# Global Exponential Stability of Discrete-Time Multidirectional Associative Memory Neural Network with Variable Delays

Authors: Min Wang; Tiejun Zhou; Xiaolan Zhang
Journal: ISRN Discrete Mathematics
 (2012)

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## Abstract

A discrete-time multidirectional associative memory neural networks model with varying time delays is formulated by employing the semidiscretization method. A sufficient condition for the existence of an equilibrium point is given. By calculating difference and using inequality technique, a sufficient condition for the global exponential stability of the equilibrium point is obtained. The results are helpful to design global exponentially stable multidirectional associative memory neural networks. An example is given to illustrate the effectiveness of the results.


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## Body

## 1. Introduction

The multidirectional associative memory (MAM) neural networks were first proposed by the Japanese scholar M. Hagiwara in 1990 [1]. The MAM neural networks have found wide applications in areas of speech recognition, image denoising, pattern recognition, and other more complex intelligent information processing. So they have attracted the attention of many researchers [2–7]. In [5], we proposed a mathematical model of multidirectional associative memory neural network with varying time delays as follows, which consists of the m fields, and there are nk neurons in the field k (k=1,2,…,m):
(1.1)dxkidt=Iki-akixki(t)+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj(t-τpjki(t))),
where k=1,2,…,m, i=1,2,…,nk, xki(t) denote the membrane voltage of the ith neuron in the field k at time t, aki>0 denote the decay rate of the ith neuron in the field k, fpj(·) is a neuronal activation function of the ith neuron in the field k, wpjki is the connection weight from the jth neuron in the field p to the ith neuron in the field k, Iki is the external input of the ith neuron in the field k, and τpjki(t) is the time delay of the synapse from the j neuron in the field p to the ith neuron in the field k at time t. We studied the existence of an equilibrium point by using Brouwer fixed-point theorem and obtained a sufficient condition for the global exponential stability of an equilibrium point by constructing a suitable Lyapunov function.However, discrete-time neural networks are more important than their continuous-time counterparts in applications of neural networks. One can refer to [8–10] in order to find out the research significance of discrete-time neural networks. To the best of our knowledge, few studies have considered the stability of discrete-time MAM neural networks.In this paper, we first formulate a discrete-time analogues of the continuous-time network (1.1), and in a next study the existence and the global exponential stability of an equilibrium point for the discrete-time MAM neural network.


## 2. Discrete-Time MAM Neural Network Model and Some Notations

In this section we formulate a discrete-time MAM neural network model with time-varying delays by employing the semidiscretization technique [8].LetN=∑k=1m‍nk, ℤ denote the integers set, ℤ+={1,2,3,…}, ℤ0+={0,1,2,…}, and ℤ[a,b]={a,a+1,…,b} where a,b∈ℤ.Leth be a fixed positive real number denoting a uniform discretionary step size and [u] denote the integer part of the real number u. If t∈[nh,(n+1)h)(n∈ℤ0+), then [t/h]=n. By replacing the time t of the network (1.1) with nh, we can formulate the following approximation of the network (1.1):
(2.1)dxkidt=Iki-akixki(t)+∑p=1,p≠km‍∑j=1np‍wpjki×fpj(xpj(nh-[τpjki(nh)h]h))
for t∈[nh,(n+1)h)(n∈ℤ0+). Denote [τpjki(nh)/h]=κpjki(n), we rewrite (2.1) as follows:
(2.2)dxkidt+akixki(t)=Iki+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj(nh-κpjki(n)h)).
Multiplying both sides of (2.2) by eakit, we obtain
(2.3)d(xkieakit)dt=eakit(Iki+∑p=1,p≠km‍∑j=1np‍wpjki×fpj(xpj(nh-κpjki(n)h))).
Integrating the (2.3) over [nh,t)(t≤(n+1)h), we have
(2.4)xki(t)eakit-xki(nh)eakinh=eakit-eakinhaki×(Iki+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj(nh-κpjki(n)h))).
Letting t→(n+1)h in (2.4), we obtain
(2.5)xki((n+1)h)=xki(nh)e-akih+1-e-akihaki×(Iki+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj(nh-κpjki(n)h))).
If we adopt the notations
(2.6)xki(n)≜xki(nh),αki(n)≜e-akih,wpjki≜wpjki1-e-akihaki,Iki≜Iki1-e-akihaki,
then we obtain the discrete-time analogue of the continuous-time network (1.1) as follows:
(2.7)xki(n+1)=Iki+αkixki(n)+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj(n-κpjki(n)))
for n∈ℤ0+, k∈ℤ[1,m], i∈ℤ[1,nk]. Obviously, 0<αki<1.Throughout this paper, for anyk∈ℤ[1,m], p∈ℤ[1,m](p≠k), i∈ℤ[1,nk], j∈ℤ[1,np], we assume that the neuronal activation functions fki and the time delays sequences κpjki(n) satisfy the following conditions, respectively:(H1)
There existLki>0 such that |fki(x)-fki(y)|≤Lki|x-y| for each x,y∈ℝ,(H2)
0<κ¯pjki=supn∈ℤ0+κpjki(n)<+∞.The initial conditions associated with (2.7) are of the form
(2.8)xki(n)=φki(n),
where k∈ℤ[1,m], i∈ℤ[1,nk], n∈ℤ[-κki,0], κki=max1≤p≤m,p≠kmax1≤j≤npκ¯pjki.For convenience sake, setcol(bki)=(b11,…,b1n1,b21,…,b2n2,…,bm1,…,bmnm)T, x=col(xki), f(x)=col(fki(xki)). For any matrixes U=(uij) and V=(vij), we use the notation U≥V to mean that uij≥vij for all i, j, and |U|=(|uij|). Let matrix A=diag(1-α11,…,1-α1n1,1-α21,…,1-αm1,…,1-αmnm), L=diag(L11,…,L1n1,L21,…,Lm1,…,Lmnm),
(2.9)W=(O11W12⋯W1mW21O22⋯W2m⋯⋯⋯⋯Wm1Wm2⋯Omm),
where
(2.10)Wkp=(wp1k1wp2k1⋯wpnpk1wp1k2wp2k2⋯wpnpk2⋯⋯⋯⋯wp1knkwp2knk⋯wpnpknk),Okk are zero matrixes(k,p∈ℤ[1,m]).


