I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence

$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial M) $$

I want to understand the connecting homomorphism. My question is:

In general, are there any geometric descriptions of the "cycles" in $KO$ theory and the connecting homomorphism?