A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a (group) divisible design. Divisible design graphs were introduced in 2011 as a generalization of $(v,k,λ)$-graphs. Here we describe four new infinite families that can be obtained from the symplectic strongly regular graph $Sp(2e,q)$ ($q$ odd, $e\geq 2$) by modifying the set of edges. To achieve this we need two kinds of spreads in $PG(2e-1,q)$ with respect to the associated symplectic form: the symplectic spread consisting of totally isotropic subspaces and, when $e=2$, a special spread consisting of lines which are not totally isotropic. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power $q$ is a major result of this paper. We have included relevant back ground from finite geometry, and when $q=3,5$ and $7$ we worked out all possible special spreads.