Multistability and Instability of Competitive Neural Networks with Mexican-Hat-Type Activation Functions

Abstract

We investigate the existence and dynamical behaviors of multiple equilibria for competitive neural networks with a class of general Mexican-hat-type activation functions. The Mexican-hat-type activation functions are not monotonously increasing, and the structure of neural networks with Mexican-hat-type activation functions is totally different from those with sigmoidal activation functions or nondecreasing saturated activation functions, which have been employed extensively in previous multistability papers. By tracking the dynamics of each state component and applying fixed point theorem and analysis method, some sufficient conditions are presented to study the multistability and instability, including the total number of equilibria, their locations, and local stability and instability. The obtained results extend and improve the very recent works. Two illustrative examples with their simulations are given to verify the theoretical analysis.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Xiaobing Nie 1,2 and Jinde Cao 1,3 and Shumin Fei 2

Academic Editor:Weinian Zhang

1, Department of Mathematics, and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210096, China
2, School of Automation, Southeast University, Nanjing 210096, China
3, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 2 January 2014; Accepted 9 April 2014; 6 May 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the past decades, some famous neural network models, including Hopfield neural networks, cellular neural networks, Cohen-Grossberg neural networks, and bidirectional associative memory neural networks, had been proposed in order to solve some practical problems. It should be mentioned that in the above network models only the neuron activity is taken into consideration. That is, there exists only one type of variables, the state variables of the neurons in these models. However, in a dynamical network, the synaptic weights also vary with respect to time due to the learning process, and the variation of connection weights may have influences on the dynamics of neural network. Competitive neural networks (CNNs) constitute an important class of neural networks, which model the dynamics of cortical cognitive maps with unsupervised synaptic modifications. In this model, there are two types of state variables: that of the short-term memory (STM) describing the fast neural activity and that of long-term memory (LTM) describing the slow unsupervised synaptic modifications. The CNNs can be written in the following form: [figure omitted; refer to PDF] where i=1,2,...,N , j=1,2,...,P,_xi (t) is the neuron current activity level, _fj (_xj (t)) is the output of neurons, _mij (t) is the synaptic efficiency, _yj is the constant external stimulus, _aij represents the connection weight between the i th neuron and the j th neuron, _Bi is the strength of the external stimulus, _Ii is the constant input, and _αi >0 and _βi ...5;0 denote disposable scaling constants.

After setting _Si (t)=^∑j=1P_mij (t)_yj =^yT_mi (t) , where y =^{(_y1 ,_y2 ,...,_yP )T} and _mi (t)=^{(_mi1 (t),_mi2 (t),...,_miP (t))T} and assuming the input stimulus y to be normalized with unit magnitude ^|y|2 =^y12 +...+^yP2 =1 , then the above networks are simplified as [figure omitted; refer to PDF]

The qualitative analysis of neural dynamics plays an important role in the design of practical neural networks. To solve problems of optimization and signal processing, neural networks have to be designed in such a way that, for a given external input, they exhibit only one globally stable state (i.e., monostability). This matter has been treated in [1-7]. On the other hand, if neural networks are used to analyze associative memories, the coexistence of multiple locally stable equilibria or periodic orbits is required (i.e., multistability or multiperiodicity), since the addressable memories or patterns are stored as stable equilibria or stable periodic orbits. In monostability analysis, the objective is to derive conditions that guarantee that each network contains only one steady state, and all the trajectories of the network converge to it, whereas in multistability analysis, the networks are allowed to have multiple equilibria or periodic orbits (stable or unstable). In general, the usual global stability conditions are not adequately applicable to multistable networks.

Recently, the multistability or multiperiodicity of neural networks has attracted the attention of many researchers. In [8, 9], based on decomposition of state space, the authors investigated the multistability of delayed Hopfield neural networks and showed that the n -neuron neural networks can have ²ⁿ stable orbits located in ²ⁿ subsets of ^...n . Cao et al. [10] extended the above method to the Cohen-Grossberg neural networks with nondecreasing saturated activation functions with two corner points. In [11, 12], the multistability of almost-periodic solution in delayed neural networks was studied. Kaslik and Sivasundaram [13, 14] firstly revealed the effect of impulse on the multistability of neural networks. In [15-17], high-order synaptic connectivity was introduced into neural networks and the multistability and multiperiodicity were considered, respectively, for high-order neural networks based on decomposition of state space, Cauchy convergence principle, and inequality technique. In [18-22], the authors indicated that under some conditions, there exist ³ⁿ equilibria for the n -neuron neural networks and ²ⁿ of which are locally exponentially stable. In [23], the Hopfield neural networks with nondecreasing piecewise linear activation functions with 2r corner points were considered. It was proved that under some conditions, the n -neuron neural networks can have and only have ^(2r+1)n equilibria, ^(r+1)n of which are locally exponentially stable and others are unstable. In [24], the multistability of neural networks with k+m step stair activation functions was discussed based on an appropriate partition of the n -dimensional state space. It was shown that the n -neuron neural networks can have ^(2k+2m-1)n equilibria, ^(k+m)n of which are locally exponentially stable. In particular, the case of k=m was previously discussed in [25]. For more references, see [26-32] and references therein.

