Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences

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1 Applied athematics and Computation 6 (27) Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences. Rahman a, *,. Jackiewicz b, B.D. Welfert b a Department of athematics and Statistics, University of North Florida, 4567 St. Johns Bluff Rd. South, Jacksonville, FL 32224, USA b Department of athematics and Statistics, P.O. Box 527, Tempe, A 527-4, USA Abstract This paper extends the results of synaptically generated wave propagation through a network of connected excitatory neurons to a continuous model, defined by a Fredholm Volterra integro-differential equation (FVIDE), which includes memory effects of the past in the propagation. Stochastic approximation and numerical simulations are discussed. Ó 26 Elsevier Inc. All rights reserved. Keywords: Fredholm Volterra integro-differential equation; Euler Hermite method; Stochastic approximation; Error; Neural network. Introduction Noisy systems can be modelled in several different ways: For example, Langevin s equation describes a linear physical system to which white noise is added, and the linear theory for it has been extended to nonlinear stochastic differential equations with additive white noise, see []. This approach is based on additive white noise. Another approach was derived from work of Bogoliubov, see [9,] on averaging nonlinear oscillatory systems. In this system parameters are allowed to be random processes, and methods based on averaging and ergodic theory provide useful predictions from the model. This approach is referred to as being based on parametric noise. A third approach is to derive models, such as arkov Chains, for the systems state variable as being random variables, and then use the method of probability theory; for analysis, see []. Nonlinear Fredholm integro differential equation arises whenever the connections of excitable neurons are distributed according to the normal probability distribution and it is appropriate to take kernel as a Gaussian, or normal distribution of influence from any site. Gaussian functions, or normal distributions, arise naturally in neuroscience model, see [5] and [6]. It is known from the central limit theorem of probability theory that most random processes, when correctly scaled have (approximately) normal distributions. If we include some * Corresponding author. addresses: mrahman@unf.edu (. Rahman), jackiewi@math.asu.edu (. Jackiewicz), bdw@asu.edu (B.D. Welfert) /$ - see front matter Ó 26 Elsevier Inc. All rights reserved. doi:.6/j.amc

2 74. Rahman et al. / Applied athematics and Computation 6 (27) 73 2 memory effects of the past and then Fredholm integro-differential equation becomes Fredholm Volterra integro-differential equation. The purpose of this paper is to apply a random perturbation method to Fredholm Volterra integro differential equation that has a parametric noise, and demonstrate that for relatively small variance of noise the propagation sustains. We have shown this result using perturbation method in Fredholm Volterra integro differential equation described in [2]. 2. odel Consider the initial-value problem for the Fredholm Volterra integro differential equation of convolution type _h a ðs; tþ ¼ þ cos hðs; tþþfðtþ þl R t Kðt t ; yðt =ÞÞ R kðs s Þ h _ a ðs ; t Þds dt ðþ f a ðs; t; yðt=þ; h a ; aþ h a ðs; Þ ¼gðsÞ; 6 s 6,t P with a given smooth initial function g and _ h a ðs ; t Þ d dt h a ðs ; t Þ. n is vector of random processes that satisfies certain natural conditions to be listed below. The kernel K(t, n) is defined by Kðt; nþ ¼AðnÞ a e a t ; ð2þ where A(n) = n with a >. In applications n = y(t/) and is a small parameter. It measures the ratio of time scales between noise (fast) and the system response. The strength of the connection is defined by kðsþ ¼pffiffiffiffiffiffiffiffiffiffi 2pr 2 e s2 r 2 with r >. F(t) is the forcing function. < l is small, represents dimensionless parameter, reflecting the strength of connection between neurons. n takes values in a measurable space Y R N. We suppose that n satisfies the following conditions: It is a stationary arkov processes. It is ergodic process (in the sense of Birkhoff) having ergodic measure, say q, iny. It satisfies strong mixing conditions. k is a measurable function of Y with respect to q. When these conditions are satisfied we can obtain a useful approximations to the solution h a ðs; tþ ¼ p h a ðs; tþþ ffiffi ~ha ðs; tþþerror and d ~ h a ðs; tþ ¼f ha ðt; s; h a ; aþ ~ h a ðs; tþdt þ Bðs; tþdw; ð5þ p where derivation B(s,t) is shown in Section 4 and dw ¼ ffiffiffiffi dt Nð; Þ and ha ðs; tþ solves corresponding averaged system is _h a ðs; tþ ¼ þ cos hðs; tþþfðtþ þl R t Kðt t Þ R kðs s Þ _ ha ðs ; t Þds dt ð6þ f a ðs; t; h a ; aþ h a ðs; Þ ¼gðsÞ; 6 s 6,t P with initial function g and _ ha ðs ; t Þ d dt h a ðs ; t Þ. ð3þ ð4þ

