  Contemporary multidimensional cosmological theories predict different
variations of fundamental physical constants in course of the cosmological
evolution. On the basis of the QSO spectra analysis, we show that the
fine-structure constant \alpha=e^2/(\hbar c) and the proton-to-electron mass
ratio \mu=m_p/m_e reveal no statistically significant variation over the last
90% of the lifetime of the Universe. At the 2\sigma significance level, the
following upper bounds are obtained for the epoch corresponding to the
cosmological redshifts z ~ 3 (i.e., ~ 10 Gyr ago):
  |\Delta\alpha/\alpha| < 0.00016 and |\Delta\mu/\mu| < 0.00022.
  The corresponding upper limits to the time-average rates of the constant
variations are |d\alpha/(\alpha dt)| < 1.6\times 10^{-14} yr^{-1} and
|d\mu/(\mu dt)| < 2.2\times10^{-14} yr^{-1}. These limits serve as criteria for
selection of those theoretical models which predict \alpha and \mu variation
with the cosmological time. In addition, we test a possible anisotropy of the
high-redshift fine splitting over the celestial sphere, which might reveal a
non-equality of \alpha values in causally disconnected areas of the Universe.
