This work develops and tests the hypothesis that similarity of meaning derives from substitution regularities in the linguistic environment, represented topographically in the brain. We develop a general mathematical theory of semantic space models and use this to motivate a set of new methods for high-dimensional space construction. We then develop a low-dimensional topographic map model of the lexicon from statistical considerations and apply both models to a range of semantic priming experiments. We show that both models capture a large number of previously un-modelled semantic relations. We model association! semantic relatedness and their interaction, and offer a new theory of associative priming. We then demonstrate that, contrary to previous findings, the models also replicate graded and mediated priming effects. Mediated priming is of theoretical importance to memory models because its existence has been taken as evidence for spreading activation and against compound cue models. We show how a semantic space account representing only substitutability relations can account for mediated priming without making any specific architectural assumptions.