Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 615274, pptx

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 615274, 17 pages doi:10.1155/2011/615274 Research Article Hamming Star-Convexity Packing in Information Storage Mau-Hsiang Shih and Feng-Sheng Tsai Department of Mathematics, National Taiwan Normal University, 88 Section 4, Ting Chou Road, Taipei 11677, Taiwan Correspondence should be addressed to Feng-Sheng Tsai, fstsai@abel.math.ntnu.edu.tw Received 8 December 2010; Accepted 16 December 2010 Academic Editor: Jen Chih Yao Copyright q 2011 M H. Shih and F S. Tsai. 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. A major puzzle in neural networks is understanding the information encoding principles that implement the functions of the brain systems. Population coding in neurons and plastic changes in synapses are two important subjects in attempts to explore such principles. This forms the basis of modern theory of neuroscience concerning self-organization and associative memory. Here we wish to suggest an information storage scheme based on the dynamics of evolutionary neural networks, essentially reflecting the meta-complication of the dynamical changes of ne urons as well as plastic changes of synapses. The information storage scheme may lead to the development of a complete description of all the equilibrium states fixed points of Hopfield networks, a space- filling network that weaves the intricate structure of Hamming star-convexity, and a plasticity regime that encodes information based on algorithmic Hebbian synaptic plasticity. 1. Introduction The study of memory includes two important components: the storage component of memory and the systems component of memory 1, 2. The first is concerned with exploring the molecular mechanisms whereby memory is stored, whereas the second is concerned with analyzing the organizing principles that mediate brain systems to encode, store, and retrieve memory. The first neurophysiological description about the systems component of memory was proposed by H ebb 3. His postulate reveals a principle of learning, which is often summarized as “the connections between neurons are strengthened when they fire simultaneously.” The Hebbian concept stimulates an intensive effort to promote the building of associative memory models of the brain 4–9. Also, it leads to the development of a LAMINART model matching in laminar visual cortical circuitry 10, 11,thedevelopmentof 2 Fixed Point Theory and Applications an Ising model used in statistical physics 12–15, and the study of constrained optimization problems such as the famous traveling salesman problem 16. However, since it was initiated by Kohonen and Anderson in 1972, associative memory has remained widely open in neural networks 17–21. It generally includes questions concerning a description of collective dynamics and computing with attractors in neural networks. Hence the central question 22: “given an arbitrary set of prototypes of 01-strings of length n, is there any recurrent network such that the set of all equilibrium states of this network is exactly the set of those prototypes?” Many attempts have been made to tackle this problem. For instance, using the method of energy minimization, Hopfield in 1982 constructed a network of nerve cells whose dynamics tend toward an equilibrium state when the retrieval operation is