It is well known that the type of activation functions plays a very important role in the multistability analysis of neural networks. In the abovementioned and most existing works, the activation functions employed in multistability analysis were mainly focused on sigmoidal activation functions and nondecreasing saturated activation functions, which are all monotonously increasing. In this paper, we will consider a class of continuous Mexican-hat-type activation functions, which are defined as follows (see Figure 1): [figure omitted; refer to PDF] where _pi , _ri , _qi , _ui , _li,1 , _li,2 , _ci,1 , _ci,2 are constants with -∞<_pi <_ri <_qi <+∞ , _li,1 >0 , and _li,2 <0 , i=1,2,...,N . In particular, when _pi =-1 , _ri =1 , _qi =3 , _ui =-1 , _li,1 =1 , _li,2 =-1 , _ci,1 =0 , and _ci,2 =2 (i=1,2,...,N) , the above activation functions _fi reduce to the following special activation functions employed in [33]: [figure omitted; refer to PDF]

Figure 1: Mexican-hat-type activation functions (3).

[figure omitted; refer to PDF]

It is necessary to point out that the Mexican-hat-type activation functions are not monotonously increasing, which are totally different from sigmoidal activation functions and nondecreasing saturated activation functions. Hence, the results and methods mentioned above cannot be applied to neural networks with activation functions (3). Very recently, the multistability and instability of Hopfield neural networks with activation functions (4) were studied in [33]. Inspired by [33], in this paper, we will investigate the multistability and instability of CNNs with activation functions (3). It should be noted that the structure of CNNs differs from and is more complex than that in [33]. Moreover, the activation functions (3) employed in this paper are more general than activation functions (4). More precisely, the contributions of this paper are three-fold as follows.

Firstly, we define four index subsets and present sufficient condition under which the CNNs with activation functions (3) have multiple equilibria, by tracking the dynamics of each state component and applying fixed point theorem. The index subsets are defined in terms of maximum and minimum values, which are different from and less restrictive than those given in [33]. Furthermore, we discuss the exact existence of equilibria for CNNs.

Secondly, based on some analysis method, we analyze the dynamical behaviors of each equilibrium point for CNNs, including local stability and instability. The dynamical behaviors of such system are much more complex than those of Hopfield neural networks considered in [33], due to the complexity of the networks structure and generality of activation functions.

Thirdly, specializing the model and activation functions to those in [33], we show that the obtained results extend and improve the very recent works in [33].

Finally, two examples with their simulations are given to verify and illustrate the validity of the obtained results.

2. Main Results

Firstly, we define the four index subsets as follows: [figure omitted; refer to PDF] where _vi =_fi (_ri ) . It is easy to see that _ui ...4;_fi (x)...4;_vi for x∈... .

Remark 1.

In this paper, the index subsets are defined in terms of maximum and minimum values, which are different from those given in [33], where they are defined in terms of absolute values. In general, our conditions are less restrictive, which have been shown in [16].

Remark 2.

The inequality _di -(_aii +^αi-1_βiBi )_li,1 <0 holds for all i∈_...2 .

Proof.

By the definition of index subset _...2 , _ui =_fi (_pi ) and _vi =_fi (_ri ) , we obtain [figure omitted; refer to PDF] It follows from (6) and (7) that [figure omitted; refer to PDF] Noting that _pi <_ri and substituting _fi (_pi )=_li,1pi +_ci,1 and _fi (_ri )=_li,1ri +_ci,1 into (8), we can derive that _di -(_aii +^αi-1_βiBi )_li,1 <0 (i∈_...2 ) .

Remark 3.

The inequality _di -(_aii +^αi-1_βiBi )_li,2 >0 holds for all i∈_...2 ∪_...3 .

Proof.

From Remark 2, we get _aii +^αi-1_βiBi >0 (i∈_...2 ) . Thus, inequality _di -(_aii +^αi-1_βiBi )_li,2 >0 holds for all i∈_...2 , due to _li,2 <0 .

By the definition of index subset _...3 and equalities _ui =_fi (_qi ) , _vi =_fi (_ri ) , we get [figure omitted; refer to PDF] which implies that -_diri +(_aii +^αi-1_βiBi )_fi (_ri )>-_diqi +(_aii +^αi-1_βiBi )_fi (_qi ) . By using equalities _fi (_ri )=_li,2ri +_ci,2 , _fi (_qi )=_li,2qi +_ci,2 and noting that _ri <_qi , the inequality _di -(_aii +^αi-1_βiBi )_li,2 >0 (i∈_...3 ) can be proved easily.

