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Oscillatory interference models propose a mechanism by which the spatial firing pattern of grid cells can arise from the interaction of multiple oscillators that shift in relative phase. These models produce aspects of the physiological data such as the phase precession dynamics observed in grid cells. However, existing oscillatory interference models did not predict the in-field DC shifts in the membrane potential of grid cells that have been observed during intracellular recordings in navigating animals. Here, we demonstrate that DC shifts can be generated in an oscillatory interference model when half-wave rectified oscillatory inputs are summed by a leaky integrate-and-fire neuron with a long membrane decay constant (100 ms). The non-linear mean of the half-wave rectified input signal is reproduced in the grid cell's membrane potential trace producing the DC shift within field. For shorter values of the decay constant integration is more effective if the input signal, comprising input from 6 head direction selective populations, is temporally spread during in-field epochs; this requires that the head direction selective populations act as velocity controlled oscillators with baseline oscillations that are phase offset from one another. The resulting simulated membrane potential matches several properties of the empirical intracellular recordings, including: in-field DC-shifts, theta-band oscillations, phase precession of both membrane potential oscillations and grid cell spiking activity relative to network theta and a stronger correlation between DC-shift amplitude and firing-rate than between theta-band oscillation amplitude and firing-rate. This work serves to demonstrate that oscillatory interference models can account for the DC shifts in the membrane potential observed during intracellular recordings of grid cells without the need to appeal to attractor dynamics.