Numerical solution of a Fredholm integro-differential equation modelling _ h-neural networks

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1 Available online at Applied Mathematics and Computation 195 (2008) Numerical solution of a Fredholm integro-differential equation modelling _ h-neural networks Z. Jackiewicz a, M. Rahman b, *, B.D. Welfert a a Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, United States b Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, United States Abstract We propose several approaches to the numerical solution of a new Fredholm integro-differential equations modelling neural networks. A solution strategy based on expansions onto standard cardinal basis functions and collocation is presented. Comparative numerical experiments illustrate specific advantages and drawbacks of the different approaches and are used to motivate alternate strategies. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fredholm integro-differential equation; Pseudospectral methods; Piecewise linear approximation; Polynomial approximation; Rational approximation; Collocation; Gaussian quadrature; Neural networks 1. Introduction We consider the initial-value problem for the integro-differential equation of convolution type o t hðx; tþ ¼f ðhðx; tþ; tþþl R 1 Kðx yþo thðy; tþdy; ð1:1þ hðx; 0Þ ¼gðxÞ; 6 x 6 1; t P 0, with a given smooth function f and an initial function g. Problem (1.1) is a continuous analog of a discrete model of transmission line in neural networks with h-synapses _ during sustained bursting activity [7,8]. The angle hðx; tþ represents the phase of the signal at time t associated with a neuron located at the point x. The function f represents external inputs and potential effects. The convolution integral accounts for either inhibitory (K < 0) or excitatory (K > 0) influence of neighboring neurons. In this work we only consider the latter case, i.e., the kernel K(s) is assumed to be a nonnegative integrable function defined on R, and normalized such that R 1 KðsÞds ¼ 1. We shall mainly be interested in the Gaussian kernel KðsÞ ¼ p 1 r ffiffiffi e ðs=rþ2 for s 2 R ð1:2þ p * Corresponding author. addresses: jackiewi@math.la.asu.edu (Z. Jackiewicz), mrahman@unf.edu (M. Rahman), bdw@math.la.asu.edu (B.D. Welfert) /$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi: /j.amc

2 524 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) and the exponential kernel ( 1 r KðsÞ ¼ e s=r for s P 0; 0 for s < 0; ð1:3þ for some r > 0. These kernels have been widely considered in related applications, most recently in the modelling of the spread of diseases [12]. Note that as r! 0 both kernels (1.2) and (1.3) tend to the Dirac function, in which case the problem (1.1) reduces to an ordinary differential equation f ðhðx; tþ; tþ o t hðx; tþ ¼ 1 l for each x 2 R. Thus l < 1 can be viewed as an excitatory parameter. Kernels of the form (1.2) are associated to bidirectional influence while those of the form (1.3) correspond to unidirectional connectivity. In both cases the influence is stronger between neurons that are close to each other. A forward Euler time discretization of (1.1) with a stepsize Dt yields h iþ1 ðxþ ¼F ðxþþl i ¼ 0; 1;...; with Kðx yþh iþ1 ðyþdy F ðxþ ¼h i ðxþþdtf ðh i ðxþ; idtþ l Kðx yþh i ðyþdy and h i ðxþ ¼hðx; idtþ. Eq. (1.4) is a Fredholm integral equation of the second kind. A solution strategy for (1.1) for both kernels, based on expansions of hðx; tþ in terms of cardinal basis functions and collocation at a grid x 0 < x 1 < < x n is presented in Section 2. The resulting differential system involves a matrix representing the contribution of the integral term in (1.1). Piecewise linear, polynomial and rational pseudospectral approximations are investigated in Sections 3 5. In Section 6, the above approaches are applied to a specific example and their relative merits are compared. Finally, in Section 7 some concluding remarks are given and plans for future research briefly outlined. 2. Pseudospectral approximation Since ~ hðx; tþ ¼hðx=r; tþ satisfies (1.1) with KðsÞ ¼p 1 ffiffi p e s2 or KðsÞ ¼e s for s P 0, w.l.o.g we assume r =1. Consider an approximation hðx; tþ Xn h j ðtþ j ðxþ ð2:1þ j¼0 of the solution to (1.1), where f j g 06j6n x 0 < < x n, which satisfy 1 if i ¼ j; j ðx i Þ¼ 0 otherwise; ð1:4þ are Lagrange cardinal basis functions associated to a grid and h j ðtþ ¼hðx j ; tþ. Substituting (2.1) into (1.1) and collocating at x = x i yields h 0 i ðtþ ¼f ðh iðtþ; tþþl Xn a i;j h 0 j ðtþ with a i;j ¼ j¼0 ð2:2þ Kðx i yþ j ðyþdy; 0 6 i; j 6 n: ð2:3þ

