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PubMed2001

An efficient and robust statistical modeling approach to discover differentially expressed genes using genomic expression profiles.

We have developed a statistical regression modeling approach to discover genes that are differentially expressed between two predefined sample groups in DNA microarray experiments. Our model is based on well-defined assumptions, uses rigorous and well-characterized statistical measures, and accounts for the heterogeneity and genomic complexity of the data. In contrast to cluster analysis, which attempts to define groups of genes and/or samples that share common overall expression profiles, our modeling approach uses known sample group membership to focus on expression profiles of individual genes in a sensitive and robust manner. Further, this approach can be used to test statistical hypotheses about gene expression. To demonstrate this methodology, we compared the expression profiles of 11 acute myeloid leukemia (AML) and 27 acute lymphoblastic leukemia (ALL) samples from a previous study (Golub et al. 1999) and found 141 genes differentially expressed between AML and ALL with a 1% significance at the genomic level. Using this modeling approach to compare different sample groups within the AML samples, we identified a group of genes whose expression profiles correlated with that of thrombopoietin and found that genes whose expression associated with AML treatment outcome lie in recurrent chromosomal locations. Our results are compared with those obtained using t-tests or Wilcoxon rank sum statistics.

Acute Disease↗
PubMed1999

Robustness of supersensitivity to small signals in nonlinear dynamical systems.

Nonlinear dynamical systems possessing an invariant subspace can display interesting dynamical behavior, such as on-off intermittency and bubbling. This Rapid Communication shows that a class of such systems has amazing features of (1) supersensitivity to small input signals and (2) robustness of the supersensitivity in the presence of noise. These features make the systems very promising as small signal detectors.

Journal Article↗
PubMed2000

Robust synchronization of chaotic systems.

The question of robustness of synchronization with respect to small arbitrary perturbations of the underlying dynamical systems is addressed. We present examples of chaos synchronization demonstrating that normal hyperbolicity is a necessary and sufficient condition for the synchronization manifold to be smooth and persistent under small perturbations. The same examples, however, show that in real applications normal hyperbolicity is not sufficient to give quantitative bounds for deformations of the synchronization manifold, i.e., even in the case of normal hyperbolicity two almost identical systems may cause large synchronization errors.

Animals↗
PubMed2000

Symmetry-breaking on-off intermittency under modulation: robustness of supersensitivity, resonance, and information gain

Nonlinear dynamical systems possessing an invariant subspace in the phase space and chaotic or stochastic motion within the subspace often display on-off intermittency close to the threshold of stability of the subspace. In a class of symmetric systems, the intermittency is symmetry breaking [Ying-Cheng Lai, Phys. Rev. E 53, R4267 (1996)]. We report interesting and practically important universal behavior of robustness of supersensitivity, resonance, and information gain in this class of systems when subjected to a weak modulation. While intermittent loss of synchronization may be harmful to the application of high-quality synchronization of coupled chaotic systems, the features reported here may lead to interesting application of on-off intermittency.

Journal Article↗
PubMed2001

Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling.

The phenomenon of slow switching in populations of globally coupled oscillators is discussed. This characteristic collective dynamics, which was first discovered in a particular class of the phase oscillator model, is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population. We argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom. We also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators. Although a heteroclinic loop might seem to arise only exceptionally, we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony. We argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle-node bifurcations. Slow switching arises when a system with a heteroclinic loop is weakly perturbed. This will be demonstrated with a vector model by applying weak noises. Other types of weak symmetry-breaking perturbations can also cause slow switching.

Journal Article↗
PubMed2001

Noise robust estimates of correlation dimension and K2 entropy.

Using Gaussian kernels to define the correlation sum we derive simple formulas that correct the noise bias in estimates of the correlation dimension and K2 entropy of chaotic time series. The corrections are only based on the difference of correlation dimensions for adjacent embedding dimensions and hence preserve the full functional dependencies on both the scale parameter and embedding dimension. It is shown theoretically that the estimates, which are derived for additive white Gaussian noise, are also robust for moderately colored noise. Simulations underline the usefulness of the proposed correction schemes. It is demonstrated that the method gives satisfactory results also for non-Gaussian and dynamical noise.

Journal Article↗
PubMed2002

Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors.

Chaotic attractors arising in physical systems are often nonhyperbolic. We compare two sources of nonhyperbolicity: (1) tangencies between stable and unstable manifolds, and (2) unstable dimension variability. We study the effects of noise on chaotic attractors with these nonhyperbolic behaviors by investigating the scaling laws for the Hausdorff distance between the noisy and the deterministic attractors. Whereas in the presence of tangencies, interactive noise yields attractor deformations, attractors with only dimension variability are robust, despite the fact that shadowing is grossly violated.

