  We consider running coupling constant and beta-function in holographic models
supported by Einstein-dilaton-Maxwell action for heavy and light quarks. To
obtain the dependence of the running coupling constant $\alpha$ on temperature
and chemical potential we impose boundary conditions on the dilaton field that
depend on the position of the horizon. We use two types of boundary conditions:
a simple boundary condition with the dilaton field vanishing at the horizon,
and a boundary condition that ensures agreement with lattice calculations of
string tension between quarks at zero chemical potential. The location of the
first-order phase transitions in $(T,\mu)$-plane does not depend on the dilaton
boundary conditions for light and heavy quarks. At these phase transitions, the
functions $\alpha$ and $\beta$ undergo jumps depending on temperature and
chemical potential. We also show that for the second boundary conditions the
running coupling decreases with increasing temperature and the dependence on
temperature and chemical potential for both light and heavy quarks are actually
specified by functions of one variable, demonstrating in this sense auto-model
behavior.
