Analysis and Computational Approximation of a Forward-Backward Equation Arising in Nerve Conduction

Transcription

1 Analysis and Computational Approximation of a Forward-Backward Equation Arising in Nerve Conduction P.M. Lima, M.F. Teodoro, N.J.Ford and P.M. Lumb Abstract This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a solution defined on the whole real axis, which tends to given values at ±.The numerical algorithms, developed previously by the authors for linear problems, were upgraded to deal with the case of nonlinear problems on unbounded domains. Numerical results are presented and discussed. 1 Introduction The present paper is concerned with a nonlinear MTFDE of the form RCv (t) = f (v(t)) + v(t τ) + v(t + τ) 2v(t), (1) where R and C are constants and f is a given function, as described below. We are interested in a solution of (1), increasing on ], [, which satisfies the conditions P.M. Lima Dep. Matemática/CEMAT, Instituto Superior Técnico, Universidade Técnica de Lisboa,Av.Rovisco Pais, Lisboa,Portugal, plima@math.ist.utl.pt M.F. Teodoro, Dep. Matemática, Instituto Politécnico de Setúbal, Estefanilha, Setúbal, Portugal mteodoro@est.ips.pt N.J. Ford, Dep. Mathematics, University of Chester,Parkgate Road, Chester CH4BJ, UK njford@chester.ac.uk P.M. Lumb, Dep. Mathematics, University of Chester, Parkgate Road, Chester CH4BJ,UK p.lumb@chester.ac.uk 1

2 2 P.M. Lima, M.F. Teodoro, N.J.Ford and P.M. Lumb lim v(t) = 0, lim t v(t) = 1, v(0) = 0.5. (2) t The problem (1)-(2) was analysed in [1], [2] and [3], where its physical meaning is explained in detail. The unknown v represents the transmembrane potential at a node in a myelinated axon. The function f reflects the current-voltage model, which is given by f (v) = bv(v a)(1 v), (3) where b > 0 and 0 < a < 1/2. R and C are respectively the nodal resistivity and the nodal capacity. The mathematical formulation can be derived from an electric circuit model which assumes the so-called pure saltatory conduction (PSC). This means that the myelin has such high resistance and low capacitance that it completely insulates the membrane; therefore, if a node is sufficiently stimulated and its transmembrane potential reaches a certain threshold level, ionic currents are generated which excite the neighbouring node. As a consequence, this node also attains the mentioned threshold potential. In this way, the process propagates across the nerve axon, giving the impression of an excitation jumping node to node. In the cited work of Chi, Bell and Hassard the problem (1)-(2) was thoroughly investigated, both from the analytical and numerical point of view. A computational algorithm was proposed and the first mumerical results (as far as we are aware) have been obtained for a nonlinear MTFDE. In the present paper, we continue the analytical and numerical investigation of equation (1), called the discrete Fitzhugh-Nagumo equation. In section 2, we analyse the asymptotic behaviour of its solutions at infinity. In section 3, we discuss computational methods for its numerical approximation. In section 4, we present some numerical results and we finnish with some conclusions in section 5. 2 Asymptotic Approximation at Infinity Since in the next section we describe computational methods for the numerical solution of the given problem, it is necessary to study the asymptotic behaviour of the required solution at infinity, so that we can define the domain where this solution needs to be computed. An extensive analysis of this behaviour has been provided in [3], so here we will just recall the main results from that paper. Let us first consider the case where t. According to the conditions (2), v( ) = 0, so that in order to linearise equation (1) about this point, we first use the Taylor expansion for function f : f (v) = f (0) + v f (0) + v2 2 f (0) + O(v 3 ) = v f (0) + v2 2 f (0) + O(v 3 ), where f is given by (3).

3 Forward-Backward Equation in Nerve Conduction 3 As usual, in order to obtain a characteristic equation for (1) at we must replace f by the main term of its Taylor expansion and assume that v has the form w 1 (t) = ε 1 e λ(t+l), (4) where L is a sufficiently large parameter and ε 1 is an estimate for v 1 ( L). In this way we obtain the equation λ + 2 f (0) 2cosh(λτ) = 0. (5) This equation has two real roots; since we are interested in a function w 1 that tends to 0 at, we choose the positive one, which we denote by λ 1. The case where t can be handled in an analogous way. In this case, we have the following Taylor expansion for f : f (v) = (v 1) f (1) + (v 1)2 2 Moreover, as t +, we assume that v has the form f (1) + O((v 1) 3 ). w 2 (t) = 1 ε 2 e λ(t L), (6) where ε 2 is an estimate of 1 v 2 (L). In this way we obtain the characteristic equation λ + 2 f (1) 2cosh(λτ) = 0 (7) In this case we choose the negative root of the characteristic equation λ 2, in order to have w 2 (t) 1, as t +. Now we have obtained two representations of the solution of our problem, (4) and (6), which can be used to approximate the solution, for t < L and t > L, respectively, where L is a sufficiently large number. According to the form of equation (1), L must be a multiple of the delay τ; in our computations we have used L = 2τ or L = 3τ, which is large enough to obtain a reasonable accuracy. These representations of the solution are used in the computational methods to replace the boundary conditions (2). In the next section we will show how this can be achieved. 3 Computational Methods 3.1 Numerical Methods for Linear Boundary Value Problems Boundary value problems for linear mixed-type functional differential equations (MTFDE) have been developed in [5], [6], [7] and [9]. In these papers we have considered equations of the form

