The aim of this text is to elucidate the oscillating patterns which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency. By exploiting the properties of underlying integrable systems, we can complement the KAM picture and go beyond the classical results about gyrokinetics. The purely magnetic situation was addressed before. We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field.