The memristor is a novel semiconductor device equipped with a memory due to the change of its electrical resistance. In this way, it may mimic the behavior of a synapse in the human brain. We analyze a model of memristors that are coupled with an electric network consisting of various electronic devices. While nonlinear drift-diffusion equations describe the motion of charged particles within the memristor, the node potentials in the network follow the Kirchhoff law, i.e. ordinary differential equations and algebraic constraints. The coupling between them results in a system of partial differential-algebraic equations. The existence analysis employs entropy methods crucially.