On the stability of the Cartesian product of a neural ring and an arbitrary neural network
© Ivanov et al.; licensee Springer. 2014
Received: 23 March 2014
Accepted: 6 June 2014
Published: 18 July 2014
The stability of a system of neural networks connected to a ring has been studied extensively throughout recent years. Our main contribution within this work states that the stability region in the parameter space of a discrete-time model can be extended by breaking such a ring provided that there is a sufficiently large number of networks. Also, it has been shown that for a small ring, paradoxical values may appear within its parameter space for which such a ring is stable; meanwhile, corresponding linear configuration is unstable.
Many neural networks of artificial or natural origin, including the brain net, have a ring structure . The stability of a ring neural network with delayed interactions has been studied in recent works such as [2–5]. In particular,  examined the breaking of a ring neural network into a linear neural network which gives an extended stability region in the parameter space provided that there is a sufficiently large number of neurons at the ring neural network. In this paper, we take such an approach to address the related question dealing with a discrete-model of ring consisting of identical (maybe complicated) networks. We characterize closely what happens with the stability of such rings after they are broken.
This paper is structured as follows. In Section 2, formal definitions of the Cartesian product of neural networks, ring and linear configuration of a network are stated. In Section 3, it is proven that by breaking a sufficiently large ring of neural networks, it does not lose its stability. Also an example of a small torus neural network, i.e. a ring consisting of small neural rings, is given. Hence, after two consecutive cut transformations, it yields a grid configuration. We show that there is a small region within the parameter space resulting in loss of stability in the breaking of the ring neural network. Such parameter values will be called paradoxical.
2 The Cartesian product of neural networks
where n is the number of neurons in the network, is a damping factor of neuron self-oscillations, is a delay in the damping process of neuron self-oscillations, is a delay in the neuron interactions (). Entries of the matrix represent interaction forces among n different neurons, thus that every entry at the principal diagonal of A will be zero. For every j (), the j th component of is the state of the j th neuron at the moment s. The entry of the matrix A is the force of action from the v th neuron to the j th neuron. We proceed to give formal definitions to neural networks and the Cartesian product of networks as follows.
Definition 1 A neural network is an ordered 5-tuple , where , (), . We call (1) the defining equation of the network . We say that two neural networks are compatible if and only if they have the same γ, k, m. Given two compatibles networks and , we define their Cartesian product as the neural network , where the Kronecker sum operation ⊕ is defined as follows: , having ⊗ as the Kronecker product operation, and , stand for the unit matrices of orders n, r, respectively.
where () are entries of .
It is not hard to see that for any given neural network , its matrix A can be seen as a weighted directed graph with a set of vertices and a set of directed edges E defined as follows: if an entry of A is nonzero, then and weighted by . Such a graph does not depend on γ, k, m.
For any given pair , of compatible networks, their Cartesian products and are isomorphic in the sense that one defining equation can be obtained from another by a straightforward permutation of components.
Now, let us consider the following example of ring and linear configurations of networks, both playing a crucial role in our main results.
We state the following key property of the Kronecker sum.
If () is a full list of eigenvalues of an matrix and () is the corresponding list for an matrix , then the eigenvalues of are given by (, ).
3 The stability of a ring of neural networks
Our main purpose is to study the stability of a ring and linear configuration of a neural network. Hence, we proceed to state straightforwardly stability definitions for the defining equation (1). We say that this equation is stable (asymptotically stable) if and only if every solution has a bounded norm (the sequence tends to zero as ). Quite often stability requirements of a system are adjusted [15, 16], we will state the following definitions along these lines. Given ρ is a positive real number, we say that equation (1) is ρ-stable (ρ-asymptotically stable) if and only if the sequence is bounded (the sequence tends to zero as ). Equations that are not ρ-stable (asymptotically ρ-stable) will be called ρ-unstable (asymptotically ρ-unstable). We should notice that when , (asymptotic) ρ-stability is equivalent to the usual Lyapunov notion of (asymptotic) stability. Furthermore, stability cones [17, 18] for stability analysis of (1) will be extensively used in our analysis. Stability cones for stability analysis of differential delayed matrix equations were introduced in .
Theorem 2 Let , and be compatible neural networks, obeying the condition , then for every , there exists such that for all , if is ρ-stable, then is asymptotically ρ-stable.
We proceed with such a construction by an exhaustive case analysis over a and b.
CASE 1.1: There exists j () such that lies outside the ρ-stability cone for the given values of k, m. Then the point (see (3)) lies outside the ρ-stability cone, therefore the network is ρ-unstable for every . So we can put in the conclusion of the theorem.
CASE 1.2: There exists j () such that lies outside the ρ-stability cone. Let us use the fact that approaches when , being the integral part of z. We conclude from (3) that there exists an such that for every , the point lies outside the ρ-stability cone. Therefore the network is ρ-unstable for every .
CASE 1.3: For all j (), both and lie inside the ρ-stability cone or on its boundary. Since (recall that ), all the points (see (4)) lie inside the line segment with the endpoints , (see (5), (6)). But the section of the ρ-stability cone at the level has the property of being convex, hence all the points (, ) lie inside the ρ-stability cone. Therefore the neural network is asymptotically ρ-stable. This enables one to put in the conclusion of the theorem.
CASE 2: , . This case is similar to CASE 1.
CASE 3.1: There exists j () such that lies outside the ρ-stability cone. If , then . Hence by (3) there exists such that for every , the point lies outside the ρ-stability cone. Therefore the network is ρ-unstable for every .
CASE 3.2: There exists j () such that lies outside the ρ-stability cone. This case is similar to CASE 3.1, the only difference being in using instead of .