3 . Rahman et al. / Applied athematics and Computation 6 (27) KðtÞ ¼ Kðt; yþqðdyþ q Kðt; y Þþq 2 Kðt; y 2 Þ: Y ð7þ The problem () is a continuous analog of a discrete Voltage Controlled Oscillator Neuron (VCON) model with perturbed kernel of transmission line in neural networks discussed in [5,6]. The angle h a (s,t) represents the phase at time t associated with a neuron located at the point s, and the integral appearing in () corresponds to the influence of the neighboring neuron along with some some memory effects of the past. It is shown by numerical experimentation that solution can be expanded using Central Limit Theorem for arkov processes (see [2,3]): In particular h a ¼ p h a þ ffiffi ~ha þ error; ðþ where d ~ h a ¼ f ha ðt; s; h a ; aþ ~ h a dt þ BðtÞdw; ð9þ p where, B(t) shown in Section 4 and dw ¼ ffiffiffiffi dt Nð; Þ. A solution strategy for () is based on the use of Gaussian rules and interpolation to a regular grid on bounded interval for some final time as we used in our earlier paper [4] and we also used random path described by 2-state transition probability matrix. In Section 3, we gave numerical method to solve the perturbed model. In Section 4, we gave experimental results. Finally, we investigate the asymptotic behavior of the model as!. 3. Simulating stochastic process Noise can arise in the network in a variety of ways: The cells have irregularities in their metabolism and in their membrane properties. The chemical bath in which they reside fluctuates. Input signal arriving from other cells or from outside the body are irregular, etc. We will use two state arkov process to generate this noise. Thus consider the case where y(t) = y(t/) takes values in a countable set Y ¼fy ; y 2 g; ðþ where we choose y =, y 2 =. The transition probability matrix P(,y i,y j ), where i,j =,2 can be represented by the 2 2 matrix P such that P i,j = P(,y i,y j ) denotes the probability y() =y j when starting with y() = y i. Let P ¼ P! P! ðþ P! P! ¼ a a b b ; ð2þ where < a,b < and << 2. The matrix P satisfies the following properties (see [5]). P is nonnegative (its entries are probabilities). P is stochastic, i.e., Pe = e (for sure one of the y j is the outcome of the transition from y i. P is irreducible, i.e., any state y j can be eventually be reached in a finite number of steps with nonzero probabilities. P is aperiodic (or acyclic, e.g., not a permutation). This implies that jk 2 j <.

