  Resistive-switching -- the current-/voltage-induced electrical resistance
change -- is at the core of memristive devices, which play an essential role in
the emerging field of neuromorphic computing. This study is about resistive
switching in a Mott-insulator, which undergoes a thermally driven
metal-to-insulator transition. Two distinct switching mechanisms were reported
for such a system: electric-field-driven resistive switching and electrothermal
resistive switching. The latter results from an instability caused by Joule
heating. Here, we present the visualization of the reversible resistive
switching in a planar V$_2$O$_3$ thin-film device using high-resolution
wide-field microscopy in combination with electric transport measurements. We
investigate the interaction of the electrothermal instability with the
strain-induced spontaneous phase-separation in the V$_2$O$_3$ thin film at the
Mott-transition. The photomicrographs show the formation of a narrow metallic
filament with a minimum width $\lesssim$ 500\,nm. Although the filament
formation and the overall shape of the current-voltage characteristics (IVCs)
are typical of an electrothermal breakdown, we also observe atypical effects
like oblique filaments, filament splitting, and hysteretic IVCs with
sawtooth-like jumps at high currents in the low-resistance regime. We were able
to reproduce the experimental results in a numerical model based on a
two-dimensional resistor network. This model demonstrates that resistive
switching, in this case, is indeed electrothermal and that the intrinsic
heterogeneity is responsible for the atypical effects. This heterogeneity is
strongly influenced by strain, thereby establishing a link between switching
dynamics and structural properties.
