  The redshift range z=4-6 marks a transition phase between primordial and
mature galaxy formation in which galaxies considerably increase their stellar
mass, metallicity, and dust content. The study of galaxies in this redshift
range is therefore important to understand early galaxy formation and the fate
of galaxies at later times. Here, we investigate the burstiness of the recent
star-formation history (SFH) of 221 $z\sim4.5$ main-sequence galaxies at log(M)
> 9.7 by comparing their ultra-violet (UV) continuum, H$\alpha$ luminosity, and
H$\alpha$ equivalent-width (EW). The H$\alpha$ properties are derived from the
Spitzer [3.6$\mu$m]-[4.5$\mu$m] broad-band color, thereby properly taking into
account model and photometric uncertainties. We find a significant scatter
between H$\alpha$ and UV-derived luminosities and star-formation rates (SFRs).
About half of the galaxies show a significant excess in H$\alpha$ compared to
expectations from a constant smooth SFH. We also find a tentative
anti-correlation between H$\alpha$ EW and stellar mass, ranging from
1000$\r{A}$ at log(M) < 10 to below 100$\r{A}$ at log(M) > 11. Consulting
models suggests that most $z\sim4.5$ galaxies had a burst of star-formation
within the last 50 Myrs, increasing their SFRs by a factor of > 5. The most
massive galaxies on the other hand might decrease their SFRs, and may be
transitioning to a quiescent stage by z=4. We identify differential dust
attenuation (f) between stars and nebular regions as the main contributor to
the uncertainty. With local galaxies selected by increasing H$\alpha$ EW
(reaching values similar to high-z galaxies), we predict that f approaches
unity at $z>4$ consistent with the extrapolation of measurements out to z=2.