4 76. Rahman et al. / Applied athematics and Computation 6 (27) 73 2 The above conditions guarantee the existence of a unit vector q > such that q T e ¼ ð3þ and lim k! P k = eq T. Thus q is is the unique positive left eigen vector associated to k = satisfying (3). Itis natural to consider the spectral decomposition (see [3]) and thus!! b a a a P ¼ þð a bþ ¼ P þð a bþp 2 b a b with P 2 i ¼ P i, P i P j =. Then P k ¼ðk P þ k 2 P 2 Þ k ¼ k k P k þ kk 2 P k 2 ¼ P þð a bþ k P 2! b a ¼ eq T : Thus q ¼ b and q 2 ¼ a and in state with probability 4. An Euler Hermite approach a b. This tells us such a system, in the long run, will be in state with probability, irrespective of the initial state in which the system started. a In this section we will describe numerical method for where integral term R is approximated by the Gauss Hermite quadrature rule for the kernel k defined by (). Then using () in () we have _h a ðs; tþ ¼ þ cos h a ðs; tþþfðtþ þl Kðt t ; yðt l =ÞÞ pffiffiffiffiffi 2p r h a ðs; Þ ¼gðsÞ; e ðs s Þ2 r 2 _ h a ðs ; t Þds dt ð4þ 6 s 6,t P. Using change of variables s s ¼ g, r s ¼ s rg, ds ¼ rdg, (4) takes the form, _h a ðs; tþ ¼ þ cos h a ðs; tþþfðtþ þ p l ffiffiffiffiffi Kðt t ; yðt =ÞÞ e g2 ha _ ðs rg; t Þdg dt 2p h a ðs; Þ ¼gðsÞ; ð5þ 6 s 6,t P. To approximate (5) we restrict the range of the variable s to the symmetric interval [ A,A], where A is large enough, and and introduce the grid s j = A +(j )Ds,j =,2,3,...,2N +. Then s = A,s N+ =,s 2N+ = A. Then (5) for s = s j, j =,...,2N +, and l ¼ p l ffiffiffi takes the form, 2p _h a ðs j ; tþ ¼ þ cos h a ðs j ; tþþfðtþ þl Kðt t ; yðt =ÞÞ e g2 ha _ ðs j rg; t Þdg dt ð6þ h a ðs j ; Þ ¼gðs j Þ; j =,...,2N +. We discretize by the Gauss Hermite quadrature rule e x2 dx ¼ Xn w k f ðx k Þþ n! pffiffiffi p 2 n ð2nþ! f ð2nþ ðnþ ð7þ k¼ 6 n 6. The rule (7) is convergent if the function f satisfies the inequality jf ðxþj 6 ex2 jxj þq

5 for some q >, compare [7]. oreover, if f has bounded derivatives of any order then the errors of (7) tend to zero as n!with spectral rate. Then,! t _h a ðs j ; tþ ¼ þ cos h a ðs j ; tþþfðtþþl Kðt t ; yðt =ÞÞ XNH w k ha _ ðs j rg k ; t Þ dt k¼ ðþ h a ðs j ; Þ ¼gðs j Þ; j =,2,...,2N +. Here w,...,w NH and g,g 2,...,g NH are the weights and abscissas of the Gauss Hermite quadrature rules, which can be computed using eigen values of the tridiagonal matrices (see [7]). Now introducing the grid t i ¼ði ÞDt; Dt ¼ T ; i ¼ ; 2;...; þ on the time interval [,T]. Then for t = t i, () takes the form! ti _h a ðs j ; t i Þ¼þcos h a ðs j ; t i ÞþFðtðiÞÞ þ l Kðt i t ; yðt =ÞÞ XNH w k ha _ ðs j rg k ; t Þ dt k¼ ð9þ h a ðs j ; t Þ¼gðs j Þ; j =,2,...,2N +. This implies _h a ðs j ; t i Þ¼þcos h a ðs j ; t i ÞþFðtðiÞÞ þ l Pi h a ðs j ; t Þ¼gðs j Þ; l¼ lþ t l Kðt i t ; yðt =ÞÞ XNH k¼! w k ha _ ðs j rg k ; t Þ dt where j =,2,...,2N +,i =,2,..., +. We will approximate h _ a ðs j rg k ; t Þ by h _ a ðs j rg k ; t l Þ, if t 2 [t l,t l+ ]! _h a ðs j ; t i Þ¼þcos h a ðs j ; t i ÞþFðtðiÞÞ þ l Pi tlþ Kðt i t ; yðt =ÞÞ XNH w k ha _ ðs j rg k ; t l Þ dt l¼ t l k¼ ð2þ h a ðs j ; t Þ¼gðs j Þ; j =,2,...,2N +,i =,2,..., +. Now introducing mini-grid t l n ¼ t l þðn Þ Dt ; n ¼ ; 2;...; þ ; ¼ Then tl ¼ t l; t l ¼ t þ lþ. Then (2) takes the form _h a ðs j ; t i Þ¼þcos h a ðs j ; t i ÞþFðtðiÞÞ þ l Pi P Dt Kðt i t l n ; PNH yðtl n =ÞÞ w k ha _ ðs j rg k ; t l Þ dt l¼ n¼ k¼ h a ðs j ; t Þ¼gðs j Þ; ð22þ j =,2,...,2N +,i =,2,..., +. Discretizing h _ a ðs j ; t i Þ and h _ a ðs j rg k ; t l Þ appearing in (22) takes the form h a ðs j ;t iþ Þ h a ðs j ;t i Þ P ¼ þ cos h Dt a ðs j ; t i ÞþFðtðiÞÞ þ l i P ðkðt i t l n ; yðtl n =ÞÞ h a ðs j ; t Þ¼gðs j Þ;. Rahman et al. / Applied athematics and Computation 6 (27) PNH k¼ l¼ n¼ w k ðh a ðs j rg k ; t lþ Þ h a ðs j rg k ; t l ÞÞÞdt j =,2,...,2N +,i =,2,..., +. Then for n =,2,..., +, t l n ¼ t l n ¼ t l þðn Þ Dt ¼ ðl ÞDt þðn Þ Dt ðl Þ þ n ADt fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} index ð2þ ð23þ ð24þ

