In this study, we show how internationalization and foreign-owned firms influence synergies in the regional innovation systems of Hungary. We first distinguish three innovation system functions (knowledge exploitation, knowledge exploration, and organizational control) operating in regions and study their interactions using entropy statistics. The functions and their interactions are measured by analysing the distribution of firms in terms of geographical location, organizational size (number of employees), technologies (NACE codes of the OECD), and ownership (foreign versus domestic share in registered stock) in the 2005. Synergy is defined as mutual information among the three dimensions; a fourth dimension is added in order to bring internationalization (FDI) into the model. The factor is relevant since the four-dimensional model explains the GDP contributions to regional development in Hungary, whereas the three-dimensional model does not. We find that regional innovation systems in Hungary are self-organized differently, in relation to a relatively small number of foreign firms. These firms have a large positive effect on synergy in regions between the Hungarian capital and the Austrian border. However, FDI has negative effects on domestic synergy in the lagging eastern and southern provinces of the country.