  The dynamics of glass-forming liquids display several outstanding features,
such as two-step relaxation and dynamic heterogeneities, which are difficult to
predict quantitatively from first principles. In this work, we revisit a simple
theoretical model of the $\beta$-relaxation, i.e., the first step of the
relaxation dynamics. The model, first introduced by Cavagna et al., describes
the dynamics of the system in the neighborhood of a saddle point of the
potential energy surface. We extend the model to account for density-density
correlation functions and for the 4-point dynamic susceptibility. We obtain
analytical results for a simple schematic model, making contact with related
results for p-spin models and with the predictions of inhomogeneous
mode-coupling theory. Building on recent computational advances, we also
explicitly compare the model predictions against overdamped Langevin dynamics
simulations of a glass-forming liquid close to the mode-coupling crossover. The
agreement is quantitative at the level of single-particle dynamic properties
only up to the early $\beta$-regime. Due to its inherent harmonic
approximation, however, the model is unable to predict the dynamics on the time
scale relevant for structural relaxation. Nonetheless, our analysis suggests
that the agreement with the simulations may be largely improved if the modes'
spatial localization is properly taken into account.
