Let X be a proper scheme over the field F of functions meromorphic in an open neighborhood of zero in the complex plane. The scheme X gives rise to a proper morphism of complex analytic spaces X^h -> D* = D-{0}. It is well known that, after shrinking the disc D (and replacing X^h by its preimage), the cohomology groups H^i(X^h_t,Z) of the fiber X^h_t at a point t \in D* form a variation of mixed Hodge structures on D*, which admits a limit mixed Hodge structure. We’ll describe the weight zero subspace of this limit mixed Hodge structure in terms of the non-Archimedean analytic space X^an associated with the scheme X over the completion of the field F.