Artificial neural networks are developed for solving some complex problems in control, optimal computation, pattern recognition, information processing, and associative memory [1]-[13]. American scientist Hopfield makes a great contribution for the development of neural network. That is the implementation of neural network by simple circuit devices, resistors, capacitors and amplifiers [14]. Hopfield neural network (HNN) can mimic the human's associative memory function and accomplish optimization. The key point is the weights of HNN which are implemented by resistors for simulating neuron synapse. While the bottleneck is that linear resistors cannot reflect variability of synapse for resistance of linear resistor being invariable.
Memristor [15], [16], the arising fourth circuit device, makes it better to simulate the variability of neuron synapse. Pershin and Ventra [17] gives their experimental research results that neurons with memristors as synapses can simulate the associative memory function of a dog. Hence, memristor is the advancing spot in the present physics research. Several models of memristor have been set up and its properties have been analyzed in [18]-[21]. Based on these analyses, memristor can be used to mimic synapse in neural computing architecture [22], construct memristor bridge synapse [23] and brain combined with the conventional complementary metal oxide semiconductor (CMOS) technology [24], set memristive neural network [25], [26] and implement memristor array for image processing [27] etc.
Some researchers derive mathematical model of memristive recurrent neural network (MRNN) by replacing resistors with memristors in Hopfield and cellular neural network circuit [28]-[30]. MRNN is modeled by state-dependent switched systems by simplifying the memristance as two-valued device with different terminal voltage. With differential inclusion theory, Lyapunov-Krasovskii function and some other math tools, some sufficient conditions are derived for dynamics of MRNN, such as, convergence and attractivity [31]-[33], periodicity and dissipativity [34], dissipativity for stochastic and discrete case, global exponential almost periodicity, and complete stability [35], multi-stability [36], etc. Considering the trouble from the switching property of memristor, researchers derive some interesting results about exponential stabilization, reliable stabilization, and finite-time stabilization of MRNN by designing different state feedback controllers [37], [38] and sampled-data controller [39]. All of these results make a solid foundation for MRNN's application to associative memory.
Associative memory is a distinguished function of human brain which can be simulated by recurrent neural network (RNN). The design problem is that some given prototype patterns are to be stored by RNN, and then the stored patterns can be recalled by some prompt information. In the existing literatures [40]-[46], there are two design methods for associative memory. One is that prototype patterns are designed as multiple locally asymptotically stable equilibria and initial values are the recalling probes. Another is that a prototype pattern is designed as the unique globally asymptotically stable equilibrium point with one external input as the recalling probe. Different external inputs mean different equilibrium points, i.e., different prototype patterns.
To the best of our knowledge, the bottleneck of associative memory based on RNN is that capacity of RNN is limited and different storage task needs different RNN because resistance can not be changed. Furthermore, there are few works about associative memory based on MRNN. Hence, the contribution of this paper is obtaining a threshold voltage for memristor by simulation, presenting a novel type of MRNN with infinite number of sub neural networks, and design a program for associative memory based on MRNN. Compared with MRNN models in the existing literatures, the difference is that every coefficient of MRNN has infinite number of values, not two values. Furthermore, every coefficient can be changed by the external input. So the associative memory based on MRNN seems to solve the problem of storage capacity.
The rest of this paper is organized as the following sections. Memristor property analysis and some preliminaries are stated in Section 2. Then, some sufficient conditions are given to ensure global stability and multi-stability of MRNN by some maths tools in Section 3, respectively. Next, design procedure for associative memory based on MRNN is given in Section 4. To elucidate our results, three simulation examples are presented in Section 5. At last, conclusion is drawn in Section 6.
