  We establish an extended version of the Einstein - Maxwell - axion model by
introducing into the Lagrangian cross-terms, which contain the gradient
four-vector of the pseudoscalar (axion) field in convolution with the Maxwell
tensor. The gradient model of the axion-photon coupling is applied to
cosmology: we analyze the Bianchi-I type Universe with an initial magnetic
field, electric field induced by the axion-photon interaction, cosmological
constant and dark matter, which is described in terms of the pseudoscalar
(axion) field. Analytical, qualitative and numerical results are presented in
detail for two distinguished epochs: first, for the early Universe with
magnetic field domination; second, for the stage of late-time accelerated
expansion.