It follows from the second equation of system (2) that [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] Therefore, _Si (0)∈[^αi-1_βiui ,^αi-1_βivi ] always implies that _Si (t)∈[^αi-1_βiui ,^αi-1_βivi ] . That is, if S(0)∈^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] , then the solution S(t;S(0)) will stay in ^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] for all t...5;0 .

Let ^{(xT (t),ST (t))T} be a solution of system (2) with initial state ^{(xT (0),ST (0))T} ∈^...N ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] . In the following, we will discuss the dynamics of state components _xi (t) for i∈_...i (i=1,2,3,4) , respectively.

Lemma 4.

All the state components _xi (t) , i∈_...1 , will flow to the interval (-∞,_pi ] when t tends to +∞ .

Proof.

According to the different location of _xi (0) , there are two cases for us to discuss.

Case (i). Consider _xi (0)∈(-∞,_pi ] . In this case, if there exists some ^t* ...5;0 such that _xi (^t* )=_pi , _xi (t)...4;_pi for 0...4;t...4;^t* , then it follows from system (2) and the definition of _...1 that [figure omitted; refer to PDF] Hence, _xi (t) would never get out of (-∞,_pi ] . Similarly, we can also conclude that once _xi (_T0 )∈(-∞,_pi ] for some _T0 >0 , then _xi (t) would never escape from (-∞,_pi ] for all t...5;_T0 .

Case (ii). Consider _xi (0)∈(_pi ,+∞) . In this case, we claim that _xi (t) would monotonously decrease until it reaches the interval (-∞,_pi ] in some finite time.

In fact, when _xi (t)∈(_qi ,+∞) , noting that the definition of _...1 and _fi (_pi )=_fi (_qi ) , we obtain [figure omitted; refer to PDF] when _xi (t)∈(_ri ,_qi ] , by virtue of equalities _fi (_ri )=_li,2ri +_ci,2 and _fi (_qi )=_li,2qi +_ci,2 , we get [figure omitted; refer to PDF] when _xi (t)∈(_pi ,_ri ] , it follows from _fi (_pi )=_li,1pi +_ci,1 ,_fi (_ri )=_li,1ri +_ci,1 and system (2) that [figure omitted; refer to PDF]

In summary, wherever the initial state _xi (0) (i∈_...1 ) is located in, _xi (t) would flow to and enter the interval (-∞,_pi ] and stay in this interval forever.

Lemma 5.

All the state components _xi (t) , i∈_...3 , will flow to the interval [_ri ,_qi ] when t tends to +∞ .

Proof.

We prove it in the following three cases due to the different location of _xi (0) .

Case (i). Consider _xi (0)∈[_ri ,_qi ] . In this case, if there exists some ^t* ...5;0 such that _xi (^t* )=_ri , _ri ...4;_xi (t)...4;_qi for 0...4;t...4;^t* , then we have [figure omitted; refer to PDF] similarly, if there exists some ^t** ...5;0 such that _xi (^t** )=_qi , _ri ...4;_xi (t)...4;_qi for 0...4;t...4;^t** , then we get [figure omitted; refer to PDF]

From the above two inequalities, we know that if _xi (0)∈[_ri ,_qi ] , then _xi (t) would never get out of this interval. In the same way, we can also obtain that if there exists some _T0 >0 such that _xi (_T0 )∈[_ri ,_qi ] , then _xi (t) would stay in it for all t...5;_T0 .

Case (ii). Consider _xi (0)∈(-∞,_ri ) . When _xi (t)∈(-∞,_pi ] , from the definition of index subset _...3 , we get [figure omitted; refer to PDF] when _xi (t)∈(_pi ,_ri ) , it follows from the definition of index subset _...3 , equalities _fi (_pi )=_li,1pi +_ci,1 , _fi (_ri )=_li,1ri +_ci,1 that [figure omitted; refer to PDF] Thus, in this case, _xi (t) would monotonously increase until it reaches the interval [_ri ,_qi ] .

Case (iii). Consider _xi (0)∈(_qi ,+∞) . When _xi (t)∈(_qi ,+∞) , it follows that [figure omitted; refer to PDF] Therefore, _xi (t) would monotonously decrease until it enters the interval [_ri ,_qi ] .

In summary, wherever the initial state _xi (0) (i∈_...3 ) is located in, _xi (t) would flow to and enter the interval [_ri ,_qi ] and stay in it finally.

Lemma 6.

All the state components _xi (t) , i∈_...4 , will flow to the interval [_qi ,+∞) when t tends to +∞ .

Proof.

Similar to the proof of Lemmas 4 and 5, we will prove it in the following two cases.

