Beyond Place Cells: Using Algebraic Geometry to Determine Underlying Structure of Receptive Fields Associated with Sensory Neurons

Abstract

In 2014, John O’Keefe was jointly awarded a Nobel Prize for his 1971 discovery of place cells, a population of neurons in the hippocampus which contribute to an organism’s intuitive understanding of their position in relation to their surrounding space. Each place cell has a representative receptive field, the area in which the place cell fires when occupied by the organism, which based on their overlapping structure can provide us with information about an individual’s surroundings. The challenge of determining this overlapping structure, starting from neuronal firing patterns without context, still remains, but can be explored using algebraic geometry. Tools from algebraic geometry can describe the receptive fields of place cells to better understand the phenomena and how the signals of the brain contribute to the individual’s perception of their surroundings. To compute these relationships, we use the neural code, a binary representation of neurons firing; additionally, using Euler diagrams (a simple example being the Venn diagram) we can visualize the representation of the receptive field. Under certain constraints (i.e., connectivity), the overlap of receptive fields can be realized in up to three dimensions. This finding seems intuitive when applied to place cells since we perceive our position in respect to our surroundings in a three‐dimensional space. These same tools can be applied to other neuronal populations regardless of an intuitive dimensionality of the receptive fields. It is difficult to imagine a visual representation of all receptive fields, but with certain constraints we can construct a geometric visualization. To address this difficulty, we have developed a code using place cell data to compute the dimension of its receptive field. Then, using publicly available data sets, we used our code to determine the dimensionality of other sensory neuronal populations.