Numerical Simulation of the FitzHugh-Nagumo Equations

Abstract

The variational iteration method and Adomian decomposition method are applied to solve the FitzHugh-Nagumo (FN) equations. The two algorithms are illustrated by studying an initial value problem. The obtained results show that only few terms are required to deduce approximated solutions which are found to be accurate and efficient.

1. Introduction

The pioneering work of Hodgkin and Huxley [1], and subsequent investigations, has established that good mathematical models for the conduction of nerve impulses along an axon can be given. These models take the form of a system of ordinary differential equations, coupled to a diffusion equation. Simpler models, which seem to describe the qualitative behavior, have been proposed by FitzHugh [2] and Nagumo [3]. This paper is devoted to the study of the FitzHugh-Nagumo (FN) system: 𝑣_𝑡(𝑥,𝑡)=𝑣_𝑥𝑥()(𝑣((),𝑤𝑥,𝑡−𝑓𝑥,𝑡))−𝑤𝑥,𝑡_𝑡(𝑥,𝑡)=𝑏𝑣(𝑥,𝑡)−𝛾𝑤(𝑥,𝑡), (1.1) where 𝑏 and 𝛾 are positive constants and 𝑓(𝑣(𝑥,𝑡)) is nonlinear function. Existence and uniqueness for this system is given in 1978 by Rauch and Smoller [4], in which they showed that small solutions 𝑣(𝑥,𝑡) decay to 0 as 𝑡→∞ and large pulses produce a traveling wave. We consider the FN equations in the following form 𝑣_𝑡(𝑥,𝑡)=𝑣_𝑥𝑥()(𝑣((),𝑤𝑥,𝑡−𝑓𝑥,𝑡))−𝑤𝑥,𝑡_𝑡(𝑥,𝑡)=𝑏𝑣(𝑥,𝑡), (1.2) and the function 𝑓(𝑣(𝑥,𝑡)) is given by Mckean [5] such that: 𝑓(𝑣(()(𝑣())1𝑥,𝑡))=𝑣𝑥,𝑡−𝐻𝑥,𝑡−𝑎,0≤𝑎≤2, (1.3) where 𝐻 is the Heaviside step function 𝐻(𝑠)=0𝑠<0,1𝑠≥0. (1.4)

The exact solution of this system is given by: 𝑣(𝑥,𝑡)=𝑎𝑒^𝛼1𝑧1,𝑧≤0𝑎−𝑝^′₁𝑒^𝛼1𝑧−1𝑝^′₂𝑒^𝛼2𝑧−1𝑝^′₃𝑒^𝛼3𝑧,0≤𝑧≤𝑧₁𝑒^{−𝛼2𝑧1}−1𝑝^′₂𝑒^𝛼2𝑧+𝑒^{−𝛼3𝑧1}−1𝑝^′₃𝑒^𝛼3𝑧,𝑧₁𝑤()≤𝑧,𝑥,𝑡=𝑣_𝑥𝑥(𝑥,𝑡)−𝑣_𝑡(𝑥,𝑡)−𝑓(𝑣(𝑥,𝑡)), (1.5)(1.6) where 𝑧=𝑥+𝑐𝑡, 𝑐 is the speed of the traveling wave and 𝛼_𝑖,𝑖=1,2,3 are the zeros of the polynomial 𝑝(𝛼)=𝛼³−𝑐𝛼²𝑏−𝛼−𝑐,𝑝^′_𝑖=𝑝^′𝛼_𝑖,𝑖=1,2,3. (1.7)

A numerical scheme for FN equations [6] by collocation method and the “Hopscotch” finite difference scheme first proposed by Gordon [7], and further developed by Gourlay and McGuire [8, 9]. Other possible schemes which were considered are (i) finite difference schemes [10], (ii) Galerkin-type schemes [11], and (iii) collocation schemes with quadratic and cubic splines [6]. In this paper, we use the variational iteration and Adomian decomposition methods to find the numerical solutions of the FN equations which will be useful in numerical studies. In our numerical study we consider the case 𝑏=0.1 and 𝑎=0.3, also 𝛼𝑐=0.7122,₁𝛼=1.46192629534582,₂𝛼=−0.1639653991443764,₃𝑧=−0.5857608638090818,₁=4.5976770121482735, (1.8) with these parameters now we can use the exact travelling wave solution (1.5) to test the suggested numerical methods.

2. The Formalism

We introduce the main points of each of the two methods, where details can be found in [12–37].