Case (i). Consider _xi (0)∈[_qi ,+∞) . If there exists some ^t* ...5;0 such that _xi (^t* )=_qi , _xi (t)...5;_qi for 0...4;t...4;^t* , then [figure omitted; refer to PDF] Therefore, _xi (t) would never get out of [_qi ,+∞) . By the same method, we also get that once _xi (_T0 )∈[_qi ,+∞) for some _T0 >0 , then _xi (t) would stay in this interval for all t...5;_T0 .

Case (ii). Consider _xi (0)∈(-∞,_qi ) . In this case, we claim that _xi (t) would monotonously increase until it enters the interval [_qi ,+∞) .

In fact, when _xi (t)∈(-∞,_pi ) , we have [figure omitted; refer to PDF] when _xi (t)∈[_pi ,_ri ] , we get [figure omitted; refer to PDF] when _xi (t)∈(_ri ,_qi ) , then we obtain [figure omitted; refer to PDF]

In summary, wherever the initial state _xi (0) (i∈_...4 ) is located in, _xi (t) would flow to [_qi ,+∞) when t is big enough and stay in it forever.

Denote (-∞,_pi ]=^{(-∞,_pi ]1} ×^{[_pi ,_ri ]0} ×^{[_ri ,+∞)0} ; [_pi ,_ri ]=^{(-∞,_pi ]0} ×^{[_pi ,_ri ]1} ×^{[_ri ,+∞)0} ; [_ri ,+∞)=^{(-∞,_pi ]0} ×^{[_pi ,_ri ]0} ×^{[_ri ,+∞)1} . For any i∈_...2 , let _δi =(^δi(1) ,^δi(2) ,^δi(3) )=(1,0,0) or (0,1,0) or (0,0,1) and define [figure omitted; refer to PDF]

It is easy to see that there exist 3#...2 Ωδ -type regions, where #...2 denotes the number of elements in the set ...2 . Now, we will prove the following theorem on the existence of multiple equilibria for system (2).

Theorem 7.

Suppose that _...1 ∪_...2 ∪_...3 ∪_...4 ={1,2,...,N} . Then, system (2) with activation functions (3) has ^3#_...2 equilibria.

Proof.

Pick a region arbitrarily _Ωδ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] ; we will show that there exists an equilibrium point located in each _Ωδ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] .

Denote _...2,1 ={i∈_...2 :^δi(1) =1} , _...2,2 ={i∈_...2 :^δi(2) =1} , and _...2,3 ={i∈_...2 :^δi(3) =1} . It is easy to see that _...2,1 ∪_...2,2 ∪_...2,3 =_...2 .

Let ^{(xT (t),ST (t))T} be any solution of system (2) with initial state ^{(xT (0),ST (0))T} ∈_Ωδ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] . Then, for i∈_...2,1 , if there exists some ^t* ...5;0 such that _xi (^t* )=_pi , then we get from the definition of index subset _...2 that [figure omitted; refer to PDF] Similarly, for i∈_...2,3 , if _xi (t)...5;_qi , note that _fi (_pi )=_fi (_qi ) ; then [figure omitted; refer to PDF] and if there exists some ^t** ...5;0 such that _xi (^t** )=_ri , then [figure omitted; refer to PDF] That is, the trajectory _xi (t) with _xi (0)∈[_ri ,+∞),i∈_...2,3 , would enter and stay in the interval [_ri ,_qi ] , which implies that there does not exist any equilibria with the corresponding i th state component located in [_qi ,+∞) .

Combining with Lemmas 4-6, it can be concluded that _xi (t),i∉_...2,2 , would never escape from the corresponding interval of _Ωδ . Furthermore, denote [figure omitted; refer to PDF] Then, from (26)-(28), we can derive that if _Ωδ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] has an equilibrium point, it must be located in _Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] .

Note that any equilibrium point of system (2) is a root of the following equations: [figure omitted; refer to PDF] Equivalently, the above equations can be rewritten as [figure omitted; refer to PDF]

In _Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] , any equilibrium point of system (2) satisfies the following equations: [figure omitted; refer to PDF]