performed asynchronously 13. Furthermore, to circumvent limited capacity in storage and retrieval of Hopfield networks, Personnaz et al. in 1986 investigated the behavior of neural networks designed with the projection rule, which guarantees the errorless storage and retrieval of prototypes 23, 24. In 1987, Diederich and Opper proposed an iterative scheme to substitute a local learningrulefortheprojectionrulewhenthe prototypes are linearly independent 25, 26. This sheds light on the possibility of storing correlated prototypes in neural networks with local learning rules. In addition to the discussion on learning mechanisms for associative memory, Hopfield networks have also given a valuable impetus to basic research in combinatorial fixed point theory in neural networks. In 1992, Shrivastava et al. conducted a convergence analysis of a class of Hopfield networks and showed that all equilibrium states of these networks have a one-to-one correspondence with the maximal independent sets of certain undirected graphs 27.M ¨ uezzino ˘ glu and G ¨ uzelis¸ in 2004 gave a further compatibility condition on the correspondence between equilibrium states and maximal independent sets, which avoids spurious stored patterns in information storage and provides attractiveness of prototypes in retrieval operation 28. Moreover, the analytic approach of Shih and Ho 29 in 1999 as well as Shih and Dong 30 in 2005 illustrated the reverberating-circuit structure to determine equilibrium states in generalized boolean networks, leading to a solution of the boolean Markus-Yamabe problem and a proof of ne twork perspective of the Jacobian conjecture, respectively. More recently, we described an evolutionary neural network in which the connection strengths between neurons are highly evolved according to algorithmic Hebbian synaptic plasticity 31. To explore the influence of synaptic plasticity on the evolutionary neural network’s dynamics, a sort of driving forces from the meta-complication of the evolutionary neural network’s nodal-and-coupling activities is introduced, in contrast with the explicit construction of global Lyapunov functions in neural networks 10, 13, 32, 33 and in accordance with the limitation of finding a common quadratic Ly apunov function to control a switched system’s dynamics 34–36. A mathematical proof asserts that the ongoing changes of the evolutionary network’s nodal-and-coupling dynamics will eventually come to rest at equilibrium states 31. This result reflects, in a deep mathematical sense, that plastic changes in the coupling dynamics may appear as a mechanism for associative memory. In this respect, an information storage scheme for associative memory may be suggested as follows. It comprises three ingredients. First, based on the Hebbian learning rule, establish a primitive neural network whose equilibrium states contain the prototypes and derive a common qualitative property P from all the domains of attraction of equilibrium states. Second, determine a merging process that merges the domains of attraction of equilibrium states such that each merging domain contains exactly one prototype and that preserves the property P. Third, based on algorithmic Hebbian synaptic plasticity, probe Fixed Point Theory and Applications 3 a plasticity regime that guides the evolution of the primitive neural network such that each vertex in