  A striking feature of the timelike nucleon electromagnetic form factors,
investigated in $e^+e^- \to N\bar N$ annihilation reactions, is the modulation
by local structures of small magnitude and oscillatory form, showing up above
$N\bar N$ threshold. Starting from an isospin decomposition of the proton and
neutron form factors it is shown that such structures are the natural
consequence of the interference of a large and a small amplitudes, resulting in
a sinusoidal behavior as a function of the "invariant energy" if the relative
phase shift varies with energy. Thus, periodic oscillations superimposed on a
smooth background will be observed. In this scenario, an equal size of the
modulation for neutron and proton discovered by recent BESIII data evidently
implies the particular isoscalar or isovector nature of these local structures,
or their orthogonal interference, hence specifies their origin as excited
vector mesons whose widths are tied to the modulation frequency.We clarify that
the phase difference of modulation between neutron and proton as BESIII data
found, but not the modulation itself, is the evidence of an imaginary part of
the timelike nucleon electromagnetic form factors, which is associated with the
rescattering processes.