## 3. The Existence and Global Exponential Stability of an Equilibrium Point

In this section, we will give two theorems about the existence and the global exponential stability of an equilibrium point of the discrete-time MAM neural network (2.7).Lemma 3.1.
If0<b<1, 0≤x≤1-b, then bex≤x+b.Proof.
Define a functiong(t)=1-t+lnt (0<t≤1). From g'(t)=(1-t)/t>0 for 0<t<1, we know that the function g(t) is increasing on the interval (0,1]. Therefore, g(b)<g(1)=0 for 0<b<1. So we have 1-b<-lnb.
Define a functionh(x)=bex-x-b for x≥0 again. Obviously h'(x)=bex-1, h′′(x)=bex. From h′′(x)>0, the h'(x) is increasing. Because there exists an unique critical number x0=-lnb, we know that h'(x)<h'(-lnb)=0 for 0≤x≤-lnb. It shows that h(x) is deceasing on [0,-lnb]. So we have h(x)=bex-x-b≤h(0)=0. In view of 1-b<-lnb, we obtain bex≤x+b for 0≤x≤1-b.With a similar method of [5], we can prove the following Theorem 3.2.Theorem 3.2.
Suppose that all the neuronal activation functionsfki(·) (k∈ℤ[1,m], i∈ℤ[1,nk]) are continuous and the condition (H1) holds. If B=A-|W|L is a nonsingular M matrix, then there exists an equilibrium point of the discrete-time MAM neural network (2.7).Proof.
Letβki=∑p=1,≠km∑j=1np|wpjkifpj(0)|/(1-αki)+|Iki|/(1-αki). Obviously, βki≥0. Denote β=col(βki), r=col(rki)=(A-|W|L)-1Aβ. Because Aβ≥0 and B=A-|W|L is a nonsingular M matrix, by Lemma A3 in [11], we have r=(A-|W|L)-1Aβ≥0. That is rki≥0 for any k∈ℤ[1,m], i∈ℤ[1,nk]. From the definition of r, we have (E-A-1|W|L)r=β. Therefore,
(3.1)11-αki[∑p=1,p≠km‍∑j=1np‍|wpjki|Lpjrpj]+βki=rki.
LetΩ={x=col(xki)∣xki∈[-rki,rki]} with a norm ∥x∥=max1≤k≤mmax1≤i≤nk{|xki|}. Obviously, Ω is a bounded closed compact subset. Define a function F:Ω→ℝN as F(x)=col(Fki(x)), where
(3.2)Fki(x)=11-αki[∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj)+Iki].
From the condition (H1) and (3.1), we have
(3.3)|Fki(x)|≤11-αki[∑p=1,p≠km∑j=1np‍|wpjki|(Lpj|xpj|+|fpj(0)|)+|Iki|]≤11-αki[∑p=1,p≠km∑j=1np‍|wpjki|(Lpjrpj+|fpj(0)|)+|Iki|]=11-αki[∑p=1,p≠km‍∑j=1np‍|wpjki|Lpjrpj]+βki=rki.
Thus F(x) is a self-map from Ω to Ω. By Brouwer fixed-point theorem, there exists at least a x*∈Ω, such that F(x*)=x*. That is
(3.4)xki*=Iki+αkixki*+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj*).
Therefore x* is an equilibrium point of the MAM neural network (2.7).Next we prove the global exponential stability of the equilibrium point of the discrete-time MAM neural network (2.7).Theorem 3.3.
Suppose that all the neuronal activation functionsfki(·)(k∈ℤ[1,m],i∈ℤ[1,nk]) are continuous and the conditions (H1) and (H2) hold. If B=A-|W|L is a nonsingular M matrix, then the equilibrium point of the MAM neural network (2.7) is global exponential stable.Proof.
Letx*=col(xki*) be an equilibrium point for the MAM neural network (2.7), x(n)=col(xki(n)), be an arbitrary solution of (2.7). Set u(n)=col(uki(n)), where
(3.5)uki(n)=xki(n)-xki*
for k∈ℤ[1,m], i∈ℤ[1,nk]. Define functions