the merging domain will tend toward the unique prototype underlying the dynamics of the resulting evolutionary neural network. Our point of departure is the convexity packing lurking behind Hopfield networks. We consider the domain of attraction in which every initial state in the domain tends toward the equilibrium state asynchronously. For the asynchronous operating mode, each trajectory in the phase space can represent as one of the “connected” paths between the initial state and the equilibrium state when it is measured by the Hamming metric. It provides a clear map that all the domains of attraction in Hopfield networks are star-convexity-like and that the phase space can be filled with those star-convexity-like domains. And it applies to frame a primitive Hopfield network that might consolidate an insight of exploring a plasticity regime in the information storage scheme. 2. Information Storage of Hopfield Networks Let {0, 1} n denote the binary code consisting of all 01 strings of fixed length n,andletX  {x 1 ,x 2 , ,x p } be an arbitrary set of prototypes in {0, 1} n . For each positive integer k,let k  {1, 2, ,k}. Using the formal neurons of McCulloch and Pitts 37,wecanconstruct a Hopfield network of n coupled neurons, namely, 1, 2, ,n,whose synaptic strengths are listed in an array, denoted by the matrix A a ij  n×n , and defined on the basis of the Hebbian learning rule, that is, a ij  p  s1 x s i x s j for every i, j ∈ n . 2.1 The firing state of each neuron i is denoted by x i  1, whereas the quiescent state is x i  0. The function is the Heaviside function: u1foru ≥ 0, otherwise 0, which describes an instantaneous unit pulse. The dynamics of the Hopfield network is encoded by the function F f 1 ,f 2 , ,f n ,where f i  x   ⎛ ⎝ n  j1 a ij x j − b i ⎞ ⎠ 2.2 encodes the dynamics of neuron i, x x 1 ,x 2 , ,x n  is a vector of state variables in the phase space {0, 1} n ,andb i ∈ is the threshold of neuron i for each i ∈ n . For every x, y ∈{0, 1} n ,definethe vectorial distance between x and y 38, 39, denoted as dx, y,tobe d  x, y   ⎛ ⎜ ⎜ ⎜ ⎝   x 1 − y 1   . . .   x n − y n   ⎞ ⎟ ⎟ ⎟ ⎠ . 2.3 4 Fixed Point Theory and Applications For every x, y ∈{0, 1} n , define the order relation x ≤ y by x i ≤ y i for each i ∈ n ;thechain interval between x and y, denoted as Cx, y,tobe C  x, y    z ∈ { 0, 1 } n ; d  z, y  ≤ d  x, y  . 2.4 Note that Cx, yCy, x,andthenotationCx, y means that Cx, y \{x}.TheHamming metric ρ H on {0, 1} n is defined to be ρ H  x, y   #  i ∈ n ; x i /  y i  2.5 for every x, y ∈{0, 1} n 40.Denotebyγx, y a chain joining x and y with the minimum Hamming distance, meaning that γ  x, y    x, u 1 ,u 2 , ,u r−1 ,y  , 2.6 where ρ H u i ,u i1 1fori  0, 1, ,r − 1withu 0  x, u r  y,andρ H x, u 1 ρ H u 1 ,u 2  ··· ρ H u r−1 ,yρ H x, y.ThenwehaveCx, y  γx, y, where the union is taken over all chains joining x and y with the minimum Hamming distance. Denote by ·, · the Euclidean scalar product in n .Asetofelementsy α in {0, 1} n , where α runs through some index set in I, is called orthogonal if y α ,y β   0foreachα, β ∈ I with α /  β.TwosetsY and Z in {0, 1} n are called mutually orthogonal if y, z  0foreach y ∈ Y and z ∈ Z. Given a set Y  {y 1 ,y 2 , ,y q } in {0, 1} n ,wedefinethe01-span of Y , denoted as 01-spanY, to be the set consists o f all elements of the form τ 1 y 1 τ 2 y 2 ··· τ q y q , where τ i ∈{0, 1} for each i ∈ q . We assume that x i /  0foreachi ∈ p .Foreachi ∈ p ,define N 1 x i   x s ∈ X;  x s ,x i  /  0  2.7 and define recursively N j1 x i   x s ∈ X;  x s ,x k  /  0forsomex k ∈ j x i  2.8 for each j ∈ . Clearly, for each i ∈ p we have N 1 x i ⊂ N 2 x i ⊂ N 3 x i ⊂··· , 2.9 and thereby there exists a smallest positive integer, denoted as si,suchthat N si x i  N sij x i for each j ∈ . 