We use ab initio electronic structure calculations within the generalized gradient approximation (GGA+U) to density functional theory to determine the microscopic exchange interactions in the series of orthorhombic rare-earth manganites, o-RMnO₃. Our motivation is to construct a model Hamiltonian (excluding effects due to spin-orbit coupling), which can provide an accurate description of the magnetism in these materials. First, we consider TbMnO₃, which exhibits a spiral magnetic order at low temperatures. We map the exchange couplings in this compound onto a Heisenberg Hamiltonian and observe a clear deviation from the Heisenberg-like behavior. We consider first the coupling between magnetic and orbital degrees of freedom as a potential source of non-Heisenberg behavior in TbMnO₃, but conclude that it does not explain the observed deviation. We find that higher order magnetic interactions (biquadratic and four-spin ring couplings) should be taken into account for a proper treatment of the magnetism in TbMnO₃ as well as in the other representatives of the o-RMnO₃ series with small radii of the R cation.