2 Memristor Recurrent Neural Network Model 2.1 Memristor and Its PropertyThe definition of memristor [15] is a functional relation between charge
$ v(t)=\Big(R_{\rm on}\frac{w(t)}{D}+R_{\rm off}\Big(1-\frac{w(t)}{D}\Big)\Big)i(t) $ | (1) |
$ \frac{dw(t)}{dt}=\mu_V\frac{R_{\rm on}}{D}i(t)\label{B} $ | (2) |
where
The
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Figure 1
The curve of |
From Fig. 1, it shows that a memristor will not change its resistance unless the terminal voltage exceeds a certain threshold value
$ \begin{align}\label{M1} R(w)=&\; \left\{\begin{array}{ll} R(w, u),&u>V_T\\ R_w,& u<V_T \end{array}\right. \end{align} $ | (3) |
where
$ \begin{align}\label{CH} T_w=\frac{\Phi_D}{V_AR_{\rm off}^2}\big[(R(w_0))^2-(R(w))^2\big] \end{align} $ | (4) |
where
Remark 1: Simulation shows that there exists a threshold voltage for the memristor, i.e., memristance can be changed by terminal voltage with amplitude value being greater than threshold value. The result is consistent with the theoretical analysis in [20]. This property of memristor can reflect variability of neuron synapses. Furthermore, it makes memristor being suitable for constructing neural network with coefficients which can be changed according to our needs.
2.2 ModelIn this section, we will firstly present the mathematical model for MRNN, and then give some concepts and lemmas in order to obtain our main results. MRNN is modelled by the following differential equation systems:
$ \begin{align}\label{SM} \frac{dx_i(t)}{dt}\;= &-c_ix_i(t)+\sum\limits_{j=1}^{n}a_{ij}(u_i)f(x_j(t)) \nonumber\\&+ \sum\limits_{j=1}^{n}b_{ij}(u_i)f(x_j(t-\tau(t)))+u_i \end{align} $ | (5) |
where
$ \begin{align}\label{eq-1} |f(r_1)-f(r_2)|\leq \mu|r_1-r_2| \end{align} $ | (6) |
where
According to circuit theory and the property of memristor, it results that there exist some constants
$ \begin{align}\label{inter}\left\{ \begin{array}{ll} \underline{a}_{ij}\leq a_{ij}(u_i)\leq \overline{a}_{ij}\\ \underline{b}_{ij}\leq b_{ij}(u_i)\leq \overline{b}_{ij}. \end{array} \right. \end{align} $ | (7) |
Remark 2: Compared with these models in [31]-[38], the difference of MRNN (5) is that coefficients
Let
$ \phi(\vartheta) = (\phi _{1} (\vartheta ), \phi_{2} (\vartheta ), \ldots, \phi _{n} (\vartheta ))^T $ |
where
Definition 1 [48]: The equilibrium point
$ \left\|x(t;t_0, \phi, u)-x^*\right\| \leq \beta ||\phi-x^*||_{\infty}\exp\{-\alpha (t-t_{0})\} $ |
where
Lemma 1 [49]: Let