6 7. Rahman et al. / Applied athematics and Computation 6 (27) 73 2 so that y tl n ¼ yððl Þ þ nþþ and thus (24) takes the form h a ðs j ; t iþ Þ¼h a ðs j ; t i ÞþDtðþcos h a ðs j ; t i ÞþFðtðiÞÞÞ P þ ldt i P Kðt i t l n ; yððl Þ þ nþþ PNH w k ðh a ðs j rg k ; t lþ Þ h a ðs j rg k ; t l Þ dt l¼ n¼ k¼ h a ðs j ; t Þ¼gðs j Þ; ð25þ j =,2,...,2N +,i =,2,..., +. Thus using (25) Euler Hermite method takes the form P h a ðs j ; t iþ Þ¼h a ðs j ; t i ÞþDt þ cos h a ðs j ; t i ÞþFðtðiÞ Þþl Dt i P Kðt i t l n ; yðn þ ðl ÞÞÞ l¼ n¼l PNH ð26þ w k ðh a ðs j rg k ;;t lþ Þ h a ðs j rg k ; t l ÞÞ k¼ h a ðs j ; t Þ¼gðsðjÞÞ; j ¼ ; 2;...; 2N þ ; i ¼ ; 2;...; : We have to approximate h a (s j rg k,t l+ ) and h a (s j rg k,t l ) using the values h a (s j,t l ) and h a (s j,t l+ ) defined at the grid points s j.if s j rg k 6 s then for any t we use h a ðs j rg k ; tþ ¼h a ðs j ; tþ: If sðþ < s j g k <6 s 2Nþ then we use piecewise linear interpolation h a ðs j rg k ; tþ ¼hðs j ; l þ Þþðh a ðs j ; l þ Þ h a ðs j ; lþþ s j rg k s j ; Ds where s j rg k 2 (x q,s q ]. This corresponds to the index q defined by s j rg q ¼ ceil k s þ : Dt If s j rg k P s 2N+ = A, we choose h a ðs j rg k ; tþ ¼h a ðs 2Nþ ; tþ: We also generate random sequence y using random number in ATLAB generated function rand for two state transition matrix (see Section 2) given by () with initial state y() =. Since t i = (i )Dt,i =,2,..., +, +2,..., +. Then t þ ¼ Dt ¼ T ¼ T. Thus for i =,2,..., *, if rand < a and y(i) = then y(i + ) = ; if rand 6 a and y(i) = then y(i + ) = ; if rand < b and y(i) = then y(i + ) = ; if rand 6 b and y(i) = then y(i +)=. 5. Stochastic approximation In this section we derived explicit expression for Stochastic Correction Equation defined by (9) to estimate the asymptotic behavior of h a ð p h a þ ffiffi ~ha Þ ¼ oðþ ð27þ as!, where h a and h a are the solution of () and (6) respectively and ~ h a is the solution of the Stochastic Correction Eq. (9), for more detail see [3], is given by B 2 ðtþ ¼ b aða þ bþ gðt; s; h a; y ; aþ 2 ½2 a bš; ð2þ