4 4 P.M. Lima, M.F. Teodoro, N.J.Ford and P.M. Lumb x (t) = α(t)x(t) + β(t)x(t 1) + γ(t)x(t + 1), (8) where x is the unknown function, α, β, γ are known functions. MTFDEs of the considered form contain deviating (advanced and delayed) arguments and for this reason are known also as forward-backward equations. The authors of [4] have developed a new approach to the analysis of the autonomous case. They have analysed MTFDEs as boundary value problems (BVP), that is, they have considered the problem of finding a differentiable solution on a certain real interval [0,k 1], given its values on the intervals [ 1,0] and (k 1,k]. Assuming that such a solution exists they have introduced a numerical algorithm to compute it. In [9], a numerical algorithm based on the collocation method was proposed for the solution of such BVPs. In [6, 7] these methods were extended to the non-autonomous case and a new algorithm, based on the least squares method, was introduced. In [5], a new numerical algorithm was proposed, based on the decomposition of the solution into a growing and a decaying component. This approach, which is based on the analytical results of [8], provides a way of reducing the illconditioning of the boundary value problem. The algorithm developed in [9] and [6], which will be applied to the solution of the present problem, is based on the so-called ODE approach: the solution is sought as the sum of two terms, one of which is defined from the initial data (boundary conditions and equation); the other one must be computed as a linear combination of known basis functions (usually splines). Using the method of steps, the problem of computing this term can be reduced to the solution of a boundary value problem (BVP) for a k-th order ODE (k is the length of the interval where the solution must be computed). This last problem can be solved by standard methods of numerical analysis, such as the collocation or the finite elements method. As shown in [6], the error of these methods, when applied to linear equations of the type of (8) on a bounded interval, is of order h 2. The numerical method described in the present paper, although based on this approach, has two new features: 1) it allows us to deal with problems on the whole real axis (where the boundary conditions are given at infinity) and 2) nonlinear equations are considered. In the next sections we will explain how to handle this case. 3.2 Numerical Solution by the Newton Method Once we know the approximate solution of the equation for t L and t L (as described in Sec.2), the problem is reduced to a BVP on [ L,L], where L is a multiple of τ. The nonlinear problem can be reduced to a sequence of linear problems by means of the Newton method. In the i-th iteration of the Newton method, we have to solve a linear equation of the form:

5 Forward-Backward Equation in Nerve Conduction 5 RCv i+1(t) f (v i )(v i+1 (t) v i (t)) L(v i+1 (t)) = f (v i (t)), t [ L,L], (9) where L(v(t)) = v(t + τ) + v(t τ) 2v(t). We search for a monotone solution v i+1 which satisfies the boundary conditions v i+1 (t) = w 1 (t), v i+1 (t) = w 2 (t), t [ L τ, L]; t [L,L + τ]. (10) where w 1 and w 2 are given by (4) and (6), respectively. In order to compute an initial approximation v 0, which enables the convergence of the Newton iteration process, we need the values of λ 1,λ 2,τ, ε 1 and ε 2. These values can be obtained by solving a system of five nonlinear equations: λ F (0) 2cosh(λ 1 τ) = 0 λ F (1) 2cosh(λ 2 τ) = 0 lim t 0 v(t) = 1/2; lim t 0+ v(t) = 1/2; lim t 0 v (t) = lim t 0+ v (t). The values of v in this system are computed, using the method of steps, and assuming that v satisfies the obtained asymptotic expansions, when t < L and t > L. More precisely, if v(t) is defined at [ L τ, L] by (10) then we can define it on the interval [ L, L + τ] and on the following intervals using the recurrence formula: (11) v(t + τ) = 2v(t) + RCv (t) v(t τ) + g(v(t)), (12) where g(v) = f (v), if v is the solution of the nonlinear equation (1); g(v) = f (v i ) f (v i )(v v i ), if v is the solution of the Newton iterates (9) (here v i is the preceding iterate). In the same way, starting from the definition of v at [L,L+τ] by (10), this function can be defined on [L τ,l] by the backwards formula: v(t τ) = 2v(t) + RCv (t) v(t + τ) + g(v(t)). (13) The system (11) is solved again at each iterate of the Newton method, in order to update the parameter values. 4 Numerical Results The algorithm was implemented in the form of a MATLAB code. In this section we present and discuss some numerical results. The test case considered in our computations is one of the cases discussed in [3]. We consider equation (1), with R = C = 1, where f is defined by (3), with a = 0.05,

