Shil'nikov homoclinic chaos is intimately related to type-III intermittency in isolated rabbit arteries: role of nitric oxide

Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification

In this paper, we show how the phenomenon of intermittency observed in systems with turbulent flow-sound interaction is related to the formation of homoclinic orbits in the phase space. Such orbits that emerge via the intersection of the stable and unstable manifold of an equilibrium configuration result from interactions that happen at multiple spatial/temporal scales associated with turbulent convection and wave propagation. Through a quantification of the time spent by the dynamics in the aperiodic states using recurrence plots, we show how the presence of homoclinic orbits in the dynamics may be convincingly demonstrated, which is often not possible through a visual inspection of the phase space of the attractor.

Skin blood flow dynamics and its role in pressure ulcers

Pressure ulcers are a significant healthcare problem affecting the quality of life in wheelchair bounded or bed-ridden people and are a major cost to the healthcare system. Various assessment tools such as the Braden scale have been developed to quantify the risk level of pressure ulcers. These tools have provided an initial guideline on preventing pressure ulcers while additional assessments are needed to improve the outcomes of pressure ulcer prevention. Skin blood flow function that determines the ability of the skin in response to ischemic stress has been proposed to be a good indicator for identifying people at risk of pressure ulcers. Wavelet spectral and nonlinear complexity analyses have been performed to investigate the influences of the metabolic, neurogenic and myogenic activities on microvascular regulation in people with various pathological conditions. These findings have contributed to the understanding of the role of ischemia and viability on the development of pressure ulcers. The purpose of the present review is to provide an introduction of the basic concepts and approaches for the analysis of skin blood flow oscillations, and present an overview of the research results obtained so far. We hope this information may contribute to the development of better clinical guidelines for the prevention of pressure ulcers.

Multiple factors influence calcium synchronization in arterial vasomotion

The intercellular synchronization of spontaneous calcium (Ca(2+)) oscillations in individual smooth muscle cells is a prerequisite for vasomotion. A detailed mathematical model of Ca(2+) dynamics in rat mesenteric arteries shows that a number of synchronizing and desynchronizing pathways may be involved. In particular, Ca(2+)-dependent phospholipase C, the intercellular diffusion of inositol trisphosphate (IP(3), and to a lesser extent Ca(2+)), IP(3) receptors, diacylglycerol-activated nonselective cation channels, and Ca(2+)-activated chloride channels can contribute to synchronization, whereas large-conductance Ca(2+)-activated potassium channels have a desynchronizing effect. Depending on the contractile state and agonist concentrations, different pathways become predominant, and can be revealed by carefully inhibiting the oscillatory component of their total activity. The phase shift between the Ca(2+) and membrane potential oscillations can change, and thus electrical coupling through gap junctions can mediate either synchronization or desynchronization. The effect of the endothelium is highly variable because it can simultaneously enhance the intercellular coupling and affect multiple smooth muscle cell components. Here, we outline a system of increased complexity and propose potential synchronization mechanisms that need to be experimentally tested.

Modeling Ca2+ signaling in the microcirculation: intercellular communication and vasoreactivity

A network of intracellular signaling pathways and complex intercellular interactions regulate calcium mobilization in vascular cells, arteriolar tone, and blood flow. Different endothelium-derived vasoreactive factors have been identified and the importance of myoendothelial communication in vasoreactivity is now well appreciated. The ability of many vascular networks to conduct signals upstream also is established. This phenomenon is critical for both short-term changes in blood perfusion as well as long-term adaptations of a vascular network. In addition, in a phenomenon termed vasomotion, arterioles often exhibit spontaneous oscillations in diameter. This is thought to improve tissue oxygenation and enhance blood flow. Experimentation has begun to reveal important aspects of the regulatory machinery and the significance of these phenomena for the regulation of local perfusion and oxygenation. Mathematical modeling can assist in elucidating the complex signaling mechanisms that participate in these phenomena. This review highlights some of the important experimental studies and relevant mathematical models that provide the current understanding of these mechanisms in vasoreactivity.

Calcium dynamics and signaling in vascular regulation: computational models

Calcium is a universal signaling molecule with a central role in a number of vascular functions including in the regulation of tone and blood flow. Experimentation has provided insights into signaling pathways that lead to or affected by Ca(2+) mobilization in the vasculature. Mathematical modeling offers a systematic approach to the analysis of these mechanisms and can serve as a tool for data interpretation and for guiding new experimental studies. Comprehensive models of calcium dynamics are well advanced for some systems such as the heart. This review summarizes the progress that has been made in modeling Ca(2+) dynamics and signaling in vascular cells. Model simulations show how Ca(2+) signaling emerges as a result of complex, nonlinear interactions that cannot be properly analyzed using only a reductionist's approach. A strategy of integrative modeling in the vasculature is outlined that will allow linking macroscale pathophysiological responses to the underlying cellular mechanisms.

