  In this paper we present an attempt to estimate the redshift evolution of the
molecular to neutral gas mass ratio within galaxies (at fixed stellar mass).
For a sample of five nearby grand design spirals located on the Main Sequence
(MS) of star forming galaxies, we exploit maps at 500 pc resolution of stellar
mass and star formation rate ($M_{\star}$ and SFR). For the same cells, we also
have estimates of the neutral ($M_{\rm HI}$) and molecular ($M_{\rm H_2}$) gas
masses. To compute the redshift evolution we exploit two relations: {\it i)}
one between the molecular-to-neutral mass ratio and the total gas mass ($M_{\rm
gas}$), whose scatter shows a strong dependence with the distance from the
spatially resolved MS, and {\it ii)} the one between
$\log(M_{\rm{H_2}}/M_{\star})$ and $\log(M_{\rm{HI}}/M_{\star})$. For both
methods, we find that $M_{\rm H_2}$/$M_{\rm HI}$ within the optical radius
slightly decreases with redshift, contrary to common expectations of galaxies
becoming progressively more dominated by molecular hydrogen at high redshifts.
We discuss possible implications of this trend on our understanding of the
internal working of high redshift galaxies.
