Spatial cognition relies on an internal map-like representation of space provided by hippocampal place cells, which in turn are thought to rely on grid cells as a basis. Spatial Semantic Pointers (SSP) have been introduced as a way to represent continuous spaces and positions via the activity of a spiking neural network. In this work, we further develop SSP representation to replicate the firing patterns of grid cells. This adds biological realism to the SSP representation and links biological findings with a larger theoretical framework for representing concepts. Furthermore, replicating grid cell activity with SSPs results in greater accuracy when constructing place cells. Improved accuracy is a result of grid cells forming the optimal basis for decoding positions and place cell output. Our results have implications for modelling spatial cognition and more general cognitive representations over continuous variables.