Informational dynamics of vasomotion in microvascular networks: a review

Vasomotion refers to spontaneous oscillation of small vessels observed in many microvascular beds. It is an intrinsic phenomenon unrelated to cardiac rhythm or neural and hormonal regulation. Vasomotion is found to be particularly prominent under conditions of metabolic stress. In spite of a significant existent literature on vasomotion, its physiological and pathophysiological roles are not clear. It is thought that modulation of vasomotion by vasoactive substances released by metabolizing tissue plays a role in ensuring optimal delivery of nutrients to the tissue. Vasomotion rhythms exhibit a great variety of temporal patterns from regular oscillations to chaos. The nature of vasomotion rhythm is believed to be significant to its function, with chaotic vasomotion offering several physiological advantages over regular, periodic vasomotion. In this article, we emphasize that vasomotion is best understood as a network phenomenon. When there is a local metabolic demand in tissue, an ideal vascular response should extend beyond local microvasculature, with coordinated changes over multiple vascular segments. Mechanisms of information transfer over a vessel network have been discussed in the literature. The microvascular system may be regarded as a network of dynamic elements, interacting, either over the vascular anatomical network via gap junctions, or physiologically by exchange of vasoactive substances. Drawing analogies with spatiotemporal patterns in neuronal networks of central nervous system, we ask if properties like synchronization/desynchronization of vasomotors have special significance to microcirculation. Thus the contemporary literature throws up a novel view of microcirculation as a network that exhibits complex, spatiotemporal and informational dynamics.

Dynamics of a three-variable nonlinear model of vasomotion: comparison of theory and experiment

The effects of pharmacological interventions that modulate Ca(2+) homeodynamics and membrane potential in rat isolated cerebral vessels during vasomotion (i.e., rhythmic fluctuations in arterial diameter) were simulated by a third-order system of nonlinear differential equations. Independent control variables employed in the model were [Ca(2+)] in the cytosol, [Ca(2+)] in intracellular stores, and smooth muscle membrane potential. Interactions between ryanodine- and inositol 1,4,5-trisphosphate-sensitive intracellular Ca(2+) stores and transmembrane ion fluxes via K(+) channels, Cl(-) channels, and voltage-operated Ca(2+) channels were studied by comparing simulations of oscillatory behavior with experimental measurements of membrane potential, intracellular free [Ca(2+)] and vessel diameter during a range of pharmacological interventions. The main conclusion of the study is that a general model of vasomotion that predicts experimental data can be constructed by a low-order system that incorporates nonlinear interactions between dynamical control variables.

Shilnikov homoclinic orbit bifurcations in the Chua's circuit

We analytically describe the complex scenario of homoclinic bifurcations in the Chua's circuit. We obtain a general scaling law that gives the ratio between bifurcation parameters of different nearby homoclinic orbits. As an application of this theoretical approach, we estimate the number of higher order subsidiary homoclinic orbits that appear between two consecutive lower order subsidiary orbits. Our analytical finds might be valid for a large class of dynamical systems and are numerically confirmed in the parameter space of the Chua's circuit.

Basic structures of the Shilnikov homoclinic bifurcation scenario

We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits.

Role of the endothelium on arterial vasomotion

It is well-known that cyclic variations of the vascular diameter, a phenomenon called vasomotion, are induced by synchronous calcium oscillations of smooth muscle cells (SMCs). However, the role of the endothelium on vasomotion is unclear. Some experimental studies claim that the endothelium is necessary for synchronization and vasomotion, whereas others report rhythmic contractions in the absence of an intact endothelium. Moreover, endothelium-derived factors have been shown to abolish vasomotion by desynchronizing the calcium signals in SMCs. By modeling the calcium dynamics of a population of SMCs coupled to a population of endothelial cells, we analyze the effects of an SMC vasoconstrictor stimulation on endothelial cells and the feedback of endothelium-derived factors. Our results show that the endothelium essentially decreases the SMCs calcium level and may move the SMCs from a steady state to an oscillatory domain, and vice versa. In the oscillatory domain, a population of coupled SMCs exhibits synchronous calcium oscillations. Outside the oscillatory domain, the coupled SMCs present only irregular calcium flashings arising from noise modeling stochastic opening of channels. Our findings provide explanations for the published contradictory experimental observations.

Deterministic nonlinear characteristics ofin vivoblood flow velocity and arteriolar diameter fluctuations

We have performed a nonlinear analysis of fluctuations in red cell velocity and arteriolar calibre in the mesenteric bed of the anaesthetized rat. Measurements were obtained under control conditions and during local superfusion with NG-nitro-L-arginine (L-NNA, 30 microM) and tetrabutylammonium (TBA, 0.1 mM), which suppress NO synthesis and block Ca2+ activated K+ channels (KCa), respectively. Time series were analysed by calculating correlation dimensions and largest Lyapunov exponents. Both statistics were higher for red cell velocity than diameter fluctuations, thereby potentially differentiating between global and local mechanisms that regulate microvascular flow. Evidence for underlying nonlinear structure was provided by analysis of surrogate time series generated from the experimental data following randomization of Fourier phase. Complexity indices characterizing time series under control conditions were in general higher than those derived from data obtained during superfusion with L-NNA and TBA.