Stability of MRNN is the foundation for its application to associative memory. So we discuss global stability and multi-stability of MRNN in the following subsections. Firstly, we analyze the differences between MRNN and traditional RNN. Traditional RNN is described by, for
$ \begin{align}\label{SM-2} \frac{dx_i(t)}{dt}\;= &-c_ix_i(t)+\sum\limits_{j=1}^{n}a_{ij}f(x_j(t)) \nonumber\\&+ \sum\limits_{j=1}^{n}b_{ij}f(x_j(t-\tau(t)))+u_i \end{align} $ | (8) |
where
Discussion: According to above analysis, coefficients
This subsection discusses global stability of MRNN (5). By using comparative principle and the existing stability criteria, it derives some sufficient conditions for global stability of (5). The following activation function will be adopted in the rest of the paper
$ \begin{align}\label{f} f (r)=&\; \left\{\begin{array}{ll} 4k-3,&r\in \big[4k-3, +\infty\big)\\ 2r-(4k-3), &r\in \big[4k-5, 4k-3\big)\\ \ldots, \\ 2r-5, &r\in \big[3, 5\big)\\ 1, &r\in \big[1, 3\big)\\ r, &r\in \big(-1, 1\big)\\ -1, &r\in \big(-3, -1\big]\\ 2r+5, &r\in \big(-5, -3\big]\\ \ldots, \\ 2r+4k-3, &r\in \big(3-4k, 5-4k\big]\\ 3-4k,&r\in \big(-\infty, 3-4k\big].\end{array}\right. \end{align} $ | (9) |
Obviously,
Lemma 2: If the following three differential systems
$ \begin{equation}\label{sys-1} \dot{y}(t)=g_1(y(t)) \end{equation} $ | (10) |
$ \begin{equation}\label{sys-2} \dot{y}(t)=g_2(y(t)) \end{equation} $ | (11) |
$ \begin{equation}\label{sys-3} \dot{y}(t)=g_3(y(t)) \end{equation} $ | (12) |
have one common equilibrium point
Proof: Take the same initial value
$ \begin{align} &\;g_1(y)\leq g_2(y) \leq g_3(y)\nonumber\\ &\;\int_{t_0}^{t}g_1(y)\leq \int_{t_0}^{t}g_2(y) \leq \int_{t_0}^{t}g_3(y)\nonumber\\ &\;y_1(t)\leq y_2(t)\leq y_3(t). \end{align} $ | (13) |
Hence,
$ \begin{align*} &|y_1(t)|\leq \|\phi_1\|\exp\{-\alpha_1(t-t_0)\}\\ &|y_3(t)|\leq \|\phi_3\|\exp\{-\alpha_3(t-t_0)\}. \end{align*} $ |
So there must exist
$ \begin{align*} |y_2(t)|\leq \|\phi_2\|\exp\{-\alpha_2(t-t_0)\} \end{align*} $ |
is valid, i.e., (11) is globally exponentially stable.
Because the external inputs
Lemma 3 [48]: If for
By (5) and (7), we have
$ \begin{align}\label{SM-1} \frac{dx_i(t)}{dt}\;= &-c_ix_i(t)+\sum\limits_{j=1}^{n}\overline{a}_{ij}(u_i)f(x_j(t)) \nonumber\\&+ \sum\limits_{j=1}^{n}\overline{b}_{ij}(u_i)f(x_j(t-\tau(t)))+u_i \end{align} $ | (14) |
and
$ \frac{dx_i(t)}{dt}\;= -c_ix_i(t)+\sum\limits_{j=1}^{n}{\underline{a}_{ij}}(u_i)f(x_j(t)) \nonumber\\~~~~~~~~+ \sum\limits_{j=1}^{n}{\underline{b}_{ij}}(u_i)f(x_j(t-\tau(t)))+u_i $ | (15) |
for
Theorem 1: If coefficients of neural networks (14) and (15) satisfy that
