Periodic Spatiotemporal Patterns Research Articles

Periodic patterns in a ring of delay-coupled oscillators

We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.

Mining periodic patterns in spatio-temporal sequences at different time granularities1

With the advancement of technology, it is now easy to collect the location information of mobile users over time. Spatio-temporal data mining techniques were proposed in the literature for the extraction of patterns from spatio-temporal data. However, current techniques can only extract patterns of the finest time granularity, and therefore overlooks potential patterns available at coarser time granularities. In this work, we propose two techniques to allow mining at different time granularities. Experimental results show that the proposed techniques are indeed effective and efficient for mining periodic spatio-temporal patterns at different time granularities.

Learning of Spatiotemporal Patterns in Ising-Spin Neural Networks: Analysis of Storage Capacity by Path Integral Methods

We encode periodic spatiotemporal patterns in Ising-spin neural networks, using the simple learning rule inspired by the spike-timing-dependent synaptic plasticity. It is then found that periodically oscillating spin neurons successfully reproduce phase differences of the encoded periodic patterns. The storage capacity of this associative memory neural network is enhanced with an adequate level of asymmetry in synapse connections. To understand the properties of these nonequilibrium retrieval states of the neural network, we carry out an analysis based on a path integral method. The relation of a dynamic crosstalk term to time-persistent oscillation of a correlation function well explains the enhancement of the storage capacity in spite of our approximation on nonpersistent terms. We investigate the accuracy of this approximation further by detailed comparison with numerical simulations.

Spatiotemporal learning in analog neural networks using spike-timing-dependent synaptic plasticity

Incorporating the spike-timing-dependent synaptic plasticity (STDP) into a learning rule, we study spatiotemporal learning in analog neural networks. First, we study learning of a finite number of periodic spatiotemporal patterns by deriving the dynamics of the order parameters. When a pattern is retrieved successfully, the order parameters exhibit periodic oscillation. Analyzing this oscillation of the order parameters, we elucidate the relation of the STDP time window to the properties of the retrieval state; the phase of the Fourier transform of the STDP time window determines the retrieval frequency and the time average of the STDP time window crucially affects the storage capacity. We also evaluate the stability of the order parameter oscillation and identify the retrieval state that is stable in single-pattern learning but unstable in multiple-pattern learning even when the retrieval state is independent of a pattern number. To examine the further applicability of the STDP-based learning rule, we also study learning of nonperiodic spatiotemporal Poisson patterns. Our numerical simulations demonstrate that the Poisson patterns are memorized successfully not only in analog neural networks but also in spiking neural networks.

Rhythmic Episodes of Subthreshold Membrane Potential Oscillations in the Rat Inferior Olive NucleiIn Vivo

In vitro studies of inferior olive neurons demonstrate that they are intrinsically active, generating periodic spatiotemporal patterns. These self-generated patterns of activity extend the role of olivary neurons beyond that of a deliverer of teaching or error signals. However, autorhythmicity or patterned activity of complex spikes in the cerebellar cortex was observed in only a few studies. This discrepancy between the self-generated rhythmicity in the inferior olive observed in vitro and the sporadic reports on rhythmicity of complex spikes can be reconciled by recording intracellularly from inferior olive neurons in situ. To this end, we recorded intracellularly from olivary neurons of anesthetized rats. We demonstrate that, in vivo, olivary neurons show both slow and fast rhythmic processes. The slow process (0.2-2 Hz) is expressed as rhythmic transitions from quiescent periods to periods of fast rhythm, manifested as subthreshold oscillations of 6-12 Hz. Spikes, if they occur, are locked to the depolarized phase of these subthreshold oscillations and, therefore, hold and transfer rhythmic information. The transient nature of these oscillatory epochs accounts for the difficulties to uncover them by prolonged recordings of complex spikes activity in the cerebellar cortex.

Coexistence of synchronized and desynchronized patterns in coupled chaotic dynamical systems

We analyse the synchronous chaos and its stability in coupled map lattices and coupled nonlinear oscillators. The existence of desynchronized patterns such as standing wave like spatiotemporal periodic patterns within the critical size limit is identified for some sets of random initial conditions. The emergence of such patterns is explained using the concept of controlling of chaos. In the case of coupled map lattices, an expression for standing wave like pattern is given in terms of unstable periodic orbits of the isolated map. Finally, the role of initial conditions on the existence of multiple stable states in both coupled map lattices and coupled oscillators is studied.

