We model cyclic voltammetry experiments on supported lipid films where a non-trivial dependence of the capacitance on the applied voltage is observed. Previously, based on a mean-field treatment of the Flory-Huggins type, under the assumption of strongly screened electrostatic interactions, it has been hypothesized that peaks in the capacitance-vs-voltage profiles correspond to a sequence of structural or phase transitions within the interface. To examine this hypothesis, in this study we use both mean-field calculations and Monte Carlo simulations where the electrostatic effects due to the varying electric potential and the presence of salt are accounted for explicitly. Our main focus is on the structure of the film and the desorption-readsorption phenomena. These are found to be driven by a strong competition for the progressively charged-up (hydrophobic) surface between lipid hydrocarbon tails and the electrode counterions (cations). As the surface charge density is raised, the following phase phenomena within the interface are clearly observed: (i) a gradual displacement of the monolayer from the surface by the counterions, leading to complete monolayer desorption and formation of an electric double layer by the surface, (ii) a transformation of the monolayer into a bilayer upon its desorption, (iii) in the case of zwitterionic (or strongly polar) lipid head groups, the desorption is followed by the bilayer readsorption to the electrode via interaction with the electric double layer and release of the excess counterions into the bulk solution. We argue then that the voltammetry peaks are associated with a stepwise process of formation of layers of alternating charge: electric double layer -upon film desorption, triple or multi-layer -upon film readsoption. * Corresponding author, E-mail: a.brukhno@leeds.ac.uk † Also known as Andrei V. Broukhno 1 arXiv:1007.3004v2 [cond-mat.soft]