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We have conducted particle-in-cell simulations of laser pulse propagation through neutral He, including the effects of tunneling ionization, within the parameter regime of the l'OASIS experiments [1, 2] at the Lawrence Berkeley National Lab (LBNL). The simulations show the theoretically predicted [3] blue shifting of the laser frequency at the leading edge of the pulse. The observed blue shifting is in good agreement with the experimental data. These results indicate that such computations can be used to accurately simulate a number of important effects related to tunneling ionization for laser-plasma accelerator concepts, such as steepening due to ionization-induced pump depletion, which can seed and enhance instabilities. Our simulations show self-modulation occurring earlier when tunneling ionization is included then for a pre-ionized plasma.