Where can a place cell put its fields? Let us count the ways

A hippocampal place cell exhibits multiple firing fields within and across environments. What factors determine the configuration of these fields, and could they be set down in arbitrary locations? We conceptualize place cells as performing evidence combination across many inputs and selecting a threshold to fire. Thus, mathematically they are perceptrons, except that they act on geometrically organized inputs in the form of multiscale periodic grid-cell drive, and external cues. We analytically count which field arrangements a place cell can realize with structured grid inputs, to show that many more place-field arrangements are realizable with grid-like than one-hot coded inputs. However, the arrangements have a rigid structure, defining an underlying response scaffold. We show that the “separating capacity” or spatial range over which all potential field arrangements are realizable equals the rank of the grid-like input matrix, which in turn equals the sum of distinct grid periods, a small fraction of the unique grid-cell coding range. Learning different grid-to-place weights beyond this small range will alter previous arrangements, which could explain the volatility of the place code. However, compared to random inputs over the same range, grid-structured inputs generate larger margins, conferring relative robustness to place fields when grid input weights are fixed.