812 research outputs found
Chimera States in a Two-Population Network of Coupled Pendulum-Like Elements
More than a decade ago, a surprising coexistence of synchronous and
asynchronous behavior called the chimera state was discovered in networks of
nonlocally coupled identical phase oscillators. In later years, chimeras were
found to occur in a variety of theoretical and experimental studies of chemical
and optical systems, as well as models of neuron dynamics. In this work, we
study two coupled populations of pendulum-like elements represented by phase
oscillators with a second derivative term multiplied by a mass parameter
and treat the first order derivative terms as dissipation with parameter
. We first present numerical evidence showing that chimeras do
exist in this system for small mass values . We then proceed to explain
these states by reducing the coherent population to a single damped pendulum
equation driven parametrically by oscillating averaged quantities related to
the incoherent population
Breathers in the weakly coupled topological discrete sine-Gordon system
Existence of breather (spatially localized, time periodic, oscillatory)
solutions of the topological discrete sine-Gordon (TDSG) system, in the regime
of weak coupling, is proved. The novelty of this result is that, unlike the
systems previously considered in studies of discrete breathers, the TDSG system
does not decouple into independent oscillator units in the weak coupling limit.
The results of a systematic numerical study of these breathers are presented,
including breather initial profiles and a portrait of their domain of existence
in the frequency-coupling parameter space. It is found that the breathers are
uniformly qualitatively different from those found in conventional spatially
discrete systems.Comment: 19 pages, 4 figures. Section 4 (numerical analysis) completely
rewritte
Analog Realization of Arbitrary One-Dimensional Maps
An increasing number of applications of a one-dimensional (1-D) map as an information processing element are found in the literature on artificial neural networks, image processing systems, and secure communication systems. In search of an efficient hardware implementation of a 1-D map, we discovered that the bifurcating neuron (BN), which was introduced elsewhere as a mathematical model of a biological neuron under the influence of an external sinusoidal signal, could provide a compact solution. The original work on the BN indicated that its firing time sequence, when it was subject to a sinusoidal driving signal, was related to the sine-circle map, suggesting that the BN can compute the sine-circle map. Despite its rich array of dynamical properties, the mathematical description of the BN is simple enough to lend itself to a compact circuit implementation. In this paper, we generalize the original work and show that the computational power of the BN can be extended to compute an arbitrary 1-D map. Also, we describe two possible circuit models of the BN: the programmable unijunction transistor oscillator neuron, which was introduced in the original work as a circuit model of the BN, and the integrated-circuit relaxation oscillator neuron (IRON), which was developed for more precise modeling of the BN. To demonstrate the computational power of the BN, we use the IRON to generate the bifurcation diagrams of the sine-circle map, the logistic map, as well as the tent map, and then compare them with exact numerical versions. The programming of the BN to compute an arbitrary map can be done simply by changing the waveform of the driving signal, which is given to the BN externally; this feature makes the circuit models of the BN especially useful in the circuit implementation of a network of 1-D maps
Self-Organization in a Parametrically Coupled Logistic Map Network: A Model for Information Processing in the Visual Cortex
In this paper, a new model seeking to emulate the way the visual cortex processes information and interacts with subcortical areas to produce higher level brain functions is described. We developed a macroscopic approach that incorporates salient attributes of the cortex based on combining tools of nonlinear dynamics, information theory, and the known organizational and anatomical features of cortex. Justifications for this approach and demonstration of its effectiveness are presented. We also demonstrate certain capabilities of this model in producing efficient sparse representations and providing the cortical computational maps
Discrete Breathers
Nonlinear classical Hamiltonian lattices exhibit generic solutions in the
form of discrete breathers. These solutions are time-periodic and (typically
exponentially) localized in space. The lattices exhibit discrete translational
symmetry. Discrete breathers are not confined to certain lattice dimensions.
Necessary ingredients for their occurence are the existence of upper bounds on
the phonon spectrum (of small fluctuations around the groundstate) of the
system as well as the nonlinearity in the differential equations. We will
present existence proofs, formulate necessary existence conditions, and discuss
structural stability of discrete breathers. The following results will be also
discussed: the creation of breathers through tangent bifurcation of band edge
plane waves; dynamical stability; details of the spatial decay; numerical
methods of obtaining breathers; interaction of breathers with phonons and
electrons; movability; influence of the lattice dimension on discrete breather
properties; quantum lattices - quantum breathers. Finally we will formulate a
new conceptual aproach capable of predicting whether discrete breather exist
for a given system or not, without actually solving for the breather. We
discuss potential applications in lattice dynamics of solids (especially
molecular crystals), selective bond excitations in large molecules, dynamical
properties of coupled arrays of Josephson junctions, and localization of
electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see
also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm
Corticonic models of brain mechanisms underlying cognition and intelligence
The concern of this review is brain theory or more specifically, in its first part, a model of the cerebral cortex and the way it:(a) interacts with subcortical regions like the thalamus and the hippocampus to provide higher-level-brain functions that underlie cognition and intelligence, (b) handles and represents dynamical sensory patterns imposed by a constantly changing environment, (c) copes with the enormous number of such patterns encountered in a lifetime bymeans of dynamic memory that offers an immense number of stimulus-specific attractors for input patterns (stimuli) to select from, (d) selects an attractor through a process of “conjugation” of the input pattern with the dynamics of the thalamo–cortical loop, (e) distinguishes between redundant (structured)and non-redundant (random) inputs that are void of information, (f) can do categorical perception when there is access to vast associative memory laid out in the association cortex with the help of the hippocampus, and (g) makes use of “computation” at the edge of chaos and information driven annealing to achieve all this. Other features and implications of the concepts presented for the design of computational algorithms and machines with brain-like intelligence are also discussed. The material and results presented suggest, that a Parametrically Coupled Logistic Map network (PCLMN) is a minimal model of the thalamo–cortical complex and that marrying such a network to a suitable associative memory with re-entry or feedback forms a useful, albeit, abstract model of a cortical module of the brain that could facilitate building a simple artificial brain. In the second part of the review, the results of numerical simulations and drawn conclusions in the first part are linked to the most directly relevant works and views of other workers. What emerges is a picture of brain dynamics on the mesoscopic and macroscopic scales that gives a glimpse of the nature of the long sought after brain code underlying intelligence and other higher level brain functions. Physics of Life Reviews 4 (2007) 223–252 © 2007 Elsevier B.V. All rights reserved
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