2.1. The Variational Iteration Method (VIM)

The VIM is the general Lagrange method, in which an extremely accurate approximation at some special point can be obtained, but not an analytical solution. To illustrate the basic idea of the VIM we consider the following general partial differential equation: 𝐿_𝑡𝑢(𝑥,𝑡)+𝐿_𝑥𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)+𝑔(𝑡,𝑥)=0, (2.1) where 𝐿_𝑡 and 𝐿_𝑥 are linear operators of 𝑡 and 𝑥 respectively, and 𝑁 is a nonlinear operator. According to the VIM, we can expressed the following correction functional in 𝑡-, and 𝑥-directions, respectively, as follows:𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡_𝑡0𝜆𝐿_𝑠𝑢_𝑛(𝑥,𝑠)+𝐿_𝑥+𝑁𝑢_𝑛()()𝑢𝑥,𝑠+𝑔𝑥,𝑠𝑑𝑠,_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑥_𝑥0𝜇𝐿_𝑠𝑢_𝑛(𝑡,𝑠)+𝐿_𝑡+𝑁𝑢_𝑛(𝑡,𝑥)+g(𝑡,𝑠)𝑑𝑠, (2.2a)(2.2b)

where 𝜆 and 𝜇 are general Lagrange multipliers, which can be identified optimally via the variational theory, and 𝑢_𝑛(𝑥,𝑡) are restricted variations which mean that 𝛿𝑢_𝑛(𝑥,𝑡)=0. By this method, it is required first to determine Lagrange multipliers 𝜆 and 𝜇 that will be identified optimally. The successive approximations 𝑢_𝑛+1(𝑥,𝑡), 𝑛≥0 of the solution 𝑢(𝑥,𝑡) will be readily obtained upon using the determined Lagrange multipliers and any selective function 𝑢₀(𝑥,𝑡). Consequently, the solution is given by 𝑢(𝑥,𝑡)=lim_𝑛→∞𝑢_𝑛(𝑥,𝑡). (2.3)

The above analysis yields the following theorem.

Theorem 2.1. The VIM solution of the partial differential equation (2.1) can be determined by (2.3) with the iterations (2.2a) or (2.2b).

2.2. Adomian Decomposition Method (ADM)

Applying the inverse operator 𝐿⁻¹(⋅)=∫^𝑡₀(⋅)𝑑𝑡 to both sides of (2.1) and using the initial condition, we get 𝑢₀()(),𝑢𝑥,𝑡=𝑢𝑥,0_𝑛+1(𝑥,𝑡)=^𝑡₀−𝐿_𝑥𝑢_𝑛(𝑥,𝑡)−𝐴_𝑛−𝑔(𝑥,𝑡)𝑑𝑡,𝑛≥0, (2.4) where the nonlinear operator 𝑁(𝑢)=∑^∞_𝑛=0𝐴_𝑛 is the Adomian polynomial determined by 𝐴_𝑛=1𝑑𝑛!𝑑𝜆^𝑛𝑛∞_𝑖=0𝜆^𝑖𝑢_𝑖(𝑥,𝑡)_𝜆=0,𝑛=0,1,2…. (2.5)

We next decompose the unknown function 𝑢(𝑥,𝑡) by a sum of components defined by the following decomposition series 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑢_𝑛(𝑥,𝑡). (2.6)

The above analysis yields the following theorem

Theorem 2.2. The ADM solution of the partial differential equation (2.1) can be determined by the series (2.6) with the iterations (2.4).

3. Applications

We solve the FN equations using the two methods VIM and ADM.

3.1. The VIM for the FN Equations

Consider the FN equations in the form 𝑣_𝑡−𝑣_𝑥𝑥(𝑣)𝑤+𝑓+𝑤=0,_𝑡−𝑏𝑣=0. (3.1)

Then the VIM formulae take the forms 𝑣_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)+^𝑡₀𝜆𝑣_𝑠(𝑥,𝑠)−𝑣_𝑥𝑥()((()𝑤𝑥,𝑠+𝑓𝑣𝑥,𝑠))+𝑤𝑥,𝑠𝑑𝑠,_𝑛+1(𝑥,𝑡)=𝑤_𝑛(𝑥,𝑡)+^𝑡₀𝜇(𝑤_𝑠(𝑥,𝑠)−𝑏𝑣(𝑥,𝑠)𝑑𝑠, (3.2) where 𝑣₀(𝑥,𝑡)=𝑣(𝑥,0), 𝑤₀(𝑥,𝑡)=𝑤(𝑥,0) and 𝑛≥0. This yields the stationary conditions 𝜆^′(𝑠)=0,𝜆+1|_𝑠=𝑡=0,𝜇^′(𝑠)=0,𝜇+1_𝑠=𝑡=0, (3.3)

Hence, the Lagrange multipliers are 𝜆(𝑠)=𝜇(𝑠)=−1. (3.4)

