Simultaneous localisation and mapping (SLAM) algorithms are commonly used in robotic systems for learning maps of novel environments. Brains also appear to learn maps, but the mechanisms are not known and it is unclear how to infer these maps from neural activity data. We present BrainSLAM; a method for performing SLAM using only population activity (local field potential, LFP) data simultaneously recorded from three brain regions in rats: hippocampus, prefrontal cortex, and parietal cortex. This system uses a convolutional neural network (CNN) to decode velocity and familiarity information from wavelet scalograms of neural local field potential data recorded from rats as they navigate a 2D maze. The CNN's output drives a RatSLAM-inspired architecture, powering an attractor network which performs path integration plus a separate system which performs `loop closure' (detecting previously visited locations and correcting map aliasing errors). Together, these three components can construct faithful representations of the environment while simultaneously tracking the animal's location. This is the first demonstration of inference of a spatial map from brain recordings. Our findings expand SLAM to a new modality, enabling a new method of mapping environments and facilitating a better understanding of the role of cognitive maps in navigation and decision making.