Journal Article↗
PubMed2002

Multiscale dynamics and robust critical scaling in a continuum current sheet model.

We analyze the self-organized critical behavior of a continuum running avalanche model. We demonstrate that over local interaction scales, the model behavior is affected by low-dimensional chaotic dynamics that plays the role of the primary noise source. With the help of scale-free avalanches, the uncertainty associated with chaos is distributed over a variety of intermediate scales and thus gives rise to spatiotemporal fluctuations that are characterized by power-law distribution functions. We show that globally, the continuum model displays structurally stable critical scaling that can be observed in a finite region in the control parameter space. In this region, the system exhibits a power-law critical divergence of the integrated response function over a broad range of dissipation rates. The observed behavior involves a remarkably stable spatial configuration. We explain the robust features of the model by the adjustable dynamics of its global loading-unloading cycle, which allows maintaining the long-term stationary state without affecting the intrinsic avalanche dynamics.

Journal Article↗
PubMed2002

Square spiral photonic crystals: robust architecture for microfabrication of materials with large three-dimensional photonic band gaps.

We provide a blueprint for microfabricating photonic crystals with very large and robust three-dimensional photonic band gaps (PBG's). These crystals are based on interleaving polygonal spiral posts and can be efficiently manufactured on a large scale in a one-step process using the recently introduced technique of glancing angle deposition. We classify various families of spiral photonic crystals based on (i) the parent three-dimensional (3D) point lattice to which they are most closely related, (ii) the crystallographic axis of the parent lattice around which the spiral posts wind, and (iii) the set of points of the parent lattice which the spiral arms connect or nearly connect. We obtain optimal geometries for the spiral photonic crystals by detailed mapping of the size and location of the PBG within a multidimensional parameter space which characterizes the shape of each spiral post. For the optimum PBG, the spiral arms and elbows may deviate significantly from the points of the original point lattice. The largest 3D PBG's are obtained for square spiral posts that wind around the [001] axis of diamond (or face centered cubic) lattice and in which the spiral arm segments approximately connect either the fifth or first nearest-neighbor points of the parent lattice. In the case of silicon posts (with a dielectric constant of 11.9) in an air background, whose arm segments nearly connect fifth nearest-neighbor point of the diamond lattice, the full PBG can be as large as 16% of the gap center frequency. For the corresponding air posts in a silicon background, the maximum PBG is 24% of the gap center frequency. We compute both the total electromagnetic density of states and the electromagnetic field distributions near the PBG. It is suggested the PBG in an optimized structure is highly tolerant to various forms of disorder that may arise during the manufacturing process.

Journal Article↗
PubMed2002

Robustness of the in-degree exponent for the World-Wide Web.

We consider a stochastic model for directed scale-free networks following power laws in the degree distributions in both incoming and outgoing directions. In our model, the number of vertices grow geometrically with time with a growth rate p. At each time step, (i) each newly introduced vertex is connected to a constant number of already existing vertices with the probability linearly proportional to in-degree distribution of a selected vertex, and (ii) each existing vertex updates its outgoing edges through a stochastic multiplicative process with mean growth rate of outgoing edges g and its variance sigma(2). Using both analytic treatment and numerical simulations, we show that while the out-degree exponent gamma(out) depends on the parameters, the in-degree exponent gamma(in) has two distinct values, gamma(in)=2 for p>g and 1 for p<g, independent of different parameters values. The latter case has logarithmic correction to the power law. Since the vertex growth rate p is larger than the degree growth rate g for the World-Wide Web (WWW) nowadays, the in-degree exponent appears robust as gamma(in)=2 for the WWW.

Journal Article↗
PubMed2003

Robustness and enhancement of neural synchronization by activity-dependent coupling.

We study the synchronization of two model neurons coupled through a synapse having an activity-dependent strength. Our synapse follows the rules of spike-timing dependent plasticity. We show that this plasticity of the coupling between neurons produces enlarged frequency-locking zones and results in synchronization that is more rapid and much more robust against noise than classical synchronization arising from connections with constant strength. We also present a simple discrete map model that demonstrates the generality of the phenomenon.

Animals↗
PubMed2003

Robustness and perturbation in the modeled cascade heart rate variability.

In this study, numerical experiments are conducted to examine the robustness of using cascade to describe the multifractal heart rate variability (HRV) by perturbing the hierarchical time scale structure and the multiplicative rule of the cascade. It is shown that a rigid structure of the multiple time scales is not essential for the multifractal scaling in healthy HRV. So long as there exists a tree structure for the multiplication to take place, a multifractal HRV and related properties can be captured by using the cascade. But the perturbation of the multiplicative rule can lead to a qualitative change. In particular, a multifractal to monofractal HRV transition can result after the product law is perturbed to an additive one at the fast time scale. We suggest that this explains the similar HRV scaling transition in the parasympathetic nervous system blockade.

