^′′+𝑢(𝑘−𝑢)(𝑢−1)=0. (3.1) The highest linear term 𝑢^′′ is now given by 𝑢^′′=𝑐₁exp[(3𝑝+𝑐)𝜂]+⋯𝑐₂,𝑢exp[4𝑝𝜂]+⋯₃=𝑐₃exp[3𝑐𝜂]+⋯𝑐₄=𝑐exp[3𝑝𝜂]+⋯³exp[(3𝑐+𝑝)𝜂]+⋯𝑐₄.exp[4𝑝𝜂]+⋯ (3.2) Balancing the highest order of Exp-function in (3.2), we have 3𝑝+𝑐=3𝑐+𝑝, and this gives 𝑝=𝑐. Using the same method, we can also obtain that 𝑞=𝑑. Wu and He [33] systematically studied the choice of the values of the parameters, and revealing the solution very weakly depends upon the values of the parameters. An illustrating example of Dodd–Bullough–Mikhailov equation was given.

Wu and He considered the following three cases.

Case 1. 𝑝=𝑐=1, 𝑞=𝑑=1. Case 2. 𝑝=𝑐=2, 𝑞=𝑑=2. Case 3. 𝑝=𝑐=2, 𝑞=𝑑=1.
All cases led to the equivalent result. Bekir and Boz [34] pointed out that 𝑝=𝑐=1 and 𝑞=𝑑=1 are valid for most nonlinear partial differential equations.

For simplicity, we set 𝑝=𝑐=1 and 𝑞=𝑑=1, so (2.3) reduces to 𝑎𝑢(𝜂)=¹exp(𝜂)+𝑎₀+𝑎₋₁exp(−𝜂)exp(𝜂)+𝑏₀+𝑏₋₁exp(−𝜂). (3.3) Substituting (3.3) into (3.1), and by the help of Mathematica, we have −1𝐴[𝐶₃𝑒^3𝜂+𝐶₂𝑒^2𝜂+𝐶₁𝑒^𝜂+𝐶₀+𝐶₋₁𝑒^−𝜂+𝐶₋₂𝑒^−2𝜂+𝐶₋₃𝑒^−3𝜂]=0, (3.4) where 𝐴=(𝑒^𝜂+𝑏₀+𝑏₋₁𝑒^−𝜂)³,𝐶₃=𝑘𝑎₁−𝑎²₁−𝑘𝑎²₁+𝑎³₁,𝐶₂=𝑘𝑎₀−𝜔²𝑎₀−𝜗𝑎₀−2𝑎₀𝑎₁−2𝑘𝑎₀𝑎₁+3𝑎₀𝑎²₁−2𝑘𝑎₁𝑏₀+𝜗𝑎₁𝑏₀+𝜔²𝑎₁𝑏₀−𝑎²₁𝑏₀−𝑘𝑎²₁𝑏₀,𝐶₁=𝑘𝑎₋₁−2𝜗𝑎₋₁−4𝜔²𝑎₋₁−𝑎²₀−𝑘𝑎²₀−2𝑎₋₁𝑎₁−2𝑘𝑎₋₁𝑎₁+3𝑎²₀𝑎₁+3𝑎₋₁𝑎²₁+2𝑘𝑎₁𝑏₋₁+2𝜗𝑎₁𝑏₋₁+4𝜔²𝑎₁𝑏₋₁−𝑎²₁𝑏₋₁−𝑘𝑎²₁𝑏₋₁+2𝑘𝑎₀𝑏₀−𝜔²𝑎₀𝑏₀−𝜗𝑎₀𝑏₀−2𝑎₀𝑎₁𝑏₀−2𝑘𝑎₀𝑎₁𝑏₀+𝑘²𝑎₁𝑏²₀+𝜗𝑎₁𝑏²₀−𝜔²𝑎₁𝑏²₀,𝐶₀=−2𝑎₋₁𝑎₀−2𝑘𝑎₋₁𝑎₀+𝑎³₀+6𝑎₋₁𝑎₀𝑎₁+2𝑘𝑎₀𝑏₋₁+6𝜔²𝑎₀𝑏₋₁−2𝑎₀𝑎₁𝑏₋₁−2𝑘𝑎₀𝑎₁𝑏₋₁+2𝑘𝑎₋₁𝑏₀+3𝜔²𝑎₋₁𝑏₀−𝑎²₀𝑏₀−𝑘𝑎²₀𝑏₀−2𝑎₋₁𝑎₁𝑏₀−2𝑘𝑎₋₁𝑎₁𝑏₀+2𝑘𝑎₁𝑏₋₁𝑏₀−3𝜔²𝑎₁𝑏₋₁𝑏₀+3𝜗𝑎₁𝑏₋₁𝑏₀+𝑘𝑎₀𝑏²₀,𝐶₋₁=−𝑎²₋₁−𝑘𝑎²₋₁+3𝑎₋₁𝑎²₀+3𝑎²₋₁𝑎₁+2𝑘𝑎₋₁𝑏₋₁−2𝜗𝑎₋₁𝑏₋₁+4𝜔²𝑎₋₁𝑏₋₁−𝑎²₀𝑏₋₁−𝑘𝑎²₀𝑏₋₁−2𝑎₋₁𝑎₁𝑏₋₁−2𝑘𝑎₋₁𝑎₁𝑏₋₁+𝑘𝑎₁𝑏²₋₁−4𝜔²𝑎₁𝑏²₋₁+2𝜗𝑎₁𝑏²₋₁−2𝑎₋₁𝑎₀𝑏₀−2𝑘𝑎₋₁𝑎₀𝑏₀+2𝑘𝑎₀𝑏₋₁𝑏₀+𝜔²𝑎₀𝑏₋₁𝑏₀+𝜔𝑎₀𝑏₋₁𝑏₀+𝑘𝑎₋₁𝑏²₀−𝜔²𝑎₋₁𝑏²₀−𝜗𝑎₋₁𝑏²₀,𝐶₋₂=3𝑎²₋₁𝑎₀−2𝑎₋₁𝑎₀𝑏₋₁−2𝑘𝑎₋₁𝑎₀𝑏₋₁+𝑘𝑎₀𝑏²₋₁−𝜔²𝑎₀𝑏²₋₁+𝜗𝑎₀𝑏²₋₁−𝑎²₋₁𝑏₀−𝑘𝑎²₋₁𝑏₀+2𝑘𝑎₋₁𝑏₋₁𝑏₀+𝜔²𝑎₋₁𝑏₋₁𝑏₀−𝜗𝑎₋₁𝑏₋₁𝑏₀,𝐶₋₃=𝑎²₋₁−𝑎²₋₁𝑏₋₁−𝑘𝑎²₋₁𝑏₋₁+𝑘𝑎₋₁𝑏²₋₁. (3.5)