In the subset region _{∏i∈...2,2} [_pi ,_ri ]×_{∏i∈...2,3 ∪...3} [_ri ,_qi ] , define a map Γ as follows: [figure omitted; refer to PDF] For i∈_...2,2 , substituting _fi (_pi )=_li,1pi +_ci,1 into (6) and noting that _di -(_aii +^αi-1_βiBi )_li,1 <0 (Remark 2), we get that [figure omitted; refer to PDF] Similarly, substituting _fi (_ri )=_li,1ri +_ci,1 into (7) results in [figure omitted; refer to PDF] For i∈_...2,3 , note that -_dipi >-_diqi , _fi (_pi )=_fi (_qi )=_li,2qi +_ci,2 , and _fi (_ri )=_li,2ri +_ci,2 ; it follows from (6)-(7) and _di -(_aii +^αi-1_βiBi )_li,2 >0 (Remark 3) that [figure omitted; refer to PDF] For i∈_...3 , note that _vi =_fi (_ri )=_li,2ri +_ci,2 and _ui =_fi (_qi )=_li,2qi +_ci,2 ; we can get from the definition of index subset _...3 that [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] Then combining inequality _di -(_aii +^αi-1_βiBi )_li,2 >0 and (38) together gives [figure omitted; refer to PDF] Therefore, the map Γ maps a bounded and closed set into itself. Applying Brouwer's fixed point theorem, there exists one ^{(x_i1 * ,x_i2 * ,...,x_is * )T} ∈_{∏i∈...2,2} [_pi ,_ri ]×_{∏i∈...2,3 ∪...3} [_ri ,_qi ] such that [figure omitted; refer to PDF] where _i1 ,_i2 ,...,_is represent the elements of index subset _...2,2 ∪_...2,3 ∪_...3 .

Then for i∈_...1 ∪_...2,1 , define [figure omitted; refer to PDF] By virtue of the definition of index subsets _...1 and _...2 , we have _{lim...ξ[arrow right]-∞} ..._Fi (ξ)=+∞ and [figure omitted; refer to PDF] Therefore, there exists the unique ^xi* ∈(-∞,_pi ) such that _Fi (^xi* )=0 .

Similarly, for i∈_...4 , define [figure omitted; refer to PDF] and we can also derive that there exists the unique ^xi* ∈(_qi ,+∞) such that _Fi (^xi* )=0 .

Denote ^{(x* T ,S* T )T} =(^x1* ,^x2* ,...,^xN* , _β1^α1-1_f1 (^x1* ) , _β2^α2-1_f2 (^x2* ),...,_βN^αN-1_fN (^xN* )^)T . It is easy to see that ^{(x* T ,S* T )T} is the equilibrium point located in subset _Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] , which is also the equilibrium point located in subset _Ωδ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] . It should be noted that, by the definition of _...1 -_...4 , any state component of ^x* cannot touch the boundary of _Ωδ . That is, ^x* is located in the interior of _Ωδ (see Remark 8). Therefore, system (2) has ^3#_...2 equilibria.

Remark 8.

^{x *} is located in the interior of _Ωδ .

Proof.

Note that ^x* satisfies the following equations: [figure omitted; refer to PDF] In the following, we prove that ^xi* ...0;_pi , i∈_...1 . Otherwise, from the definition of _...1 , we have [figure omitted; refer to PDF] that is a contradiction. Similarly, from the definition of _...2 -_...4 , we can also get [figure omitted; refer to PDF] That is, ^x* is located in the interior of _Ωδ .

Remark 9.

Suppose that _...1 ∪_...2 ∪_...3 ∪_...4 ={1,2,...,N} . Furthermore, if the following conditions [figure omitted; refer to PDF] hold, then system (2) with activation functions (3) can have and only have ^3#_...2 equilibria.

Proof.

From the proof of Theorem 7, we only need to prove that the fixed point of Γ is unique. In fact, suppose that there exists another fixed point ^{(x_i1 ** ,x_i2 ** ,...,x_is ** )T} ∈_{∏i∈...2,2} [_pi ,_ri ]×_{∏i∈...2,3 ∪...3} [_ri ,_qi ] such that [figure omitted; refer to PDF] Denote the index _il such that [figure omitted; refer to PDF] If _il ∈_...2,2 , from (_ailil +^{α_il -1}_βilBil )_{lil ,1} -_dil >0 (Remark 2) and (47), we have [figure omitted; refer to PDF] If _il ∈_...2,3 ∪_...3 , from _dil -(_ailil +^{α_il -1}_βilBil )_{lil ,2} >0 (Remark 3) and (48), we have [figure omitted; refer to PDF] By the above two inequalities, we can deduce that ^xi* =^xi** for all i∈_...2,2 ∪_...2,3 ∪_...3 . That is, the fixed point of map Γ is unique.

Theorem 10.

Suppose that _...1 ∪_...2 ∪_...3 ∪_...4 ={1,2,...,N} .

(i) If the following conditions [figure omitted; refer to PDF]

hold for all i∈_...2,3 ∪_...3 , then system (2) with activation functions (3) has ^2#_...2 locally stable equilibria.

(ii) Furthermore, if the following conditions [figure omitted; refer to PDF]

hold for all i∈_...2,2 , then the other ^3#_...2 -^2#_...2 equilibria are unstable.

Proof.

In the following, we will discuss the dynamical behaviors of ^3#_...2 equilibria in two cases, respectively.

Case (i) . Consider _...2,2 =∅ . In this case, _Ωδ and _Ω~δ can be rewritten as [figure omitted; refer to PDF]

From (53)-(54), we can choose a sufficiently small number η>0 such that [figure omitted; refer to PDF] hold for all i∈_...2,3 ∪_...3 .

