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Ultra-luminous X-ray sources (ULXs) are those X-ray sources located away fromthe centre of their host galaxy with luminosities exceeding the Eddington limitof a stellar-mass black hole ($L_X>10^{39}\;{\rm erg\,s}^{-1}$). The discoveryof X-ray pulsations in some of these objects (e.g. M82~X-2) suggests that acertain fraction of the ULX population may have a neutron star accretor. Wepresent systematic modelling of low- and intermediate-mass X-ray binaries(LMXBs and IMXBs; donor-star mass range $0.92$--$8.0$~M$_{\odot}$ andneutron-star accretors) to explain the formation of this sub-population ofULXs. Using MESA, we explored the allowed initial parameter space of binarysystems consisting of a neutron star and a low- or intermediate-mass donor starthat could explain the observed properties of ULXs. Our simulations take intoaccount beaming effects, stellar rotation, general angular momentum losses, anda detailed and self-consistent calculation of the mass-transfer rate. We studythe conditions that lead to dynamical stability of these systems, which dependsstrongly on the response of the donor star to mass loss. Using two values forthe initial neutron star mass ($1.3$~M$_{\odot}$ and $2.0$~M$_{\odot}$), wepresent two sets of mass-transfer calculation grids. We find that LMXBs/IMXBscan produce NS-ULXs with typical time-averaged isotropic-equivalent X-rayluminosities of $10^{39}$--$10^{41}\;{\rm erg\,s}^{-1}$ on a timescale up to$\sim\!1.0\;{\rm Myr}$ for the lower luminosities. We also estimate theirlikelihood of detection, the types of white-dwarf remnants left behind by thedonors, and the total amount of mass accreted by the neutron stars. We alsocompare our results to the observed pulsating ULXs. Our results suggest that alarge subset of the observed pulsating ULX population can be explained byLMXBs/IMXBs undergoing a super-Eddington mass-transfer phase.

The origin of pulsating ultra-luminous X-ray sources: Low- and intermediate-mass X-ray binaries containing neutron star accretors