We study the impact of deviations from the Kennicutt-Schmidt relation (quantified by the `burstiness parameter $kappa_s$), gas metallicity ($Z$), and density ($n$) on the observed [OIII]88$mu$m/[CII]158$mu$m surface brightness ratios ($Sigma_{[OIII]}/Sigma_{[CII]}$) in nine galaxies at $zapprox6-9$. We first discuss possible biases in the measured $Sigma_{[OIII]}/Sigma_{[CII]}$ ratios by comparing the data with zoom-in cosmological simulations, and then use a Markov Chain Monte Carlo algorithm to derive the best fit values of ($kappa_s, Z, n$). We find that (i) the strongest dependence of $Sigma_{[OIII]}/Sigma_{[CII]}$ is on $kappa_s$; (ii) high ratios identify starburst galaxies with short gas depletion times ($t_{dep}=6-49,rm Myr$); (iii) a secondary dependence on density is found, with $Sigma_{[OIII]}/Sigma_{[CII]}$ anticorrelating with $n$ as a result of the lower [OIII] critical density, (iv) the ratio only weakly depends on $Z$. The nine galaxies are significantly enriched (Z=0.2-0.5 $Z_odot$), and dense ($n=10^{1-3} {rm cm}^{-3}$). This lends further support to the starburst scenario in which a rapid enrichment of the interstellar medium is expected.
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