Solving the system (3.5) simultaneously using Mathematica, we obtain the following results.

Case 1. 1𝜔=±√2,𝜗=−2𝑘−12,𝑎₋₁=0,𝑎₀=0,𝑎₁=1,𝑏₋₁=0. (3.6)Case 2. 𝑘𝜔=±√2,𝜗=2𝑘−𝑘²2,𝑎₋₁=𝑘𝑏₋₁,𝑎₀=12(𝑘𝑏₀±𝑘²𝑏²₀−4𝑘²𝑏₋₁𝑎),₁=0,𝑏₋₁≠0,𝑏²₀−𝑏₋₁≥0, (3.7) where 𝑏₋₁ and 𝑏₀ are free parameters.Case 3. 𝜔=±𝑘−1√2,𝜗=1−𝑘²2,𝑎₋₁=0,𝑎₀=𝑘𝑏₀,𝑎₁=1,𝑏₋₁=0,𝑏₀≠0, (3.8) where 𝑏₀ is a free parameter. For Case 1, we obtain the following solution of (1.1) by substituting (3.6) into (3.3): 𝑢₁1(𝑥,𝑡)=1+𝑏₀𝑒^{√−(1/2)𝑥+((2𝑘−1)/2)𝑡}. (3.9) For Case 2, we have 𝑢₂(𝑥,𝑡)=(1/2)(𝑘𝑏₀±𝑘²𝑏²₀−4𝑘²𝑏₋₁)+𝑘𝑏₋₁𝑒^−𝜂𝑒^𝜂+𝑏₀+𝑏₋₁𝑒^−𝜂, (3.10) where 𝜂=(𝑘/√2)𝑥+((2𝑘−𝑘²)/2)𝑡 or 𝜂=−(𝑘/√2)𝑥+((2𝑘−𝑘²)/2)𝑡. Case 3 leads to the following exact solution: 𝑢₃𝑒(𝑥,𝑡)=^𝜂+𝑘𝑏₀𝑒^𝜂+𝑏₀, (3.11) where 𝜂=((𝑘−1)/√2)𝑥+((1−𝑘²)/2)𝑡 or 𝜂=−((𝑘−1)/√2)𝑥+((1−𝑘²)/2)𝑡.

