  In the framework of the slave-boson approach to the $t-t'-t''-J$ model, it is
found that for electron-doped high-$T_c$ cuprates, the staggered
antiferromagnetic (AF) order coexists with superconducting (SC) order in a wide
doping level ranged from underdoped to nearly optimal doping at the mean-field
level. In the coexisting phase, it is revealed that the spin response is
commensurate in a substantial frequency range below a crossover frequency
$\omega_{c}$ for all dopings considered, and it switches to the incommensurate
structure when the frequency is higher than $\omega_{c}$. This result is in
agreement with the experimental measurements. Comparison of the spin response
between the coexisting phase and the pure SC phase with a
$d_{x^{2}-y^{2}}$-wave pairing plus a higher harmonics term (DP+HH) suggests
that the inclusion of the two-band effect is important to consistently account
for both the dispersion of the spin response and the non-monotonic gap behavior
in the electron-doped cuprates.
