If a j = 1 n , j = 1 , ⋯ , n , (i.e., P = 1 n e _ e _ T ) we obtain

p ( z k ) = 1 n ∑ j = 1 n     z k j − 1 = 1 n 1 − z k n 1 − z k = 0

for k = 1 , ⋯ , n − 1 . Parseval’s identity yields

∑ k = 1 n − 1 | Φ ( z k ) | 2 = ∑ k = 0 n − 1 | Φ ( z k ) | 2 = ∑ k = 0 n − 1 | ( V H φ ( y _ ) ) k | 2 = φ ( y _ ) T V V H φ ( y _ ) = φ ( y _ ) T φ ( y _ ) = ∑ j = 1 n | φ ( y j ) | 2 .

Then (3.23) reduces to

σ Δ t 2 = Δ t n ∑ k = 1 n − 1 | Φ ( z k ) | 2 = Δ t n ∑ j = 1 n | φ ( y j ) | 2 ≈ Δ t ∫ Y     φ 2 ( y ) ρ Δ t ( d y ) ,     as     n → ∞ .

Assume now that the probability y i → y j , j ≠ i , is independent of j but distinct from the probability y i → y i , i.e.,

a 1 = 1 − a Δ t ,   a 2 = ⋯ = a n = a Δ t n − 1 ,

with 0 < a < 1 / Δ t . Then

p ( z ) = 1 − a Δ t + a Δ t n − 1 ( z + ⋯ + z n − 1 ) = 1 − a Δ t + a Δ t n − 1 z − z n 1 − z

for z ≠ 1 . In particular

p ( z k ) = 1 − a Δ t + a Δ t n − 1 z k − 1 1 − z k = 1 − n a Δ t n − 1 ,

for k = 1 , ⋯ , n − 1 . Therefore, by Parseval again,

σ Δ t 2 = 2 − n a Δ t n − 1 n a n − 1 1 n ∑ j = 1 n | φ ( y j ) | 2 ≈ 2 − Δ t a a ∫ Y     φ 2 ( y ) ρ Δ t ( d y )     as     n → ∞ ≈ 2 a ∫ Y     φ 2 ( y ) ρ Δ t ( d y )     as     Δ t → 0. (3.24)

A case of particular interest in the study of the transmission of a signal around a cyclic biological chain corresponds to

a 1 = 1 − a Δ t ,   a 2 = a Δ t ,   a 3 = ⋯ = a n = 0 ,

with 0 < a < 1 / Δ t . The quantity a 2 represents the transition probability of a state y j to the next state y j + 1 . Then p ( z ) = 1 − a Δ t + a z Δ t , and

1 − | p ( e i y ) | 2 | 1 − p ( e i y ) | 2 = 1 − ( 1 − a Δ t + a Δ t ) 2 − ( a Δ t ) 2 ( a Δ t − a Δ t ) 2 + ( a Δ t ) 2 = 1 − a Δ t a Δ t ,

for 0 < y < 2 π . Therefore

σ Δ t 2 = 1 − a Δ t a 1 n ∑ j = 1 n | φ ( y j ) | 2 (3.25)

≈ 1 − a Δ t a ∫ Y     φ 2 ( y ) ρ Δ t ( d y )     as     n → ∞

≈ 1 a ∫ Y     φ 2 ( y ) ρ Δ t ( d y ) ,     as     Δ t → 0. (3.26)

For n = 2 , (3.25) reduces to 1 − Δ t a a φ ( y 1 ) 2 + φ ( y 2 ) 2 2 . Note that (3.26) is half

of (3.24), as could be expected from a less dispersive signal.

In the following section our approach to modeling randomness in dynamical systems is through allowing parameters in a system to be a random process. This approach will determine when solution of this type of stochastic problem do or do not persist when the system is perturbed.

