Stability of fully asynchronous discrete-time discrete-state dynamic networks
TL;DR: This work considers networks of a large number of neurons, whose dynamics are fully asynchronous with overlapping updating, and derives conditions on the initialization of the networks, which ensures convergence to fixed points only.
Abstract: We consider networks of a large number of neurons (or units, processors, ...), whose dynamics are fully asynchronous with overlapping updating. We suppose that the neurons take a finite number of states (discrete states), and that the updating scheme is discrete in time. We make no hypotheses on the activation function of the neurons; the networks may have multiple cycles and basins. We derive conditions on the initialization of the networks, which ensures convergence to fixed points only. Application to a fully asynchronous Hopfield neural network allows us to validate our study.
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Journal Article•
TL;DR: In this article, the behavior of 1D asynchronous cellular automata with two states and nearest neighbors is investigated using a statistical approach with macroscopic parameters and the results can be used as a guideline for the research of suitable models according to robustness criteria.
Abstract: Cellular Automata (CA) are a class of discrete dynamical systems that have been widely used to model complex systems in which the dynamics is specified at local cell-scale. Classically, CA are run on a regular lattice and with perfect synchronicity. However, these two assumptions have little chance to truthfully represent what happens at the microscopic scale for physical, biological or social systems.One may thus wonder whether CA do keep their behavior when submitted to small perturbations of synchronicity.This work focuses on the study of one-dimensional (1D) asynchronous CA with two states and nearest-neighbors. We define what we mean by ``the behavior of CA is robust to asynchronism'' using a statistical approach with macroscopic parameters.and we present an experimental protocol aimed at finding which are the robust 1D elementary CA. To conclude, we examine how the results exposed can be used as a guideline for the research of suitable models according to robustness criteria.
113 citations
18 Jul 2010
TL;DR: It is shown how the theoretical algorithms give rise to computer programs that produce true topological chaos, then the applications of this approach are proposed in the area of information security.
Abstract: This paper introduces a new notion of chaotic algorithms. These algorithms are iterative and are based on so-called chaotic iterations. Contrary to all existing studies on chaotic iterations, we are not interested in stable states of such iterations but in their possible unpredictable behaviors. By establishing a link between chaotic iterations and the notion of Devaney's topological chaos, we give conditions ensuring that these kind of algorithms produce topological chaos. This leads to algorithms that are highly unpredictable. After presenting the theoretical foundations of our approach, we are interested in its practical aspects. We show how the theoretical algorithms give rise to computer programs that produce true topological chaos, then we propose applications in the area of information security.
54 citations
TL;DR: It proves that the effects of boundary conditions on neural networks do not depend on the updating iteration mode under the hypothesis of synaptic weight symmetry and presents a new general mathematical approach based on the use of a projectivity matrix in order to simplify the problem.
37 citations
Dissertation•
15 Oct 2008
TL;DR: In this article, the authors focus on the influence of conditions of bord on the performance of reseaux in the context of regulation biologiques. But their main object is to find the elements determinant les bords of these reseaux.
Abstract: Dans cette these, nous nous interessons a l'influence des conditions de bords dans les reseaux d'automates booleens a seuil, qui sont des objets mathematiques discrets classiquement utilises pour modeliser les systemes de regulation biologiques L'objectif est de mettre en evidence que les elements determinant les bords de ces reseaux, que l'on peut rapprocher dans le contexte biologique de potentiels electriques, d'hormones ou encore de micro-ARN, sont des composants d'un systeme dont l'effet peut etre primordial sur le comportement de ce dernier Cet objectif est atteint en suivant deux axes distincts Le premier consiste a montrer les liens entre l'influence des conditions de bord et les transitions de phase emergeant du comportement asymptotique de reseaux theoriques, a savoir des reseaux d'automates cellulaires Le deuxieme axe se focalise sur les systemes biologiques reels et developpe l'idee selon laquelle les conditions de bord dans ces systemes ont une influence particuliere sur les bassins d'attraction des systemes dynamiques par lesquels ces systemes sont modelises
17 citations
Cites background from "Stability of fully asynchronous dis..."
...Enfin, il convient de remarquer que les mesures invariantes deviennent, dans le cas déterministe [102, 79, 8, 20], des mesures de Dirac situées sur des points fixes....
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TL;DR: A formulation of the basins of fixed point states of fully asynchronous discrete-time discrete-state dynamic networks is given to point out the different behaviors between synchronous and asynchronous modes.