6 6 P.M. Lima, M.F. Teodoro, N.J.Ford and P.M. Lumb b = 15. In this case, by solving system (11), with L = 2τ, we obtain λ 1 = 5.98, λ 2 = 6.35, ε 1 = , ε 2 = , τ = The graphics of some Newton iterates of the solution are displayed in Fig first iterate second iterate seventh iterate Fig. 1 Graphics of some Newton iterates v i for the considered test case For comparison, we have solved the problem numerically using two finite intervals: with L = 2τ and L = 3τ. In Table 1, we display the results with L = 2τ. We denote by N the number of gridpoints at each subinterval of length τ in the collocation method, and by h the corresponding stepsize, so that h = τ/n. Let v h the approximate solution, obtained with stepsize h. Then, as a measure of the error we use the euclidean 2-norm ε h = v h v 2h 2. As an estimate of the convergence order, as usual, we take p = log 2 (ε h) log 2 (ε 2h ). N ε p ε p E E E E E E E E Table 1 Estimates of errors and convergence orders, for the solution (first two columns) and for its derivative (last two columns), in the case L = 2τ. In Table 2, we display analogous results, obtained with L = 3τ. In the overall scheme, the error depends upon the choices of h and L. When we solve the problem numerically on [ L,L] we assume that v( L) = ε 1, according

7 Forward-Backward Equation in Nerve Conduction 7 to (4), and v(l) = 1 ε 2, according to (6). Taking into account the linearisation process the error of this approximation is of order ε1 2, in the first case, and ε2 2, in the second. Therefore the choice of L imposes a limitation on the overall accuracy of the scheme and limits the benefit of reducing the error of the collocation method, which is O(h 2 ), beyond the point at which the error in the numerical scheme is smaller than the error introduced by the choice of L. Since the values of ε 1 and ε 2 are not known apriori, the value of L must be ajusted experimentally, and it depends on the values of a and b. For the case illustrated by tables 1 and 2, a choice of L = 3τ yields a sufficiently small error that the convergence rate of the collocation method is recovered in the estimates. N ε p ε p E E E E E E E E Table 2 Estimates of errors and convergence orders, for the solution (first two columns) and for its derivative (last two columns), in the case L = 3τ. In both cases, the numerical results suggest that the convergence order of the collocation method is 2, which is in agreement with the theoretical results obtained in [6] and [7]. An important issue that can be analysed from the numerical results for equation (1) is the dependence of the propagation speed of the nerve impulses on the parameters a,b of the function f. As remarked in [3], the propagation speed is inversely proportional to the delay τ, which is computed by our algorithm. In Table 3, we observe how this parameter varies with a and b. Table 3 The dependence of τ on a and b. a b = 10 b = 12 b = 14 b = From Table 3 we conclude that the propagation speed (inverse of τ ) increases with b and when a tends to 0, which is in agreement with previously obtained results (see [3]).

8 8 P.M. Lima, M.F. Teodoro, N.J.Ford and P.M. Lumb 5 Conclusions The numerical experiments have shown that the Newton method provides a fast iterative scheme and with convergence from a good initial approximation. The numerical results also suggest that the collocation method provides second order convergence, as was observed earlier for linear problems. Comparing with numerical algorithms used by other authors to solve similar problems (in particular, [3]), the present algorithm has the advantage that the convergence of the iteration process does not require the use of the continuation method (where the algorithm is first applied to a test problem with known exact solution and then this problem is transformed into the target one, by smoothly changing a certain parameter). In the near future we intend to improve the algorithm and to carry out new numerical investigations of the problem. Acknowledgements M.F. Teodoro acknowledges financial support from FCT, through grant SFRH/BD/37528/2007. The research of N. Ford and P. Lumb has been supported by a University of Chester International Research Excellence Award, funded under the Santander Universities scheme. References 1. Bell, J: Behaviour of some models of myelinated axons. IMA Journal of Mathematics Applied in Medicine and Biology 1, (1984). 2. Bell, J. and Cosner, C: Threshold conditions for a diffusive model of a myelinated axon. J. Math. Biol. 18, (1983). 3. Chi, H., Bell, J. and Hassard, B.: Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory. J. Math. Biol. 24, (1986). 4. Ford,N. and Lumb, P.: Mixed-type functional differential equations: a numerical approach. J. Comput. Appl. Math. 229, (2009). 5. Ford,N., Lumb, P., Lima, P.,Teodoro, F.: The numerical solution of forward-backward differential equations: Decomposition and related issues. J. Comput. Appl. Math. 234, (2010). 6. Lima, P., Teodoro, F., Ford, N. and Lumb,P.: Analytical and numerical investigation of mixedtype functional differential equations. J. Comput. Appl. Math. 234, (2010). 7. Lima, P., Teodoro, F., Ford N. and Lumb P. : Finite element solution of a linear mixed-type functional differential equation. Numerical Algorithms 55, (2010). 8. Mallet-Paret, J. and Verduyn Lunel, S.: Mixed-type functional differential equations, holomorphic factorization and applications. Proc. of Equadiff 2003, Inter. Conf. on Diff. Equations, HASSELT 2003, World Scientific, Singapore, (2005). 9. Teodoro, F., Lima, P., Ford, N. and Lumb, P.: New approach to the numerical solution of forward-backward equations. Front. Math. China 4, N.1, (2009).