Complex Oscillatory Phenomena Research Articles

KurSL: Model of Anharmonic Coupled Oscillations Based on Kuramoto Coupling and Sturm–Liouville Problem

Physiological signaling is often oscillatory and shows nonlinearity due to complex interactions of underlying processes or signal propagation delays. This is particularly evident in case of brain activity which is subject to various feedback loop interactions between different brain structures, that coordinate their activity to support normal function. In order to understand such signaling in health and disease, methods are needed that can deal with such complex oscillatory phenomena. In this paper, a data-driven method for analyzing anharmonic oscillations is introduced. The KurSL model incorporates two well-studied components, which in the past have been used separately to analyze oscillatory behavior. The Sturm–Liouville equations describe a form of a general oscillation, and the Kuramoto coupling model represents a set of oscillators interacting in the phase domain. Integration of these components provides a flexible framework for capturing complex interactions of oscillatory processes of more general form than the most commonly used harmonic oscillators. The paper introduces a mathematical framework of the KurSL model and analyzes its behavior for a variety of parameter ranges. The significance of the model follows from its ability to provide information about coupled oscillators’ phase dynamics directly from the time series. KurSL offers a novel framework for analyzing a wide range of complex oscillatory behaviors, such as the ones encountered in physiological signals.

Cross-sensory cuing drives cross-frequency neural coupling, dramatically altering performance of a taxing visual-detection task

Functional networks are comprised of neuronal ensembles bound through synchronization across multiple intrinsic oscillatory frequencies. Various coupled interactions between brain oscillators have been described (e.g., phase–amplitude coupling), but with little evidence that these interactions actually influence perceptual sensitivity. Here, electroencephalographic recordings were made during a sustained-attention task to demonstrate that cross-frequency coupling, driven by cross-sensory cuing, has significant consequences for perceptual outcomes (i.e., whether participants detect a near-threshold visual target). Our results reveal that phase-detection relationships at higher frequencies are entirely dependent on the phase of lower frequencies, such that higher frequencies alternate between periods when their phase is strongly predictive of visual-target detection and periods when their phase has no influence whatsoever. These data thus bridge the crucial gap between complex oscillatory phenomena and perceptual outcomes. Accounting for cross-frequency coupling between lower (i.e., delta and theta) and higher frequencies (e.g., beta and gamma), we show that visual-target detection fluctuates dramatically as a function of pre-stimulus phase, with performance swings of as much as 80%.

Contributions of Mathematical Modeling of Beta Cells to the Understanding of Beta-Cell Oscillations and Insulin Secretion

Mathematical modeling of pancreatic beta cells has contributed significantly to the understanding of the mechanisms involved in glucose-stimulated insulin secretion (GSIS). Early models of insulin secretion built in the 1970s were phenomenological with little biological foundation for the proposed mechanisms. In the 1980s, models focused on identifying the regulation of bursting electrical activity known to be important for insulin secretion. The main result was to reject proposed mechanisms as new data emerged, but important results of the role of cell-to-cell coupling were also established. New models have been proposed that provide possible explanations for the occurrence of various patterns of bursting and calcium oscillations. In addition, modeling has played an important role in comparing competing effects of calcium on both NADH and adenosine 3'-5'-cyclic monophosphate levels. Models including modern cell biological results of the regulation of insulin containing granules and cell heterogeneity have appeared, providing updated versions of the early models proposed in the 1970s. These models, when coupled to electrophysiological- and calcium-based ones, have the prospect to aid in understanding the overall picture of GSIS. In addition, they might be useful for estimating in vivo beta-cell functioning. Beta-cell modeling will likely move closer to clinical applications, where it can be expected to play an important role, as it has and will, in understanding the complex oscillatory phenomena observed in beta cells and islets.

The importance of matrix quality in fragmented landscapes: Understanding ecosystem collapse through a combination of deterministic and stochastic forces

Increasing fragmentation of natural landscapes due to anthropogenic forces are changing the ecological structure of many systems. The corpus of literature has addressed the fact that there are many complex mechanisms that may lead to population and ecosystem collapse. Here, we propose an additional mechanism that combines the deterministic framework of complex oscillatory phenomena with the stochastic framework based on the spatial phenomena of island biogeography and metapopulation theory. Combining these two frameworks, which we believe exist in disturbed natural systems, we construct a situation in which collections of competitors are controlled by specialist natural enemies. Species are isolated in fragmented habitats separated by a matrix of decreasing ecological quality, which will determine immigration potential. In this system, species are exposed to stochastic extinction of spatial immigration potential and deterministic extinction of standard oscillatory frameworks. Through the combination of these extinction events, we see that many natural systems may become increasingly unstable and subject to unexpected consequences with changes to the matrix environment. The combined effect of these two extinctions pathways highlights the need to maintain high immigration rates in order to offset possible ecosystem collapse in fragmented systems. The results highlight the importance of landscape conservation efforts, especially towards maintaining matrix environments with greater ecological integrity.

