  We design and study splitting integrators for the temporal discretization of
the stochastic FitzHugh--Nagumo system. This system is a model for signal
propagation in nerve cells where the voltage variable is solution of a
one-dimensional parabolic PDE with a cubic nonlinearity driven by additive
space-time white noise. We first show that the numerical solutions have finite
moments. We then prove that the splitting schemes have, at least, the strong
rate of convergence $1/4$. Finally, numerical experiments illustrating the
performance of the splitting schemes are provided.
