A new mathematical model for time-dependent electroencephalography (EEG) is developed and analysed. Evolution with time is introduced into the standard EEG model by considering dipolar sources with time-dependent moments and source positions. Dimensional analysis shows the validity of the quasi-stationary approximation for all tissues of the humain head. Non-linear systems of differential equations based on gating particles are used to model the postsynaptic current at the neuron level which, in turn, yields the dipolar source term of the boundary value problem. The wellposedness of the forward time-dependent EEG problem is proved by the subtraction approach for moments with L 2-regularity in time and continuous source trajectories. Numerical results explain the pipeline from the simulation of the postsynaptic current up to the potential recorded at the electrodes in a 2D circular configuration and on the three-dimensional realistic head model of a neonate. The inverse source problem is formulated with the help of a time-dependent non-linear measurement operator and identifiability and stability results are proven. It is numerically solved by the Minimum Norm Estimate and the computation of the involved Lead Field Matrix is explained for the particular case of the subtraction approach. The reconstruction of the trajectory of a moving source point with time-dependent moment illustrates the approach for the inverse problem in the 2D configuration.