Neuromorphic computation is based on memristors, which function equivalently to neurons in brain structures. These memristors can be made more efficient and tailored to neuromorphic devices by using ferroelastic domain boundaries as fast diffusion paths for ionic conduction, such as of oxygen, sodium, or lithium. In this paper, we show that the local memristor generates a second, unexpected feature, namely, weak magnetic fields that emerge from moving ferroelastic needle domains and vortices. The vortices appear near ferroelastic “junctions” that are common when the external stimulus is a combination of electric fields and structural phase transitions. Many ferroelastic materials show such phase transitions near room temperatures so that device applications display a “multiferroic” scenario where the memristor is driven electrically and read magnetically. Our computer simulation study of an elastic spring model suggests magnetic fields in the order of 10−7 T, which opens the way for a fundamentally new way of running neuromorphic devices. The magnetism in such devices emerges entirely from intrinsic displacement currents and not from any intrinsic magnetism of the material.