To compare our results with those obtained in [16], we set 𝑏₀=1 in (3.9). Equation (3.9) becomes 𝑢₁₍₁₎1(𝑥,𝑡)=1+𝑒^{√−(1/2)𝑥+((2𝑘−1)/2)𝑡}=121(1+tanh[2√2(𝑥−2𝑘−1√2𝑡)]). (3.12) In (3.10), if we set 𝑏₀=2, 𝑏₋₁=1, and 𝜂=(𝑘/√2)𝑥+((2𝑘−𝑘²)/2)𝑡, (3.10) becomes 𝑢₂₍₁₎(𝑥,𝑡)=𝑘+𝑘𝑒^−𝜂𝑒^𝜂+2+𝑒^−𝜂=𝑘1+𝑒^𝜂=𝑘2𝑘(1−tanh[2√2(𝑥−𝑘−2√2𝑡)]). (3.13) In (3.11), if we set 𝑏₀=1 and 𝜂=−((𝑘−1)/√2)𝑥+((1−𝑘²)/2)𝑡, (3.11) becomes 𝑢₃₍₁₎(𝑥,𝑡)=𝑘+𝑒^𝜂1+𝑒^𝜂=𝑘+12+1−𝑘2tanh[𝑘−12√2(𝑥−𝑘+1√2𝑡)], (3.14) These are kink solutions obtained by the tanh-coth method in [16].

In (3.9), (3.10), and (3.11), if we set parameters as follows: (1) 𝑏₀=−1; (2) 𝑏₀=−2,𝑏₋₁=1, and 𝜂=(𝑘/√2)𝑥+((2𝑘−𝑘²)/2)𝑡; (3) 𝑏₀=−1, and 𝜂=−((𝑘−1)/√2)𝑥+((1−𝑘²)/2)𝑡, respectively, we have 𝑢₁₍₂₎1(𝑥,𝑡)=1−𝑒^{√−(1/2)𝑥+(2𝑘−1/2)𝑡}=121(1+coth[2√2(𝑥−2𝑘−1√2𝑢𝑡)]),₂₍₂₎(𝑥,𝑡)=−𝑘+𝑘𝑒^−𝜂𝑒^𝜂−2+𝑒^−𝜂=𝑘1−𝑒^𝜂=𝑘2𝑘(1−coth[2√2(𝑥−𝑘−2√2𝑢𝑡)]),₃₍₂₎(𝑥,𝑡)=𝑘−𝑒^𝜂1−𝑒^𝜂=𝑘+12+1−𝑘2coth[𝑘−12√2(𝑥−𝑘+1√2𝑡)]. (3.15) These are the traveling solutions obtained by the tanh-coth method in [16].

The other three traveling solutions 𝑢₇(𝑥,𝑡), 𝑢₈(𝑥,𝑡), and 𝑢₉(𝑥,𝑡) in [16] are equivalent with 𝑢₁₍₂₎(𝑥,𝑡), 𝑢₂₍₂₎(𝑥,𝑡), and 𝑢₃₍₂₎(𝑥,𝑡), respectively.

He and Wu [26] compared the method with the Sinh-function and the Tanh-function methods, and found a single generalized solitonary solution including all solutions obtained by Yomba using the subequation method.

4. Discussions and Conclusions

The Exp-function method leads to generalized solitary solutions with some free parameters involving the known solutions in open literature. The free parameters might imply some physically meaningful results in biological process. Considering the generalized solution 𝑢₁(𝑥,𝑡) expressed in (3.9), in case 𝑏₀=1, it turns out to a special solution in [16]; the physical understating of the special solution was given in [16]. Of course we can set 𝑏₀ equal to other values, resulting in different solitary shapes. The free parameter 𝑏₀ might be also relative to initial conditions.