  We consider a nonlinear reaction--diffusion equation in a domain consisting
of two bulk regions connected via small channels periodically distributed
within a thin layer. The height and the thickness of the channels are of order
$\epsilon$, and the equation inside the layer depends on the parameter
$\epsilon$. We consider the critical scaling of the diffusion coefficients in
the channels and nonlinear Neumann-boundary condition on the channels' lateral
boundaries. We derive effective models in the limit $\epsilon \to 0 $, when the
channel-domain is replaced by an interface $\Sigma$ between the two
bulk-domains. Due to the critical size of the diffusion coefficients, we obtain
jumps for the solution and its normal fluxes across $\Sigma$, involving the
solutions of local cell problems on the reference channel in every point of the
interface $\Sigma$.
