Volatile memory devices relying on the Mott-type insulator-to-metal transition of vanadium oxide (VO₂) are widely utilized in the field of neuromorphic computing. Such devices, however, are realized in a nanoscale geometry, where the switching relies on the self-heating of an ultrasmall spot as well as the presence of extremely high electric fields in the active region. In this paper, we investigate the interplay of such nanoscale thermal and nonlinear electronic phenomena by investigating the temperature and voltage dependent conduction properties of our custom-designed VO₂ devices, where a V-shaped electrode focuses the switching to an ultrasmall single-spot active region. This simplified spatial structure of the active volume facilitates the device modeling and the identification of physical mechanisms behind the phase transition. We find that purely thermal or electronic effects fail to describe the device operation, however, according to our finite element simulations, a combined electronic and thermal model provides a precise description of the device characteristics. These results facilitate the understanding as well as the thermal and electronic design of novel VO₂-based neuronal devices.