Animals↗
PubMed2003

Fractal properties of robust strange nonchaotic attractors in maps of two or more dimensions.

We consider the existence of robust strange nonchaotic attractors in a simple class of quasiperiodically forced systems. Rigorous results are presented demonstrating that the resulting attractors are strange in the sense that their box-counting dimension D0 is larger than their information dimension D1 by 1 (i.e., D(0)=D(1)+1). We also show how this property is manifested in numerical experiments.

Journal Article↗
PubMed2003

Soliton "molecules": robust clusters of spatiotemporal optical solitons.

We show how to generate robust self-sustained clusters of soliton bullets-spatiotemporal (optical or matter-wave) solitons. The clusters carry an orbital angular momentum being supported by competing nonlinearities. The "atoms" forming the "molecule" are fully three-dimensional solitons linked via a staircaselike macroscopic phase. Recent progress in generating atomic-molecular coherent mixing in the Bose-Einstein condensates might open potential scenarios for the experimental generation of these soliton molecules with matter waves.

Journal Article↗
PubMed2003

Robust soliton clusters in media with competing cubic and quintic nonlinearities.

Systematic results are reported for dynamics of circular patterns ("necklaces"), composed of fundamental solitons and carrying orbital angular momentum, in the two-dimensional model, which describes the propagation of light beams in bulk media combining self-focusing cubic and self-defocusing quintic nonlinearities. Semianalytical predictions for the existence of quasistable necklace structures are obtained on the basis of an effective interaction potential. Then, direct simulations are run. In the case when the initial pattern is far from an equilibrium size predicted by the potential, it cannot maintain its shape. However, a necklace with the initial shape close to the predicted equilibrium survives very long evolution, featuring persistent oscillations. The quasistable evolution is not essentially disturbed by a large noise component added to the initial configuration. Basic conclusions concerning the necklace dynamics in this model are qualitatively the same as in a recently studied one which combines quadratic and self-defocusing cubic nonlinearities. Thus, we infer that a combination of competing self-focusing and self-defocusing nonlinearities enhances the robustness not only of vortex solitons but also of vorticity-carrying necklace patterns.

Journal Article↗
PubMed2003

Robustness of retrieval properties against imbalance between long-term potentiation and depression of spike-timing-dependent plasticity.

Spike-timing-dependent plasticity (STDP) has recently been shown in some physiological studies. STDP depends on the precise temporal relationship of presynaptic and postsynaptic spikes. Many authors have indicated that a precise balance between long-term potentiation (LTP) and long-term depression (LTD) of STDP is significant for a stable learning. However, a situation in which the balance is maintained precisely is inconceivable in the brain. Using a method of the statistical neurodynamics, we show robust retrieval properties of spatiotemporal patterns in an associative memory model against the imbalance between LTP and LTD. When the fluctuation of LTD is assumed to obey a Gaussian distribution with mean 0 and variance delta(2), the storage capacity takes a finite value even at large delta. This means that the balance between LTP and LTD of STDP need not be maintained precisely, but must be maintained on average. Furthermore, we found that the basin of attraction becomes smaller as delta increases while an initial critical overlap remains unchanged.

Animals↗
PubMed2004

Robustness of the inverse cascade in two-dimensional turbulence.

We study quasisteady inverse cascades in unbounded and bounded two-dimensional turbulence driven by time-independent injection and dissipated by molecular viscosity. It is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy-range energy spectrum to be steeper than k(-5) (ruling out a direct cascade) unless 1-r<<1. A direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales (1-r<<1). These facts underlie the robustness of the Kolmogorov-Kraichnan k(-5/3) inverse cascade, which is readily observable in numerical simulations without an accompanying direct enstrophy cascade. We numerically demonstrate an instance where the k(-5/3) inverse-cascading range is realizable with 79% of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region. As equilibrium is approached, the respective logarithmic slopes -alpha and -beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy alpha+beta approximately 8. These results are consistent with recent theoretical predictions.

Journal Article↗
PubMed2004

Design and robustness of delayed feedback controllers for discrete systems.

We study a matrix form of time-delay feedback control in the context of discrete time maps of high dimension. In almost all cases where standard proportional feedback control methods can achieve control, time-delay feedback controllers containing only static elements can be designed to achieve identical linear stability properties. Analysis of an example involving a ring of coupled maps that can be controlled at only two sites demonstrates that the time-delay controller equivalent to a standard optimal controller can be equally robust in the presence of noise, except at special points in parameter space where the uncontrolled system has a mode with Floquet multiplier exactly equal to 1. Numerical simulations confirm the results of the analysis.

Journal Article↗