3 In matrix form (2.2) becomes ði laþh 0 ðtþ ¼F ðhðtþ; tþ; where A ¼ða i;j Þ 06i;j6n and h 0 ðtþ f ðh 0 ðtþ; tþ HðtÞ ¼ ; F ðhðtþ; tþ ¼ : h n ðtþ f ðh n ðtþ; tþ The matrix la represents the spatial connection between the values hðx i ; tþ. We make the following remarks. The problem (1.1) is translation invariant in the spatial direction x, i.e., if hðx; tþ is a solution then h c ðx; tþ hðx þ c; tþ is also a solution with initial condition h c ðx; 0Þ ¼gðx þ cþ. This follows from the relations: o t h c ðx; tþ ¼o t hðx þ c; tþ ¼f ðhðx þ c; tþ; tþþl ¼ f ðh c ðx; tþ; tþþl Kðx þ c yþo t hðy; tþdy Kðx yþo t hðy þ c; tþdy ¼ f ðh c ðx; tþ; tþþl Kðx yþo t h c ðy; tþdy: This implies that the range of points x j used can be shifted to accommodate specific numerical requirements. Because of the infinite domain of integration the approximation (2.1) must be used in extrapolation mode, i.e., for values outside the interval ½x 0 ; x n Š. This places restrictions on the type of basis functions that can be considered in order to guarantee convergence of the integral in (1.1) as well as to avoid numerical difficulties associated with their behavior at x ¼1. In the next Sections 3 5 we will show how the choice of basis functions j affects the determination and characteristics of the matrix A. 3. Piecewise linear approximation Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) We consider here nodal basis functions j which are piecewise linear in ½x 0 ; x n Š and extended outside this interval as follows: 8 >< aðxþ; < x 6 x 0 ; x 0 ðxþ ¼ 1 x x 1 x 0 ; x 0 6 x 6 x 1 ; >: 0; x 1 6 x < þ1; 8 0; < x 6 x j ; x x >< j x j x j ; x j 6 x 6 x j ; j ðxþ ¼ x jþ1 x ð3:1þ x jþ1 x j ; x j 6 x 6 x jþ1 ; >: 0; x jþ1 6 x < þ1 for j ¼ 1; 2;...; n 1 (hat functions), and 8 0; < x 6 x n ; >< x x n ðxþ ¼ n x n x n ; x n 6 x 6 x n ; >: bðxþ; x n 6 x < þ1; where a and b are polynomials such that aðx 0 Þ¼bðx n Þ¼1. The basis functions j ; j ¼ 0; 1;...; n, are graphed in Fig. 1. These functions were also used in [9], where the integral appearing in (1.1) was approximated by Gauss Hermite or Gauss Laguerre quadrature rules for the kernel (1.2) or (1.3), respectively, and the resulting ð2:4þ

