Sharp wave-ripples (SPW-Rs) are widely studied oscillatory patterns observed during sleep and awake immobility in the hippocampal formation of most mammals [Buzsáki, 2015]. During SPW-Rs behavioral sequences are replayed compressed in time [Wilson & McNaughton, 1994; Diba & Buzsáki, 2007], which led to the hypothesis that they are mediating memory transfer from the hippocampus to the neocortex [Buzsáki, 1989]. SPW-Rs can be internally generated in the highly recurrent CA3 network [Maier et al., 2003], but their incidence during sleep is affected by slow oscillations in the cortex [Sirota et al., 2003, Sullivan et al., 2011; Zutshi & Buzsáki 2023], appearing with much higher probability during the UP states of the entorhinal cortex [Kajikawa et al., 2022]. Here we investigate the interplay between cortical inputs and the prevalence of sharp waves (SPWs) in a threshold linear rate model of the CA3 network. The model contains one excitatory population (P) and two inhibitory populations. One of the inhibitory populations represents PV+ basket cells (B) and the other a hypothetical population of anti-SPW cells (A). The model can show bistability between a regime corresponding to a SPW state (highly active P and B) and a non-SPW state, where only the A population is active, suppressing the other two [Evangelista et al., 2020]. Cortical states were modeled as noisy input to all three populations, which could trigger transitions between the two stable states. We first characterized the changes to the fixpoint structure as a function of cortical inputs. We derive analytical conditions for how cortical inputs modify the firing rate of active populations during SPW and non-SPW states, and describe the bifurcations. This allows us to derive conditions under which cortical UP states increase the rate of SPW in this model. We confirm our solutions by running simulations of the stochastic rate model. Our results allow us to constrain the parameters of the model such that it generates more SPWs during UP states, thus aligning it with experimental observations. The obtained conditions also generate experimentally testable predictions on the connection strength between each of the studied CA3 populations, as well as on their respective inputs.