  We apply Godsil-McKay switching to the symplectic graphs over $\mathbb{F}_2$
with at least 63 vertices and prove that the 2-rank of (the adjacency matrix
of) the graph increases after switching. This shows that the switched graph is
a new strongly regular graph with parameters $(2^{2\nu}\!-1, 2^{2\nu-1},
2^{2\nu-2},2^{2\nu-2})$ and 2-rank $2\nu+2$ when $\nu\geq 3$. For the
symplectic graph on $63$ vertices we investigate repeated switching by computer
and find many new strongly regular graphs with the above parameters for $\nu=3$
with various 2-ranks. Using these results and a recursive construction method
for the symplectic graph from Hadamard matrices, we obtain several graphs with
the above parameters, but different 2-ranks for every $\nu\geq 3$.