The Euler method to approximate the analytic solution of the IVP

{ x ˙ ( t ) = f ( t , x , y ( t / ε ) ) , x ( 0 ) = x 0 . (4.29)

We can derive an entire family of discrete numerical methods (including the Euler method) by truncating the Taylor series and utilizing Taylor’s Theorem. First, we rewrite the Taylor series expansion in differential form,

x ( t + h ) = x ( t ) + h 1 ! x ′ ( t ) + h 2 2 ! y ″ ( t ) + ⋯ + h n n ! y ″ ( t ) + ⋯ = x ( t ) + h f ( t , x , y ( t / ε ) ) + h 2 2 f ′ ( t , x , y ( t / ε ) ) f ( t , x , y ( t / ε ) ) + ⋯   + h 3 6 ( f ″ ( f , f ) + f ′ ( f ′ ( f ) ) ) + ⋯ .

The Euler method is found by truncating Taylor series at the first derivative, giving

x ( t + h ) = x ( t ) + h f ( t , x , y ( t / ε ) ) + o ( h 2 ) , (4.30)

where o ( h 2 ) is the error term, or Taylors remainder term, which is of order h 2 . So if we define x n = x ( t + h ) , and x n + 1 = x ( t + h ) , we obtain

x n + 1 = x n + h f ( n , x n , y ( n / ε ) ) ,

which has local error term (error at each step) of o ( h 2 ) , giving a global error o ( h ) .

Numerical schemes for solving SDE can be classified into either explicit or implicit methods. Explicit methods compute approximations that are dependent on previous approximations only, whereas the implicit methods compute approximations that are dependent on previous and current approximations. The Euler method presented earlier is also known as the explicit Euler method. The explicit Euler method is defined as

x i + 1 = x i + h f ( t i , x i , y i / ε ) , (4.31)

where x i = x ( t i ) and x i + 1 = x ( t i + h ) . Here h is the step size. This method has a global error ≈ o ( h ) . The implicit method is defined as

x i + 1 = x i + h f ( t i , x i + 1 , y ( i + 1 ) / ε ) , (4.32)

where x i = x ( t i ) , and x i + 1 = x ( t i + h ) . This method has a global error ≈ o ( h ) .

We write the algorithm for the following equation, which we will be used as a test equation since it has an analytic solution.

{ x ˙ ( t ) = y ( t / ε ) , x ( 0 ) = 0 , (4.33)

over [ 0,1 ] and then

x ( t + h ) = x ( t ) + h y ( t / ε ) + o ( h ) . Now,

1) choose a step h = ( 1 − 0 ) / N . Set x n = 0 + n h , n = 0 : N .

2) Generate approximation x n to from the following recursion:

x n = 0 + h ∑ k = 0 n − 1 y ( k h / ε ) = h ∑ k = 0 n − 1 y ( k h / ε ) = h n y ¯ + h ∑ l = 0 h ( n − 1 ) / ε ( y ( l ) − y ¯ ) = h n y ¯ + H ε 1 1 / ε ∑ l = 0 h ( n − 1 ) / ε ( y ( l ) − y ¯ ) ,

for n = 0 : N − 1 .

As ε → 0 ,

x n → h n y ¯ + H ε W ( h n ) + E r r o r ,

where h = H ε in our numerical simulation and W is realized by the normally distributed random number whose expectation and variance are 0 and 1, respectively. A computer simulation of the distribution of solution of x ( 1, ε ) using Euler without max step size, ode15s without max step, and ode15s using max step size are given in Figure 1.

We consider a stochastic process x ε ( t ) is defined by the differential equation

{ x ˙ ( t ) = − t x ( y ( t ε ) − 0.5 ) ≡ f ( t , ε , x , y ) , x ( 0 ) = 1 , (4.34)

with

Y = { 0,1 } ,

P = ( 0.8 0.2 0.1 0.9 ) ,

Q = ( − 0.2 0.2 0.1 − 0.1 ) ,

and limiting distribution

ρ = { 1 / 3 , 2 / 3 }

We want to solve (4.34) numerically over time interval [ 0,1 ] using

1) Forward Euler with smaller time scale.

