We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. For one-dimensional biased random walks, we derive exact results which tighten bounds for entropy production extrema obtained with martingale theory and reveal a symmetry between the distribution of the maxima and minima of entropy production. Furthermore, we show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Marčenko–Pastur distribution of random-matrix theory. Using this result, we obtain efficient estimates for the extreme-value statistics of stochastic transport processes from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results with numerical simulations of stochastic models of molecular motors.

The crosstalk between organs plays a crucial role in physiological processes. This coupling is a dynamical process, it must cope with a huge variety of rhythms with frequencies ranging from milliseconds to hours, days, seasons. The brain is a central hub for this crosstalk. During sleep, automatic rhythmic interrelations are enhanced and provide a direct insight into organ dysfunctions, however their origin remains a difficult issue, in particular in sleep disorders. In this study, we focus on EEG, ECG, and airflow recordings from polysomnography databases. Because these signals are non-stationary, non-linear, noisy, and span wide spectral ranges, a time-frequency analysis, based on wavelet transforms, is more appropriate to handle this complexity. We design a wavelet-based extraction method to identify the characteristic rhythms of these different signals, and their temporal variability. These new constructs are combined in pairs to compute their wavelet-based time-frequency complex coherence. These time-frequency coherence maps highlight the occurrence of a slowly modulated coherence pattern in the frequency range [0.01–0.06] Hz, which appears in both obstructive and central apnea. A preliminary exploration of a large database from the National Sleep Research Resource with respiration disorders, such as apnea provides some clues on its relation with autonomic cardio-respiratory coupling and brain rhythms. We also observe that during sleep apnea episodes (either obstructive or central), the cardiopulmonary coherence (in particular respiratory sinus-arrhythmia) in the frequency range [0.1–0.7] Hz strongly diminishes, suggesting a modification of this coupling. Finally, comparing time-averaged coherence with heart rate variability spectra in different apnea episodes, we discuss their common trait and their differences.

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