(3.6)Fpj(z)=fpj(z+xpj*)-fpj(xpj*),
where p∈ℤ[1,m], j∈ℤ[1,np]. Obviously, Fpj(0)=0 and from the condition (H1), we have
(3.7)|Fpj(z)|≤Lpj|z|
for any z∈ℝ. By (3.5), the MAM neural network (2.7) is reduced to the form
(3.8)uki(n+1)=αkiuki(n)+∑p=1,p≠km‍∑j=1np‍wpjkiFpj(upj(n-κpjki(n))),
where k∈ℤ[1,m], i∈ℤ[1,nk]. Obviously, there exists an equilibrium point u*=0 of the system (3.8). From (2.8) and (3.5), the initial conditions associated with (3.8) are of the form
(3.9)uki(n)=ψki(n)=φki(n)-xki*,
where k∈ℤ[1,m], i∈ℤ[1,nk], n∈ℤ[-κki,0]. Let ψ=col(ψki(n)), ∥ψ∥=maxk∈ℤ[1,m]maxi∈ℤ[1,nk]sup-κki≤n≤0|ψki(n)|.
BecauseB=A-|W|L is a nonsingular M matrix, then there exist constants ξki>0 (i∈ℤ[1,nk], k∈ℤ[1,m]) such that
(3.10)(1-αki)ξki-∑p=1,p≠km∑j=1np‍|wpjki|Lpjξpj>0.
Define the functions(3.11)Hki(λ)=(1-αki-λ)ξkie-λ-∑p=1,p≠km‍∑j=1np‍|wpjki|Lpjξpjeλκ-pjki,
where λ∈ℝ+, k∈ℤ[1,m], i∈ℤ[1,nk]. Apparently Hki(λ) is strictly monotone decreasing and continuous function. In view of (3.10), it is clear that Hki(0)>0,Hki(1-αki)<0. Therefore there exist λki∈(0,1-αki) such that Hki(λki)=0(k∈ℤ[1,m], i∈ℤ[1,nk]). Taking α=mink∈ℤ[1,m]mini∈ℤ[1,nk]{λki}<mink∈ℤ[1,m]mini∈ℤ[1,nk]{1-αki}, we have
(3.12)Hki(α)=(1-αki-α)ξkie-α-∑p=1,p≠km‍∑j=1np‍|wpjki|Lpjξpjeακ-pjki≥0
for i∈ℤ[1,nk], k∈ℤ[1,m].
Setyki(n)=eαnuki(n). By calculating Δ|yki(n)|=|yki(n+1)|-|yki(n)| along the solutions of system (3.8), we have
(3.13)Δ|yki(n)|=eα(n+1)|uki(n+1)|-eαn|uki(n)|≤αki-e-ααkieα(n+1)|uki(n+1)|+eαnαki×∑p=1,p≠km‍∑j=1np‍|wpjki||Fpj(upj(n-κpjki(n)))|.
By using the inequality (3.7), we have
(3.14)Δ|yki(n)|≤αki-e-ααkieα(n+1)|uki(n+1)|+eαnαki×∑p=1,p≠km∑j=1np‍|wpjki|Lpj|upj(n-κpjki(n))|=αki-e-ααki|yki(n+1)|+1αki∑p=1,p≠km∑j=1np‍|wpjki|×Lpjeακpjki(n)|ypj(n-κpjki(n))|≤αki-e-ααki|yki(n+1)|+1αki∑p=1,p≠km∑j=1np‍|wpjki|×Lpjeακ-pjki|ypj(n-κpjki(n))|≤(αkieα-1)αkieα|yki(n+1)|+1αki∑p=1,p≠km∑j=1np‍|wpjki|×Lpjeακ-pjkisupn-κ-pjki≤s≤n|ypj(s)|.
By Lemma 3.1, we have
(3.15)Δ|yki(n)|≤(α+αki-1)αkieα|yki(n+1)|+1αki∑p=1,p≠km∑j=1np‍|wpjki|×Lpjeακ-pjkisupn-κ-pjki≤s≤n|ypj(s)|.
Let ξ_=mink∈ℤ[1,m]mini∈ℤ[1,nk]{ξki}, ξ¯=maxk∈ℤ[1,m]maxi∈ℤ[1,nk]{ξki} and l0=(1+δ)∥ψ∥/ξ_, where δ is a positive constant. Therefore, when s∈ℤ[-κki,0], we have
(3.16)|yki(s)|=eαs|ψki(s)|≤∥ψ∥<ξkil0
for i∈ℤ[1,nk], k∈ℤ[1,m]. We assert that
(3.17)|yki(n)|<ξkil0
for n∈ℤ+, i∈ℤ[1,nk] and k∈ℤ[1,m]. If the assertion is false, then there exist k, i, and a minimum time t0∈ℤ+ such that |yki(t0+1)|≥ξkil0, Δ|yki(t0)|≥0, and |ypj(n)|≤ξpjl0 when n∈ℤ[-κki,t0]. From (3.12) and (3.15), and noticed that α+αki-1<0, we obtain
(3.18)Δ|yki(t0)|≤1αki[(α+αki-1)ξkie-α+∑p=1,p≠km‍∑j=1np‍|wpjki|Lpjξpjeακ-pjki]l0<0.
It conflicts with Δ|yki(t0)|≥0. Therefore,
(3.19)|yki(n)|<ξkil0
for n∈ℤ+. Then we have
(3.20)|uki(n)|<ξkil0e-αn≤(1+δ)ξ¯ξ_∥ψ∥e-αn=M∥ψ∥e-αn,
where M=(1+δ)ξ¯/ξ_>1. So the zero solution of the system (3.8) is global exponential stable; thus the equilibrium point of the discrete-time MAM neural network (2.7) is global exponential stable.