General soliton solutions for nonlinear dispersive waves in convective type instabilities

The two-dimensional Ginzburg–Landau equation (GLE) is obtained from basic equations by a linear stability analysis. This equation governs the evolution of slowly varying envelopes of periodic spatio-temporal patterns related to Rayleigh–Bénard convective instabilities. In addition, the phase instabilities of the complex GLE (CGLE) with quintic and space-dependent cubic terms modelling the Eckhaus and zigzag convective instabilities are reported. We find soliton solution classes to the elliptic and hyperbolic CGLE, by applying the function transformation method. The two-dimensional CGLE is transformed to a sine-Gordon equation, a sinh-Gordon equation and other equations, which depend only on one function χ. The general solution of the equation in χ leads to a general soliton solution of the two-dimensional CGLE. The obtained solutions contain some interesting specific solutions such as plane solitons, N multiple solitons and propagating breathers. We also discuss the soliton stability of the CGLE.

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

We present a short survey on the biological modeling, dynamics analysis, and numerical simulation of nonlocal spatial effects, induced by time delays, in diffusion models for a single species confined to either a finite or an infinite domain. The nonlocality, a weighted average in space, arises when account is taken of the fact that individuals have been at different points in space at previous times. We discuss and compare two existing approaches to correctly derive the spatial averaging kernels, and we summarize some of the recent developments in both qualitative and numerical analysis of the nonlinear dynamics, including the existence, uniqueness (up to a translation), and stability of traveling wave fronts and periodic spatio-temporal patterns of the model equations in unbounded domains and the linear stability, boundedness, global convergence of solutions and bifurcations of the model equations in finite domains.

Complex patterns in three-dimensional excitable media with advection

Wave propagation in three-dimensional excitable media subject to homogeneous Neumann boundary conditions and subject to irrotational or rotational advective fields which satisfy the no-penetration condition is studied numerically. It is shown that the velocity field annihilates spiral waves and results in complex periodic spatio-temporal patterns which are characterized by either planar or curved fronts depending on the flow compressibility, number and location of stagnation points, rotation and straining. For sinusoidal velocity fields with a frequency equal to one, almost planar fronts which resemble those found in two-dimensional excitable media are found, whereas, when the frequency is increased, several curved fronts propagate at about the same speed and surround each other when they approach an edge of the domain. It is also shown that the main effect of increasing the frequency of the velocity field on the concentrations at fixed monitor locations is to decrease the period of the spikes of the activator’s concentration.

Linear stability analysis of retrieval state in associative memory neural networks of spiking neurons.

We study associative memory neural networks of the Hodgkin-Huxley type of spiking neurons in which multiple periodic spatiotemporal patterns of spike timing are memorized as limit-cycle-type attractors. In encoding the spatiotemporal patterns, we assume the spike-timing-dependent synaptic plasticity with the asymmetric time window. Analysis for periodic solution of retrieval state reveals that if the area of the negative part of the time window is equivalent to the positive part, then crosstalk among encoded patterns vanishes. Phase transition due to the loss of the stability of periodic solution is observed when we assume fast function for direct interaction among neurons. In order to evaluate the critical point of this phase transition, we employ Floquet theory in which the stability problem of the infinite number of spiking neurons interacting with function is reduced to the eigenvalue problem with the finite size of matrix. Numerical integration of the single-body dynamics yields the explicit value of the matrix, which enables us to determine the critical point of the phase transition with a high degree of precision.

SPATIOTEMPORAL PERIODIC PATTERNS OF A TWO-DIMENSIONAL SYMMETRICALLY COUPLED MAP LATTICES

Aim: The spatiotemporal periodic pattern of a two-dimensional symmetrically coupled map lattice is constructed. Method: Without solving the modeling equations, a series of spatiotemporal periodic orbits in coupled map lattices are deduced by known orbits of one-dimensional coupled map lattices with lower spatial period. The stability of the deduced orbits is analyzed. Results: The L2×L2 Jacobian matrices can be simplified as diagonal matrices of a few 2×2 matrices. Conclusion: The stability of constructed orbits can never be better than that of the original ones.