2.10 It is readily seen that for each i ∈ p and for each x j ∈ N si x i ,wehave N si x i  N sj x j , 2.11 Fixed Point Theory and Applications 5 and clearly, for every i, j ∈ p , exactly one of the following conditions holds: N si x i  N sj x j or N si x i ∩ N sj x j  ∅. 2.12 According to 2.8 and 2.12, we can pick all distinct sets N 1 ,N 2 , ,N q from {N s1 x 1 , N s2 x 2 , ,N sp x p } and obtain the orthogonal partition of X,thatis,N i and N j are mutually or- thogonal for every i /  j and X   i∈ q N i .Foreachk ∈ q ,define ξ k  ⎛ ⎜ ⎜ ⎜ ⎝ max  x i 1 ; x i ∈ N k  . . . max  x i n ; x i ∈ N k  ⎞ ⎟ ⎟ ⎟ ⎠ . 2.13 Then we have the orthogonal set {ξ 1 ,ξ 2 , ,ξ q } generated by the orthogonal partition of X, which is denoted as GopX. Using the “orthogonal partition,” we can give a complete description of the equi- librium states of the Hopfield network encoded by 2.1 and 2.2 with ultra-low thresh- olds. Theorem 2.1. Let X be a set consisting of nonzero vectors in {0, 1} n , and let the function F be defined by 2.1 and 2.2 with 0 <b i ≤ 1 for each i ∈ n .Then,FixF01-spanGopX. Proof. Let X  {x 1 ,x 2 , ,x p } and let GopX{ξ 1 ,ξ 2 , ,ξ q }. By orthogonality of GopX, 1 −  q i1 ξ i j is0or1foreachj ∈ n . Thus the point GopX,definedby  Gop  X     1 − q  i1 ξ i 1 , 1 − q  i1 ξ i 2 , ,1 − q  i1 ξ i n  , 2.14 lies in {0, 1} n .LetU 0  C0, GopX and U i  C0,ξ i  for each i ∈ q . Note that the sets U i and U j are mutually orthogonal for every i /  j.Letξ   q i1 α i ξ i for α i ∈{0, 1} and i ∈ q . We prove now that Fξξ by showing that F  x  ∈ C  x, ξ  for each x ∈ U 0  q  i1 α i U i . 2.15 Let x  u 0   q i1 α i u i where u i ∈ U i for i  0, 1, ,q.SinceX ∩ C0,ξ k N k for each k ∈ q , 6 Fixed Point Theory and Applications we have F  x   ⎛ ⎝ p  i1  x i T u 0  x i  q  j1 p  i1  α j  x i T u j  x i  − b ⎞ ⎠  ⎛ ⎝ q  j1  x i ∈N j  α j  x i T u j  x i  − b ⎞ ⎠ ≤ q  j1 α j ξ j . 2.16 Thus we need only consider the case Fx ν  0andξ ν  1forsomeν ∈ n . Under the case, there exists r ∈ q such that α r  1andξ r ν  1, so that F  x  ν ≥ ⎛ ⎝  x i ∈N r  x i T u r  x i ν − b ν ⎞ ⎠ ≥ ⎛ ⎝  x i ∈N r u r ν  x i ν  2 − b ν ⎞ ⎠ . 2.17 Since Fx ν  0, we have x ν  u r ν  0. This implies that dFx,ξ ≤ dx, ξ,thatis, Fx ∈ Cx, ξ. We turn now to prove that Fx /  x for each x / ∈ 01-spanGopX. To accomplish this, we first show that { 0, 1 } n   α i ∈{0,1},i∈ q U 0  q  i1 α i U i . 2.18 Let x ∈{0, 1} n .Weassociatetoeachi ∈ q apoint z i   x 1 ξ i 1 ,x 2 ξ i 2 , ,x n ξ i n  2.19 and put z 0  x −  q i1 z i .Thenforeachi ∈ q ,thereexistα i ∈{0, 1} such that z i ∈ α i U i .Since for each k ∈ n z 0 k  x k − q  i1 x k ξ i k ≤ 1 − q  i1 ξ i k , 2.20 we have z 0 ∈ U 0 ,proving2.18.Thuseachx / ∈ 01-spanGopX canbewrittenas x  u 0   q i1 α i u i ,whereα i ∈{0, 1}, u i ∈ U i for i  0, 1, ,q and, further, we have either u 0 /  0orthereexistsr ∈ q such that α r  1andu r /  ξ r . Fixed Point Theory and Applications 7 Case 1 u 0 /  0. Then there exists ν ∈ n such that u 0 ν  1andx k ν  0foreachk ∈ p .This implies that x ν  u 0 ν  q  i1 α i u i ν  1, F  x  ν  ⎛ ⎝ q  j1  x i ∈N j  α j  x i T u j  x i ν  − b ν ⎞ ⎠  0, 2.21 proving Fx /  x. Case 2. There exists r ∈ q such that α r  1andu r /  ξ r .Then C  0,ξ r  ∩  X \ C  0,u r  ∩  X \ C  0,ξ r − u r  /  ∅. 