4 526 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Fig. 1. Piecewise linear basis functions. systems of differential equations were then integrated by the explicit Euler method. Stochastic approximations to the perturbed Fredholm Volterra integro-differential equations are investigated in [10] Exponential kernel (1.3) For the kernel (1.3) the resulting coefficients a i;j from (2.3) can be evaluated explicitly. The matrix A is lower triangular. Interestingly, the matrix A does not depend on the function b(x). On the other hand the choice of a(x) is critical is obtaining coefficients a i;0 which are uniformly bounded as x 1 x 0! 0. This is the case if a(x) = 1 (the definition of 0 then resembles that of j ; j ¼ 1;...; n 1, by introducing a fictitious point x ¼), but not otherwise (e.g., if a(x) is simply the linear extrapolation of 0 ðxþ from the interval ½x 0 ; x 1 Š), see Appendix A. The choice aðxþ ¼1 ð¼ bðxþ) also has the advantage that the matrix A is nonnegative and stochastic, since j ðxþ P 0 ) a i;j P 0 and X n j¼0 j ðxþ ¼1 ) Xn j¼0 a i;j ¼ Z xi e y xi dy ¼ 1: The entries of A can be interpreted as connection probabilities between the discrete values hðx i ; tþ. As a result the matrix I la from (2.4) is guaranteed to be nonsingular for l < 1 and any distribution of points x 0 ; x 1 ;...; x n Gaussian kernel (1.2) For the kernel (1.2) the coefficients a i;j must be evaluated numerically, e.g., using a m-point Hermite quadrature rule a i;j ¼ 1 pffiffiffi e y2 j ðx i yþdy Xm w H k p jðx i g H k Þ; ð3:2þ k¼1 where g H k and w H k represent the nodes and corresponding weights of the Gaussian quadrature rule, which can be computed using eigenvalue theory of tridiagonal matrices [4]. These values are listed in Table 1 in the case m =8. For larger values of m and jg k j the weights become very small. For large ji jj the function j typically vanishes in a neighborhood of x i which includes all x i g H k, k ¼ 1; 2;...; m, for lower values of m, so that a i;j ¼ 0. Therefore the matrix A is expected to have a banded structure. In general the precise sparsity pattern of A depends on m as well on the specific distribution of the points x i. However note that if x n i ¼ x i for i ¼ 0; 1;...; n and bðxþ ¼að xþ then Table 1 Nodes and weights of Hermite quadrature (m = 8) g H k ± ± ± ± w H k

5 a n i;n j ¼ Xm k¼1 ¼ Xm k¼1 w H k n jðx n i g H k Þ¼Xm w H k jðx i g H k Þ¼a i;j: k¼1 w H k jð ð x i g H Xm k ÞÞ ¼ k¼1 w H m k jðx i g H m k Þ Because w H k P 0, P m k¼1 wh k ¼ 1, and the basis functions j are nonnegative the matrix A obtained by applying the quadrature is again nonnegative and stochastic, and can also be interpreted in terms of connection probabilities. The theory of (stochastic) nonnegative matrices [2] then implies that the system matrix I la is nonsingular for l < Polynomial approximation To improve the accuracy of the approximation (2.1) we now assume that the trial functions j are standard Lagrange interpolating polynomials at the nodes x 0 ; x 1 ;...; x n of degree n. The polynomial j can be written in barycentric form j ðxþ ¼ b j=ðx x j Þ P n b ð4:1þ k=ðx x k Þ for ¼ 0; 1;...; n and x 6¼ x k, k ¼ 0; 1;...; n, with 1=b j ¼ Yn ðx j x k Þ; ;k6¼j [5, p. 245]. A typical basis function j is graphed in Fig. 2. It turns out that specific properties of Hermite polynomial make the determination of the matrix A simpler in the case of the kernel (1.2) than in the case of the kernel (1.3) Gaussian kernel (1.2) Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Let H j ðxþ denote the Hermite polynomial of degree j, 06 j 6 n. Then X n H i ðx k Þa j;k ¼ p 1 ffiffiffi p for 0 6 i; j 6 n, i.e., HA T ¼ V ¼ p 1 ffiffiffi p ¼ð2x j Þ i e ðxj yþ2 X n e ðxj yþ2 H i ðyþdy ðsee Appendix BÞ H i ðx k Þ k ðyþdy ð4:2þ Fig. 2. Polynomial basis function.

