![]() Moreover, we have developed a general method to quantify the separability of point clusters in high-dimensional state spaces 11. In this context, we have also analyzed the micro-structure of cortical activity during sleep and found that it reflects respiratory events and the state of daytime vigilance 10. Indeed, we have confirmed this hypothesis in former work 8, where we applied a previously developed method for analyzing and comparing spatiotemporal cortical activation patterns 9. It is then a natural hypothesis that each sleep stage corresponds to a different cluster within EEG state space, and that the trajectory of the random walk moves to the corresponding cluster whenever a new sleep stage is entered. The momentary amplitudes of an N-channel EEG recording represent a point in an N-dimensional state space and the ongoing time series of vectorial amplitudes defines a random walk within this high-dimensional space. In this case, multi-channel electroencephalographic (EEG) recordings offer a convenient way to quantify the ongoing changes in the brain over long periods, but also with high temporal resolution 6, 7. ![]() Such transitions typically show up in the time series by a sudden (or gradual) change of the statistical properties of the fluctuating state variables.Ī typical example of such mode-switching behavior is the sleep cycle in humans and other mammals, where the brain is passing through a sequence of seemingly distinct sleep stages 5. The underlying processes can often be quantified in the form of multivariate time series and a mathematical analysis of the time series can be used to predict future states of the system, or simply to better understand its internal dynamics 1, 2, 3.Īlthough in simple physical systems, state variables fluctuate around a fixed mean value and with a fixed variance (as in the case of local pressure variations in a gas at equilibrium), complex systems often have multiple dynamical attractors 4, i.e., a set of qualitatively different modes of behavior, between which the system will occasionally switch. Many complex systems, such as the earth’s crust, the weather, biological organisms, or the stock market, show continuous fluctuations of their internal state variables, even in the absence of external perturbations. Based on the hyper-parameters, we finally perform a pairwise similarity analysis between the different sleep stages, using a quantitative measure for the separability of data clusters in multi-dimensional spaces. Moreover, we find that the hyper-parameters are not piece-wise constant, as the traditional hypnograms would suggest, but show rising or falling trends within and across sleep stages, pointing to an underlying continuous rather than sub-divided process that controls human sleep. It turns out that also the hyper-parameters have characteristic, sleep-stage-dependent distributions, which can be exploited for a simple Bayesian sleep stage detection. Next, we perform a super-statistical analysis by computing hyper-parameters, such as the standard deviation, kurtosis, and skewness of the raw signal distributions, within subsequent 30-second epochs. We first demonstrate the heterogeneity of the random process by showing that each sleep stage has a characteristic distribution and temporal correlation function of the raw EEG signals. To address this question, we consider one-channel EEG signals as heterogeneous random walks: stochastic processes controlled by hyper-parameters that are themselves time-dependent. It is however unclear whether this subdivision into discrete stages with sharply defined boundaries is truly reflecting the dynamics of human sleep. In clinical practice, human sleep is classified into stages, each associated with different levels of muscular activity and marked by characteristic patterns in the EEG signals.
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