  Starting from \textit{ab-initio} calculations, we derive a five-band Hubbard
model to describe the CuO$_2$ chains of LiCu$_2$O$_2$. This model is further
simplified to a low-energy effective Heisenberg model with nearest-neighbor
(NN) $J_1$, and next-nearest-neighbor (NNN) $J_2$ interactions, combining
perturbation theory, exact diagonalization calculations and Density Matrix
Renormalization Group results. For realistic parameters we find the
corresponding values of these interactions. The obtained effective model is
consistent with a spiral-magnetic ground state as experimentally observed.
Using symmetry arguments, the spiral state is a sufficient condition for the
ferroelectricity observed in the system.