7 . Rahman et al. / Applied athematics and Computation 6 (27) where derivation of g(t,s,h a,y,a) is obtained using f ðt; s; h a ; y ; aþ, f ðt; s; h a ; y 2 ; aþ and f ðt; s; h a ; aþ, such that q g(t,s,h a,y,a)+q 2 g(t,s,h a,y 2,a) =, with q + q 2 =, where and gðt; s; h a ; y ; aþ f ðt; s; h a ; aþ f ðt; s; h a ; y ; aþ ð29þ gðt; s; h a ; y 2 ; aþ f ðt; s; h a ; aþ fðt; s; h a ; y 2 ; aþ: Thus using (7), (6) takes the form f ðt; s; h a ; aþ ¼ þ cos h a ðs; tþþfðtþ þ l ðq Kðt t ; y Þþq 2 Kðt t ; y 2 ÞÞ kðs s Þ hðs _ ; t Þds dt : ð3þ ð3þ Thus again using (2), we obtain f ðt; s; h a ; aþ ¼ þ cos h a ðs; tþþfðtþ þ l q y a e a ðt t Þ þ q 2 y 2 a e a ðt t Þ kðs s Þ _ ha ðs ; t Þds dt : Then using y = y and y = y 2,in() we have f ðt; s; h a ; y ; aþ ¼ þ cos t h a ðs; tþþfðtþþl Kðt t ; y Þ kðs s Þ _ ha ðs ; t Þds dt and ¼ þ cos h a ðs; tþþf ðtþ f ðt; s; h a ; y 2 ; aþ ¼ þ cos h a ðs; tþþf ðtþþl ¼ þ cos h a ðs; tþþf ðtþþl Thus using (33) and (3), (29) becomes, Kðt t ; y 2 Þ q y 2 a e a ðt t Þ gðt; s; h a ; y ; aþ ¼ þ cos h a ðs; tþþfðtþ þ l q y a e a ðt t Þ þ q 2 y 2 a e a ðt t Þ q 2 y 2 a e a ðt t Þ kðs s Þ _ ha ðs ; t Þds kðs s Þ _ ha ðs ; t Þds kðs s Þ _ ha ðs ; t Þds cos h a ðs; tþ FðtÞ l y a e a ðt t Þ kðs s Þ _ ha ðs ; t Þds ¼ l kðs s Þ _ ha ðs ; t Þds dt dt dt : dt dt ð32þ ð33þ ð34þ ð35þ and thus using (34), (3), and (3) becomes, gðt; s; h a ; y 2 ; aþ ¼ þ cos h a ðs; tþþfðtþ þ l q y a e a ðt t Þ þ q 2 y 2 a e a ðt t Þ ¼ l cos h a ðs; tþ F ðtþ l ðq 2 Þy 2 a e a ðt t Þ y 2 a e a ðt t Þ kðs s Þ _ ha ðs ; t Þds kðs s Þ _ ha ðs ; t Þds kðs s Þ _ ha ðs ; t Þds dt : dt dt ð36þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bð2 a bþ t Bðs; tþ ¼ l q aða þ bþ 2 y 2 a e a ðt t Þ kðs s Þ _ ha ðs ; t Þds and thus the Stochastic correction equation

