In many transition metal oxides the electrical resistance is observed to undergo dramatic changes induced by large biases. In magnetite, Fe$_3$O$_4$, below the Verwey temperature, an electric field driven transition to a state of lower resistance was recently found, with hysteretic current-voltage response. We report the results of pulsed electrical conduction measurements in epitaxial magnetite thin films. We show that while the high- to low-resistance transition is driven by electric field, the hysteresis observed in $I-V$ curves results from Joule heating in the low resistance state. The shape of the hysteresis loop depends on pulse parameters, and reduces to a hysteresis-free "jump" of the current provided thermal relaxation is rapid compared to the time between voltage pulses. A simple relaxation time thermal model is proposed that captures the essentials of the hysteresis mechanism.