## 4. An Example

Consider the following discrete-time MAM neural network with three fields:(4.1)x11(n+1)=I11+α11x11(n)+w2111f21(x21(n-κ2111(n)))+w3111f31(x31(t-κ3111)),x12(n+1)=I12+α12x12(n)+w2112f21(x21(n-κ2112(n)))+w3112f31(x31(t-κ3112)),x21(n+1)=I21+α21x21(n)+w1121f11(x11(n-κ1121(n)))+w1221f12(x12(n-κ1221(n)))+w3121f31(x31(n-κ3121(n))),x31(n+1)=I31+α31x31(n)+w1131f11(x11(n-κ1131(n)))+w1231f12(x12(n-κ1231(n)))+w2131f21(x21(n-κ2131(n))),
where the neuronal signal decay rates α11=0.1, α12=0.2, α21=0.2, α31=0.1, the external inputs I11=1, I21=1, I21=1, I31=1, and connection weights w2111=0.2, w3111=0.3, w2112=0.2, w2131=0.4, w1121=0.25, w1221=-0.2, w3121=0.3, w1131=-0.2, w1231=0.2, w2131=0.3. The neuronal activation functions fki(x)=x, the time delays κpjki(n)=5+sin((π/2)n).Obviously, signal transfer functionsfki(x) are continuous, and they satisfy the condition (H1), and the constant Lki=1. The time delays κpjki(n) satisfy the condition (H2), and κki=6.By calculating, we have(4.2)B=A-|W|L=(0.90-0.2-0.300.8-0.2-0.4-0.25-0.20.8-0.3-0.2-0.2-0.30.9).
It is easy to verify that the matrix B is a nonsingular M matrix. Then by Theorems 3.2 and 3.3, there exists an equilibrium point which is globally exponentially stable for the MAM neural network (4.1).The numerical simulation is given in Figure1 in which ten cases of initial values are taken at random. From Figure 1, we can know that the MAM neural network (4.1) converges globally exponentially to the equilibrium point (x11*,x12*,x21*,x31*)T≈(2.1735,2.68,1.9635,1.8782)T no matter what it starts from the initial states.Figure 1
The globally exponential stability of the equilibrium of the MAM network (4.1) with 10 cases random initial values.


## 5. Conclusions

In this paper, we have formulated a discrete-time analogue of the continuous-time multidirectional associative memory neural network with time-varying delays by using semidiscretization method. Some sufficient conditions for the existence and the global exponential stability of an equilibrium point have been obtained. Our results have shown that the discrete-time analogue inherits the existence and global exponential stability of equilibrium point for the continuous-time MAM neural network.



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