6 528 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) with H i;j ¼ H i ðx j Þ; V i;j ¼ð2x j Þ i ð4:3þ for 0 6 i; j 6 n. The matrices H (Vandermonde-like, in the sense of [6, Chapter 22])andV (Vandermonde) are nonsingular because the points x i are distinct. They are related through a unit lower triangular matrix T such that H = TV and with inverse T ¼jTj (see Appendix C). Consequently, A T ¼ H V ¼ V jt jv shows that k = 1 is the only eigenvalue of A (with multiplicity n + 1), independently of the particular choice of x i s. Therefore the system (2.4) is a regular ordinary differential system if and only if l 5 1. When l = 1 the mass matrix in (2.4) has a two-dimensional kernel and the system must be constrained with two conditions on the right-hand side. These conditions can be shown to constrain HðtÞ; 0 6 i 6 n, such that the interpolant to f ðhðtþþ remains of degree at most n 2. The matrix A is computed by solving the (primal) system (4.2) using a variation of Björck and Pereyra s algorithm [3] described in [6, pp ] at a cost of Oð3n 3 Þ operations (for general nodes x j s and without counting divisions by 2; this is a slight gain over a direct factorization and forward/backward elimination assuming H is explicitly available). This algorithm produces good results despite the extreme ill-conditioning of H for any choice of points x j. Despite P n j¼0 a i;j ¼ 1 for i ¼ 0; 1;...; n, the unboundedness of j, in particular for large jxj s, implies that a i;j can become unbounded as n increases. Indeed, for j n=2, j then exhibits large variations around y = x 0 (see Fig. 2), where e ðx 0 yþ 2 is not small, so that ja 0;bn=2c j is expected to be large Exponential kernel (1.3) The shifting formula (B.4) does not hold for Laguerre polynomials, but a similar formula can be determined numerically using a 5-point recurrence relation, see (D.3), and used to derive a system of the form (4.3) for A X n L i ðx k Þa j;k ¼ ¼ ¼ Z xj X n e y x j Z xj 0 ¼ Xj ¼ w 0;i ðx j Þ e y xj L i ðyþdy e y L i ðx j yþdy w k;i ðx j Þ L i ðx k Þ k ðyþdy 0 e y L k ðyþdy ðsee Appendix DÞ for 0 6 i; j 6 n, i.e., LA T ¼ W ð4:4þ with L i;j ¼ L i ðx j Þ; W i;j ¼ w 0;i ðx j Þ; 0 6 i; j 6 n: The matrices L and W are related by a unit lower triangular matrix S such that L = SW (see last remark at the end of Appendix D). This implies that W is nonsingular, and, as above, that A has unique (multiple) eigenvalue 1. Thus the system (2.4) is regular for l <1. The cost of computing the matrix A is O 3 2 n3 for the determination of the matrix W and, as above, Oð3n 3 Þ for solving the system (4.4). To avoid large values of L i ðx j Þ for x j 0 (since all zeros of L i are positive for i > 0) it may be suitable to shift the interval ½x 0 ; x n Š to ½0; x n x 0 Š, see the remarks at the end of Section 2.