Proof: Because the activation
Let
$ \begin{align*} &\;z(t)=(x_1(t))-x_1^{\star}, x_2(t)-x_2^{\star}, \ldots, x_n(t)-x_n^{\star})\\ &\;\overline{f}(z_i(t))=f(x_i(t)+x_i^{\star})-f(x_i^{\star}). \end{align*} $ |
Hence,
$ \begin{align} \dot{z}_i(t)= &-c_iz_i(t)+\sum\limits_{i=1}^na_{ij}(u_i)\overline{f}(z_i(t))\nonumber\\ &+\sum\limits_{i=1}^na_{ij}(u_i)\overline{f}(z_i(t-\tau(t))). \end{align} $ | (46) |
Let
$ \begin{align}\label{V-1} D^{+}V_i(t)\leq&-c_iV_i(t)+\sum\limits_{i=1}^n|a_{ij}(u_i)|V_j(t)\nonumber\\ &+\sum\limits_{i=1}^n|b_{ij}(u_i)|V_j(t-\tau(t)). \end{align} $ | (17) |
Let
$ \begin{align} \Psi(t)=&-c_iV_i(t)+\sum\limits_{i=1}^n|a_{ij}(u_i)|V_j(t)\nonumber\\ &+\sum\limits_{i=1}^n|b_{ij}(u_i)|V_j(t-\tau(t)). \end{align} $ | (18) |
Since
$ \begin{align*} &\;\underline{a}_{ij}\leq a_{ij}(u_i)\leq \overline{a}_{ij}, &\;\underline{b}_{ij}\leq b_{ij}(u_i)\leq\overline{b}_{ij} \end{align*} $ |
then
$ \begin{align*} &\;|\underline{a}_{ij}|\leq |a_{ij}(u_i)|\leq |\overline{a}_{ij}|, &\;|\underline{b}_{ij}|\leq |a_{ij}(u_i)|\leq|\overline{b}_{ij}| \end{align*} $ |
or
$ \begin{align*} &\;|\overline{a}_{ij}|\leq |a_{ij}(u_i)|\leq |\underline{a}_{ij}|, &\;|\overline{b}_{ij}|\leq |a_{ij}(u_i)|\leq|\underline{b}_{ij}|. \end{align*} $ |
So
$ \begin{align}\label{neq-1} \Psi(t)\leq&-c_iV_i(t)+\sum\limits_{i=1}^n|\overline{a}_{ij}|V_j(t)\nonumber\\ &+\sum\limits_{i=1}^n|\overline{b}_{ij}|V_j(t-\tau(t)) \end{align} $ | (19) |
or
$ \begin{align}\label{neq-2} \Psi(t)\leq &-c_iV_i(t)+\sum\limits_{i=1}^n|\underline{a}_{ij}|V_j(t)\nonumber\\ &+\sum\limits_{i=1}^n|\underline{b}_{ij}|V_j(t-\tau(t)). \end{align} $ | (20) |
According to the condition of Theorem
Remark 3: When
Multi-stability of RNN means that RNN has coexisting multi attractors. Memory patterns can be stored by these attractors. Memory capacity of RNN is up to the number of attractors. Another factor affecting memory is the activation function
$ \begin{align}\label{O1} \Omega_k=&\; \Big\{\prod\limits_{i=1}^n \ell^{(i)}, ~~\ell^{(i)}=\big(-\infty, -(4k-3)\big]~{\rm or}\nonumber\\ &\; ~~~ \big(-(4k-3), -(4k-5)\big]~{\rm or}~\ldots~{\rm or}~ \big(-3, -1\big]~{\rm or}\nonumber\\ &\; ~~~~ \big(-1, 1\big)~{\rm or}~\big[1, 3\big)~{\rm or}~\ldots~{\rm or}~\nonumber\\ &\; ~~~~ \big[4k-5, 4k-3\big)~{\rm or}~\big[4k-3, +\infty\big)\Big\}. \end{align} $ | (21) |
But there are limited number of equilibrium points and output patterns for RNN with these two kinds of activation functions. Hence, we discuss multi-stability of MRNN with the activation function (9).
