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Asynchronous updating of coupled maps leads to synchronization.

We investigate the spatiotemporal dynamics of two dimensional coupled map lattices, in the strong coupling phase, evolving under updating rules incorporating varying degrees of asynchronicity. Interestingly, we observe that parallel updates never allow synchronization among the sites, while asynchroncity has the effect of opening up windows in parameter space where the synchronized dynamics gains stability. As asynchronicity increases, the parameter range supporting synchronization gets rapidly wider. Detailed numerics, including bifurcation diagrams and patterns formed en route to synchronization, are reported. We also attempt a mean-field analysis of the system in order to try and account for the stability of the spatiotemporal fixed point under asynchronous updates. (c) 2000 American Institute of Physics.

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Numerical studies on the dynamics of the globally coupled maps with sequential updating

A system of globally coupled logistic maps with sequential updating is analyzed numerically. It is found that deterministic asynchronous updating schemes may have dramatical influences on the dynamical behaviors of globally coupled systems. Transitions from spatiotemporal chaos to spatially organized states are observed as the coupling parameter varies. It is shown that the model system may exhibit a variety of collective properties such as clustering, traveling wave patterns, and spatial bifurcation cascades.

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Synchronization and propagation of bursts in networks of coupled map neurons.

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.

Action Potentials↗

Parameter estimation in spatially extended systems: the Karhunen-Lóeve and Galerkin multiple shooting approach.

Parameter estimation for spatiotemporal dynamics for coupled map lattices and continuous time domain systems is shown using a combination of multiple shooting, Karhunen-Loéve decomposition and Galerkin's projection methodologies. The resulting advantages in estimating parameters have been studied and discussed for chaotic and turbulent dynamics using small amounts of data from subsystems, availability of only scalar and noisy time series data, effects of space-time parameter variations, and in the presence of multiple time scales.

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Spatiotemporal patterns of an evoked network oscillation in neocortical slices: coupled local oscillators.

We have discovered an evoked network oscillation in rat neocortical slices and have examined its spatiotemporal patterns with voltage-sensitive dye imaging. The slices (visual and auditory cortices) were prepared in a medium of low calcium, high magnesium and with sodium replaced by choline to reduce the excito-toxicity and sodium loading. After slicing, the choline was washed out while normal calcium, magnesium, and sodium concentrations were restored. The oscillation was evoked by a single electrical shock to slices bathed in normal artificial cerebral spinal fluid (ACSF). The oscillation was organized as an all-or-none epoch containing 4-13 cycles at a central frequency approximately 25 Hz. The activity can be reversibly blocked by 6-cyano-7-nitroquinoxalene-2,3-dione (CNQX). 2-amino-5-phosphonopentanoic acid (APV), and atropine but not by bicuculline, indicating polysynaptic excitatory mechanisms. Voltage-sensitive dye imaging showed high-amplitude oscillation signals in superficial and middle cortical layers. Spatiotemporally, the oscillations were organized as waves, propagating horizontally along cortical laminar. Each oscillation cycle was associated with one wave propagating in space. The waveforms were often different at different locations (e.g., extra cycles), suggesting the co-existence of multiple local oscillators. For different cycles, the waves often initiated at different locations, suggesting that local oscillators are competing to initiate each oscillation cycle. Overall our results suggest that this cortical network oscillation is organized at two levels: locally, oscillating neurons are tightly coupled to form local oscillators, and globally the coupling between local oscillators is weak, allowing abrupt spatial phase lags and propagating waves with multiple initiation sites.

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Spatiotemporal consequences of relaxation time scales in threshold-coupled systems.

We study a class of models incorporating threshold-activated coupling on a lattice of chaotic elements. In such systems, the relaxation time allowed between chaotic updates determines the intrinsic driving rate due to the local chaos, and we show that there exists an inverse cascade from fixed spatial profiles to spatiotemporal chaos, as the relaxation time grows shorter. We analyze how this spectrum of spatiotemporal transitions arises from the competing time scales of the local chaos and the propagation of coupling.