Let ^{(xT (t),ST (t))T} be any a solution of system (2) with initial state ^{(xT (0),ST (0))T} ∈_Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] . From Lemmas 4-6 and the proof of Theorem 7, we get that there exists some _T0 ...5;0 such that ^{(xT (_T0 ),ST (_T0 ))T} ∈_Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] . By the positive invariance of _Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] , we know that the solution ^{(xT (t),ST (t))T} ∈_Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] for all t...5;_T0 , and its dynamics can be described by [figure omitted; refer to PDF] We get from model (60) that [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] then by using (61)-(62), we can derive that [figure omitted; refer to PDF] Similarly, we have [figure omitted; refer to PDF]

Let σ>0 be an arbitrary real number and [figure omitted; refer to PDF] then [figure omitted; refer to PDF] In the following, we will prove that [figure omitted; refer to PDF] If (67) is not true, then there exists some i∈_...2,3 ∪_...3 and ^t* >_T0 such that either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] For the first case, it follows from (58) and (63) that [figure omitted; refer to PDF] that is a contradiction. For the second case, by using (59) and (64), we get [figure omitted; refer to PDF] that is also a contradiction. So (67) holds; that is, |_xi (t)-^xi* |...4;_l0^e-ηt , |_Si (t)-^Si* |...4;_l0^e-ηt hold for all t...5;_T0 and i∈_...2,3 ∪_...3 .

From (61), we derive that [figure omitted; refer to PDF] which implies that _{lim...t[arrow right]+∞} ...|_Si (t)-^Si* |=0 (i∈_...1 ∪_...2,1 ∪_...4 ) . Thus, for any [varepsilon]>0 , there exists a constant _T1 ...5;_T0 such that [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] That is, _{lim...t[arrow right]+∞} ...|_xi (t)-^xi* |=0 (i∈_...1 ∪_...2,1 ∪_...4 ) .

To sum up, for all i=1,2,...,N , we have _{lim...t[arrow right]+∞} ...|_xi (t)-^xi* |=0 , _{lim...t[arrow right]+∞} ...|_Si (t)-^Si* |=0 . That is, the equilibrium point ^{(x* T ,S* T )T} is locally stable in _Ωδ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] .

Case (ii) . Consider _...2,2 ...0;∅ . In this case, let ^{(xT (t),ST (t))T} be a solution of system (2) with initial condition ^{(xT (0),ST (0))T} nearby ^{(x* T ,S* T )T} ∈_Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] . Without loss of generality, we assume that ^{(xT (t),ST (t))T} ∈_Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] for all t...5;0 . In fact, we can conclude that if the solution ^{(xT (t),ST (t))T} gets out of region _Ω~δ ×^∏i=1N [^αi-1_βiui ,^αi-1_βivi ] , the equilibrium point ^{(x* T ,S* T )T} must be unstable according to the definition of instability. Thus, system (2) can be rewritten as [figure omitted; refer to PDF] It follows from model (76) that [figure omitted; refer to PDF]

Pick an initial state ^{(xT (0),ST (0))T} of system (2) such that [figure omitted; refer to PDF] In the following, we claim that [figure omitted; refer to PDF] holds for all t...5;0 . In fact, if (79) is not true, then there exists some _t0 ...5;0 such that [figure omitted; refer to PDF] then there exists some index _i0 such that one of the following three cases holds:

(a) _i0 ∈_...2,3 ∪_...3 and |_xi0 (_t0 )-^{x_i0 *} |=_{max...i∈...2,2} ...|_xi (_t0 )-^xi* | ,

(b) _i0 ∈_...2,3 ∪_...3 and |_Si0 (_t0 )-^{S_i0 *} |=_{max...i∈...2,2} ...|_xi (_t0 )-^xi* | ,

(c) _i0 ∈_...2,2 and |_Si0 (_t0 )-^{S_i0 *} |=_{max...i∈...2,2} ...|_xi (_t0 )-^xi* | .

For the case (a), it follows from (53) and (77) that [figure omitted; refer to PDF] For the case (b), by using (54) and (77), we get that [figure omitted; refer to PDF] Similarly, for the case (c), by means of (56) and (77), we have [figure omitted; refer to PDF]

Meanwhile, denote _i1 ∈_...2,2 such that |_xi1 (_t0 )-^{x_i1 *} |=_{max...i∈...2,2} ...|_xi (_t0 )-^xi* | ; then we obtain from (55) and (77) that [figure omitted; refer to PDF] Thus, combining inequalities (79) and (84) together gives [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] holds for all t...5;0 .

Therefore, there must exist an index _i2 ∈_...2,2 and an increasing time sequence {_tk^}k=1∞ with _{lim...t[arrow right]+∞} ..._tk =+∞ such that |_xi2 (_tk )-^{x_i2 *} |...5;_{max...i∈...2,2} ...|_xi (_t0 )-^xi* | . That is, _xi2 (t) would not converge to ^{x_i2 *} when t tends to +∞ . In other words, the equilibrium point ^x* is unstable.

