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We consider the variation of tropical Hodge structure (TVHS) associated tofamilies of tropical varieties. The family of the tropical intermediateJacobians of the associated tropical Hodge structure defines a bundle oftropical Jacobians, whose sections we call the tropical normal functions. Wedefine formal sequential derivatives of these functions on the base withrespect to the natural Gauss-Manin connection as the Hodge theoretic invariantsdetecting tropical cycles in the fibers. The associated invariants which aredefined inductively are the higher Abel-Jacobi invariants in the tropicalcategory. They naturally identify the tropical Bloch-Beilinson filtration onthe tropical Chow group. We examine this construction on the moduli of tropicalcurves with marked points, in order to study the tropical tautological classesin the tautological ring of $\mathcal{M}_{g,n}^{\text{trop}}$. The expectationis the nontriviality of these cycles could be examined with less complexity inthe tropical category. The construction is compatible with the tropicalizationfunctor on the category of schemes, and the aforementioned procedure will alsoprovide an alternative way to examine the relations in the tautological ring of$\mathcal{M}_{g,n}$ in the schemes category.

Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles