A logarithmic interpretation of Edixhoven's jumps for Jacobians

Let $A$ be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the N\'eron model of $A$ that measures the behaviour of the N\'eron model under tame base change. We interpret the jumps in this filtration in terms of lattices of logarithmic differential forms in the case where $A$ is the Jacobian of a curve $C$, and we give a compact explicit formula for the jumps in terms of the combinatorial reduction data of $C$.