7 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Because the exponential kernel (1.3) has a fatter tail compared to the Gaussian kernel (1.2) the size of some of the off-diagonal coefficients of A grows even faster than in the case of the Gaussian kernel Evaluation of A via numerical quadrature The coefficients of the matrix A can also be approximated using a numerical Gaussian quadrature rule such as (3.2). More specifically, a i;j ¼ KðyÞ j ðx i yþdy b j X m k¼1 w k =ðx i x j g k Þ P n ¼0 b =ðx i x g k Þ : ð4:5þ The approximated matrix A can be evaluated in Oð9mn 2 Þ operations. For m n=2 the quadrature rule becomes exact for the polynomials j of degree n. The resulting cost of O 9 2 n3 is higher than that of the method exposed above in the case of the kernel (1.2), but comparable in the case of the kernel (1.3). In practice, however, a smaller value of m may be sufficient to obtain a good approximation of the matrix I la, especially for small values of l. 5. Rational approximation To obtain an approximation of h which combines the normal spectral accuracy of the polynomial approach and the boundedness of the coefficients in the matrix A obtained with the piecewise linear approach we now consider a linear rational pseudospectral method [1] based on the choice of n + 1 rational basis functions j (x) of the form (4.1) but with b j ¼ðÞ j for ¼ 0; 1;...; n. The Euclidean form of j is j ðxþ ¼p j ðxþ=qðxþ with and p j ðxþ ¼ðÞ j qðxþ ¼ Yn Y n ;k6¼j ðx x k Þ Xn ðx x k Þ ðþ k x x k : For n odd q has degree n 1 and, ðnþ=2 X j ðxþ ðþ j x ðx 2k x 2kþ1 Þ as x!1. This case leads to numerical instability and is not considered further. For n even the polynomial q has degree n so that lim jðxþ ¼ðÞ j : x!1 It is also known that q does not vanish in R [11]. A typical basis function j is graphed in Fig. 3. The coefficients of the matrix A must be determined by using a numerical quadrature rule (4.5), where b j ¼ðÞ j. 6. Numerical experiments In this section we present results of numerical experiments using the approaches developed in Sections 3 5. We have tested these methods on the problem (1.1) with f ðh; tþ ¼x þ cos h ð6:1þ

8 530 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Fig. 3. Rational basis function (n even). and ( 2; t 6 10; x ¼ 1; t > 10: ð6:2þ The function f defined by (6.1) with x given by (6.2) is typical of Voltage Controlled Oscillators (VCOs): the (piecewise) constant term x corresponds to the center frequency of oscillations and the term cos h represents a controlling voltage. Note that a saddle-node bifurcation occurs for the model (1.1) with l = 0 when x =1 [7]. We set l ¼ 0:2 (r = 1). The initial function is set to gðxþ ¼3px=T with ½x 0 ; x n Š¼½ T ; T Š, with T =100. For the piecewise linear approximation discussed in Section 3 a uniform grid of n þ 1 ¼ 101 points in ½ T ; T Š is chosen. Contour plots of cos hðx; tþ are shown in Fig. 4. We can observe the propagation of the initial wave. For large t the solution tends to a steady-state corresponding to cos h ¼. For a given x the specific value of hðx; 1Þ ¼ ð2k þ 1Þp depends on the global state of the solution at t = 10. At t = 20 the solution looks like a staircase, see Fig. 5. The polynomial pseudospectral solution described in Section 4 leads to instability on the uniform grid and we consider instead the Chebyshev grid on the interval ½ T ; T Š defined by x i ¼ T cos pi n ; ð6:3þ i ¼ 0; 1;...; n. Contour plots of cosðhðx; tþþ for polynomial solution are shown in Fig. 6. The corresponding solution for t = 10 and t = 20 is presented in Fig. 7. We can only observe slow initial convergence to the staircase solution for moderate values of n (n 6 30). This approximation is oscillatory, compare right graph in Fig. 7, and deteriorates rapidly for larger values of n, compare also Fig Fig. 4. Contour plots of cos hðx; tþ for the piecewise linear approximation on a grid uniformly distributed in [00,100] (n = 100).

9 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Fig. 5. Piecewise linear solution for the kernel (1.2) at t = 10 (left) and t = 20 (right) with n = Fig. 6. Contour plots of cos hðx; tþ for the polynomial approximation on the Chebyshev grid (6.3) (n = 30). The results of numerical experiments with pseudospectral method based on rational approximation discussed in Section 5 on the uniform grid with n = 100 are presented in Figs. 8 and 9. Although the contour plots of cosðhðx; tþþ are not as smooth as that corresponding to piecewise linear interpolation in Fig. 4 and there are also small oscillations visible on the right graph of Fig. 9, the overall accuracy of the method based on rational approximation is better than that based on piecewise linear approximation or polynomial approximation for n P 30. This is illustrated in Fig. 10 where the accuracy of the three schemes developed in Sections 3 5 is compared for n

10 532 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Fig. 7. Polynomial solution on the Chebyshev grid (6.3) for the kernel (1.2) at t = 10 (left) and t = 20 (right) with n = Fig. 8. Contour plots of cos hðx; tþ for the rational approximation on the uniform grid (n = 100). 7. Concluding remarks We proposed and compared different approaches to the numerical solution of Fredholm integro-differential equations modelling neural networks. All these approaches are based on pseudo spectral collocation methods with piecewise linear, polynomial or rational approximations. Numerical experiments demonstrate that the method based on rational approximation on the uniform grid is most accurate for the number of grid points n P 30 and that the method based on polynomial approximation on uniform or Chebyshev grid becomes unstable and leads to the loss of accuracy for large values of n, compare Fig. 10.