Substituting these values of Lagrange multipliers into the functional correction (3.2) gives the iterations formulae 𝑣_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)+^𝑡₀𝜆𝑣_𝑠(𝑥,𝑠)−𝑣_𝑥𝑥()(𝑣(()𝑤𝑥,𝑠+𝑓𝑥,𝑠))+𝑤𝑥,𝑠𝑑𝑠,_𝑛+1(𝑥,𝑡)=𝑤_𝑛(𝑥,𝑡)+^𝑡₀𝜇(𝑤_𝑠(𝑥,𝑠)−𝑏𝑣(𝑥,𝑠)𝑑𝑠. (3.5)

We start with initial approximations as follows 𝑣₀(𝑥,𝑡)=𝑎𝑒^𝛼1𝑥1,𝑥≤0,𝑎−𝑝₁′𝑒^𝛼1𝑥−1𝑝₂′𝑒^𝛼2𝑥−1𝑝₃′𝑒^𝛼3𝑥,0≤𝑥≤𝑧₁,𝑒^{−𝛼2𝑧1}−1𝑝₂′𝑒^𝛼2𝑥+𝑒^{−𝛼3𝑧1}−1𝑝₃′𝑒^𝛼3𝑥,𝑧₁𝑤≤𝑥,₀(𝑥,𝑡)=𝑣₀(𝑥,𝑡)_𝑥𝑥−𝑣₀(𝑥,𝑡)_𝑡−𝑓𝑣₀(𝑥,𝑡), (3.6) and then the first iterations are𝑣₁(𝑥,𝑡)=𝑣11,𝑧≤0,𝑣12,0≤𝑧≤𝑧₁,𝑣13𝑧₁≤𝑧,𝑣11=0.312355𝑒^1.46193𝑥(0.960445+𝑡),𝑣12=𝑒^{−0.749726𝑥}𝑒^0.585761𝑥(1.45816−0.170279𝑡)+𝑒^2.21165𝑥(−0.000361868−0.000376771𝑡)+𝑒^0.163965𝑥(−1.1578+0.483011𝑡),𝑣13=𝑒^{−0.749726𝑥}𝑒^0.163965𝑥(15.9522−6.65492𝑡)+𝑒^0.585761𝑥(),𝑤−1.64071+0.191596𝑡₁(𝑥,𝑡)=𝑤11,𝑧≤0𝑤12,0≤𝑧≤𝑧₁𝑤13𝑧₁≤𝑧𝑤11=0.03𝑒^1.46193𝑥(0.960445+𝑡),𝑤12=𝑒^{−0.749726𝑥}𝑒^0.749726𝑥+𝑒^0.16396𝑥(0.277531−0.11578𝑡)+𝑒^2.21165𝑥(−0.000034755−0.0000361868𝑡)+𝑒^0.585761𝑥(−1.24868+0.145816𝑡),𝑤13=𝑒^{𝑥−0.749726}𝑒^0.585761𝑥(1.405−0.164071𝑡)+𝑒^0.163965𝑥(−3.82383+1.59522𝑡), (3.7) and so on.

The VIM produces the solutions 𝑣(𝑥,𝑡),𝑤(𝑥,𝑡) as follows 𝑣(𝑥,𝑡)=lim_𝑛→∞𝑣_𝑛(𝑥,𝑡),𝑤(𝑥,𝑡)=lim_𝑛→∞𝑤_𝑛(𝑥,𝑡), (3.8) where 𝑣_𝑛(𝑥,𝑡),𝑤_𝑛(𝑥,𝑡), will be determined in a recursive manner.

3.2. The ADM for the FN Equations

Consider the FN equations in the following form: 𝐿𝑣=𝑣_𝑥𝑥−𝑓(𝑣)−𝑤,𝐿𝑤=𝑏𝑣, (3.9)(3.10) where 𝐿(⋅)=𝜕(⋅)/𝜕𝑡. Operating by 𝐿⁻¹(⋅)=∫^𝑡₀(⋅)𝑑𝑡 on both sides of (3.9), we get 𝑣(𝑥,𝑡)=𝑣(𝑥,0)+^𝑡₀𝑣_𝑥𝑥()(𝑣(()𝑤()()+𝑥,𝑡−𝑓𝑥,𝑡))−𝑤𝑥,𝑡𝑑𝑡,𝑥,𝑡=𝑤𝑥,0^𝑡₀(𝑏𝑣(𝑥,𝑡)𝑑𝑡. (3.11)

