  Gamma Ray Burst prompt emission is believed to originate from electrons
accelerated in a highly relativistic outflow. "Internal shocks" due to
collisions between shells ejected by the central engine is a leading candidate
for electron acceleration. While synchrotron radiation is generally invoked to
interpret prompt gamma-ray emission within the internal shock model,
synchrotron self-Compton (SSC) is also considered as a possible candidate of
radiation mechanism. In this case, one would expect a synchrotron emission
component at low energies, and the naked-eye GRB 080319B has been considered as
such an example. In the view that the gamma-ray lightcurve of GRB 080319B is
much more variable than its optical counterpart, in this paper we study the
relative variability between the synchrotron and SSC components. We develop a
"top-down" formalism by using observed quantities to infer physical parameters,
and subsequently to study the temporal structure of synchrotron and SSC
components of a GRB. We complement the formalism with a "bottom-up" approach
where the synchrotron and SSC lightcurves are calculated through a Monte-Carlo
simulations of the internal shock model. Both approaches lead to the same
conclusion. Small variations in the synchrotron lightcurve can be only
moderately amplified in the SSC lightcurve. The SSC model therefore cannot
adequately interpret the gamma-ray emission properties of GRB 080319B.