Lemma 4 [49]: For the given integer
$ \begin{align} &\;a_{ii}+b_{ii} -(4k-3)\sum\limits_{j=1, j\neq i}^n\Big(|a_{ij}+b_{ij}|\Big)-|u_i|>c_i\label{th2-1} \end{align} $ | (22) |
$ \begin{align} &\;a_{ii}+b_{ii} +\sum\limits_{j=1, j\neq i}^n\Big(|a_{ij}+b_{ij}|\Big)+\frac{|u_i|}{(4k-3)}\nonumber\\ &\;~~~~~~~~~~~~~~~~<\Big(1+\frac{2}{(4k-3)}\Big)c_i,\label{th2-2} \end{align} $ | (23) |
then RNN with the activation function (9) has
Let
Theorem 2: If the following inequalities are valid
$ \begin{align} &\;\underline{a}_{ii}+\underline{b}_{ii} -(4k-3)\sum\limits_{j=1, j\neq i}^n\Big(|\breve{a}_{ij}|+|\breve{b}_{ij}|\Big)-|u_i|>c_i \end{align} $ | (24) |
$ \begin{align} &\;\overline{a}_{ii}+\overline{b}_{ii} +\sum\limits_{j=1, j\neq i}^n\Big(|\breve{a}_{ij}|+|\breve{b}_{ij}|\Big)+\frac{|u_i|}{(4k-3)}\nonumber\\ &\;~~~~~~~~~~~~~~~~<(1+\frac{2}{(4k-3)})c_i \end{align} $ | (25) |
then for
Proof: In order to prove multi-stability of (5), it is sufficient to verify whether conditions (22) and (23) are valid or not. For
$ \begin{align*} &a_{ii}(u_i)+b_{ii}(u_i)\\&\;\quad-(4k-3)\sum\limits_{j=1, j\neq i}^n\Big(|a_{ij}(u_i)+b_{ij}(u_i)|\Big)-|u_i|\\ &\;\geq \underline{a}_{ii}+\underline{b}_{ii}-(4k-3)\sum\limits_{j=1, j\neq i}^n\Big(|a_{ij}(u_i)+b_{ij}(u_i)|\Big)-|u_i|\\ &\;\geq \underline{a}_{ii}+\underline{b}_{ii} -(4k-3)\sum\limits_{j=1, j\neq i}^n\Big(|\breve{a}_{ij}|+|\breve{b}_{ij}|\Big)-|u_i|\\ &\;>c_i. \end{align*} $ |
And then
$ \begin{align*} &\;(1+\frac{2}{(4k-3)})c_i\\ &\;>\overline{a}_{ii}+\overline{b}_{ii} +\sum\limits_{j=1, j\neq i}^n\Big(|\breve{a}_{ij}|+|\breve{b}_{ij}|\Big)+\frac{|u_i|}{(4k-3)}\\ &\;\geq a_{ii}(u_i)+b_{ii}(u_i)+\sum\limits_{j=1, j\neq i}^n\Big(|\breve{a}_{ij}|+|\breve{b}_{ij}|\Big)+\frac{|u_i|}{(4k-3)}\\ &\;\geq a_{ii}(u_i)+b_{ii}(u_i)+\sum\limits_{j=1, j\neq i}^n\Big(|a_{ij}(u_i)|+|b_{ij}(u_i)|\Big) \\ &\;\quad+\frac{|u_i|}{(4k-3)}. \end{align*} $ |
Hence, (22) and (23) are valid for
Remark 4: In fact, (24) and (25) are minimum value and maximum value of (22) and (23), respectively. Hence, we generalize the systematic method [50], [51] to analyzing multi-stability of MRNN. Compared with results in [49], the conditions are more conservative. But MRNN has infinite number of sub neural networks, i.e, globally exponentially stable equilibrium points of MRNN (5) are infinite times
Based on the above analysis, we discuss associative memory design method based on MRNN (5). Memory patterns are described by bipolar value
Synthesis Problem: There are
Design procedure:
