Memristors are memory resistors promising a rapid integration into future memory technologies. However, progress is still critically limited by a lack of understanding of the physical processes occurring at the nanoscale. Here we correlate device electrical characteristics with local atomic structure, chemistry and temperature. We resolved a single conducting channel that is made up of a reduced phase of the as-deposited titanium oxide. Moreover, we observed sufficient Joule heating to induce a crystallization of the oxide surrounding the channel, with a peculiar pattern that finite element simulations correlated with the existence of a hot spot close to the bottom electrode, thus identifying the switching location. This work reports direct observations in all three dimensions of the internal structure of titanium oxide memristors.