Population synthesis is used to model the number of neutron stars in globular clusters that are observed as low-mass X-ray sources and millisecond radio pulsars. The dynamical interactions between binary and single stars in a cluster are assumed to take place only with a continuously replenished “background” of single stars whose properties keep track of the variations in parameters of the cluster as a whole and the evolution of single stars. We use the hypothesis that the neutron stars forming in binary systems from components with initial masses of ∼8–12 M _⊙ during the collapse of degenerate O-Ne-Mg cores through electron captures do not acquire a high space velocity. The remaining neutron stars (from single stars with masses >8 M _⊙ or from binary components with masses >12 M _⊙) are assumed to be born with high space velocities. According to this hypothesis, a sizeable fraction of the forming neutron stars remain in globular clusters (about 1000 stars in a cluster with a mass of 5 × 10⁵ M _⊙). The number of millisecond radio pulsars forming in such a cluster in the case of accretion-driven spinup in binary systems is found to be ∼10, in agreement with observations. Our modeling also reproduces the observed shape of the X-ray luminosity function for accreting neutron stars in binary systems with normal and degenerate components and the distribution of spin periods for millisecond pulsars.