2.22 Indeed, if the left hand side of 2.22 is empty, then for every x i ∈ N r  X ∩ C0,ξ r ,exactly one of the following conditions holds: x i ∈ C  0,u r  or x i ∈ C  0,ξ r − u r  . 2.23 Divide the set N r into two subsets: x i ∈ M 1 if x i ∈ C  0,u r  , x i ∈ M 2 if x i ∈ C  0,ξ r − u r  . 2.24 Then, by the construction of ξ r ,wehaveM 1 /  ∅ and M 2 /  ∅.Nowletx σ ∈ M 1 and x η ∈ M 2 . Since M 1 and M 2 are mutually orthogonal, we get N sσ x σ ⊂ M 1 and N sη x η ⊂ M 2 .This con-tradicts N sσ x σ  N sη x η  N r , 2.25 proving 2.22. Therefore, there exist x k ∈ C  0,ξ r  ∩  X \ C  0,u r  ∩  X \ C  0,ξ r − u r  2.26 and k 1 ,k 2 ∈ n with u r k 1  1andξ r − u r  k 2  1suchthatx k k 1  x k k 2  1. Since ξ r − u r  k 2  1, 8 Fixed Point Theory and Applications u i k 2  0fori  0, 1, ,qand x i k 2  0foreachx i / ∈ N r . This implies that x k 2  u 0 k 2  q  i1 α i u i k 2  0, F  x  k 2 ≥ ⎛ ⎝  x i ∈N r  x i T u r  x i k 2 − b k 2 ⎞ ⎠ ≥  x k k 1 u r k 1 x k k 2 − b k 2   1, 2.27 revealing Fx /  x,provingTheorem 2.1. 3. Domains of Attraction and Hamming Star-Convex Building Blocks By analogy with the notion of star-convexity in vector spaces, a set U in {0, 1} n is said to be Hamming star-convex if there exists a point y ∈ U such that Cx, y ⊂ U for each x ∈ U.We call y a star-center of U. Let X be a set in {0, 1} n ,andletΛ X denote the collection o f all 01-spanY,whereY is an orthogonal set consisting of nonzero vectors in {0, 1} n ,suchthatX ⊂ 01-spanY.Then Λ X /  ∅.Indeed,iftheorder“≤”onΛ X is defined by A ≤ B if and only if A ⊂ B,thenΛ X , ≤ becomes a partially ordered set and there exists an orthogonal set Y such that 01-spanY is minimal in Λ X .WecallsuchY the kernel of X. A labeling procedure for establishing the kernel Y of X is described as follows. Let X  {x 1 ,x 2 , ,x p } in {0, 1} n .IfX  {0},thenY  {y}, where y /  0, is the kernel of X. Otherwise, define the labelings λ i   x 1 i ,x 2 i , ,x p i  for each i ∈ n 3.1 and pick all distinct nonzero labelings v 1 ,v 2 , ,v q from λ 1 ,λ 2 , ,λ n . Then the orthogonal set Y  {y 1 ,y 2 , ,y q }, given by y i j  1ifλ j  v i ,otherwisey i j  0foreachi ∈ q and j ∈ n ,isthekernelofX see Figure 1. Note that since the computation of the kernel can be implemented by radix sort, its computational complexity is in Θpn. Let Y  {y 1 ,y 2 , ,y q } be the kernel of X. We associate to each y k ∈ Y an integer nk ∈ ,twosetsofnodes V k   v k,l ; l ∈ nk  ,W k   w k,j ; y k j  1,j ∈ n  , 3.2 and a set of edges E k such that G k V k ∪ W k ,E k  is a simple, connected, and bipartite graph with color classes V k and W k . The graph-theoretic notion and terminologies can be found in 41.Foreachj ∈ n ,putu k,l j  1ifv k,l and w k,j are a djacent, otherwise 0. Let G  {G 1 ,G 2 , ,G q } and denote by BipY, G the collection of all vectors u k,l constructed by the bipartite graphs in G see Figure 1. Fixed Point Theory and Applications 9 n(1)=4 n(2)=2 n(3)=3 Bipartite graphs v 1,1 v 1,2 v 1,3 v 1,4 v 3,1 v 3,2 v 3,3 v 2,1 v 2,2 w 1,1 w 1,3 w 1,8 w 1,9 w 1,12 w 2,4 w 2,7 w 2,15 w 2,16 w 3,10 w 3,11 w 3,13 x 1 x 2 x 3 (1, 0, 0) ( 0, 0, 0 ) (0, 0, 0)(0, 0, 0) (0, 0, 0) (1, 1, 