A surrogate data analysis is presented, which is based on the fluctuations of the "entropy" $S$ defined in the natural time-domain [Phys. Rev. E {\bf 68}, 031106, 2003]. This entropy is not a static one as, for example, the Shannon entropy. The analysis is applied to three types of time-series, i.e., seismic electric signals, "artificial" noises and electrocardiograms, and "recognizes" the non-Markovianity in all these signals. Furthermore, it differentiates the electrocardiograms of healthy

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... grams of healthy humans from those of the sudden cardiac death ones. If $\delta S$ and $\delta S_{shuf}$ denote the standard deviation when calculating the entropy by means of a time-window sweeping through the original data and the "shuffled" (randomized) data, respectively, it seems that the ratio $\delta S_{shuf}/\delta S$ plays a key-role. The physical meaning of $\delta S_{shuf}$ is investigated.