Step 1: Use vectors
Step 2: For the desired memory vectors, do the following:
1)
2) Take
Step 3: If
$ \begin{align*} &|a_{ij}|_{\max}=\max\limits_{1\leq \delta \leq q+1}|a_{ij}^{\delta}|\\ &|b_{ij}|_{\max}=\max\limits_{1\leq \delta \leq q+1}|b_{ij}^{\delta}|. \end{align*} $ |
Remark 5: Compared with the work in [41], we do not require that
Example 1: Consider the following MRNN with activation function
$ \begin{eqnarray}\label{ex1} \left\{\begin{array}{l} \dot{x}_1(t)=-{x}_1(t)+a_{11}f({x}_1(t))+a_{11}f({x}_2(t))\\ ~~~~~~~~~~+b_{11}f({x}_1(t-0.1))\\ ~~~~~~~~~~+b_{12}f({x}_2(t-0.1))+0.8\\ \dot{x}_2(t)=-{x}_2(t)+a_{21}f({x}_1(t))+a_{22}f({x}_2(t))\\ ~~~~~~~~~~+b_{21}f({x}_1(t-0.2))\\ ~~~~~~~~~~+b_{22}f({x}_2(t-0.2))+0.4 \end{array}\right. \end{eqnarray} $ | (26) |
where
$ \begin{align*} &-3\leq a_{11}\leq -2, ~ \frac{1}{3}\leq a_{12}\leq \frac{1}{2}\\ &-\frac{1}{3}\leq a_{21}\leq \frac{1}{2}, ~ -3\leq a_{22}\leq -2\\ &0.9\leq b_{11}\leq 1, ~ -\frac{1}{4}\leq b_{12}\leq \frac{1}{4}\\ &\frac{1}{4}\leq b_{21}\leq \frac{1}{2}, ~ 0.6\leq b_{22}\leq 1\\ &c_1=c_2=4.8.\\ \end{align*} $ |
According to Theorem 1, every sub neural network of MRNN (26) is globally exponentially stable. Let
![]() |
Figure 2
Transient behaviors of |
![]() |
Figure 3
Transient behaviors of |
![]() |
Figure 4
Phase plot of |
Example 2: Consider a MRNN with activation function
$ \begin{eqnarray}\label{ex2} \left\{\begin{array}{l} \dot{x}_1(t)=-{x}_1(t)+a_{11}f({x}_1(t))+a_{11}f({x}_2(t))\\ ~~~~~~~~~~+b_{11}f({x}_1(t-0.1))\\ ~~~~~~~~~~+b_{12}f({x}_2(t-0.1))+0.05\\ \dot{x}_2(t)=-{x}_2(t)+a_{21}f({x}_1(t))+a_{22}f({x}_2(t))\\ ~~~~~~~~~~+b_{21}f({x}_1(t-0.2))\\ ~~~~~~~~~~+b_{22}f({x}_2(t-0.2))-0.04 \end{array}\right. \end{eqnarray} $ | (27) |
where
$ \begin{align*} &\;0.5\leq a_{11}\leq 0.7, ~0.01\leq a_{12}\leq 0.02\\ &\;-0.02\leq a_{21}\leq -0.01, ~0.4\leq a_{22}\leq 0.6\\ &\;0.4\leq b_{11}\leq 0.5, ~-0.02\leq b_{12}\leq -0.01\\ &\;0.005\leq b_{21}\leq 0.01, ~0.6\leq b_{22}\leq 0.7. \end{align*} $ |
According to Theorem
![]() |
Figure 5
Transient behaviors of |
Example
![]() |
Figure 6 Three letters "I, L, U" and number "7" being presented by gray map. |
These four desired patterns can be denoted by memory vectors
$ \begin{align*} &\;\alpha^1=(1, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 1)\\ &\;\alpha^2=(5, 5, 5, 5, 1, 1, 1, 5, 1, 1, 1, 5)\\ &\;\alpha^3=(5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 5, 5)\\ &\;\alpha^4=(5, 1, 1, 1, 5, 1, 1, 5, 5, 5, 5, 5). \end{align*} $ |
The objective is to design one