Abstract: This paper gives a formulation of the basins of fixed point states of fully asynchronous discrete-time discrete-state dynamic networks. That formulation provides two advantages. The first one is to point out the different behaviors between synchronous and asynchronous modes and the second one is to allow us to easily deduce an algorithm which determines the behavior of a network for a given initialization. In the context of this study, we consider networks of a large number of neurons (or units, processors, etc.), whose dynamic is fully asynchronous with overlapping updates . We suppose that the neurons take a finite number of discrete states and that the updating scheme is discrete in time. We make no hypothesis on the activation functions of the nodes, so that the dynamic of the network may have multiple cycles and/or basins. Our results are illustrated on a simple example of a fully asynchronous Hopfield neural network.
8 citations
Cites background or methods from "Stability of fully asynchronous dis..."
...• 011: The last two components are not updated....
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...This model corresponds to the most general one which incorporates the sequential, parallel and block-sequential cases....
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References
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TL;DR: A model of a system having a large number of simple equivalent components, based on aspects of neurobiology but readily adapted to integrated circuits, produces a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size.
Abstract: Computational properties of use of biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.
16,652 citations
"Stability of fully asynchronous dis..." refers background or result in this paper
...• Can the theorem generalize Hopfield’s results [ 12 ], [13] or Goles et al. [18] results? — No, Theorem 4.1 gives all the initial iterates which make the neural network globally convergent whatever its dynamic, so initial states which leads to the global...
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...Hopfield [ 12 ] has shown the convergence to stable state for symmetric neural networks with nonnegative diagonal operating in serial mode....
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TL;DR: A model for a large network of "neurons" with a graded response (or sigmoid input-output relation) is studied and collective properties in very close correspondence with the earlier stochastic model based on McCulloch - Pitts neurons are studied.
Abstract: A model for a large network of "neurons" with a graded response (or sigmoid input-output relation) is studied. This deterministic system has collective properties in very close correspondence with the earlier stochastic model based on McCulloch - Pitts neurons. The content- addressable memory and other emergent collective properties of the original model also are present in the graded response model. The idea that such collective properties are used in biological systems is given added credence by the continued presence of such properties for more nearly biological "neurons." Collective analog electrical circuits of the kind described will certainly function. The collective states of the two models have a simple correspondence. The original model will continue to be useful for simulations, because its connection to graded response systems is established. Equations that include the effect of action potentials in the graded response system are also developed.
6,042 citations
Book•
01 Jul 1988
TL;DR: In this article, a model for a large network of "neurons" with a graded response (or sigmoid input-output relation) is studied, which has collective properties in very close correspondence with the earlier stochastic model based on McCulloch--Pitts neurons.
Abstract: A model for a large network of "neurons" with a graded response (or sigmoid input--output relation) is studied. This deterministic system has collective properties in very close correspondence with the earlier stochastic model based on McCulloch--Pitts neurons. The content-addressable memory and other emergent collective properties of the original model also are present in the graded response model. The idea that such collective properties are used in biological systems is given added credence by the continued presence of such properties for more nearly biological "neurons." Collective analog electrical circuits of the kind described will certainly function. The collective states of the two models have a simple correspondence. The original model will continue to be useful for simulations, because its connection to graded response systems is established. Equations that include the effect of action potentials in the graded response system are also developed.
5,734 citations
01 May 1999
TL;DR: A model of a system having a large number of simple equivalent components, based on aspects of neurobiology but readily adapted to integrated circuits, produces a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size.
Abstract: Computational properties of use to biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.
2,865 citations
01 Jan 1989
TL;DR: This book focuses on numerical algorithms suited for parallelization for solving systems of equations and optimization problems, with emphasis on relaxation methods of the Jacobi and Gauss-Seidel type.
Abstract: This book focuses on numerical algorithms suited for parallelization for solving systems of equations and optimization problems Emphasis on relaxation methods of the Jacobi and Gauss-Seidel type, and issues of communication and synchronization Topics covered include: Algorithms for systems of linear equations and matrix inversion; Herative methods for nonlinear problems; and Shortest paths and dynamic programming
1,423 citations
Additional excerpts
...In the numerical analysis community, asynchronous algorithms have been addressed [4], [ 5 ], [7], [8], and [22], however, these algorithms do not apply to discrete-state neural networks where multiple basins of attraction are vital, because of the strong hypotheses the continuous-state model requires....
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