  Objective. The distribution of equal states (DES) quantifies amplitude
fluctuations in biomedical signals. However, under certain conditions, such as
a high resolution of data collection or special signal processing techniques,
equal states may be very rare, whereupon the DES fails to measure the amplitude
fluctuations. Approach. To address this problem, we develop a novel threshold
DES (tDES) that measures the distribution of differential states within a
threshold. To evaluate the proposed tDES, we first analyze five sets of
synthetic signals generated in different frequency bands. We then analyze sleep
electroencephalography (EEG) datasets taken from the public PhysioNet. Main
results. Synthetic signals and detrend-filtered sleep EEGs have no neighboring
equal values; however, tDES can effectively measure the amplitude fluctuations
within these data. The tDES of EEG data increases significantly as the sleep
stage increases, even with datasets covering very short periods, indicating
decreased amplitude fluctuations in sleep EEGs. Generally speaking, the
presence of more low-frequency components in a physiological series reflects
smaller amplitude fluctuations and larger DES. Significance.The tDES provides a
reliable computing method for quantifying amplitude fluctuations, exhibiting
the characteristics of conceptual simplicity and computational robustness. Our
findings broaden the application of quantitative amplitude fluctuations and
contribute to the classification of sleep stages based on EEG data.