0) (1, 1, 0) (1, 1, 0) (1, 1, 0) ( 0, 1, 1 ) ( 0, 1, 1 )(1, 0, 0)(1, 0, 0) (0, 1, 1) (0, 1, 1) (0, 1, 1) y 1 y 2 y 3 G 1 G 2 G 3 u 1,1 u 1,2 u 1,3 u 1,4 u 2,1 u 2,2 u 3,1 u 3,2 u 3,3 Bip(Y, G) y 1 y 2 y 3 v 1 =(0, 1, 1), v 2 =(1, 1, 0), v 3 =(1, 0, 0) The kernel determined by labelings Labelings Non-zero labelings Figure 1: A schematic illustration of the generation of the kernel Y and BipY, G. Denote by FixF the set of all equilibrium states fixed points of F and denote by D GS ξ the domain of attraction of the equilibrium state ξ underlying Gauss-Seidel iteration a particular mode of asynchronous iteration x i  t  1   f i  x 1  t  1  , ,x i−1  t  1  ,x i  t  , ,x n  t  3.3 for t  0, 1, and i ∈ n . Theorem 3.1. Let X be a subset of {0, 1} n ,andletY  {y 1 ,y 2 , ,y q } be the kernel of X. Associate to each Bip  Y, G    u k,l ; k ∈ q ,l∈ nk  3.4 10 Fixed Point Theory and Applications a function F defined by 2.2 with a ij  q  k1 nk  l1 u k,l i u k,l j for each i, j ∈ n 3.5 and 0 <b i ≤ 1 for each i ∈ n .Then i X ⊂ FixF; ii for each ξ ∈ FixF, the domain of attraction D GS ξ is Hamming star-convex with ξ as a star-center. Proof. For each k ∈ q ,sinceG k is simple, connected, and bipartite with color classes V k and W k ,wehave N sk,l u k,l  N sk,j u k,j for each l, j ∈ nk . 3.6 It follows from the orthogonality of Y that  u k,l ; l ∈ nk  ⊂ N sk,j u k,j ⊂ C  0,y k  3.7 for each k ∈ q and j ∈ nk .Furthermore,sinceG k is connected for each k ∈ q ,wehave max  u k,l j ; l ∈ nk   y k j for each j ∈ n . 3.8 This implies that GopBipY, G  Y,andbyTheorem 2.1,wehave Fix  F   01-span  Gop  Bip  Y, G   ⊃ X, 3.9 proving i.Toproveii, we first show that for each i ∈ q and α i ∈{0, 1}, C  0,  Y   q  i1 α i C  0,y i  ⊂ D GS  q  i1 α i y i  , 3.10 where Y1 −  q i1 y i 1 , 1 −  q i1 y i 2 , ,1 −  q i1 y i n .LetU denote the set in the left hand side of 3.10,andletx ∈ U, y   q i1 α i y i ,andz ∈ Cx, y. Split z into two parts: z   z 1 − q  i1 z 1 y i 1 ,z 2 − q  i1 z 2 y i 2 , ,z n − q  i1 z n y i n   q  i1  z 1 y i 1 ,z 2 y i 2 , ,z n y i n  . 3.11 Then the first part of z lies in C0, Y,andthesecondpartofz lies in  q i1 α i C0,y i .This shows that U is Hamming star-convex with y as a star-center, that is, C  x, y  ⊂ U for each x ∈ U. 3.12 [...]... Systems, Man, and Cybernetics, vol 13, no 5, pp 815–826, 1983 33 E Goles and S Mart´nez, Neural and Automata Networks: Dynamical Behavior and Applications, vol ı 58 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990 Fixed Point Theory and Applications 17 34 T Ando and M.-H Shih, “Simultaneous contractibility,” SIAM Journal on Matrix Analysis and Applications, ... Fogelman-Souli´ and G Weisbuch, “Random iterations of threshold networks and associative e memory,” SIAM Journal on Computing, vol 16, no 1, pp 203–220, 1987 16 Fixed Point Theory and Applications 9 D J Willshaw, O P Buneman, and H C Longuet-Higgins, “Non-holographic associative memory,” Nature, vol 222, no 5197, pp 960–962, 1969 10 S Grossberg, “Nonlinear neural networks: principles, mechanisms, and architectures,”... , y / ∅ and {v1 , v2 , , vk } ∩ C x, yα / ∅ We may assume that v1 , v2 , , vs ∈ C xα , y and vs 1 , vs 2 , , vk ∈ C x, yα , where s ∈ Æ k Then, by 4.2 , 4.3 , and the induction hypothesis, there exist u1 , u2 , , us ∈ C xα , y and us 1 , us 2 , , uk ∈ C x, yα such that C x, y C u1 , v 1 ∪ C u2 , v 2 ∪ · · · ∪ C uk , v k , 4.12 C