2) evaluate x ¯ + ε x ˜ , where

x ¯ ˙ = − t x ( 1 3 y 1 + 2 3 y 2 − 0.5 ) ≡ f ¯ ( t , ε , x ) , (4.35)

and

d x ˜ = f ¯ x ( t , ε , x ) x ˜ d t + σ ( t ) d W . (4.36)

3) compare 2. with 1., where x ¯ ( t ) and x ˜ ( t ) are the solutions of (4.35) (using large time scale) and (4.36) respectively. And σ ( t ) is computed using (2.15) along with ρ , f , f ¯ , and d W = d t N ( 0,1 ) . Comparison of the averaged system to the perturbed system as well as error convergence are illustrated in Figure 2 and Figure 3.

We consider a stochastic process x ε ( t ) is defined by the differential equation

{ x ˙ ( t ) = 1 / 2 + cos ( y ( t ε ) ) cos ( x ) ≡ f ( t , ε , x , y ) , x ( 0 ) = 1 , (4.37)

with

Y = { 0,1 } ,

P = ( 0.8 0.2 0.1 0.9 ) ,

Q = ( − 0.2 0.2 0.1 − 0.1 ) ,

and limiting distribution

ρ = { 1 / 3 , 2 / 3 }

We want to solve (4.37) numerically over time interval [ 0,1 ] using

1) Forward Euler with smaller time scale.

2) evaluate x ¯ + ε x ˜ (using smaller time scale, h), where

x ¯ ˙ = 1 / 2 + ( 1 3 cos 0 + 2 3 cos 1 ) cos ( x ¯ ) ≡ f ¯ ( t , ε , x ) , (4.38)

and

d x ˜ = f ¯ x ( t , ε , x ) x ˜ d t + σ ( t ) d W . (4.39)

3) compare 2. with 1., where x ¯ ( x ) and x ˜ ( t ) is the solution of (4.38) (using large time scale), and (4.39) respectively. And σ ( t ) is computed using (2.15) along with ρ , f , f ¯ and d W = d t N ( 0,1 ) . Comparison of the averaged system to the perturbed system as well as error convergence are illustrated in Figure 4 and Figure 5.

We consider a stochastic process x ε ( t ) is defined by the differential equation

{ x ˙ ( t ) = 1 + y ( t ε ) − ( 3 + y ( t ε ) ) cos ( x ) ≡ f ( t , ε , x , y ) x ( 0 ) = 1 , (4.40)

with

Y = { 0 , 1 } ,

P = ( 0.8 0.2 0.1 0.9 ) ,

Q = ( − 0.2 0.2 0.1 − 0.1 ) ,

and limiting distribution

ρ = { 1 / 3 , 2 / 3 } .

We want to solve (4.40) numerically over time interval [ 0,1 ] using

1) Forward Euler with smaller time scale.

2) evaluate x ¯ + ε x ˜ (using smaller time scale, h), where

x ¯ ˙ = 1 + y − ( 3 + y ) cos ( x ) ≡ f ¯ ( t , ε , x ) . (4.41)

and

d x ˜ = f ¯ x ( t , ε , x ) cos ( x ¯ ) x ˜ d t + σ ( t ) d W . (4.42)

3) compare 2. with 1. where x ¯ ( t ) and x ˜ ( t ) is the solution of (4.41) (using large time scale) and (4.42) respectively. And σ ( t ) is computed using (2.15) along with ρ , f , f ¯ , and d W = d t N ( 0,1 ) . Comparison of the averaged system to the perturbed system as well as error convergence are illustrated in Figure 6 and Figure 7.

