  We consider the variation of tropical Hodge structure (TVHS) associated to
families of tropical varieties. The family of the tropical intermediate
Jacobians of the associated tropical Hodge structure defines a bundle of
tropical Jacobians, whose sections we call the tropical normal functions. We
define formal sequential derivatives of these functions on the base with
respect to the natural Gauss-Manin connection as the Hodge theoretic invariants
detecting tropical cycles in the fibers. The associated invariants which are
defined inductively are the higher Abel-Jacobi invariants in the tropical
category. They naturally identify the tropical Bloch-Beilinson filtration on
the tropical Chow group. We examine this construction on the moduli of tropical
curves with marked points, in order to study the tropical tautological classes
in the tautological ring of $\mathcal{M}_{g,n}^{\text{trop}}$. The expectation
is the nontriviality of these cycles could be examined with less complexity in
the tropical category. The construction is compatible with the tropicalization
functor on the category of schemes, and the aforementioned procedure will also
provide an alternative way to examine the relations in the tautological ring of
$\mathcal{M}_{g,n}$ in the schemes category.

Algebraic Geometry;Combinatorics