ui , v i ∩ C uj , v j ∅ Fixed Point Theory and Applications. .. inequalities: βi − γi ≤ αi − γi for each i ∈ Æ q 4.18 14 Fixed Point Theory and Applications Figure 2: The proof given in Theorem 4.1 reveals a merging process of the Hamming star-convexity packing Here the 6-cube is filled with 3 nonoverlapping Hamming star-convex sets with star-centers spatially distributed in three vertices To show 4.18 , fix i ∈ Æ q and consider only two cases Case 1 αi 1 Since zi ∈ C 0,... states of Hopfield networks with ultra-low thresholds It provides a basis for the building of a primitive Hopfield network Fixed Point Theory and Applications 15 whose equilibrium states contain the prototypes A common qualitative property, namely, the Hamming star-convexity, can be deduced from all those domains of attraction of equilibrium states and a merging process, which preserves the Hamming star-convexity,... valid for every x, y ∈ {0, 1}n with ρH x, y with ρH x, y p 1 Choose α so that xα / yα , and use the complemented notation: 0 1, 1 0 Then, we have C x, y C xα , y ∪ C x, yα , C xα , y ∩ C x, yα where xα x1 , , xα−1 , xα , xα 1 , , xn and yα ∅, y1 , , yα−1 , y α , yα 1 , , yn 4.2 4.3 12 Fixed Point Theory and Applications ∅ or {v1 , v2 , , vk } ∩ C x, yα ∅ We may assume Case 1 {v1 , v2 , .. .Fixed Point Theory and Applications 11 Combining 3.12 with 2.15 shows that x1 , , xi−1 , fi x , xi 1 , , xn ∈ C x, y ⊂ U 3.13 for each i ∈ Æ n and x ∈ U Since Fix F ∩ U {y} by Theorem 2.1, inclusion 3.10 follows immediately from 3.13 Now, by combining 3.10 with 2.18 , we obtain C 0, Á Y q q αi C 0, yi αi yi DGS i 1 3.14 i 1 for each i ∈ Æ q and αi ∈ {0, 1}, proving... regime that guides the dynamics of the evolutionary coupling states such that x t converges and x t ∈ Δ for every t 0, 1, The plasticity regime, even when insoluble in the information storage scheme by assigning A 0 to be the matrix of synaptic strengths of the primitive Hopfield network given in Theorem 3.1 and Δ to be the Hamming star-convex set given in Theorem 4.1, is a guide to understand and explain... storage and retrieval of associative memory Acknowledgment This work was supported by the National Science Council of the Republic of China References 1 E R Kandel, “The molecular biology of memory storage: a dialog between genes and synapses,” in Nobel Lectures, Physiology or Medicine 1996–2000, H Jornvall, Ed., pp 392–439, World Scientific, Singapore, 2003 2 B Milner, L R Squire, and E R Kandel, “Cognitive... work? Learning, attention, and groupings by the laminar circuits of visual cortex,” Spatial Vision, vol 12, no 2, pp 163–185, 1999 12 J Hertz, A Krogh, and R G Palmer, Introduction to the Theory of Neural Computation, Santa Fe Institute Studies in the Sciences of Complexity Lecture Notes, I, Addison-Wesley, Redwood City, Calif, USA, 1991 13 J J Hopfield, “Neural networks and physical systems with emergent . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 615274, 17 pages doi:10.1155/2011/615274 Research Article Hamming Star-Convexity. Behavior and Applications, vol. 58 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. Fixed Point Theory and Applications 17 34 T. Ando and M H Fogelman-Souli ´ e and G. Weisbuch, “Random it erations of threshold networks and as sociative memory,” SIAM Journal on Computing, vol. 16, no. 1, pp. 203–220, 1987. 16 Fixed Point Theory and Applications 9