Quantitative modelling of voltage oscillations and other oscillatory phenomena with the current burst model

AbstractThe current burst model (CBM) has been extended to include more complex oscillatory phenomena of the Si electrode in aqueous HF. In particular it accounts for additional effects inherent to the voltage oscillations, i.e. capacitive effects, very small oxide thickness, etc. A powerful Monte‐Carlo implementation now allows to simulate all presently known aspects of electrodes. Besides current oscillations under potentiostatic conditions, the CBM can now quantitatively produce voltage oscillations and other more complex phenomena like driven oscillations in close agreement with experiments. New data extraction routines provide information about, e.g., correlation lengths and thus bridge the gap to general non‐linear pattern formation theories. The CBM is far simpler than competing theories, but is nevertheless the only model presently capable of quantitatively simulating all aspects of oscillatory phenomena of the Si electrode. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24 h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER–TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light–dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.

From simple to complex oscillatory behavior in metabolic and genetic control networks.

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.

Mathematical modeling of complex oscillatory phenomena during CO oxidation over Pd zeolite catalysts

A mathematical model, which simulates the complicated dynamic behavior experimentally observed during CO oxidation over Pd zeolite catalysts is presented. It describes the coupling of reaction rate oscillations, generated by various parts of the inhomogeneous catalytic layer through the gas phase. It can be shown, that the resulting dynamic behavior depends upon the difference between natural frequencies of local oscillators and the strength of coupling, which is defined mostly by the degree of conversion. Chaotic behavior could be identified under the condition of weak coupling for local oscillators with widely different natural frequencies. In the range of strong coupling the phenomenon of phase death has been obtained. A special type of intermittency chaos (“on–off” chaos) was observed in a small region of parameters under the conditions of strong coupling.

Bursting, Chaos and Birhythmicity Originating from Self-modulation of the Inositol 1,4,5-trisphosphate Signal in a Model for Intracellular Ca2 +Oscillations

We investigate the various types of complex Ca2+oscillations which can arise in a model based on the mechanism of Ca2+-induced Ca2+release (CICR), that takes into account the Ca2+- stimulated degradation ofinositol 1,4,5-trisphosphate (InsP3) by a 3-kinase.This model was previously proposed in the course of an investigation ofplausible mechanismscapable of generating complex Ca2+oscillations(Borghans et al. , 1997 ). Besides simple periodic behavior, this model for cytosolic Ca2+oscillations in nonexcitable cells shows complex oscillatory phenomena like bursting or chaos. We show that the model also admits a coexistence between two stable regimes of sustained oscillations (birhythmicity). The occurrence of the sevarious modes of oscillatory behavior is analysed by means of bifurcation diagrams. Complex oscillations are characterized by means of Poincaré sections, power spectra and Lyapounov exponents. The results point to the role of self-modulation of the InsP3signal by 3-kinase as a possible source for complex temporal patterns in Ca2+signaling.

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.

Finding complex oscillatory phenomena in biochemical systems. An empirical approach.

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.

Temporal self-organization in biochemical systems: periodic behavior vs. chaos.

The patterns of temporal self-organization in regulated biochemical systems are examined. Simple periodic oscillations are the most frequent type of such organization, as exemplified by glycolytic oscillations in yeast and muscle and by the periodic synthesis of adenosine 3',5'-cyclic monophosphate in Dictyostelium discoideum amoebas. These phenomena originate, respectively, from the periodic operation of the product-activated phosphofructokinase and adenylate cyclase reactions. The analysis of a model for a multiply regulated biochemical system shows more complex oscillatory phenomena, e.g., the coexistence between two stable periodic regimes for the same set of parameter values (birhythmicity) and chaos. The latter phenomenon of aperiodic oscillations occurs in a narrow range of parameter values and is much less frequent than simple or complex periodic behavior. It is suggested that a sufficient condition for the occurrence of birhythmicity and chaos in a regulated biological system subjected to a constant environment (i.e., in the absence of periodic forcing) may be the simultaneous presence and interaction of two mechanisms capable of producing oscillations.