  We study a one-dimensional discrete nonlinear Schr\"odinger model with
hopping to the first and a selected N-th neighbor, motivated by a helicoidal
arrangement of lattice sites. We provide a detailed analysis of the
modulational instability properties of this equation, identifying distinctive
multi-stage instability cascades due to the helicoidal hopping term.
Bistability is a characteristic feature of the intrinsically localized breather
modes, and it is shown that information on the stability properties of weakly
localized solutions can be inferred from the plane-wave modulational
instability results. Based on this argument, we derive analytical estimates of
the critical parameters at which the fundamental on-site breather branch of
solutions turns unstable. In the limit of large N, these estimates predict the
emergence of an effective threshold behavior, which can be viewed as the result
of a dimensional crossover to a two-dimensional square lattice.