In the examples above, directly simulated solution x ε ( t ) with smaller time scale matches more closely to the solution x ¯ ( t ) + ε x ˜ ( t ) , (where x ¯ ( t ) solved using larger time scale and x ˜ ( t ) is the solution of d x ˜ = f ¯ x ( t , y , x ¯ ) x ˜ d t + σ ( t ) d w , using Euler with large time scale H) as ε is decreased, the convergence seems to take place. Using E | x ε ( t ) − ( x ¯ ( t ) + ε x ˜ ( t ) ) | , where E denotes the expected value, leads the concept of strong convergence. A method is said to have strong order of convergence equal to m if there exists a constants K such that

E | x ε ( t ) − ( x ¯ ( t ) + ε x ˜ ( t ) ) | ≤ K ( ε ) m , (4.43)

and ε is sufficiently small. It can be shown that perturbation method has strong order of convergence m = 1 . In our numerical tests, we will focus on the error at the end point t = t f i n a l , so let

E r r o r s t r o n g = E | x ε ( t ) − ( x ¯ ( t ) + ε x ˜ ( t ) ) | . (4.44)

If the bound in (4.43) holds with m = 1 at any fixed point in [ 0, t f i n a l ] , then it certainly holds at the end point, so we have

E r r o r s t r o n g ≤ K ε 1 , (4.45)

for sufficiently small ε . It is shown that perturbation method has strong order of convergence m = 1 . While experimenting the error E r r o r s t r o n g , we implicitly assumed that number of other sources of error are negligible, including error arising from approximating an expected value by sample mean, inherent error in the random generator, and 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 for 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. The sampling error decays like 1 / n , where n is the number of sample paths used. A study in ( [19] ) 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.

Although the definition of strong convergence involves an expected value, it has implications for individual simulations. The Markov inequality says that if a random variable ξ has a finite expected value, then for any a > 0 the probability that | ξ | ≥ a is bounded above by E | ξ | / a , that is,

P ( | ξ | > a ) ≤ E | ξ | a .

Hence taking a = ε 1 / 2 , we see that perturbation method’s strong convergence of order m = 1 is

P ( | x ε ( t ) − ( x ¯ ( t ) + ε x ˜ ( t ) ) | ≥ ε 1 / 2 ) ≤ K ε 1 / 2 ,

or, equivalently,

P ( | x ε ( t ) − ( x ¯ ( t ) + ε x ˜ ( t ) ) | < ε 1 / 2 ) ≥ 1 − K ε 1 / 2 .

This shows that the error at a fixed point in [ 0, t f i n a l ] is small with probability close to 1.

Stochastic approximations for the process x ˙ ( t ) with parametric noise can be used to analyze aspects of noise. The result of the analysis is an approximation of the form x ( t ) ≈ x ¯ ( t ) + ε x ˜ ( t ) + o ( ε ) for x ( t ) of the system. This can be used to evaluate the impact of parametric noise in the neural network. First of all, the system can be averaged. The system for x ¯ may or may not be analytically solvable but we develop numerical method for its solution. Second, the next order term solves a linear system forced by a Gaussian process, whose statistics depends on the nature of noise in the model.

Our systems are characterized by some system components which combine very fast and very slow behavior. These systems require adaptable step-size, as only in certain phases they require very small step size. It is important to use integration method that allows an efficient step size control. A system is called stiff when integrated with an explicit algorithm and a local error tolerance 10 − n , the step size of the algorithm is forced down to below a value indicated by the local error estimate due to constraints imposed on it by the limited size of the numerical stable region. Ode15s is a variable-order solver based on numerical differentiations formulas, optionally uses the backward differentiations formulas (also known a Gear’s method) like ode113, ode15s is multi-step solver. If one suspects that the problem is stiff or if ode45 fails or is very inefficient, try ode15s. But this slow vs. fast time scale problem, if we do not use max step size for the ode15s, the method does not give the correct result which is reflected in Figure 1.

Future work will address systems involving noisy neural network and the impact of noise on a node’s information processing capability which is determined by its signal-to-noise ratio which can be estimated by spectral methods.