CASE 3.3: For all j (), both and lie inside the ρ-stability cone or on its boundary. This case is similar to CASE 1.3, the only difference being in using instead of .
CASE 4: , . This case is similar to CASE 3. Hence, our proof is completed. □
Considering the semigroup structure of all neural networks for γ, k, m fixed, it is not hard to see that the neural network its identity element, the fact which entails that such a structure is really a commutative monoid. By replacing by ℰ in Theorem 2, we obtain an interesting consequence.
Theorem 3 Let and be compatible neural networks, obeying that , then for every , there exists such that for all if is ρ-stable, then is asymptotically ρ-stable.
A similar result to this corollary for a continuous-time neural network model was shown in . We do remark that our main Theorem 2 states that in the case , the breaking of large ring neural networks extends the asymptotic stability domain in the parameter space providing a sufficiently large size. The latter is crucial to it, in fact it is no longer true when the number of networks in such a ring is not large enough. We will state adequate definitions and an example to support this issue.
Definition 2 Let , and be pairwise compatible neural networks. Consider to be a point in the ab-plane; we call it paradoxical for both transformations and , if the network is asymptotically stable, and is unstable.
In connection with the above investigations, some open problems arise. For example, in  a detailed analysis of appearance and disappearance of paradoxical points in a continuous-time model of neural ring networks was performed. Consequently, natural directions for future research are the analysis of these phenomena in our discrete-time model of neural networks. Moreover, in the future, we intend to examine relevant issues in neural networks with distributed delays.
This work was supported by grants from the Ministry of Education of Russia, Chelyabinsk State Pedagogical University, Russia, and by the grant from Fondecyt No. 1130112, Chile.
- Vishwanathan A, Bi GQ, Zeringue HC: Ring-shaped neuronal networks: a platform to study persistent activity. Lab Chip 2011, 11(6):1081–1088. 10.1039/c0lc00450bView ArticleGoogle Scholar
- Kaslik E: Dynamics of a discrete-time bidirectional ring of neurons with delay. In Proceedings of Int. Joint Conf. on Neural Networks. IEEE Comput. Soc., Los Alamitos; 2009:1539–1546. Atlanta, Georgia, USA, 14–19 June 2009Google Scholar
- Kaslik E, Balint S: Complex and chaotic dynamics in a discrete-time delayed Hopfield neural network with ring architecture. Neural Netw. 2009, 22(10):1411–1418. 10.1016/j.neunet.2009.03.009View ArticleGoogle Scholar
- Yuan Y, Campbell SA: Stability and synchronization of a ring of identical cells with delayed coupling. J. Dyn. Differ. Equ. 2004, 16: 709–744. 10.1007/s10884-004-6114-yMathSciNetView ArticleMATHGoogle Scholar
- Khokhlova TN, Kipnis MM: The breaking of a delayed ring neural network contributes to stability: the rule and exceptions. Neural Netw. 2013, 48: 148–152.View ArticleMATHGoogle Scholar
- Kaslik E, Balint S: Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network. Chaos Solitons Fractals 2007, 34: 1245–1253. 10.1016/j.chaos.2006.03.107MathSciNetView ArticleMATHGoogle Scholar
- Idels L, Kipnis M: Stability criteria for a nonlinear nonautonomous system with delays. Appl. Math. Model. 2009, 33(5):2293–2297. 10.1016/j.apm.2008.06.005MathSciNetView ArticleMATHGoogle Scholar
- Arbib M (Ed): The Handbook of Brain Theory and Neural Networks. 2nd edition. MIT Press, Cambridge; 2002.Google Scholar
- Ivanov SA, Kipnis MM: Stability analysis of discrete-time neural networks with delayed interactions: torus, ring, grid, line. Int. J. Pure Appl. Math. 2012, 78(5):691–709.Google Scholar
- Imrich W, Klavžar S, Rall DF: Graphs and Their Cartesian Products. AK Peters, Wellesley; 2008.MATHGoogle Scholar
- Brualdi RA, Cvetković DM: A Combinatorial Approach to Matrix Theory and Its Applications. CRC Press, Boca Raton; 2008.View ArticleMATHGoogle Scholar
- Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge; 1994.MATHGoogle Scholar
- Brouwer AE, Haemers WH: Spectra of Graphs. Springer, Berlin; 2011.MATHGoogle Scholar
- Cvetković DM, Doob M, Sachs H: Spectra of Graphs - Theory and Applications. 3rd edition. Wiley, New York; 1998.MATHGoogle Scholar
- Chestnov VN:Synthesis -controllers for multidimensional systems with given accuracy and degree of stability. Autom. Remote Control 2011, 72(10):2161–2175. 10.1134/S0005117911100134MathSciNetView ArticleMATHGoogle Scholar
- Gryazina EN, Polyak BT: Stability regions in the parameter space: D -decomposition revisited. Automatica 2006, 42(1):13–26. 10.1016/j.automatica.2005.08.010MathSciNetView ArticleMATHGoogle Scholar
- Kipnis MM, Malygina VV: The stability cone for a matrix delay difference equation. Int. J. Math. Math. Sci. 2011., 2011: Article ID 860326Google Scholar
- Ivanov SA, Kipnis MM, Malygina VV: The stability cone for a difference matrix equation with two delays. ISRN Appl. Math. 2011., 2011: Article ID 910936Google Scholar
- Khokhlova TN, Kipnis MM, Malygina VV: Stability cone for linear delay differential matrix equation. Appl. Math. Lett. 2011, 24: 742–745. 10.1016/j.aml.2010.12.020MathSciNetView ArticleMATHGoogle Scholar
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