8 . Rahman et al. / Applied athematics and Computation 6 (27) 73 2 Using (29), (2), takes the form B 2 bð2 a bþ t ðs; tþ ¼ l q aða þ bþ 2 y 2 a e a ðt t Þ kðs s Þ _ 2 ha ðs ; t Þds dt : ð37þ Thus sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bð2 a bþ t Bðs; tþ ¼ l q aða þ bþ 2 y 2 a e a ðt t Þ kðs s Þ _ ha ðs ; t Þds and thus the Stochastic correction equation becomes d h a ¼½ sin h a ðs; tþš ~ h a dt þ Bðs; tþdw: dt ð3þ ð39þ 6. Numerical results We solved Eq. () numerically over time interval [, ] using. Euler Hermite with smaller time scale. 2. Evaluate p h a ðs; tþþ ffiffi ~ha ðs; tþ, where h a ðs; tþ and ~ hðs; tþ are computed using Eqs. (6) and (39) respectively. 3. Compare 2 with where h a ðs; tþ is the solution of (6) using large time scale, H and B 2 (s,t) is also computed using the solution of (3). Fig.. Contour plot of cosh a (s,t) for the Euler Hermite quadrature method with piecewise linear interpolation of FVIDE. Fig. 2. Contour plot of cosh a (s,t) for Euler Hermite quadrature method with piecewise linear interpolation of average FVIDE.

9 . Rahman et al. / Applied athematics and Computation 6 (27) 73 2 we using Bt w/ euler 2 Using E h a ðs; tþ ð p h a ðs; tþþ ffiffi ~ha ðs; tþþ, where E denotes the expected value, leads the concept of strong convergence. Thus numerical experimentations suggests that h a ðs; tþ ¼ p h a ðs; tþþ ffiffi ~ha ðs; tþ for! for perturbed Fredholm Volterra integro-differential equation. Thus the asymptotic result for Volterra integral equations given in, [2] can extended to more general FVIDE (see Figs. 3). 7. Concluding remarks In simulation we observed, the elapsed time for solving the averaged system approximately. times that for solving the perturbed system. While experimenting the error, we implicitly assumed that number of other sources of error are negligible, including the following, for more detail, (see []): Sampling error: error arising from approximating an expected value by sample mean. Random number bias: inherent error in the random generator. Rounding error: floating point roundoff errors. For a typical computation the sampling error is likely to be the most significant of these three. In preparing the programs in these simulations we found that some experimentation is required to make the number of samples sufficiently large and the time step is sufficiently small for the predicted order of convergence to be observable. Experimentation indicates that as step size decreases, the lack of independence in the samples from a random generator typically degrades the computation before rounding errors becomes significant. Future generalization of this work can be extended in the networks of excitatory and inhibitory neurons with sparse and random connectivity. References 3 2 Fig. 3. Strong error plots using Euler Hermite and dashed line is the line with slope. [] D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, Siam Rev. 43 (3) (22) [2] A.V. Skorokhod, F.C. Hoppensteadt, Random perturbations methods with applications in science and engineering, Appl. ath. Sci. 5 (997). [3] F.C. Hoppensteadt,. Rahman, B.D. Welfert, Central limit theorems for arkov processes with applications to circular processes, Adv. Appl. Prob., submitted for publication. [4]. Jackiewicz,. Rahman, B.D. Welfert, Numerical solution of Fredholm integro-differential equation modelling neural networks, Appl. Numer. ath. 56 (26) [5] F.C. Hoppensteadt, An Introduction to the athematical Neurons, second ed., Cambridge University Press, 997.

10 2. Rahman et al. / Applied athematics and Computation 6 (27) 73 2 [6] F.C. Hoppensteadt, _ h-networks, unpublished manuscript, Arizona State University, 23. [7] P.J. Davis, P. Rabinowitz, ethods of Numerical Integration, second ed., Academic Press, New York, 94. [] B. Oksendal, Stochastic differential equations, Springer-Verlag, Berlin, 2. [9] I.I. Gikhmann, Differential equations with random functions, AS transl. 2 (973). [] R.. Kahsminski, On random perturbations determined by differential equations with a small parameter, Theo. Prob. Appl. (966) [] S. Karlin, A First Course in Stochastic Processes, Academic press, 966.

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

Available online at www.sciencedirect.com Applied Mathematics and Computation 195 (2008) 523 536 www.elsevier.com/locate/amc Numerical solution of a Fredholm integro-differential equation modelling _ h-neural