A self-organizing map of sigma–pi units

By frame of reference transformations, an input variable in one coordinate system is transformed into an output variable in a different coordinate system depending on another input variable. If the variables are represented as neural population codes, then a sigma–pi network is a natural way of coding this transformation. By multiplying two inputs it detects coactivations of input units, and by summing over the multiplied inputs, one output unit can respond invariantly to different combinations of coactivated input units. Here, we present a sigma–pi network and a learning algorithm by which the output representation self-organizes to form a topographic map. This network solves the frame of reference transformation problem by unsupervised learning.