Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a quasicoherent $O_S$–module, carrying a canonical and natural integrable connection – the "Gauss-Manin connection" (Katz-Oda 1968). Its construction following Katz and Oda uses the spectral sequence associated with the higher direct image functors $R^i\pi_\ast$ and a canonical filtration of the differential complex $\Omega_{X/k}^\ast$.
Now suppose $M$ is a motive over $S$, say in the sense of Jannsen or a 1-motive as defined by Deligne, and denote by $T_{dR}(M/S)$ its de Rham realisation. This is a quasicoherent $O_S$–module which, according to the general philosophy, must come with a canonical and natural integrable connection.
How can we construct this "Gauss-Manin connection" on $T_{dR}(M/S)$?
Just walking through the Katz-Oda paper and replacing $X$'es by $M$'s will not do, because there are no such things as $\pi_\ast$ or $\Omega_{M/k}^\ast$.
In the case $k=\mathbb C$, we can use comparison of de Rham realisation with Betti realisation (the motivic instance of de Rham's theorem, which is forced to hold in Jannsen's setting, and which proven to hold for 1–motives by Deligne) and produce the sought connextion analytically, but that is not what I want.
Best Answer
It appears that we have a precise answer to your question, given by Andreatta and Bertapelle in the paper "Universal extension crystals of 1-motives and applications" for JPAA at the link
https://doi.org/10.1016/j.jpaa.2010.11.004