Given an integrable connection on a smooth quasi-projective
algebraic surface U over a subfield k of the complex numbers, we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de Rham cohomology in terms of period integrals. These homology groups generalize the analogous groups in the same situation over curves defined by S. Bloch and H. Esnault. In dimension two, however, new features appear in this context which we explain in detail. Assuming a conjecture of C. Sabbah on the formal
classification of meromorphic connections on surfaces (known to be true if the rank is lower than or equal to 5), we prove perfectness of the period pairing in dimension two.