The ADM assumes that the unknown functions 𝑣(𝑥,𝑡) and 𝑤(𝑥,𝑡) can be expressed by an infinite series in the forms 𝑣(𝑥,𝑡)=^∞_𝑛=0𝑣_𝑛(),𝑤()𝑥,𝑡𝑥,𝑡=^∞_𝑛=0𝑤_𝑛(𝑥,𝑡), (3.12) where 𝑣_𝑛(𝑥,𝑡),and𝑤_𝑛(𝑥,𝑡) can be determined by using the recurrence relations: 𝑣_𝑛+1(𝑥,𝑡)=^𝑡₀𝑣_𝑛𝑥𝑥(𝑥,𝑡)−𝑓𝑣_𝑛(𝑥,𝑡)−𝑤_𝑛()𝑤𝑥,𝑡𝑑𝑡,_𝑛+1(𝑥,𝑡)=^𝑡₀(𝑏𝑣_𝑛(𝑥,𝑡)𝑑𝑡,𝑛=0,1,…, (3.13) where 𝑓𝑣_𝑛()=𝑣𝑥,𝑡^𝑛(𝑥,𝑡),𝑣_𝑛()𝑣𝑥,𝑡<𝑎,_𝑛(𝑥,𝑡)−1𝑣_𝑛(𝑥,𝑡)≥𝑎 (3.14)

such that 𝑣₀()(),𝑤𝑥,𝑡=𝑣𝑥,0₀(𝑥,𝑡)=𝑤(𝑥,0). (3.15)

Then the first iterations are𝑣₁(𝑥,𝑡)=0.312355𝑒^1.46193,𝑧≤0𝑡+−1+0.483011𝑒^{−0.585761𝑥}−0.170279𝑒^{−0.163965𝑥}−0.000376771𝑒^1.46193𝑥𝑡,0≤𝑧≤𝑧₁−6.65492𝑒^{−0.585761𝑥}+0.191596𝑒^{−0.163965𝑥}𝑡,𝑧₁𝑤≤𝑧₁(𝑥,𝑡)=0.03𝑒^1.46193𝑥,𝑧≤00.1−1.1578𝑒^{−0.585761𝑥}+1.45816𝑒^{−0.163965𝑥}−0.000361868𝑒^1.46193𝑥𝑡,0≤𝑧≤𝑧₁0.115.9522𝑒^{−0.585761𝑥}−1.64071𝑒^{−0.163965𝑥}𝑡,𝑧₁,≤𝑧 (3.16a)(3.16b)

and so on.

The ADM yields the solutions 𝑣(𝑥,𝑡),𝑤(𝑥,𝑡) as 𝑣(𝑥,𝑡)=^∞_𝑛=0𝑣_𝑛(𝑥,𝑡),𝑤(𝑥,𝑡)=^∞_𝑛=0𝑤_𝑛(𝑥,𝑡), (3.17) where 𝑣_𝑛(𝑥,𝑡),𝑤_𝑛(𝑥,𝑡), will be determined in a recursive manner.

4. A Test Problem for the FN Equations

We discuss the solutions of the FN equations using the two considered VIM and ADM methods.

4.1. The VIM

Solve the FN equations (1.2) using the VIM with finite iterations at time 𝑇=5. A comparison between the computed solutions and the exact solutions at different values of 𝑥 are given in Table 1. We note that the VIM solutions converge to the exact solutions specially when 𝑛 is increased. We show in Figure 1 the behavior of the VIM solutions of FN equations at time 𝑇=5. If the exact solutions are plotted on Figure 1 we will find that the VIM and exact solutions curves are indistinguishable.

Table 1 

Comparison between the exact and approximate (VIM) solutions for the FN equations at time 𝑇=5.

(a)

(b)

(c)

(d)

Figure 1 

The approximated solutions for 𝑣(𝑥,𝑡),𝑤(𝑥,𝑡) at time 𝑇=5.

4.2. The ADM

Consider the same problems and use the ADM with the same initial conditions and use the technique discussed in Section 2. A comparison between the exact solutions and ADM solutions are shown in Table 2 and it seems that the errors are very small. We show in Figure 2 the numerical solutions of the FN equations.

Table 2 

Comparison between the exact solutions and approximation solutions (ADM) for FN equations at time = 5.

(a)

(b)

(c)

(d)

Figure 2 

The approximation solutions 𝑣(𝑥,𝑡),𝑤(𝑥,𝑡).

The results listed in Table 3 are representing the maximum errors at different times of VIM and ADM which shows that the VIM is better than ADM in the solutions of FN equations.

Table 3 

The maximum errors of our suggested methods VIM and ADM.

Now we show a comparison between our schemes and other methods as shown in Table 4.

Table 4 

Comparison between VIM, ADM, and other methods by maximum errors.

It is clear that the suggested methods for solving FN equation are the best methods than all other methods. Also all other methods give the solution as a discrete solution but our methods give the solution as a function 𝑥 and 𝑡.

5. Conclusion

In this paper the solutions for the FN equations using VIM and ADM methods have been generated. All numerical results obtained using few terms of the VIM and ADM show very good agreement with the exact solutions. Comparing our results with those of previous several methods shows that the considered techniques are more reliable, powerful, and promising mathematical tools. We believe that the accuracy of the VIM and ADM recommend it to be much wider applicability and also we find that the VIM more accurate than ADM.

Acknowledgment

This paper has been supported by the research support program, King Khalid University, Saudi Arabia, Grand no. KKU-COM-11-001.