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Due to the complicated magnetic and crystallographic structures of BiFeO _3 , its magnetoelectric (ME) couplings and microscopic model Hamiltonian remain poorly understood. By employing a first-principles approach, we uncover all possible ME couplings associated with the spin-current (SC) and exchange-striction (ES) polarizations, and construct an appropriate Hamiltonian for the long-range spin-cycloid in BiFeO _3 . First-principles calculations are used to understand the microscopic origins of the ME couplings. We find that inversion symmetries broken by ferroelectric and antiferroelectric distortions induce the SC and the ES polarizations, which cooperatively produce the dynamic ME effects in BiFeO _3 . A model motivated by first principles reproduces the absorption difference of counter-propagating light beams called non-reciprocal directional dichroism. The current paper focuses on the spin-driven (SD) polarizations produced by a dynamic electric field, i.e. the dynamic ME couplings. Due to the inertial properties of Fe, the dynamic SD polarizations differ significantly from the static SD polarizations. Our systematic approach can be generally applied to any multiferroic material, laying the foundation for revealing hidden ME couplings on the atomic scale and for exploiting optical ME effects in the next generation of technological devices such as optical diodes.