In the following, we will consider the dynamical behaviors of system (2) with activation functions (4). Note that max...{-_aij ,_aij }=|_aij | and min...{-_aij ,_aij }=-|_aij | ; in this case, _...1 -_...4 reduce to the following index subsets: [figure omitted; refer to PDF] Meanwhile, conditions (47)-(48) become [figure omitted; refer to PDF]

Remark 11.

Under index subsets _...¯2 -_...¯3 , conditions (88)-(89) always hold.

Proof.

From the definition of index subset _...¯2 , we get [figure omitted; refer to PDF] Therefore, inequalities (88) and (89) hold for i∈_...2,2 and i∈_...2,3 , respectively.

Similarly, it follows from the definition of _...¯3 that [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] Therefore, condition (89) holds for i∈_...3 .

From Theorem 10 and Remark 11, we can obtain Corollary 12 as follows.

Corollary 12.

Suppose that _...¯1 ∪_...¯2 ∪_...¯3 ∪_...¯4 ={1,2,...,N} . If the following conditions [figure omitted; refer to PDF] hold, then system (2) with activation functions (4) has exactly ^3#_...¯2 equilibria, ^2#_...¯2 of them are locally stable and others are unstable.

Proof.

First of all, according to Theorem 7 and Remark 11, the exact existence of ^3#_...¯2 equilibria for system (2) with activation functions (4) can be guaranteed under condition _...¯1 ∪_...¯2 ∪_...¯3 ∪_...¯4 ={1,2,...,N} . In addition, it is easy to see that conditions (93)-(95) imply the conditions (53)-(56) with _li,1 =1 , _li,2 =-1 . From Theorem 10, we derive the result of Corollary 12.

Let _Bi =_βi =_Si (t)=0 (i=1,2,...,N) ; then system (2) is transformed into the following neural networks which are investigated in [33]: [figure omitted; refer to PDF] In this case, _...¯1 -_...¯4 turn into the following index subsets defined in [33]: [figure omitted; refer to PDF] Applying Corollary 12, we can obtain easily Corollary 13 as follows.

Corollary 13.

Suppose that _...~1 ∪_...~2 ∪_...~3 ∪_...~4 ={1,2,...,N} holds. Then system (96) with activation functions (4) has exactly ^3#_...~2 equilibria, ^2#_...~2 of them are locally stable and others are unstable.

Proof.

First of all, it follows from the definition of index subset _...~2 that (93)-(94) with _Bi =0 hold for i∈_...~2,2 and i∈_...~2,3 , respectively. From the definition of _...~3 , we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] that is, inequality (94) with _Bi =0 holds for i∈_...~3 . Inequality (95) is obvious, due to _αi >0 and _βi =0 . From Corollary 12, we derive the result of Corollary 13.

Remark 14.

In this paper, we study the multistability and instability of CNNs with activation functions (3). The models are different from and more general than those in [33], and the considered activation functions (3) are also more general than those employed in [33]. Moreover, the index subsets defined in this paper are less restrictive than those defined in [33].

Remark 15.

Compared with the results reported in [33], it can be seen that Corollary 13 above is consistent with Theorem 1 in [33]. That is, if we specialize the system and activation functions in Theorem 10 to those considered in [33], we can obtain the main result in [33]. Therefore, Theorem 10 extends and improves the main result in [33].

3. Two Illustrative Examples

For convenience, we consider the following two-dimensional CNNs: [figure omitted; refer to PDF]

Example 1.

For system (100), take _d1 =_d2 =1 , _a11 =_a22 =2 , _a12 =0.5 , _a21 =-0.5 , _B1 =0.4 , _B2 =-0.4 , _I1 =_I2 =0 , _α1 =_α2 =2 , _β1 =_β2 =1 , and [figure omitted; refer to PDF] It is easy to see that _...2 ={1,2} . In addition, by simple computations, we have [figure omitted; refer to PDF] Therefore, the conditions in Corollary 12 hold. According to Corollary 12, system (100) has exactly ³² =9 equilibria, ²² =4 equilibria are locally stable and others are unstable. In fact, by direct computations, we can obtain the nine equilibria ^{(-2.7,-1.3,-0.5,-0.5)T} , ^{(-201/80,-5/8,-1/2,-5/16)T} , ^{(-541/280,41/28,-1/2,15/56) T} , (5...12,-241...120,5...24 , ^-1...2T )^(0,0,0,0)T , (-100...361,482...361 , ^{-50...361,120...361)T} , (39...32,-701...320,25...64 , ^-1...2)T , ^{(402/281,100/281,80/281,50/281)T} , and ^{(1385/921,1102/921,457/1842,185/921)T} . From Figures 2, 3, 4, 5, and 6, it can be seen that the four equilibria ^{(-2.7,-1.3,-0.5,-0.5)T} , ^{(-541/280,41/28,-1/2,15/56)T} , ^{(39/32,-701/320,25/64,-1/2)T} , and ^{(1385/921,1102/921,457/1842,185/921)T} are locally stable. Figures 7, 8, 9, 10, and 11 confirm that the others are unstable.

