Homeostasis is one of the key concepts of physiology and the basis to understand chronic-degenerative disease and human ageing, but is difficult to quantify in clinical practice. The variability of time series resulting from continuous and non-invasive physiological monitoring is conjectured to reflect the underlying homeostatic regulatory processes, but it is not clear why the variability of some variables such as heart rate gives a favourable health prognosis whereas the variability of other variables such as blood pressure implies an increased risk factor. The purpose of the present contribution is to quantify homeostasis using time-series analysis and to offer an explanation for the phenomenology of physiological time series. Within the context of network physiology, which focusses on the interactions between various variables at multiple scales of time and space, it may be understood that different physiological variables may play distinct roles in their respective regulatory mechanisms. In the present contribution, we distinguish between regulated variables, such as blood pressure or core temperature, and physiological responses, such as heart rate and skin temperature. We give evidence that in optimal conditions of youth and health the former are characterized by Gaussian statistics, low variability and represent the stability of the internal environment, whereas the latter are characterized by non-Gaussian distributions, large variability and reflect the adaptive capacity of the human body; in the adverse conditions of ageing and/or disease, adaptive capacity is lost and the variability of physiological responses is diminished, and as a consequence the stability of the internal environment is compromised and its variability increases. Time-series analysis allows one to quantify homeostasis in the optimal conditions of youth and health and the degradation of homeostasis or homeostenosis in the adverse conditions of ageing and/or disease, and may offer an alternative approach to diagnosis in clinical practice.

Circadian rhythms become less dominant and less regular with chronic-degenerative disease, such that to accurately assess these pathological conditions it is important to quantify not only periodic characteristics but also more irregular aspects of the corresponding time series. Novel data-adaptive techniques, such as singular spectrum analysis (SSA), allow for the decomposition of experimental time series, in a model-free way, into a trend, quasiperiodic components and noise fluctuations. We compared SSA with the traditional techniques of cosinor analysis and intradaily variability using 1-week continuous actigraphy data in young adults with acute insomnia and healthy age-matched controls. The findings suggest a small but significant delay in circadian components in the subjects with acute insomnia, i.e. a larger acrophase, and alterations in the day-to-day variability of acrophase and amplitude. The power of the ultradian components follows a fractal 1/f power law for controls, whereas for those with acute insomnia this power law breaks down because of an increased variability at the 90min time scale, reminiscent of Kleitman’s basic rest-activity (BRAC) cycles. This suggests that for healthy sleepers attention and activity can be sustained at whatever time scale required by circumstances, whereas for those with acute insomnia this capacity may be impaired and these individuals need to rest or switch activities in order to stay focused. Traditional methods of circadian rhythm analysis are unable to detect the more subtle effects of day-to-day variability and ultradian rhythm fragmentation at the specific 90min time scale.

Spectra of ordered eigenvalues of finite random matrices are interpreted as a time series. Data-adaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding possible artifacts introduced by standard unfolding techniques. The fluctuation modes are scale invariant and follow different power laws for Poisson and Gaussian ensembles, which already during the unfolding allows one to distinguish the two cases.

Recently, singular value decomposition (SVD) was applied to standard Gaussian ensembles of random-matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the $\beta$-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a crossover between soft and rigid behavior. In this way, possible artifacts introduced applying unfolding techniques are avoided. By using the trend modes, we perform data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes.

Many dynamical systems from different areas of knowledge can be studied within the theoretical framework of time series, where the system can be considered as a black box, that only needs to be “listened” to. In this framework, non‐correlated series (white noise) and strongly correlated series (brownian noise or periodic series) constitute two extremes. Certain dynamical systems auto‐organize in a critical state that is characterized by 1/f or flicker noise. The family of $f^{\beta }$ noises $\text{(β≤0)}$ is fractal because fragments of the series are statistically identical to the original time series. 1/f noise $\text{(β\hspace{0.17em}=\hspace{0.17em}−1)}$ is critical because it maximizes important complexity‐related quantities as memory, information content, efficiency and fractality. 1/f noise has been observed in classical systems, but also in quantum systems, and could possibly offer a unifying bridge of understanding between the macroscopic and the quantum world. In the present article, we will discuss some examples from both worlds.