Chip-firig on graphs is a simple game with surprisingly deep connections to various parts of mathematics, and the lattice ideal of the Laplacian matrix of a graph encodes all of the combinatorics. We introduce a new family of subideals of the Laplacian lattice ideal with distinguished Gr"obner bases coming from generalized chip-firing dynamics on a graph as induced by abstract simplicial complexes on the vertex set. We will describe a conjecture generalizing a result of Postnikov and Shapiro relating the initial ideals of these binomial ideals to their power ideal deformations.