  The small (1 mm) nematode $\textit{Caenorhabditis elegans}$ (Corsi [1],
wormbook.org) has become widely used as a model organism; in particular, the
$\textit{C. elegans}$ connectome has been completely mapped, and $\textit{C.
elegans}$ locomotion has been widely studied. We describe a minimal
reaction-diffusion model for the locomotion of $\textit{C. elegans}$, using as
a framework a simplified, stylized "descending pathway" of neurons as central
pattern generator (CPG) (Xu et al., Proceedings of the National Academy of
Sciences 115, 2018). Finally, we realize a model of the required oscillations
and coupling with a network of coupled Keener (IEEE Transactions on Systems,
Man, and Cybernetics SMC-13, 1983 [3]) analog neurons. Note that Olivares et
al. (BioRxiv 710566, 2020 [4]) present a likely more realistic model more
distributed CPG. We use the simpler simulation to show that a small network of
FitzHugh-Nagumo neurons (one of the simplest neuronal models) can generate key
features of $\textit{C. elegans}$ undulation, and thus locomotion, yielding a
minimal, biomimetic model as a building block for further exploring $\textit{C.
elegans}$ locomotion.
