  We consider a system of nonlinear partial differential equations with
stochastic dynamical boundary conditions that arises in models of
neurophysiology for the diffusion of electrical potentials through a finite
network of neurons. Motivated by the discussion in the biological literature,
we impose a general diffusion equation on each edge through a generalized
version of the FitzHugh-Nagumo model, while the noise acting on the boundary is
described by a generalized stochastic Kirchhoff law on the nodes. In the
abstract framework of matrix operators theory, we rewrite this stochastic
boundary value problem as a stochastic evolution equation in infinite
dimensions with a power-type nonlinearity, driven by an additive L\'evy noise.
We prove global well-posedness in the mild sense for such stochastic partial
differential equation by monotonicity methods.
