Piecewise affine models often provide a good approximation to describe continuous
systems, but may involve a high degree of simplification. To compare solutions of the continuous
and piecewise affine models, it is important to quantify the differences between solutions
in each region of the state space. As an approach, we will use enveloping ``bands'' to
characterize continuous activation or inhibition functions, and then describe the differences
between continuous and piecewise affine solutions in terms of the width $\delta$ of these bands. As
a case study, we will consider the negative feedback loop, a classical motif in two dimensions
which results in oscillating behaviour. For this example, it is shown that the two types of models
may differ only on a compact invariant set (the interior of a limit cycle), whose diameter is a
function of the band width $\delta$. We give some generalizations to higher dimensions.

This is common work with Madalena Chaves and Camille Poignard (Inria Biocore).