The chromospheric emission estimated in the core of different lines, such as CaII H & K, Na D1 and D2, and Ha, is not always correlated between lines. In particular, the CaII H & K and Ha emission time series are anti-correlated for a few percent of the stars, contrary to what is observed on the Sun. This puzzling result has been observed for both solar-type stars and M stars. Our objective is to characterise these relationships in more detail using complementary criteria, and based on a large set of spectra obtained with HARPS for a large sample of M dwarfs. This should allow to evaluate whether or not additional processes are required to explain the observations. We analysed the time average and variability of the Ca, Na, and H{alpha} emissions for 177 M stars ranging from subspectral types M0 to M8, paying particular attention to their (anti-)correlations on both short and long timescales as well as slopes between indices. We also computed synthetic H{alpha} time series based on different assumptions of plage properties. We compared our findings with observations in order to evaluate whether or not the main observed properties could be reproduced. The statistical properties of our sample, in terms of correlations and slopes between indices at different timescales, differ from what we previously obtained for FGK stars: there are fewer stars with a null correlation, and the correlations we find show a weaker dependence on timescale. However, there can be a large dispersion from one season to another for stars with a well identified low or negative correlation. We also specify the complex relationship between the average activity levels, with a clear indication of a change in the sign of the slope from the relation between Ca and H{alpha} (and between Na and H{alpha}) for the most massive M dwarfs. In addition, we observe a change in slope in the Na-Ca relation at an intermediate activity level. At this stage, we are not able to find simple plage properties that, alone, are sufficient to reproduce the observations. However, the simulations already allow us to point out that it is not straightforward to compare the temporal variability correlation and the integrated indices. Our findings also demonstrate the need for complex activity patterns to explain some of the observations. We conclude that the relation between the three indices examined here exhibits a large diversity in behaviour over the sample studied. More detailed simulations with complex activity patterns are necessary to understand these observations. This will teach us about plage properties for this type of star.