  We develop an Euler-type particle method for the simulation of a
McKean--Vlasov equation arising from a mean-field model with positive feedback
from hitting a boundary. Under assumptions on the parameters which ensure
differentiable solutions, we establish convergence of order $1/2$ in the time
step. Moreover, we give a modification of the scheme using Brownian bridges and
local mesh refinement, which improves the order to $1$. We confirm our
theoretical results with numerical tests and empirically investigate cases with
blow-up.
