  It has been recently pointed out, that nonlocal Nambu--Jona-Lasinio models,
may present unphysical thermodynamical behavior like negative pressure and
oscillating entropy. Here we show how these thermodynamic instabilities can be
related to the analytical structure of the poles of the quark propagator in the
model. The analysis is carried out for two different regulators and we show, in
each case, how the instabilities are related to the pressence of highly
unstable poles. We also argue that the softening of these instabilities by the
inclusion of the Polyakov loop is related to the effect the latter has on the
poles of the propagator.
