Tensor decomposition of higher-order correlations by nonlinear Hebbian plasticity

Biological synaptic plasticity exhibits nonlinearities that are not accounted for by classic Hebbian learning rules. Here, we introduce a simple family of generalized, nonlinear Hebbian learning rules. We study the computations implemented by their dynamics in the simple setting of a neuron receiving feedforward inputs. We show that these nonlinear Hebbian rules allow a neuron to learn tensor decompositions of its higher-order input correlations. The particular input correlation decomposed, and the form of the decomposition, depend on the location of nonlinearities in the plasticity rule. For simple, biologically motivated parameters, the neuron learns tensor eigenvectors of higher-order input correlations. We prove that each tensor eigenvector is an attractor and determine their basins of attraction. We calculate the volume of those basins, showing that the dominant eigenvector has the largest basin of attraction. We then study arbitrary learning rules, and find that any learning rule that admits a finite Taylor expansion into the neural input and output also has stable equilibria at tensor eigenvectors of its higher-order input correlations. Nonlinearities in synaptic plasticity thus allow a neuron to encode higher-order input correlations in a simple fashion.