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 2sigma significance level, the following upper bounds are obtained for the epoch corresponding to the cosmological redshifts z ~ 3 (i.e., ~ 10 Gyr ago): |Deltaalpha/alpha| < 0.00016 and |Deltamu/mu| < 0.00022. The corresponding upper limits to the time-average rates of the constant variations are |dalpha/(alpha dt)| < 1.6times 10^{-14} yr^{-1} and |dmu/(mu dt)| < 2.2times10^{-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.
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