Figure 2: Transient behavior of _x1 in Example 1.

[figure omitted; refer to PDF]

Figure 3: Transient behavior of _x2 in Example 1.

[figure omitted; refer to PDF]

Figure 4: Transient behavior of _S1 and _S2 in Example 1.

[figure omitted; refer to PDF]

Figure 5: Phase plot of state variable ^{(_x1 ,_x2 ,_S1 )T} in Example 1.

[figure omitted; refer to PDF]

Figure 6: Phase plot of state variable ^{(_x1 ,_x2 ,_S2 )T} in Example 1.

[figure omitted; refer to PDF]

Figure 7: Transient behavior of _x1 and _x2 near the equilibrium point ^{(-201/80,-5/8,-1/2,-5/16)T} in Example 1.

[figure omitted; refer to PDF]

Figure 8: Transient behavior of _x1 and _x2 near the equilibrium point ^{(5/12,-241/120,5/24,-1/2)T} in Example 1.

[figure omitted; refer to PDF]

Figure 9: Transient behavior of _x1 and _x2 near the equilibrium point ^(0,0,0,0)T in Example 1.

[figure omitted; refer to PDF]

Figure 10: Transient behavior of _x1 and _x2 near the equilibrium point ^{(-100/361,482/361,-50/361,120/361)T} in Example 1.

[figure omitted; refer to PDF]

Figure 11: Transient behavior of _x1 and _x2 near the equilibrium point ^{(402/281,100/281,80/281,50/281)T} in Example 1.

[figure omitted; refer to PDF]

Example 2.

For system (100), take _d1 =1 , _a11 =-1 , _a12 =-0.2 , _B1 =0.2 , _d2 =_a22 =2 , _a21 =1/3 , _B2 =-0.2 , _I1 =9 , _I2 =0 , _α1 =_α2 =3 , _β1 =_β2 =1 , and [figure omitted; refer to PDF] It is easy to see that _...2 ={2} , _...4 ={1} . Herein, the parameters satisfy conditions in Remark 9 and Theorem 10: [figure omitted; refer to PDF] From Remark 9 and Theorem 10, it follows that the system has exactly three equilibria ^{(62/5,-17/5,-1,-1)T} , ^{(59/5,-1/2,-1,0)T} , and ^{(3256/295,188/59,-1,75/59)T} ; the first and the third equilibria are locally stable, while the second equilibrium point is unstable. From Figures 12, 13, 14, 15, and 16, it can be seen that the two equilibria ^{(62/5,-17/5,-1,-1)T} and ^{(3256/295,188/59,-1,75/59)T} are locally stable. Figure 17 confirms that the equilibrium point ^{(59/5,-1/2,-1,0)T} is unstable.

Figure 12: Transient behavior of _x1 in Example 2.

[figure omitted; refer to PDF]

Figure 13: Transient behavior of _x2 in Example 2.

[figure omitted; refer to PDF]

Figure 14: Transient behavior of _S1 and _S2 in Example 2.

[figure omitted; refer to PDF]

Figure 15: Phase plot of state variable ^{(_x1 ,_x2 ,_S1 )T} in Example 2.

[figure omitted; refer to PDF]

Figure 16: Phase plot of state variable ^{(_x1 ,_x2 ,_S2 )T} in Example 2.

[figure omitted; refer to PDF]

Figure 17: Transient behavior of _x1 and _x2 near the equilibrium point ^{(59/5,-1/2,-1,0)T} in Example 2.

[figure omitted; refer to PDF]

4. Conclusions

In this paper, the multistability and instability issues have been studied for CNNs with Mexican-hat-type activation functions. We showed that under some conditions, the system has ^3#_...2 equilibria, ^2#_...2 of them are locally stable and others are unstable. Two examples with their computer simulations were given to illustrate the effectiveness of the obtained results. Some thorough analyses are needed further. Here, we only treated neural networks without time delay, how about when time delays are presented? This needs to be investigated in the future.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grants no. 61203300, 61263020, and 11072059, the Specialized Research Fund for the Doctoral Program of Higher Education under Grants no. 20120092120029 and 20110092110017, the Natural Science Foundation of Jiangsu Province of China under Grants no. BK2012319 and BK2012741, the China Postdoctoral Science Foundation funded project under Grant no. 2012M511177, and the Innovation Foundation of Southeast University under Grant no. 3207012401.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

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