The impact of the Star Formation Histories on the SFR-M$_{*}$ relation
  at z$\ge$2
  In this paper we investigate the impact of different star formation histories
(SFHs) on the relation between stellar mass M$_{*}$ and star formation rate
(SFR) using a sample of galaxies with reliable spectroscopic redshift zspec>2
drawn from the VIMOS Ultra-Deep Survey (VUDS). We produce an extensive database
of dusty model galaxies, calculated starting from the new library of single
stellar population (SSPs) models presented in Cassara' et al. 2013 and weighted
by a set of 28 different SFHs based on the Schmidt function, and characterized
by different ratios of the gas infall time scale $\tau_{infall}$ to the star
formation efficiency $\nu$. The treatment of dust extinction and re-emission
has been carried out by means of the radiative transfer calculation. The
spectral energy distribution (SED) fitting technique is performed by using
GOSSIP+, a tool able to combine both photometric and spectroscopic information
to extract the best value of the physical quantities of interest, and to
consider the Intergalactic Medium (IGM) attenuation as a free parameter. We
find that the main contribution to the scatter observed in the $SFR-M_{*}$
plane is the possibility of choosing between different families of SFHs in the
SED fitting procedure, while the redshift range plays a minor role. The
majority of the galaxies, at all cosmic times, are best-fit by models with SFHs
characterized by a high $\tau_{\rm infall}/\nu$ ratio. We discuss the
reliability of the presence of a small percentage of dusty and highly star
forming galaxies, in the light of their detection in the FIR.