Spatiotemporal Patterns on a Ring Array of Electrodes

Experiments have been carried out with a ring array of iron electrodes in sulfuric acid under conditions in which high-frequency periodic and chaotic oscillations occur. All the electrodes were held at the same constant potential. The currents of the electrodes were measured independently, and therefore, the spatiotemporal patterns in current density which occurred were directly determined. Depending on the conditions of the experiments, primarily applied potential, and also solution concentration, a variety of periodic and chaotic oscillatory spatiotemporal patterns arose. The simplest case, a uniform periodic oscillation, does occur, but it is not the most common state. With a change of parameter, higher modes become important. One set of changes among pattern types with variation of potential, including an apparent period doubling, is investigated in detail. Other types of behavior including antiphase oscillations and chaotic patterns were also observed.

Periodic, quasiperiodic and chaotic spatiotemporal patterns in a tubular catalytic reactor with periodic flow reversal

Periodic, quasiperiodic and chaotic symmetric and unsymmetric spatiotemporal temperature patterns arise in the heterogeneous (two-phase) model of the catalytic reactor with periodic flow reversal. Coexistence of various types of stable solutions was observed. Chaotic solutions arising via the torus breaking route were also observed. The transitions among individual types of spatiotemporal patterns and their development are studied by means of Poincaré maps of the associated model represented as a one-parameter family of discrete dynamical systems derived from the simulation results of the original continuous model.

Wavelike patterns in one-dimensional coupled map lattices

We investigate the existence of wavelike solution for the logistic coupled map lattices for which the spatiotemporal periodic patterns can be predicted by a simple two-dimensional mapping. The existence of such wavelike solutions is proved by the implicit function theorem with constraints. We also examine the stabilities of these wave solutions under perturbations of uniform small deformation type. We show that in some specific cases these perturbations are completely general. The technique used in this paper is also applicable to investigate other space-time regular patterns.

Spatiotemporal periodic and chaotic patterns in a two-dimensional coupled map lattice system

The pattern dynamics of a two-dimensional coupled map lattice system is studied using both analytical and numerical calculations. Two interesting spatiotemporal periodic patterns are solved analytically. Their stability boundaries for small system size are obtained by linear stability analysis. As the system sizes mismatch the spatial periodicity of the patterns, a frozen random chaotic defect cluster and a slow random propagated chaotic defect string appear.

Spatiotemporal periodic pattern and propagated spatiotemporal on-off intermittency in the one-way coupled map lattice system.

One-way coupled map lattice system is investigated. An antiphase spatiotemporal period-2 state (a running wave), and an alternative type of propagated spatiotemporal on-off intermittency are found. The stability of the wave is discussed in both stationary frame and comoving frame. They are characterized by conventional Lyapunov exponent and comoving Lyapunov exponent, respectively. The stability boundaries of this state are also obtained in the parameter plane. Numerical calculation shows that the propagated spatiotemporal on-off intermittency has negative Lyapunov exponent. The distribution of the laminar phases for various sites is calculated numerically. Those sites far from the boundary site obey a power law with power exponent $\ensuremath{-}\frac{3}{2}$. This interesting phenomenon seems to be independent of the choices of local function map.

Periodic and irregular wave patterns in an open tubular gel reactor

In an open quasi-one-dimensional Belousov-Zhabotinsky reaction-diffusion system we investigated simple periodic, complex and irregular spatiotemporal concentration patterns emerging under oscillatory boundary conditions. Symmetric (identical) feed concentrations were used at each boundary of a tabular gel reactor and the length of the system was varied. For tubes shorter than 11 mm the intrinsic wavelength is larger than half the length of the tube. In this case regular pairs of pulse waves are generated. When the length of the tube was increased we found more complex spatiotemporal behavior. Small inevitable differences in the oscillation frequencies at the boundaries finally lead to irregular patterns. The Karhunen-Loeve decomposition technique was applied for an analysis of the observed patterns.

Chronotopic Lyapunov analysis. I. A detailed characterization of 1D systems

Instabilities in 1D spatially extended systems are studied with the aid of both temporal and spatial Lyapunov exponents. A suitable representation of the spectra allows a compact description of all the possible disturbances in tangent space. The analysis is carried out for chaotic and periodic spatiotemporal patterns. Singularities of the spectra and localization properties of the associated Lyapunov vectors are discussed.