11 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Fig. 9. Rational solution on the uniform grid for the kernel (1.2) at t = 10 (left) and t = 20 (right) with n = piecewise linear polynomial rational Fig. 10. Convergence of numerical approximations.

12 534 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Future work will address numerical solution of these equations by spectral methods and methods based on spline functions of high order. Acknowledgement This work was partially supported by NSF Grant DMS Appendix A For j ¼ 1; 2;...; n 1and j defined by (3.1), and K(s) given by (1.3) (r = 1) the coefficient a i;j takes the form Z xi Z y x xi a i;j ¼ e y xi j x v x j x ½xj ;x jšðyþdy þ e y xi jþ1 y v j x jþ1 x ½xj;xjþ1 ŠðyÞdy; ða:1þ j where v I denoted the characteristic function of an interval I. A direct calculation yields 8 >< 0; i < j; a i;j ¼ 1 uð h j Þ; i ¼ j; ða:2þ >: e ðxi xjþ ðuðh jþ1 Þ uð h j ÞÞ; i > j; with h j ¼ x j x j and uðhþ ¼ðe h 1Þ=h. The expression (A.2) can be verified to hold also for j = n. The second integral in (A.1) is replaced by R x i bðyþv ½xn;1ÞðyÞdy which vanishes independently of b. For j =0we ey xi obtain Z xi Z xi x a i;0 ¼ e y xi aðyþv ð;x0 ŠðyÞdy þ e y xi 1 y v x 1 x ½x0 ;x 1 ŠðyÞdy: 0 For aðyþ ¼ðx 1 yþ=ðx 1 x 0 Þ (linear extrapolation) this yields ( a i;0 ¼ 1 þ 1 h 1 ; i ¼ 0; e ðxi x 0Þ e h 1 h 1 ; i > 0: Although a i;j ; j > 0, remains bounded as max h j! 0, a i;0 does not. On the other hand, if a(x) = 1 we obtain 1; i ¼ 0; a i;0 ¼ e ðxi x0þ uðh 1 Þ; i > 0: In all cases of a(x) the matrix A is lower triangular. For a(x) = 1, the diagonal coefficients of A are f1 uð h j Þg 06j6n (with h 0 = 1). Because uð hþ > 0 for h > 0, the ODE system matrix I la is guaranteed to be nonsingular for l <1. Appendix B Hermite polynomials are defined recursively by 8 >< H 0 ðxþ ¼1; H 1 ðxþ ¼2x; >: H nþ1 ðxþ ¼2xH n ðxþ 2nH n ðxþ; n P 1: Three useful properties of Hermite polynomials are: 1. Their orthogonality with respect to the kernel (1.2) (r = 1), namely Z ( 1 1 p ffiffiffi e y2 H k ðyþh ðyþdy ¼ k!2k if k ¼ ; p 0 otherwise: ðb:1þ ðb:2þ

