Bifurcation analysis in an associative memory model

We previously reported the chaos induced by the frustration of interaction in a non-monotonic sequential associative memory model, and showed the chaotic behaviors at absolute zero. We have now analyzed bifurcation in a stochastic system, namely a finite-temperature model of the non-monotonic sequential associative memory model. We derived order-parameter equations from the stochastic microscopic equations. Two-parameter bifurcation diagrams obtained from those equations show the coexistence of attractors, which do not appear at absolute zero, and the disappearance of chaos due to the temperature effect.Comment: 19 page

Unstable Dynamics, Nonequilibrium Phases and Criticality in Networked Excitable Media

Here we numerically study a model of excitable media, namely, a network with occasionally quiet nodes and connection weights that vary with activity on a short-time scale. Even in the absence of stimuli, this exhibits unstable dynamics, nonequilibrium phases -including one in which the global activity wanders irregularly among attractors- and 1/f noise while the system falls into the most irregular behavior. A net result is resilience which results in an efficient search in the model attractors space that can explain the origin of certain phenomenology in neural, genetic and ill-condensed matter systems. By extensive computer simulation we also address a relation previously conjectured between observed power-law distributions and the occurrence of a "critical state" during functionality of (e.g.) cortical networks, and describe the precise nature of such criticality in the model.Comment: 18 pages, 9 figure

Stability and Diversity in Collective Adaptation

We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show that, although individual agents interact with their environment and other agents in a purely self-interested way, macroscopic behavior can be interpreted as game dynamics. Application to several familiar, explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors. The simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation. We also analyze the adaptation dynamics from an information-theoretic viewpoint and discuss self-organization induced by information flux between agents, giving a novel view of collective adaptation.Comment: 22 pages, 23 figures; updated references, corrected typos, changed conten

Traveling Salesman Problem

The idea behind TSP was conceived by Austrian mathematician Karl Menger in mid 1930s who invited the research community to consider a problem from the everyday life from a mathematical point of view. A traveling salesman has to visit exactly once each one of a list of m cities and then return to the home city. He knows the cost of traveling from any city i to any other city j. Thus, which is the tour of least possible cost the salesman can take? In this book the problem of finding algorithmic technique leading to good/optimal solutions for TSP (or for some other strictly related problems) is considered. TSP is a very attractive problem for the research community because it arises as a natural subproblem in many applications concerning the every day life. Indeed, each application, in which an optimal ordering of a number of items has to be chosen in a way that the total cost of a solution is determined by adding up the costs arising from two successively items, can be modelled as a TSP instance. Thus, studying TSP can never be considered as an abstract research with no real importance

The connections between the frustrated chaos and the intermittency chaos in small Hopfield networks

In a previous paper we introduced the notion of frustrated chaos occurring in Hopfield networks [Neural Networks 11 (1998) 1017]. It is a dynamical regime which appears in a network when the global structure is such that local connectivity patterns responsible for stable oscillatory behaviors are intertwined, leading to mutually competing attractors and unpredictable itinerancy among brief appearance of these attractors. Frustration destabilizes the network and provokes an erratic 'wavering' among the orbits that characterize the same network when it is connected in a non-frustrated way. In this paper, through a detailed study of the bifurcation diagram given for some connection weights, we will show that this frustrated chaos belongs to the family of intermittency type of chaos, first described by Berge et al. [Order within chaos (1984)] and Pomeau and Manneville [Commun. Math. Phys. 74 (1980) 189]. Indeed, the transition to chaos is a critical one, and all along the bifurcation diagram, in any chaotic window, the duration of the intermittent cycles, between two chaotic bursts, grows as an invert ratio of the connection weight. Specific to this regime are the intermittent cycles easily identifiable as the non-frustrated regimes obtained by altering the values of these same connection weights. We will more specifically show that anywhere in the bifurcation diagram, a chaotic window always lies between two oscillatory regimes, and that the resulting chaos is a merging of, among others, the cycles at both ends. The strength (i.e. the duration of its oscillatory phase before the chaotic burst) of the first cycle decreases while the regime tends to stabilize into the second cycle (with the strength of this second cycle increasing) that will finally get the control. Since in our study, the bifurcation diagram concerns the same connection weights responsible for the learning mechanism of the Hopfield network, we will discuss the relations existing between bifurcation, learning and control of chaos. We will show that, in some cases, the addition of a slower Hebbian learning mechanism onto the Hopfield networks makes the resulting global dynamics to drive the network into a stable oscillatory regime, through a succession of intermittent and quasiperiodic regimes. Finally, we will present a series of possible logical steps to manually construct a frustrated network. © 2002 Elsevier Science Ltd. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result