The loss of the anisotropy in the electrical conductivity in MgB$_{2}$ under pressure

MgB$_{2}$ is a multiple band superconductor, with two $\sigma $-bands and two $\pi $-bands. The $\sigma $-bands that are highly anisotropic are the responsible of the superconductivity in this compound. It has been shown that with Sc, C and Al doping the $\sigma $-bands reduce their anisotropy, but for the case of Al and C doping the bands fill up and as consequence the number of $\sigma $-carriers reduce and disappear which leads to $T_{c}$ reduction. In this work it is shown that pressure reducing the MgB$_{2}$-cell parameters which leads to an increase of the interplane $\sigma $-orbitals overlap. This leads to an increase of the $\sigma $-bands electrical conductivity in the c-direction, in other words, a reduction of the anisotropy of the $\sigma $-bands, on the other hand there is no band filling therefore no reduction of $\sigma $-carriers. This reduction as function of pressure follows a similar trend as $T_{c}$, thus showing that the anisotropy in the $\sigma $-bands could be an important factor of the high $T_{c}$ in MgB$_{2}$.