The application of methods drawn from nonlinear and stochastic dynamics to the analysis of cardiovascular time series is reviewed, with particular reference to the identification of changes associated with ageing. The natural variability of the heart rate (HRV) is considered in detail, including the respiratory sinus arrhythmia (RSA) corresponding to modulation of the instantaneous cardiac frequency by the rhythm of respiration. HRV has been intensively studied using traditional spectral analyses, e.g. by Fourier transform or autoregressive methods, and, because of its complexity, has been used as a paradigm for testing several proposed new methods of complexity analysis. These methods are reviewed. The application of time–frequency methods to HRV is considered, including in particular the wavelet transform which can resolve the time-dependent spectral content of HRV. Attention is focused on the cardio-respiratory interaction by introduction of the respiratory frequency variability signal (RFV), which can be acquired simultaneously with HRV by use of a respiratory effort transducer. Current methods for the analysis of interacting oscillators are reviewed and applied to cardio-respiratory data, including those for the quantification of synchronization and direction of coupling. These reveal the effect of ageing on the cardio-respiratory interaction through changes in the mutual modulation of the instantaneous cardiac and respiratory frequencies. Analyses of blood flow signals recorded with laser Doppler flowmetry are reviewed and related to the current understanding of how endothelial-dependent oscillations evolve with age: the inner lining of the vessels (the endothelium) is shown to be of crucial importance to the emerging picture. It is concluded that analyses of the complex and nonlinear dynamics of the cardiovascular system can illuminate the mechanisms of blood circulation, and that the heart, the lungs and the vascular system function as a single entity in dynamical terms. Clear evidence is found for dynamical ageing.
We describe an analysis of cardiac and respiratory time series recorded from 189 subjects of both genders aged 16–90. By application of the synchrosqueezed wavelet transform, we extract the respiratory and cardiac frequencies and phases with better time resolution than is possible with the marked events procedure. By treating the heart and respiration as coupled oscillators, we then apply a method based on Bayesian inference to find the underlying coupling parameters and their time dependence, deriving from them measures such as synchronization, coupling directionality and the relative contributions of different mechanisms. We report a detailed analysis of the reconstructed cardiorespiratory coupling function, its time evolution and age dependence. We show that the direct and indirect respiratory modulations of the heart rate both decrease with age, and that the cardiorespiratory coupling becomes less stable and more time-variable.
A simple mean-field idea is applicable to the pattern dynamics of large assemblies of limit-cycle oscillators with non-local coupling. This is demonstrated by developing a mathematical theory for the following two specific examples of pattern dynamics. Firstly, we discuss propagation of phase waves in noisy oscillatory media, with particular concern with the existence of a critical condition for persistent propagation of the waves throughout the medium, and also with the possibility of noise-induced turbulence. Secondly, we discuss the existence of an exotic class of patterns peculiar to non-local coupling called chimera where the system is composed of two distinct domains, one coherent and the other incoherent, separated from each other with sharp boundaries.
The onset of undamped wave propagation in noisy self-oscillatory media is identified with a Hopf bifurcation of the corresponding effective dynamical system obtained by properly renormalizing the effects of noise. We illustrate this fact on a dense array of nonlocally coupled phase oscillators for which a mean-field idea works exactly in deriving such effective dynamical equations.Unlike conservative oscillatory media, dissipative self-oscillatory media admit traveling waves without decay even in the presence of noise or other sources of randomness. This ability of wave transmission in random self-oscillatory media should be functionally relevant in a variety of living organisms for which randomness is unavoidable. For this type of systems, a critical strength of randomness is generally expected to exist such that below which the system is capable of sustaining undamped traveling waves. This critical point should be identical with the Hopf bifurcation point of an effective dynamical system obtained by properly renormalizing the effects of noise. It is reasonable to expect that a theory could be developed unambiguously on this issue in the particular case when each oscillator couples with sufficiently many oscillators, because a mean-field idea should be applicable then. In the present article, we carry out this program for nonlocally coupled phase oscillators with noise, and show how an effective dynamical equation can be derived, and how it is reduced to a small-amplitude equation near the bifurcation point. Our theory may be regarded as a natural extension of a previous theory 1) on the onset of collective oscillation for globally coupled phase oscillators with noise.Imagine an infinitely long array of nonlocally coupled oscillators which are distributed densely and subject to additive noise. By taking the continuum limit, the phases φ(x, t) of the oscillators are assumed to obey the following Langevin-typeHere ω is the natural frequency, the second term represents the nonlocal coupling, and the last term gives additive noise. The phase coupling function Γ depends only on the phase difference and satisfies the in-phase condition Γ ′ (0) < 0 1) , while its strength G depends on the distance. Specifically, we assume the form G(x) = γ exp(−γ|x|)/2 whose integral is normalized. The noise is assumed to be white Gaussian with vanishing mean, i.e., < ξ(x, t) >= 0 and < ξ(x, t)ξ(x ′ , t ′ ) >= 2Dδ(t − t ′ )δ(x − x ′ ). Since we are working with an oscillator continuum, the coupling radius typeset using PTPT E X.sty
General anesthetics are used during medical and surgical procedures to reversibly induce a state of total unconsciousness in patients. Here, we investigate, from a dynamic network perspective, how the cortical and cardiovascular systems behave during anesthesia by applying nonparametric spectral techniques to cortical electroencephalography, electrocardiogram and respiratory signals recorded from anesthetized rats under two drugs, ketamine-xylazine (KX) and pentobarbital (PB). We find that the patterns of low-frequency cortico-cardio-respiratory network interactions may undergo significant changes in network activity strengths and in number of network links at different depths of anesthesia dependent upon anesthetics used.
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