$ \begin{equation*} S(12)=\left[ \begin{array}{cccccccccccc} 1 & 5 & 5 & 5 & 1 & 5 & 5 & 5 & 1 & 5 & 5 & 5\\ 1 & 5 & 5 & 1 & 1 & 1 & 5 & 1 & 5 & 5 & 5 & 5\\ 1 & 5 & 5 & 1 & 1 & 5 & 1 & 5 & 5 & 1 & 5 & 5\\ 1 & 5 & 5 & 1 & 1 & 5 & 5 & 1 & 5 & 5 & 1 & 5\\ 5 & 1 & 1 & 5 & 5 & 1 & 1 & 1 & 1 & 1& 1 & 1\\ 1 & 5 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 5 & 1 & 5 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 5 & 1 & 5 & 5 & 5 & 1 & 5 & 5 & 1 & 5 & 1\\ 5 & 1 & 5 & 1 & 5 & 1 & 1 & 1 & 5 & 1 & 1 & 1\\ 1 & 5 & 1 & 5 & 1 & 5 & 1 & 1 & 1 & 5 & 1 & 1\\ 5 &1 & 5 & 1 & 5 & 1 & 1 &1 & 1 & 1 & 5 & 1\\ 1 & 5 & 1 & 5 & 1 & 5 & 1 & 1& 1 & 1& 5 & 1 \\ \end{array} \right] \end{equation*} $ |
is an invertible matrix. Choose
$ \begin{equation*} W=\left[ {\begin{array}{*{20}{c}} 0.5020& 0.2342 & 1.1002 & 0.0348 \\ -0.6667& 0.6667 &-0.8889 &0.4444 \\ 0.8354 &-0.0992 & 1.9891 &-0.4097 \\ -0.3333 & 0.3333 &-0.6667 & 0.6667 \\ 5.0000 &-5.0000 &10.0000 & 0.0000 \\ 5.0000 &-5.0000 &10.0000 &0.0000 \\ -12.5307 & 1.4877& -6.5031 &-0.5215 \\ -7.5307 &-3.5123&-16.5031& -0.5215 \\ 7.5307 & 3.5123& 13.1697 &-2.8119 \\ -10.0000 &10.0000&-13.3333 & 6.6667 \\ 0.0000 &-0.0000& -6.6667 & 3.3333 \\ 7.5307 & 3.5123&19.8364 &-6.1452 \\ \end{array}} \right.\\ \begin{array}{*{20}{c}} 0.1227& -0.3640 & 0.9202 & 0.5215 \\ -0.2222 & 0.4444& -0.2222 & 0 \\ 0.3449& -0.1418 & 1.1425 & 0.5215 \\ -0.0000& 0.0000& 0& 0.0000 \\ 0.0000 & 0.0000 & 10.0000 & 0 \\ 0.0000& -10.0000& 10.0000 & 0\\ -1.8405 & 5.4601& -3.8037& 12.1779 \\ 8.1595 & 5.4601 &-13.8037 & 2.1779 \\ -1.4928& -8.7935& 10.4703& -2.1779\\ -3.3333 & 6.6667 & -3.3333 & 0.0000\\ 3.3333 & 3.3333 & -6.6667& -0.0000\\ -4.8262 &-12.1268 & 17.1370 & -2.1779 \\ \end{array}\\ \left. {\begin{array}{*{20}{c}} 0.1227 & 0.9202& 0.2342& 0.7209\\ 0.0000 &-0.4444& 0.4444& -0.0000\\ 0.7894 & 1.3647& 0.1230& 1.0542\\ -0.6667& -0.0000& 0.3333& -0.3333\\ 0.0000& 10.0000& -5.0000& 5.0000\\ 0.0000& 10.0000& 5.0000& 5.0000\\ -1.8405& -3.8037& 1.4877& -5.8129\\ 8.1595&-13.8037& -3.5123&-10.8129\\ 1.8405& 7.1370& 0.1789& 10.8129\\ 0.0000& 3.3333& -3.3333& -0.0000\\ 0.0000& -3.3333& 3.3333& 0.0000\\ -8.1595 &10.4703& 6.8456& 10.8129\\ \end{array}} \right] \end{equation*} $ |
where
In this paper, we have introduced MRNN which is a family of recurrent neural networks. Some sufficient conditions are derived to assure its mono-stability and multi-stability. In the existing literature on neural network, the largest number of equilibrium points is
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