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Competing spatial and temporal instabilities in a globally coupled bistable semiconductor system near a codimension-two bifurcation

We study complex spatiotemporal dynamics in a globally coupled bistable reaction-diffusion model on a two-dimensional spatial domain. It is demonstrated that complex behavior appears near a codimension-two bifurcation point due to the competition of spatial and temporal instabilities. We derive sufficient conditions for the appearance of mixed spatiotemporal modes, and clarify the origin of a menagery of complex dynamics, such as periodic and chaotic oscillations of current filaments, low-dimensional spatiotemporal chaos including a Shil'nikov attractor, and periodic back-and-forth motion of current density fronts. Such dynamics is found in a wide range of domain sizes for square and rectangular domains. The type of dynamics is sensitive to small variations in the domain shape. We discuss and explain the differences between spatiotemporal dynamics on one-dimensional and two-dimensional domains.

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Coherent spatiotemporal patterns of ongoing activity revealed by real-time optical imaging coupled with single-unit recording in the cat visual cortex.

1. We examined the spatiotemporal organization of ongoing activity in cat visual areas 17 and 18, in relation to the spontaneous activity of individual neurons. To search for coherent activity, voltage-sensitive dye signals were correlated with the activity of single neurons by the use of spike-triggered averaging. In each recording session an area of at least 2 x 2 mm of cortex was imaged, with 124 diodes. In addition, electrical recordings from two isolated units, the local field potential (LFP) from the same microelectrodes, and the surface electroencephalogram (EEG) were recorded simultaneously. 2. The optical signals recorded from the dye were similar to the LFP recorded from the same site. Optical signals recorded from different cortical sites exhibited a different time course. Therefore real-time optical imaging provides information that is equivalent in many ways to multiple-site LFP recordings. 3. The spontaneous firing of single neurons was highly correlated with the optical signals and with the LFP. In 88% of the neurons recorded during spontaneous activity, a significant correlation was found between the occurrence of a spike and the optical signal recorded in a large cortical region surrounding the recording site. This result indicates that spontaneous activity of single neurons is not an independent process but is time locked to the firing or to the synaptic inputs from numerous neurons, all activated in a coherent fashion even without a sensory input. 4. For the cases showing correlation with the optical signal, 27-36% of the optical signal during spike occurrence was directly related to the occurrence of spontaneous spikes in a single neuron, over an area of 2 x 2 mm. In the same cortical area, 43-55% of the activity was directly related to the visual stimulus. 5. Surprisingly, we found that the amplitude of this coherent ongoing activity, recorded optically, was often almost as large as the activity evoked by optimal visual stimulation. The amplitude of the ongoing activity that was directly and reproducibly related to the spontaneous spikes of a single neuron was, on average, as high as 54% of the amplitude of the visually evoked response that was directly related to optimal sensory stimulation, recorded optically. 6. Coherent activity was detected even at distant cortical sites up to 6 mm apart.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals↗

Self-organization of chaos synchronization and pattern formation in coupled chaotic oscillators.

Pattern formation in spatiotemporal chaotic systems is investigated. Temporally chaotic and spatially ordered patterns are observed by varying the coupling strength. Spatial orderings emerge spontaneously due to self-organization of partial and nonlocal chaos synchronization, governed by various types of spatial symmetries. The first and secondary bifurcations from spatially disordered chaos to chaos with different levels of spatial orderings are observed and the scaling behaviors associated with these bifurcations are statistically analyzed.

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Propagation of desynchronous disturbances in synchronized chaotic one-way coupled map lattices.

Propagations of desynchronous perturbations in synchronization processes of spatiotemporal chaos are investigated by considering chaotic one-way coupled map lattices. Under large coupling approximation the desynchronous area in time space is analytically calculated, based on the concepts of comoving Lyapunov exponent and absolute largest Lyapunov exponent. The ideas used in this paper are expected to be applicable to synchronizations of spatiotemporally chaotic systems of coupled maps and coupled oscillators with convective instability.