13 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Their generating function e 2xy x2 ¼ X jp0 H j ðyþ x j : j! ðb:3þ 3. The shifting formula H j ðx þ yþ ¼ Xj j H k ðyþð2xþ j k : k From (B.4) and (B.2) with = 0 we get 1 p ffiffiffi p e ðxi yþ2 H j ðyþdy ¼ p 1 ffiffiffi p e y2 H j ðx i þ yþdy ¼ p 1 ffiffiffi Xj p Z j 1 ð2x i Þ j k k e y2 H k ðyþdy ¼ð2x i Þ j : A calculation based on writing e ðxi yþ2 ¼ e y2 e 2xiy x2 i and using (B.3) and (B.2) is also possible but requires a justification of the interchange of the infinite integral and summation signs. Appendix C Consider the matrices H and V defined by (4.3). From(B.4) with y = 0 we get H i;j ¼ H j ðx i Þ¼ Xj j H k ð0þð2x i Þ j k ¼ Xj k j H j k ð0þð2x i Þ k ¼ Xj k V i;k T k;j ¼ðVT Þ i;j ðb:4þ with T k;j ¼ j H j k ð0þ ¼ k ( j! ðþ k!! j k ¼ 2 even; 0; otherwise; for k ¼ 0; 1;...; j, so that H = VT. In particular T is unit upper triangular. For 0 6 i 6 j 6 n with j i odd we have ðt jt jþ i;j ¼ Xn T i;k jt k;j j¼ Xj k¼i T i;k jt k;j j¼0 since either T i;k ¼ 0(k i odd) or T k;j ¼ 0(j k odd). For j i ¼ 2p (even) we obtain! X j ðt jt jþ i;j ¼ T i;k jt k;j j¼ Xp ðþ k! j! i!! k!ðp Þ! ¼ j! X p p ðþ i!p! k¼i;k i¼2 even ¼0 ¼0 ( ¼ j! i!p! ð1 1; p ¼ 0; 1Þp ¼ 0; otherwise: This shows that T ¼jT j. Appendix D Let L j (x) denote the Laguerre polynomial of degree j, defined by the recurrence 8 L 0 ðxþ ¼1; >< L 1 ðxþ ¼ xþ1; >: L nþ1 ðxþ ¼ð2n þ 1 xþl n ðxþ n 2 L n ðxþ n P 1: ðd:1þ

14 536 Z. Jackiewicz et al. / Applied Mathematics and Computation 195 (2008) Define the functions fa i;j ðxþg 06i6j such that L j ðx yþ ¼ Xj i¼0 w i;j ðxþl i ðyþ: ðd:2þ Substituting (D.2) into the recurrence (D.1) and using (D.1) to express yl i ðyþ in terms of L i and L i1 yield the recurrence relation w i;jþ1 ¼ð2j þ 2i þ 2 xþw i;j w i;j ði þ 1Þ 2 w iþ1;j j 2 w i;j ðd:3þ for i ¼ 0; 1;...; j. This 5-point recurrence is started with w j;j ¼ðÞ j and w ;j ¼ 0 for j P 0, and w jþ1;j ¼ 0 for j P. The computation can be organized as indicated below: The recurrence (D.3) shows that w 0;j ðxþ ¼L j ðxþþpðxþ where p is a polynomial of degree less than j. References [1] R. Baltensperger, J.-P. Berrut, The linear rational collocation method, J. Comput. Appl. Math. 134 (2001) [2] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, [3] A. Björck, V. Pereyra, Solution of Vandermonde systems of equations, Math. Comput. 24 (1970) [4] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, second ed., Academic Press, New York, [5] P. Henrici, Essentials of Numerical Analysis, Wiley, New York, [6] N. Higham, Accuracy and Stability of Numerical Algorithms, second ed., SIAM, Philadelphia, [7] F.C. Hoppensteadt, An Introduction to the Mathematics of Neurons, second ed., Modelling in the Frequency Domain, Cambridge University Press, New York, [8] F.C. Hoppensteadt, Lecture Notes, Arizona State University, [9] Z. Jackiewicz, M. Rahman, B.D. Welfert, Numerical solution of a Fredholm integro-differential equation modelling neural networks, Appl. Numer. Math. 56 (2006) [10] M. Rahman, Z. Jackiewicz, B.D. Welfert, Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences, Appl. Math. Comput. 186 (2007) [11] J. Jones, B.D. Welfert, Zero-free regions for a rational function with applications, Adv. Comput. Math. 3 (1995) [12] J. Medlock, M. Kot, Spreading disease: integro-differential equations old and new, Math. Biosci. 184 (2003)