A physical model of the electron is suggested according to the basic structures of matter (BSM) hypothesis. BSM is based on an alternative concept about the physical vacuum, assuming that space contains an underlying grid structure of nodes formed of superdense subelementary particles, which are also involved in the structure of the elementary particles. The proposed grid structure is formed of vibrating nodes that possess quantum features and energy well. It is admitted that this hypothetical structure could account for the missing 'dark matter' in the universe. The signature of this dark matter is apparent in the galactic rotational curves and in the relation between masses of the supermassive black hole in the galactic center and the host galaxy. The suggested model of the electron possesses oscillation features with anomalous magnetic moment and embedded signatures of the Compton wavelength and the fine-structure constant. The analysis of the interactions between the oscillating electron and the nodes of the vacuum grid structure allows us to obtain physical meaning for some fundamental constants.