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Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses

We present a model of isodiffracting single-cycle and few-cycle ultrashort electromagnetic pulses. The model is based on exact solutions of the time-dependent paraxial wave equation with space-time coupling effects included. The spatiotemporal structure of these pulses is characterized by a scaling parameter which relates off-axis pulse shapes to the axial temporal waveforms. Depending on the spectrum a pulse may transform itself from a single-cycle pulse to a multicycle pulse along the radial coordinate. This model is also used to describe recirculating pulses in a curved mirror cavity resonator. The Gouy phase shift contributes an absolute phase that results in a pulse-to-pulse temporal instability.

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Synchronization of extended chaotic systems with long-range interactions: an analogy to Lévy-flight spreading of epidemics.

Spatially extended chaotic systems with power-law decaying interactions are considered. Two coupled replicas of such systems synchronize to a common spatiotemporal chaotic state above a certain coupling strength. The synchronization transition is studied as a nonequilibrium phase transition and its critical properties are analyzed at varying the interaction range. The transition is found to be always continuous, while the critical indexes vary with continuity with the power-law exponent characterizing the interaction. Strong numerical evidences indicate that the transition belongs to the anomalous directed percolation family of universality classes found for Lévy-flight spreading of epidemic processes.

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Spatiotemporal intermittency on fractal lattices.

The transition to turbulence via spatiotemporal intermittency is investigated for coupled maps defined on generalized Sierpinski gaskets, a class of deterministic fractal lattices. Critical exponents that characterize the onset of intermittency are computed as a function of the fractal dimension of the lattice. Windows of spatiotemporal intermittency are found as the coupling parameter is varied for lattices with a fractal dimension greater than two. This phenomenon is associated with a collective chaotic behavior of the fractal array of coupled maps.

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Pattern formation in stiff oscillatory media with nonlocal coupling: a numerical study of the hydrogen oxidation reaction on Pt electrodes in the presence of poisons.

The impact of the strength of negative (desynchronizing) global coupling (NGC) on the spatiotemporal dynamics of an electrochemical relaxation oscillator is studied numerically with a prototypical model, the electro-oxidation of hydrogen in the presence of poisons. The results are compared with recent experiments. The NGC has a destabilizing effect on the homogeneous oscillations. Both, in theory and in experiments, the basic patterns found with increasing global coupling strength are modulated oscillations, target patterns (including an asymmetric variant), and modulated pulses, the average spatial inhomogeneity during an oscillation increasing with the intensity of the NGC. It is suggested that this scenario is typical for strong relaxation oscillations, and a comparison with an electrochemical oscillator exhibiting harmonic oscillations points to the fact that the critical coupling strength, upon which the complete synchronization is destroyed, is larger for relaxation oscillations than for harmonic oscillations. In addition, the numerical simulations predicted two- and three-phase cluster patterns at high coupling strength. Also in experiments cluster patterns were observed, however only in parameter regions of the local dynamics which were different from the one investigated in this study.

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Ordering spatiotemporal chaos in complex thermosensitive neuron networks.

We have studied the effect of random long-range connections in chaotic thermosensitive neuron networks with each neuron being capable of exhibiting diverse bursting behaviors, and found stochastic synchronization and optimal spatiotemporal patterns. For a given coupling strength, the chaotic burst-firings of the neurons become more and more synchronized as the number of random connections (or randomness) is increased and, rather, the most pronounced spatiotemporal pattern appears for an optimal randomness. As the coupling strength is increased, the optimal randomness shifts towards a smaller strength. This result shows that random long-range connections can tame the chaos in the neural networks and make the neurons more effectively reach synchronization. Since the model studied can be used to account for hypothalamic neurons of dogfish, catfish, etc., this result may reflect the significant role of random connections in transferring biological information.

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