I’d say that this is the seminal paper that gives the most thorough background of how the Laplace Transform relates to Grid and Place cells and Marc Howard’s model of working memory. If you look through Howard’s other publications you’ll find a few that are a bit more philosophical and a few that are a bit more applied. Bitking had a post on this paper once, as well. Also linked.
The key mathematical difference between the Laplace Transform you’re remembering from circuit analysis and signal processing is that the one used for circuit analysis is the complex-valued transform over the full complex plane. s = sigma + i*omega. The flavor Howard uses is the real Laplace transform, which has omega = 0. So Howard’s application the transform exists only on the real line. The basis functions are the set of exponentially decreasing functions, which provide a sort of “clock” to measure time differences, and this is where all the functionality lies. The equations model any time differentiated signal input into the EC. eg. velocity. Thus I was able to compute velocity vectors for navigation. This paper linked below shows how the math was used to model a rodant’s navigation through a maze.